Research ArticleDynamic Mechanical Behavior and NumericalSimulation of Frozen Soil under Impact Loading

Dan Zhang1 Zhiwu Zhu12 and Zhijie Liu1

1School of Mechanics and Engineering Southwest Jiaotong University Chengdu 610031 China2State Key Laboratory of Frozen Soil Engineering CAREERI CAS Lanzhou 730000 China

Correspondence should be addressed to Zhiwu Zhu zzw4455163com

Received 31 March 2016 Revised 21 June 2016 Accepted 21 June 2016

Academic Editor Onome E Scott-Emuakpor

Copyright copy 2016 Dan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Split Hopkinson pressure bars (SHBP) were used to perform impact experiments on frozen soil under various impact velocities andtemperatures to analyze the effect of these parameters on the mechanical behavior of the soil Based on the Holmquist-Johnson-Cook constitutive model the dynamic mechanical properties under impact loading were analyzed The SHPB experiments offrozen soil were also simulated using the finite element analysis software LS-DYNA and the simulation results were similar tothe experimental results The temperature effect strain rate effect and the destruction process of the frozen soil as well as thepropagation process of stress waves in the incident bar transmission bar and frozen soil specimen were investigated This workprovides a good theoretical basis and technical support for frozen soil engineering applications

1 Introduction

Frozen soil a porous complex material consists of mineralgrains water ice inclusions and gas inclusions (moisture andair) the essential difference between frozen soil and meltedsoil is the presence of ice Frozen soil is widely distributedon the Earthrsquos surface a series of major construction projectswill be performed in permafrost regions and strong dynamicloadings will affect the structures of frozen soil In additionthe artificial freezing method one of the main technicaltechniques used in underground engineering overcomesthe limitations of the timbering range and depth and caneffectively prevent the deformation of adjacent water andmining The cutting process of frozen soil in the artificialmethod is essentially the processing of frozen soil underdynamic loading Therefore understanding the dynamicmechanical behavior of frozen soil is of great importance

The split Hopkinson pressure bar (SHPB) method hasplayed a significant role in the dynamic testing of materialsChen et al [1] studied the dynamic brittle and frozen brittlebehavior of frozen soil and observed oscillation and conver-gence of the strain-stress curves Zhang et al [2] establisheda phenomenological model that used thermal sensitivity to

describe the dynamic behavior of confined frozen soil Maet al [3 4] studied the dynamic behavior of artificial frozensoil which showed a dependence on both the temperatureand strain rate When the freezing temperature and impactpressure are fixed the dynamic mechanical characteristicsof artificial frozen soil under a uniaxial loading state andconfined pressure state exhibit significant differences themaximum stress under the confined pressure state is higherthan that under the uniaxial loading state which indicatesobvious stress-strengthening characteristics Liu et al [5]evaluated the dynamic stress-strain curves and dynamicparameters of frozen clay at various low temperatures watercontents and strain rates Based on systematic impact load-ing experiments of frozen soil Ma et al [6] analyzed thecharacteristics of dynamic stress-strain curves and a strainconvergence phenomenon was observed the necessity ofthis phenomenon in SHPB experiments was investigatedAt Sandia National Laboratory [7 8] in the USA SHPBtests were performed on artificial frozen soils modeledon Alaskan examples under negative lateral confinementor nearly uniaxial strain conditions Recently additionalSHPB tests were performed on undisturbed frozen soil fromAlaska and an attempt was made to describe its constitutive

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 3049097 16 pageshttpdxdoiorg10115520163049097

2 Shock and Vibration

Figure 1 A frozen soil specimen before testing

behavior using a cap plasticity model However becauseof the nonuniformity of the frozen soil in the SHPB testsand waveform dispersion effect the SHPB device data weredifficult to measure and the experimental accuracy must befurther improved In addition the entire process only took afew hundred microseconds and was applied to the frozen soilsample for only a few microseconds It was thus difficult toobserve the material failure process and describe the internalstress evolution law Because of the complexity of the internalstructure of frozen soil the development of a dynamic theoryis challenging

The numerical simulation technique has recently under-gone rapid development Through numerical simulationthe stress and strain of a unit sample can be directlydetermined and the destruction process of a sample canbe directly observed In addition the effects of interfacialfriction and parallelism tolerance can be avoided Thereforenumerical simulation can also improve test accuracy Wu etal [9] numerically simulated SHPB tests for concrete usingthe Holmquist-Johnson-Cook (HJC) constitutive model Acomparison of the stress-strain curve obtained from theSHPB test with the reconstructed curve from the numericalsimulation revealed similar mechanical behavior The failureprocess of rock subjected to combined static and dynamicloading in SHPB tests was numerically simulated by Zhu et al[10] and the strength increase factor under combined staticand dynamic loading could be predicted The test processof ceramics in SHPB tests was numerically simulated byAnderson et al [11] and the dynamic condition and lossefficacy of the ceramics were studied Chakraborty [12] usedfinite element software to simulate SHPB tests on rocksand achieved good correlation with experimental resultsHowever numerical simulation of SHPB experiments onfrozen soil has not yet been reported

Based on the existing research in this work the SHPBmethod was used to investigate the dynamic behavior ofartificial frozen soilThe tests were conducted at temperaturesofminus5∘Cminus15∘C andminus25∘Cand strain rates of 500s 750s and950s Using the finite element software LS-DYNA the effectsof the temperature and loading strain rate on the dynamicmechanical properties of frozen soil were investigated and

Table 1 Particle size distribution

Sizemm lt01 01sim025 025sim05 05sim2 gt2Proportion 32 16 25 19 8

the dynamic failuremode of frozen soil under impact loadingwas discussed

2 Experimental Studies

21 Frozen Soil Samples and Experimental Conditions Basedon one-dimensional conditions an assumption of homo-geneity and the size of the experimental equipment thedimensions of all the specimens for the tests were 12060130mm times

18mm The experimental material is the artificial frozensand Firstly the sand was crushed down by a woodenhammer Then the large soil particles would be sifted out bya sieve with a mesh size of 2mmThe remaining soil was putin an oven at 110∘C for 12 h and dehydratedThe soil was thenplaced in a sealed vessel and cooled to room temperatureTheparticle size of the soil was listed in Table 1

After that the water was mixed with the dehydrated soilto satisfy the required water content of 30 and then themixture is stored in a closed container for 24 hThe specimenswere made from the mixture by molds and smeared Vaselineon its surface in order to keep the water content of specimensunchanged Finally the well-made specimens were placed ina freezer at the prescribed temperatures (ie minus5 minus15 andminus25∘C resp) for 24 h (see Figure 1)

22 Experimental Equipment Dynamic testing of the frozensoil specimens was performed using an SHPB setup(Figure 2) An SHPB is a device used to test the dynamicmechanical behavior of materials under impact loading

The SHPB setup used in this study consisted of a strikeran incident bar a transmission bar and an acquisition systemThe material parameters of the striker and bars and thelength of the striker are important experimental parametersThe striker is made of the 35CrMnSi steel whose Youngrsquosmodulus is 191 GPa and density is 8000 kgm3 and the length

Shock and Vibration 3

Gage I Gage II

Cylinder Velocimeter

Data processingsystem

Striker

Absorb barRigid blockIncident barSpecimenTransmitted bar

Wheatstonebridge

Figure 2 Sketch of SHPB and its testing system

Incident bar Transmission barSpecimen

V1 V2

120576I

120576R

120576T

Figure 3 Testing section of SHPB

of the striker is 200mm the incident and transmission barsare made of the 7075-T6 aluminumwhose modulus is 71 GPaand density is 2100 kgm3 and the diameters of them are30mm

The incident and transmission bars are instrumentedwithstrain gauges to capture the elastic stress waves generated bythe striker bar Frozen soil specimens are placed between theincident and transmission bars The striker is launched bycompressed air An elastic compression stress wave (incidentwave) is generated by the impact of the striker bar strikingthe incident barThe incident wave is transmitted through thespecimen as a transmittedwave and is partially reflected at theinterface between the specimen and the incident bar Both theincident and reflected waves are recorded by the strain gaugeon the incident bar and the transmitted wave is recorded bythe strain gauge on the transmission bar

Assume that the stress waves propagate in both theincident and transmission bars without dispersion that is thepulses recorded at the strain gage locations represent those atthe bar ends in contact with the specimen one-dimensionalstress wave theory relates the particle velocities at both endsof specimen to the three measured strain pulses (Figure 3)

V1= 119862

119861(120576

119868minus 120576

119877)

V2= 119862

119861120576

119879

(1)

where 119862119861is the elastic bar wave speed of the bar material

and 119868 119877 and 119879 represent the incident and reflected and

transmitted pulses respectively The average engineeringstrain rate and strain in the specimen are

120576 =

V1minus V2

119871

119904

=

119862

119861

119871

119904

(120576

119868minus 120576

119877minus 120576

119879)

120576 = int

119905

0

120576 119889119905 =

119862

119861

119871

119904

int

119905

0

(120576

119868minus 120576

119877minus 120576

119879) 119889119905

(2)

where 119871119904is the initial length of the specimen The stresses at

both ends of the specimen are calculated with the followingelastic relations

120590

1=

119860

119861

119860

119878

sdot 119864

119861(120576

119868+ 120576

119877) (3)

120590

2=

119860

119861

119860

119878

sdot 119864

119861sdot 120576

119879 (4)

where119860119861and119860

119878are the cross-sectional areas of the bars and

the specimen respectively and 119864119861is Youngrsquos modulus of the

bar materialAsmentioned earlier the specimen is assumed to be stress

equilibrated in a SHPB experimentThis assumption must besatisfied in dynamic characterization of material propertiesConsequently the specimen deforms nearly uniformly suchthat the specimen response averaged over its volume is a goodrepresentative of the point-wise valid material behavior Thestress equilibration is expressed as

120590

1= 120590

2 (5)

Or from (3) and (4)120576

119868+ 120576

119877= 120576

119879 (6)

4 Shock and Vibration

2

4

6

8

10

12St

ress

(MPa

)

Strain

0000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

Pa)

0

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(b) minus15∘C

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(c) minus25∘C

Figure 4 Stress-strain curves of frozen soil under various strain rates

Equations (2) and (4) can thus be simplified as the follows

120576 = minus2

119862

119861

119871

119904

120576

119877

120576 = minus2

119862

119861

119871

119904

int

119905

0

120576

119877119889119905

120590 =

119860

119861

119860

119878

119864

119861120576

119879

(7)

Therefore once the incident reflected and transmittedsignals are measured the stress-strain data for the materialunder investigation can be obtained

23 Analysis of Experimental Results The SHPB tests wereconducted on the frozen soil at three different temperatures(minus5∘C minus15∘C and minus25∘C) and three strain rates (500s 750sand 950s)The stress-strain curves obtained after processingusing the two-wave method are presented in Figure 4 Thepeak stress and final strain increase with increasing loadingstrain rates

To more clearly illustrate the relationship between thepeak stress and strain rate the peak stress-strain rate curvesfor the three experimental temperatures are plotted inFigure 5

Figure 5 illustrates that the frozen soil exhibits an obviousstrain rate effect For a fixed temperature the peak stressincreases for higher strain rates In addition the strain rate

Shock and Vibration 5

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain rate (sminus1)

minus5∘Cminus15∘Cminus25∘C

1000900800700600500

Figure 5 Peak stress-strain rate curves for various temperatures

effect is more pronounced at lower temperature because icecrystals play a leading role during the entire crush resistanceprocess At a lower temperature the amount of ice is greaterin frozen soil the proportion of the antipressure ability inthe entire sample is thus greater and the strain rate effectbecomes more apparent

The dynamic stress-strain curves of frozen soil for afixed impact loading strain rate at various temperatures areplotted in Figure 6The peak stress increases with decreasingtemperature

To more clearly illustrate the relationship between thepeak stress and temperature the peak stress-temperaturecurves at three different strain rates are plotted in Figure 7

Figure 7 demonstrates the obvious temperature effect forthe frozen soil The peak stress increases upon increasingthe temperature for a given strain rate In addition thetemperature effect is more prominent at higher strain rate

After the impact loading experiments debris from thefractured frozen soil samples was collected Photographs ofthe fractured frozen soil samples after impact loading atminus15∘C are shown in Figure 8 It is apparent that the damageto the frozen soil was more severe at higher impact loadingstrain rates

3 Numerical Simulation Studies

The LS-DYNA program can solve geometric nonlinearity(large displacement large rotation and large strain) materialnonlinearity (dynamic models for more than 140 types ofmaterial) and contact nonlinearity (more than 50) problemsThis program is suitable for the numerical simulation ofSHPB experiments Based on the strain rate effect andtemperature effect of the frozen soil and its final destructionform the HJC model [13] was selected to simulate the SHPBimpact dynamic experiments of the frozen soil

31 HJC Material Constitutive Model The characteristics ofthe HJC constitutive model [13] reflect the dynamic responseof brittle materials such as frozen soil

120590

lowast

= [119860 (1 minus 119863) + 119861119901

lowast119873

] (1 + 119862 ln 120576

lowast

) (8)

where 119860 and 119861 are the normalized intensities 119873 and 119862 arethe pressure-hardening exponent and strain rate coefficientrespectively 120590lowast = (120590119891

1015840

119888

) is the ratio of the real equiva-lent strength and quasi-static uniaxial compressive strengthwhich is the normalized equivalent stress 119901lowast = (119901119891

1015840

119888

) isthe normalized hydrostatic force 120576lowast is the nondimensionalstrain rate a ratio of the true strain rate and reference strainrate 120576

0 and 119863 is the damage factor and is determined by the

accumulation of plastic strain which includes two parts of theequivalent plastic and plastic volumetric strain

119863 = sum

Δ120576

119901+ Δ120583

119901

120576

119891

119901+ 120583

119891

119901

(9)

Here Δ120576119901andΔ120583

119901are the equivalent plastic strain increment

and plastic volumetric strain increment respectively and 120576119891119901

and 120583119891119901are the equivalent plastic strain and plastic volumetric

strain respectively when frozen soil is broken under ordinarypressure

119901 is the actual hydrostatic pressure and can be determinedby the state equation of the curve [13] shown in Figure 9

The first stage represents the elastic compression region(119874119860 section) Here 119901 = 119870120583 where 119870 is the bulk modulus

The second stage represents the compaction deformationregion (119860119861 section) The internal pores of the frozen soilare gradually crushed and plastic volumetric damage will beproduced Here 119901 = 1198701015840120583 where 1198701015840 = (119901

1minus 119901

119888)(120583

1minus 120583

119888)

The third stage represents the region after the soliddeformation section (119861119862 section) The internal part of thefrozen soil consists entirely of crushed close-grainedmaterial

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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Shock and Vibration

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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2 Shock and Vibration

Figure 1 A frozen soil specimen before testing

behavior using a cap plasticity model However becauseof the nonuniformity of the frozen soil in the SHPB testsand waveform dispersion effect the SHPB device data weredifficult to measure and the experimental accuracy must befurther improved In addition the entire process only took afew hundred microseconds and was applied to the frozen soilsample for only a few microseconds It was thus difficult toobserve the material failure process and describe the internalstress evolution law Because of the complexity of the internalstructure of frozen soil the development of a dynamic theoryis challenging

The numerical simulation technique has recently under-gone rapid development Through numerical simulationthe stress and strain of a unit sample can be directlydetermined and the destruction process of a sample canbe directly observed In addition the effects of interfacialfriction and parallelism tolerance can be avoided Thereforenumerical simulation can also improve test accuracy Wu etal [9] numerically simulated SHPB tests for concrete usingthe Holmquist-Johnson-Cook (HJC) constitutive model Acomparison of the stress-strain curve obtained from theSHPB test with the reconstructed curve from the numericalsimulation revealed similar mechanical behavior The failureprocess of rock subjected to combined static and dynamicloading in SHPB tests was numerically simulated by Zhu et al[10] and the strength increase factor under combined staticand dynamic loading could be predicted The test processof ceramics in SHPB tests was numerically simulated byAnderson et al [11] and the dynamic condition and lossefficacy of the ceramics were studied Chakraborty [12] usedfinite element software to simulate SHPB tests on rocksand achieved good correlation with experimental resultsHowever numerical simulation of SHPB experiments onfrozen soil has not yet been reported

Based on the existing research in this work the SHPBmethod was used to investigate the dynamic behavior ofartificial frozen soilThe tests were conducted at temperaturesofminus5∘Cminus15∘C andminus25∘Cand strain rates of 500s 750s and950s Using the finite element software LS-DYNA the effectsof the temperature and loading strain rate on the dynamicmechanical properties of frozen soil were investigated and

Table 1 Particle size distribution

Sizemm lt01 01sim025 025sim05 05sim2 gt2Proportion 32 16 25 19 8

the dynamic failuremode of frozen soil under impact loadingwas discussed

2 Experimental Studies

21 Frozen Soil Samples and Experimental Conditions Basedon one-dimensional conditions an assumption of homo-geneity and the size of the experimental equipment thedimensions of all the specimens for the tests were 12060130mm times

18mm The experimental material is the artificial frozensand Firstly the sand was crushed down by a woodenhammer Then the large soil particles would be sifted out bya sieve with a mesh size of 2mmThe remaining soil was putin an oven at 110∘C for 12 h and dehydratedThe soil was thenplaced in a sealed vessel and cooled to room temperatureTheparticle size of the soil was listed in Table 1

After that the water was mixed with the dehydrated soilto satisfy the required water content of 30 and then themixture is stored in a closed container for 24 hThe specimenswere made from the mixture by molds and smeared Vaselineon its surface in order to keep the water content of specimensunchanged Finally the well-made specimens were placed ina freezer at the prescribed temperatures (ie minus5 minus15 andminus25∘C resp) for 24 h (see Figure 1)

22 Experimental Equipment Dynamic testing of the frozensoil specimens was performed using an SHPB setup(Figure 2) An SHPB is a device used to test the dynamicmechanical behavior of materials under impact loading

The SHPB setup used in this study consisted of a strikeran incident bar a transmission bar and an acquisition systemThe material parameters of the striker and bars and thelength of the striker are important experimental parametersThe striker is made of the 35CrMnSi steel whose Youngrsquosmodulus is 191 GPa and density is 8000 kgm3 and the length

Shock and Vibration 3

Gage I Gage II

Cylinder Velocimeter

Data processingsystem

Striker

Absorb barRigid blockIncident barSpecimenTransmitted bar

Wheatstonebridge

Figure 2 Sketch of SHPB and its testing system

Incident bar Transmission barSpecimen

V1 V2

120576I

120576R

120576T

Figure 3 Testing section of SHPB

of the striker is 200mm the incident and transmission barsare made of the 7075-T6 aluminumwhose modulus is 71 GPaand density is 2100 kgm3 and the diameters of them are30mm

The incident and transmission bars are instrumentedwithstrain gauges to capture the elastic stress waves generated bythe striker bar Frozen soil specimens are placed between theincident and transmission bars The striker is launched bycompressed air An elastic compression stress wave (incidentwave) is generated by the impact of the striker bar strikingthe incident barThe incident wave is transmitted through thespecimen as a transmittedwave and is partially reflected at theinterface between the specimen and the incident bar Both theincident and reflected waves are recorded by the strain gaugeon the incident bar and the transmitted wave is recorded bythe strain gauge on the transmission bar

Assume that the stress waves propagate in both theincident and transmission bars without dispersion that is thepulses recorded at the strain gage locations represent those atthe bar ends in contact with the specimen one-dimensionalstress wave theory relates the particle velocities at both endsof specimen to the three measured strain pulses (Figure 3)

V1= 119862

119861(120576

119868minus 120576

119877)

V2= 119862

119861120576

119879

(1)

where 119862119861is the elastic bar wave speed of the bar material

and 119868 119877 and 119879 represent the incident and reflected and

transmitted pulses respectively The average engineeringstrain rate and strain in the specimen are

120576 =

V1minus V2

119871

119904

=

119862

119861

119871

119904

(120576

119868minus 120576

119877minus 120576

119879)

120576 = int

119905

0

120576 119889119905 =

119862

119861

119871

119904

int

119905

0

(120576

119868minus 120576

119877minus 120576

119879) 119889119905

(2)

where 119871119904is the initial length of the specimen The stresses at

both ends of the specimen are calculated with the followingelastic relations

120590

1=

119860

119861

119860

119878

sdot 119864

119861(120576

119868+ 120576

119877) (3)

120590

2=

119860

119861

119860

119878

sdot 119864

119861sdot 120576

119879 (4)

where119860119861and119860

119878are the cross-sectional areas of the bars and

the specimen respectively and 119864119861is Youngrsquos modulus of the

bar materialAsmentioned earlier the specimen is assumed to be stress

equilibrated in a SHPB experimentThis assumption must besatisfied in dynamic characterization of material propertiesConsequently the specimen deforms nearly uniformly suchthat the specimen response averaged over its volume is a goodrepresentative of the point-wise valid material behavior Thestress equilibration is expressed as

120590

1= 120590

2 (5)

Or from (3) and (4)120576

119868+ 120576

119877= 120576

119879 (6)

4 Shock and Vibration

2

4

6

8

10

12St

ress

(MPa

)

Strain

0000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

Pa)

0

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(b) minus15∘C

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(c) minus25∘C

Figure 4 Stress-strain curves of frozen soil under various strain rates

Equations (2) and (4) can thus be simplified as the follows

120576 = minus2

119862

119861

119871

119904

120576

119877

120576 = minus2

119862

119861

119871

119904

int

119905

0

120576

119877119889119905

120590 =

119860

119861

119860

119878

119864

119861120576

119879

(7)

Therefore once the incident reflected and transmittedsignals are measured the stress-strain data for the materialunder investigation can be obtained

23 Analysis of Experimental Results The SHPB tests wereconducted on the frozen soil at three different temperatures(minus5∘C minus15∘C and minus25∘C) and three strain rates (500s 750sand 950s)The stress-strain curves obtained after processingusing the two-wave method are presented in Figure 4 Thepeak stress and final strain increase with increasing loadingstrain rates

To more clearly illustrate the relationship between thepeak stress and strain rate the peak stress-strain rate curvesfor the three experimental temperatures are plotted inFigure 5

Figure 5 illustrates that the frozen soil exhibits an obviousstrain rate effect For a fixed temperature the peak stressincreases for higher strain rates In addition the strain rate

Shock and Vibration 5

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain rate (sminus1)

minus5∘Cminus15∘Cminus25∘C

1000900800700600500

Figure 5 Peak stress-strain rate curves for various temperatures

effect is more pronounced at lower temperature because icecrystals play a leading role during the entire crush resistanceprocess At a lower temperature the amount of ice is greaterin frozen soil the proportion of the antipressure ability inthe entire sample is thus greater and the strain rate effectbecomes more apparent

The dynamic stress-strain curves of frozen soil for afixed impact loading strain rate at various temperatures areplotted in Figure 6The peak stress increases with decreasingtemperature

To more clearly illustrate the relationship between thepeak stress and temperature the peak stress-temperaturecurves at three different strain rates are plotted in Figure 7

Figure 7 demonstrates the obvious temperature effect forthe frozen soil The peak stress increases upon increasingthe temperature for a given strain rate In addition thetemperature effect is more prominent at higher strain rate

After the impact loading experiments debris from thefractured frozen soil samples was collected Photographs ofthe fractured frozen soil samples after impact loading atminus15∘C are shown in Figure 8 It is apparent that the damageto the frozen soil was more severe at higher impact loadingstrain rates

3 Numerical Simulation Studies

The LS-DYNA program can solve geometric nonlinearity(large displacement large rotation and large strain) materialnonlinearity (dynamic models for more than 140 types ofmaterial) and contact nonlinearity (more than 50) problemsThis program is suitable for the numerical simulation ofSHPB experiments Based on the strain rate effect andtemperature effect of the frozen soil and its final destructionform the HJC model [13] was selected to simulate the SHPBimpact dynamic experiments of the frozen soil

31 HJC Material Constitutive Model The characteristics ofthe HJC constitutive model [13] reflect the dynamic responseof brittle materials such as frozen soil

120590

lowast

= [119860 (1 minus 119863) + 119861119901

lowast119873

] (1 + 119862 ln 120576

lowast

) (8)

where 119860 and 119861 are the normalized intensities 119873 and 119862 arethe pressure-hardening exponent and strain rate coefficientrespectively 120590lowast = (120590119891

1015840

119888

) is the ratio of the real equiva-lent strength and quasi-static uniaxial compressive strengthwhich is the normalized equivalent stress 119901lowast = (119901119891

1015840

119888

) isthe normalized hydrostatic force 120576lowast is the nondimensionalstrain rate a ratio of the true strain rate and reference strainrate 120576

0 and 119863 is the damage factor and is determined by the

accumulation of plastic strain which includes two parts of theequivalent plastic and plastic volumetric strain

119863 = sum

Δ120576

119901+ Δ120583

119901

120576

119891

119901+ 120583

119891

119901

(9)

Here Δ120576119901andΔ120583

119901are the equivalent plastic strain increment

and plastic volumetric strain increment respectively and 120576119891119901

and 120583119891119901are the equivalent plastic strain and plastic volumetric

strain respectively when frozen soil is broken under ordinarypressure

119901 is the actual hydrostatic pressure and can be determinedby the state equation of the curve [13] shown in Figure 9

The first stage represents the elastic compression region(119874119860 section) Here 119901 = 119870120583 where 119870 is the bulk modulus

The second stage represents the compaction deformationregion (119860119861 section) The internal pores of the frozen soilare gradually crushed and plastic volumetric damage will beproduced Here 119901 = 1198701015840120583 where 1198701015840 = (119901

1minus 119901

119888)(120583

1minus 120583

119888)

The third stage represents the region after the soliddeformation section (119861119862 section) The internal part of thefrozen soil consists entirely of crushed close-grainedmaterial

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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International Journal of

Shock and Vibration 3

Gage I Gage II

Cylinder Velocimeter

Data processingsystem

Striker

Absorb barRigid blockIncident barSpecimenTransmitted bar

Wheatstonebridge

Figure 2 Sketch of SHPB and its testing system

Incident bar Transmission barSpecimen

V1 V2

120576I

120576R

120576T

Figure 3 Testing section of SHPB

of the striker is 200mm the incident and transmission barsare made of the 7075-T6 aluminumwhose modulus is 71 GPaand density is 2100 kgm3 and the diameters of them are30mm

The incident and transmission bars are instrumentedwithstrain gauges to capture the elastic stress waves generated bythe striker bar Frozen soil specimens are placed between theincident and transmission bars The striker is launched bycompressed air An elastic compression stress wave (incidentwave) is generated by the impact of the striker bar strikingthe incident barThe incident wave is transmitted through thespecimen as a transmittedwave and is partially reflected at theinterface between the specimen and the incident bar Both theincident and reflected waves are recorded by the strain gaugeon the incident bar and the transmitted wave is recorded bythe strain gauge on the transmission bar

Assume that the stress waves propagate in both theincident and transmission bars without dispersion that is thepulses recorded at the strain gage locations represent those atthe bar ends in contact with the specimen one-dimensionalstress wave theory relates the particle velocities at both endsof specimen to the three measured strain pulses (Figure 3)

V1= 119862

119861(120576

119868minus 120576

119877)

V2= 119862

119861120576

119879

(1)

where 119862119861is the elastic bar wave speed of the bar material

and 119868 119877 and 119879 represent the incident and reflected and

transmitted pulses respectively The average engineeringstrain rate and strain in the specimen are

120576 =

V1minus V2

119871

119904

=

119862

119861

119871

119904

(120576

119868minus 120576

119877minus 120576

119879)

120576 = int

119905

0

120576 119889119905 =

119862

119861

119871

119904

int

119905

0

(120576

119868minus 120576

119877minus 120576

119879) 119889119905

(2)

where 119871119904is the initial length of the specimen The stresses at

both ends of the specimen are calculated with the followingelastic relations

120590

1=

119860

119861

119860

119878

sdot 119864

119861(120576

119868+ 120576

119877) (3)

120590

2=

119860

119861

119860

119878

sdot 119864

119861sdot 120576

119879 (4)

where119860119861and119860

119878are the cross-sectional areas of the bars and

the specimen respectively and 119864119861is Youngrsquos modulus of the

bar materialAsmentioned earlier the specimen is assumed to be stress

equilibrated in a SHPB experimentThis assumption must besatisfied in dynamic characterization of material propertiesConsequently the specimen deforms nearly uniformly suchthat the specimen response averaged over its volume is a goodrepresentative of the point-wise valid material behavior Thestress equilibration is expressed as

120590

1= 120590

2 (5)

Or from (3) and (4)120576

119868+ 120576

119877= 120576

119879 (6)

4 Shock and Vibration

2

4

6

8

10

12St

ress

(MPa

)

Strain

0000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

Pa)

0

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(b) minus15∘C

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(c) minus25∘C

Figure 4 Stress-strain curves of frozen soil under various strain rates

Equations (2) and (4) can thus be simplified as the follows

120576 = minus2

119862

119861

119871

119904

120576

119877

120576 = minus2

119862

119861

119871

119904

int

119905

0

120576

119877119889119905

120590 =

119860

119861

119860

119878

119864

119861120576

119879

(7)

Therefore once the incident reflected and transmittedsignals are measured the stress-strain data for the materialunder investigation can be obtained

23 Analysis of Experimental Results The SHPB tests wereconducted on the frozen soil at three different temperatures(minus5∘C minus15∘C and minus25∘C) and three strain rates (500s 750sand 950s)The stress-strain curves obtained after processingusing the two-wave method are presented in Figure 4 Thepeak stress and final strain increase with increasing loadingstrain rates

To more clearly illustrate the relationship between thepeak stress and strain rate the peak stress-strain rate curvesfor the three experimental temperatures are plotted inFigure 5

Figure 5 illustrates that the frozen soil exhibits an obviousstrain rate effect For a fixed temperature the peak stressincreases for higher strain rates In addition the strain rate

Shock and Vibration 5

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain rate (sminus1)

minus5∘Cminus15∘Cminus25∘C

1000900800700600500

Figure 5 Peak stress-strain rate curves for various temperatures

effect is more pronounced at lower temperature because icecrystals play a leading role during the entire crush resistanceprocess At a lower temperature the amount of ice is greaterin frozen soil the proportion of the antipressure ability inthe entire sample is thus greater and the strain rate effectbecomes more apparent

The dynamic stress-strain curves of frozen soil for afixed impact loading strain rate at various temperatures areplotted in Figure 6The peak stress increases with decreasingtemperature

To more clearly illustrate the relationship between thepeak stress and temperature the peak stress-temperaturecurves at three different strain rates are plotted in Figure 7

Figure 7 demonstrates the obvious temperature effect forthe frozen soil The peak stress increases upon increasingthe temperature for a given strain rate In addition thetemperature effect is more prominent at higher strain rate

After the impact loading experiments debris from thefractured frozen soil samples was collected Photographs ofthe fractured frozen soil samples after impact loading atminus15∘C are shown in Figure 8 It is apparent that the damageto the frozen soil was more severe at higher impact loadingstrain rates

3 Numerical Simulation Studies

The LS-DYNA program can solve geometric nonlinearity(large displacement large rotation and large strain) materialnonlinearity (dynamic models for more than 140 types ofmaterial) and contact nonlinearity (more than 50) problemsThis program is suitable for the numerical simulation ofSHPB experiments Based on the strain rate effect andtemperature effect of the frozen soil and its final destructionform the HJC model [13] was selected to simulate the SHPBimpact dynamic experiments of the frozen soil

31 HJC Material Constitutive Model The characteristics ofthe HJC constitutive model [13] reflect the dynamic responseof brittle materials such as frozen soil

120590

lowast

= [119860 (1 minus 119863) + 119861119901

lowast119873

] (1 + 119862 ln 120576

lowast

) (8)

where 119860 and 119861 are the normalized intensities 119873 and 119862 arethe pressure-hardening exponent and strain rate coefficientrespectively 120590lowast = (120590119891

1015840

119888

) is the ratio of the real equiva-lent strength and quasi-static uniaxial compressive strengthwhich is the normalized equivalent stress 119901lowast = (119901119891

1015840

119888

) isthe normalized hydrostatic force 120576lowast is the nondimensionalstrain rate a ratio of the true strain rate and reference strainrate 120576

0 and 119863 is the damage factor and is determined by the

accumulation of plastic strain which includes two parts of theequivalent plastic and plastic volumetric strain

119863 = sum

Δ120576

119901+ Δ120583

119901

120576

119891

119901+ 120583

119891

119901

(9)

Here Δ120576119901andΔ120583

119901are the equivalent plastic strain increment

and plastic volumetric strain increment respectively and 120576119891119901

and 120583119891119901are the equivalent plastic strain and plastic volumetric

strain respectively when frozen soil is broken under ordinarypressure

119901 is the actual hydrostatic pressure and can be determinedby the state equation of the curve [13] shown in Figure 9

The first stage represents the elastic compression region(119874119860 section) Here 119901 = 119870120583 where 119870 is the bulk modulus

The second stage represents the compaction deformationregion (119860119861 section) The internal pores of the frozen soilare gradually crushed and plastic volumetric damage will beproduced Here 119901 = 1198701015840120583 where 1198701015840 = (119901

1minus 119901

119888)(120583

1minus 120583

119888)

The third stage represents the region after the soliddeformation section (119861119862 section) The internal part of thefrozen soil consists entirely of crushed close-grainedmaterial

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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4 Shock and Vibration

2

4

6

8

10

12St

ress

(MPa

)

Strain

0000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

Pa)

0

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(b) minus15∘C

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain000 002 004 006 008 010

500 sminus1

750 sminus1950 sminus1

(c) minus25∘C

Figure 4 Stress-strain curves of frozen soil under various strain rates

Equations (2) and (4) can thus be simplified as the follows

120576 = minus2

119862

119861

119871

119904

120576

119877

120576 = minus2

119862

119861

119871

119904

int

119905

0

120576

119877119889119905

120590 =

119860

119861

119860

119878

119864

119861120576

119879

(7)

Therefore once the incident reflected and transmittedsignals are measured the stress-strain data for the materialunder investigation can be obtained

23 Analysis of Experimental Results The SHPB tests wereconducted on the frozen soil at three different temperatures(minus5∘C minus15∘C and minus25∘C) and three strain rates (500s 750sand 950s)The stress-strain curves obtained after processingusing the two-wave method are presented in Figure 4 Thepeak stress and final strain increase with increasing loadingstrain rates

To more clearly illustrate the relationship between thepeak stress and strain rate the peak stress-strain rate curvesfor the three experimental temperatures are plotted inFigure 5

Figure 5 illustrates that the frozen soil exhibits an obviousstrain rate effect For a fixed temperature the peak stressincreases for higher strain rates In addition the strain rate

Shock and Vibration 5

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain rate (sminus1)

minus5∘Cminus15∘Cminus25∘C

1000900800700600500

Figure 5 Peak stress-strain rate curves for various temperatures

effect is more pronounced at lower temperature because icecrystals play a leading role during the entire crush resistanceprocess At a lower temperature the amount of ice is greaterin frozen soil the proportion of the antipressure ability inthe entire sample is thus greater and the strain rate effectbecomes more apparent

The dynamic stress-strain curves of frozen soil for afixed impact loading strain rate at various temperatures areplotted in Figure 6The peak stress increases with decreasingtemperature

To more clearly illustrate the relationship between thepeak stress and temperature the peak stress-temperaturecurves at three different strain rates are plotted in Figure 7

Figure 7 demonstrates the obvious temperature effect forthe frozen soil The peak stress increases upon increasingthe temperature for a given strain rate In addition thetemperature effect is more prominent at higher strain rate

After the impact loading experiments debris from thefractured frozen soil samples was collected Photographs ofthe fractured frozen soil samples after impact loading atminus15∘C are shown in Figure 8 It is apparent that the damageto the frozen soil was more severe at higher impact loadingstrain rates

3 Numerical Simulation Studies

The LS-DYNA program can solve geometric nonlinearity(large displacement large rotation and large strain) materialnonlinearity (dynamic models for more than 140 types ofmaterial) and contact nonlinearity (more than 50) problemsThis program is suitable for the numerical simulation ofSHPB experiments Based on the strain rate effect andtemperature effect of the frozen soil and its final destructionform the HJC model [13] was selected to simulate the SHPBimpact dynamic experiments of the frozen soil

31 HJC Material Constitutive Model The characteristics ofthe HJC constitutive model [13] reflect the dynamic responseof brittle materials such as frozen soil

120590

lowast

= [119860 (1 minus 119863) + 119861119901

lowast119873

] (1 + 119862 ln 120576

lowast

) (8)

where 119860 and 119861 are the normalized intensities 119873 and 119862 arethe pressure-hardening exponent and strain rate coefficientrespectively 120590lowast = (120590119891

1015840

119888

) is the ratio of the real equiva-lent strength and quasi-static uniaxial compressive strengthwhich is the normalized equivalent stress 119901lowast = (119901119891

1015840

119888

) isthe normalized hydrostatic force 120576lowast is the nondimensionalstrain rate a ratio of the true strain rate and reference strainrate 120576

0 and 119863 is the damage factor and is determined by the

accumulation of plastic strain which includes two parts of theequivalent plastic and plastic volumetric strain

119863 = sum

Δ120576

119901+ Δ120583

119901

120576

119891

119901+ 120583

119891

119901

(9)

Here Δ120576119901andΔ120583

119901are the equivalent plastic strain increment

and plastic volumetric strain increment respectively and 120576119891119901

and 120583119891119901are the equivalent plastic strain and plastic volumetric

strain respectively when frozen soil is broken under ordinarypressure

119901 is the actual hydrostatic pressure and can be determinedby the state equation of the curve [13] shown in Figure 9

The first stage represents the elastic compression region(119874119860 section) Here 119901 = 119870120583 where 119870 is the bulk modulus

The second stage represents the compaction deformationregion (119860119861 section) The internal pores of the frozen soilare gradually crushed and plastic volumetric damage will beproduced Here 119901 = 1198701015840120583 where 1198701015840 = (119901

1minus 119901

119888)(120583

1minus 120583

119888)

The third stage represents the region after the soliddeformation section (119861119862 section) The internal part of thefrozen soil consists entirely of crushed close-grainedmaterial

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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International Journal of

Shock and Vibration 5

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain rate (sminus1)

minus5∘Cminus15∘Cminus25∘C

1000900800700600500

Figure 5 Peak stress-strain rate curves for various temperatures

effect is more pronounced at lower temperature because icecrystals play a leading role during the entire crush resistanceprocess At a lower temperature the amount of ice is greaterin frozen soil the proportion of the antipressure ability inthe entire sample is thus greater and the strain rate effectbecomes more apparent

The dynamic stress-strain curves of frozen soil for afixed impact loading strain rate at various temperatures areplotted in Figure 6The peak stress increases with decreasingtemperature

To more clearly illustrate the relationship between thepeak stress and temperature the peak stress-temperaturecurves at three different strain rates are plotted in Figure 7

Figure 7 demonstrates the obvious temperature effect forthe frozen soil The peak stress increases upon increasingthe temperature for a given strain rate In addition thetemperature effect is more prominent at higher strain rate

After the impact loading experiments debris from thefractured frozen soil samples was collected Photographs ofthe fractured frozen soil samples after impact loading atminus15∘C are shown in Figure 8 It is apparent that the damageto the frozen soil was more severe at higher impact loadingstrain rates

3 Numerical Simulation Studies

The LS-DYNA program can solve geometric nonlinearity(large displacement large rotation and large strain) materialnonlinearity (dynamic models for more than 140 types ofmaterial) and contact nonlinearity (more than 50) problemsThis program is suitable for the numerical simulation ofSHPB experiments Based on the strain rate effect andtemperature effect of the frozen soil and its final destructionform the HJC model [13] was selected to simulate the SHPBimpact dynamic experiments of the frozen soil

31 HJC Material Constitutive Model The characteristics ofthe HJC constitutive model [13] reflect the dynamic responseof brittle materials such as frozen soil

120590

lowast

= [119860 (1 minus 119863) + 119861119901

lowast119873

] (1 + 119862 ln 120576

lowast

) (8)

where 119860 and 119861 are the normalized intensities 119873 and 119862 arethe pressure-hardening exponent and strain rate coefficientrespectively 120590lowast = (120590119891

1015840

119888

) is the ratio of the real equiva-lent strength and quasi-static uniaxial compressive strengthwhich is the normalized equivalent stress 119901lowast = (119901119891

1015840

119888

) isthe normalized hydrostatic force 120576lowast is the nondimensionalstrain rate a ratio of the true strain rate and reference strainrate 120576

0 and 119863 is the damage factor and is determined by the

accumulation of plastic strain which includes two parts of theequivalent plastic and plastic volumetric strain

119863 = sum

Δ120576

119901+ Δ120583

119901

120576

119891

119901+ 120583

119891

119901

(9)

Here Δ120576119901andΔ120583

119901are the equivalent plastic strain increment

and plastic volumetric strain increment respectively and 120576119891119901

and 120583119891119901are the equivalent plastic strain and plastic volumetric

strain respectively when frozen soil is broken under ordinarypressure

119901 is the actual hydrostatic pressure and can be determinedby the state equation of the curve [13] shown in Figure 9

The first stage represents the elastic compression region(119874119860 section) Here 119901 = 119870120583 where 119870 is the bulk modulus

The second stage represents the compaction deformationregion (119860119861 section) The internal pores of the frozen soilare gradually crushed and plastic volumetric damage will beproduced Here 119901 = 1198701015840120583 where 1198701015840 = (119901

1minus 119901

119888)(120583

1minus 120583

119888)

The third stage represents the region after the soliddeformation section (119861119862 section) The internal part of thefrozen soil consists entirely of crushed close-grainedmaterial

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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International Journal of

6 Shock and Vibration

0

3

6

9

12

15

18St

ress

(MPa

)

minus5∘Cminus15∘Cminus25∘C

Strain000 002 004 006 008

(a) 500s

Stre

ss (M

Pa)

0

5

10

15

20

25

Strain000 002 004 006 008

minus5∘Cminus15∘Cminus25∘C

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

pa)

Strain000 002 004 006 010008

minus5∘Cminus15∘Cminus25∘C

(c) 950s

Figure 6 Stress-strain curves of frozen soil under various temperatures

Here 119901 = 119870

1120583 + 119870

2120583

2

+ 119870

3120583

3 where 1198701 1198702 and 119870

3are

material constantsThe failure in the HJC model is mainly compression

failure For the tensile damage model of a brittle materialsuch as frozen soil volume strain failure criteria must beappended a failure strain of 0005 was used

32 Finite ElementModel Thefinite elementmodel consistedof four parts the bullet incident bar transmission barsand sample For comparison with the experimental resultsa cylinder model with a diameter of 30mm and height of18mm was used for the frozen soil The mapping meshmethod was used which is suitable for wave propagation anddynamic contact calculation The size of the mesh openingof the member bars was 4mm Considering the calculated

amount and accuracy and the size of the mesh opening of thefrozen soil sample the main part of the study was 1mm Bothmember bars and the sample used SOLID164 Automatedsingle face contact was used and the friction between deviceswas ignored

33 Selection of Material Parameters Themember bars usedthe linear elastic model similar to the experimental materialThe material parameters are listed in Table 2

The study simulated frozen soil samples using the HJCmodel The serial number of the HJC model was 111 in theLS-DYNA software program and consisted of 21 parametersin total The basic material parameters were the density 119877

0

shear modulus 119866 static compressive strength 119891119888 and tensile

strength 119879 The intensity parameters were 119860 119861 119862 119873 and

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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International Journal of

Shock and Vibration 7

6

9

12

15

18

21

24

27

Stre

ss (M

Pa)

minus24 minus18 minus12 minus6

T (∘C)

500 sminus1

750 sminus1

950 sminus1

Figure 7 Peak stress-temperature curves under various strain rates

(a) 500s (b) 750s

(c) 950s

Figure 8 Photographs of the fractured frozen soil samples after impact loading at various strain rates

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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International Journal of

8 Shock and Vibration

A

O 120583120583t120583c

Bp

p

l

pc

Figure 9 Hydrostatic pressure-volumetric strain curve

Table 2 Linear elastic material parameters of bars

Device ROkgm3 119864Pa PRBullet 8001198903 19511989011 030Incidenttransmission bars 2101198903 70011989010 030

Table 3 Initial material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

241198903 14861198909 481198906 079 160 0007 061

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 41198906 1601198906 0001

119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

SFMAXThe damage parameters were11986311198632 and 120576

119891minThepressure parameters were 119875

119888 120583119888 119875119897 120583119897 1198701 1198702 and 119870

3 The

reference strain rate was 120576

0 and the failure type was 119891

119904

In the experiments the temperature of the frozen soilwas variable 119866 of the frozen soil is the most sensitive totemperature of the four basic parameters The remainingthree basic parameters remain basically invariant with tem-perature change In this numerical simulation 119866 changedwith changing temperature The density of frozen soil is2100 kgm3 According to the available experimental data[14ndash16] 119891

119888is 90MPa 119879 is 03MPa and 119866 is in the range of

500 to 2500MPaAll the initial parameters of the HJC model which are

listed in Table 3 were obtained from the literature [13]The initial parameters were used as the standard parame-

ter set and the sensitivity of each parameter was analyzedWhen one parameter was analyzed the other parameterswere fixed The parameter was considered a sensitive param-eter if a small change in the parameter led to a large changein the result Based on repeated numerical simulations 119860

119861 119862 and 119873 were determined to be sensitive parameters ofthe HJC model for frozen soil In addition it was necessaryto determine the effects of the sensitive parameters on theconstitutive law to provide a theoretical basis and referenceguide for data fitting In this study the stress formula is asfollows

120590 =

119860

119861

119860

119878

119864

119861120576

119879 (10)

where 119860119861and 119860

119878are the same thus the trend of the stress

curve is the same as that of the transmission wave strainTo reduce the number of calculations only the trend of thetransmission wave was determined

331 Normalized Parameters 119860 and 119861 1198770 119866 119891119888 and 119879 were

replaced with the material parameter values of frozen soilThe remainder of the parameters were fixed and the value of119860was changedThe change in the transmitted wave is plottedin Figure 10

Figure 10 demonstrates that the peak stress increaseswhen the parameter 119860 increases In addition the waveformexhibits a slight difference in that the rising period becomessteep and the declining period becomes more gradual Thisresult occurs because 119860 is the cohesion strength therefore agreater value of119860 results in a greater peak stress In addition119860 is directly proportional to the damage and (1minus119863) is alwayspositive Hence the stress value increases with increasing 119860

The value of the parameter 119861 was changed separately toanalyze its effect on the transmitted wave The change of thetransmitted wave is shown in Figure 11

Figure 11 demonstrates that the peak stress increases withincreasing 119861 Its elastic stage is completely overlapped and itbegins to change at the yield point The ascent stage becomessteeper with an increase in 119861The final declining stage almostcoincides for all the values of 119861 119861 only affects the valueof the peak stress and does not control the wave shape

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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International Journal of

Shock and Vibration 9

200 250 300 3500

9

18

27

36

45

Stre

ss (M

Pa)

Time (ms)

A = 040

A = 079

A = 120

Figure 10 Transmitted waves for various values of 119860

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

B = 12

B = 16

B = 20

Figure 11 Transmitted waves for various values of 119861

because 119861 is the standard strain-hardening coefficient whichis directly proportional to the pressure term in the yieldsurface equation

332 Pressure-Hardening Exponent N The change in thetransmitted wave resulting from changing the value of N isillustrated in Figure 12

Figure 12 demonstrates that the elastic stages are primar-ily coincident In the plastic stage with increasing119873 the ris-ing slope is gradually reduced and the peak stress graduallydecreases In addition the waveform width increases withincreasing119873

200 250 300 3500

5

10

15

20

25

30

Stre

ss (M

Pa)

Time (ms)

N = 081

N = 061N = 041

Figure 12 Transmitted waves for various values of119873

200 250 300 3500

4

8

12

16

20

24

28St

ress

(MPa

)

C = 0004

C = 0007

C = 0001

Time (120583s)

Figure 13 Transmitted waves for various values of 119862

333 Strain Rate Coefficient C Thechange in the transmittedwave for various values of the parameter 119862 is shown inFigure 13

The elastic stages are observed to be almost coincidentand the slope of the yielding stage is primarily the same Onlythe peak stress increases upon increasing 119862 the wave shapeis not affected

Through sensitivity analysis of the HJC parameters theeffects of the parameters on the final waveform curve weredetermined 119860 and 119861 affect the value of peak stress119873 affectsthe value of the peak stress and pulse width and 119862 affectsthe effect of strain rate According to the effect of theseparameters and the experimental results 119860 = 12 119861 = 05119862 = 0012 and 119873 = 10 The frozen soil parameters of theHJC model are listed in Table 4

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Shock and Vibration

3

6

9

12St

ress

(Mpa

)

Strain

0012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(a) minus5∘C

4

8

12

16

20

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(b) minus15∘C

4

8

12

16

20

24

28

Stre

ss (M

pa)

0

Strain012010008006004002000

500 sminus1 Exp750 sminus1 Exp950 sminus1 Exp

500 sminus1 Sim750 sminus1 Sim950 sminus1 Sim

(c) minus25∘C

Figure 14 Comparison of experimental curves with numerical simulation curves under various strain rates

Table 4 Modified material parameters of HJC constitutive model

120588

0

kgm3 119866Pa 119891

1015840

119888

Pa 119860 119861 119862 119873

211198903 21198909 91198906 12 05 0012 10

119878max 119863

1

119863

2

120576

119891min 119879Pa 119901

119888

Pa 120583

119888

70 004 10 001 31198905 1601198906 0001119901

1

Pa 120583

1

119896

1

Pa 119896

2

Pa 119896

3

Pa 120576

0

119891

119904

0811198909 01 851198909 minus1711198909 2081198909 1119890 minus 6 0004

Among these parameters 119866 is a factor of critical influ-ence for frozen soil as it increases sharply with decreasingtemperature Three temperatures were used in the frozensoil experiments in this paper According to the existing

experimental data [16] 119866 is 500 1500 and 2500MPa whenthe temperature is minus5∘C minus15∘C and minus25∘C respectively

4 Results and Analyses

41 Strain Rate Effect and Temperature Effect The incidentreflected waves and transmitted wave were processed usingthe two-wavemethod and then the stress-strain curves werereconstructed for comparison with the SHPB experimentalcurves The first group of experiments was performed at agiven temperature and the experimental strain rates werealtered by changing the impact speed the numerical simu-lation curves are compared with the experimental curves inFigure 14

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 11

3

6

9

12

15

18St

ress

(MPa

)

Strain

0008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(a) 500s

000 003 006 0090

5

10

15

20

25

Strain

Stre

ss (M

Pa)

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(b) 750s

0

4

8

12

16

20

24

28

Stre

ss (M

Pa)

Strain012010008006004002000

minus5∘C Expminus15∘C Expminus25∘C Exp

minus5∘C Simminus15∘C Simminus25∘C Sim

(c) 950s

Figure 15 Comparison of experimental curves and numerical simulation curves under various temperatures

Figure 14 demonstrates that the peak stress and finalstrain fit well and increase with increasing strain rate Thisresult occurs because of the internal structure of frozen soilThe ice in frozen soil is a brittle material and under theconditions of high-strain-rate impact loading the damageand destruction of ice crystals play a leading role At a higherstrain rate more crack extension occurs in the same timeresulting in more energy absorption Therefore the stresspeak and final strain increase with increasing strain rate andan apparent strain rate effect is observed

The second group of experiments were performed ata given impact loading speed (thus the strain rate isconstant) and the experimental temperature was changed

A comparison of the numerical simulation curves with theexperimental curves is shown in Figure 15

Figure 15 demonstrates that the curves fit well Thepeak stress increases with decreasing temperature and thefinal strain rate is primarily the same which is called thestrain convergence phenomenon At a lower temperature agreater amount of ice remains in the frozen soil namely thecompressive capacity is higher Therefore the temperatureeffect of frozen soil is apparent

42 Homogeneity Analysis For the measurement of samplestress uniformity different studies have adopted different

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

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DistributedSensor Networks

International Journal of

12 Shock and Vibration

00

03

06

09

12

15

18

21

t998400r

1086420

120572k

Figure 16 Time history-sample stress uniformity curve

E1

E2

E3

E4

E5

E6

O

Figure 17 Location map of six points within frozen soil

methods In this work the ratio of the stress value differenceand the average value on both sides is used to measure thestress uniformity in terms of 120572

119896[17ndash19] the equation is

120572

119896=

Δ120590

119896

120590

119896

times 100 (11)

where Δ120590119896is the stress difference on both sides of the frozen

soil 120590119896is the stress average on both sides of the frozen soil

and 120572119896is the ratio between these values As 120572

119896approaches

zero the sample stress uniformity is better Generally if |120572119896| le

5 the stress distribution in a sample meets the requirementof stress uniformity In addition the nondimensional risetime 1199051015840

119903

is introduced

119905

1015840

119903

=

119905

119903

120591

119904

(12)

where 119905119903is the incident wave leading edge rise time and 120591

119904is

the time required for the stress wave to spread from the front

0

8

16

24

32

Stre

ss (M

Pa)

E1E2E3

E4E5E6

Time (120583s)260250240230220210200190

Figure 18 Stress-time curves of different points along a vertical shaftof frozen soil

facet of the sample (close to the incident bar) to the back facet(near the transmission bar) along the loading direction Therelationship between the rise time and stress homogeneity ofthe sample is shown in Figure 16The strain rate is 950s andthe temperature of the frozen soil is minus15∘C

Figure 16 demonstrates that there is a sharp shock near119905

1015840

119903

= 1 Then the curve quickly approaches zero For 1199051015840119903

ge 2the shock of the curve decreases For 1199051015840

119903

ge 3 the curve isprimarily stable and the overall level is close to zero Thesample is thought to reach a uniform stress state

43 Internal Stress Distribution of Frozen Soil Along a verti-cal axis of the frozen soil sample six points were obtained onaverage The vertical wheel base away from the center of thecircle was 08 cmThe locations of the six points are shown inFigure 17

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 13

4641e minus 161982e minus 16

minus6770e minus 17minus3336e minus 16minus5995e minus 16minus8654e minus 16minus1131e minus 15

Fringe levels

(a) 173120583119904

minus1159e minus 07minus1739e minus 07

minus2097e minus 15minus5797e minus 08

minus2319e minus 07minus2898e minus 07minus3478e minus 07

Fringe levels

(b) 191 120583119904

minus2112e minus 05minus3330e minus 05

3228e minus 06minus8947e minus 06

minus4547e minus 05minus5764e minus 05minus6982e minus 05

Fringe levels

(c) 201 120583119904

minus1082e minus 04minus1606e minus 04

minus3420e minus 06minus5580e minus 05

minus2130e minus 04minus2653e minus 04minus3177e minus 04

Fringe levels

(d) 209 120583119904

minus9685e minus 05minus1327e minus 04

minus2504e minus 05minus6094e minus 05

minus1687e minus 04minus2046e minus 04minus2405e minus 04

Fringe levels

(e) 215 120583119904

Figure 19 The internal stress distribution clouds of frozen soil

E1 is a point on the front facet of the frozen soil sampleand E6 is a point on the back facet The stress-time curves ofsix points are plotted in Figure 18

Figure 18 demonstrates that all the curves exhibit a trendthat the stress value moves down and up This result isobserved because the stress wave reflects when it is spread tothe back facet The point E1 is forced first at approximately192 120583119904 and its oscillation of the first peak is more obviousthan the other points because E1 is on the front facet withinstability Then point E2 is forced at approximately 195 120583119904Finally E6 is forced at approximately 200120583119904The propagationfrom the front facet of the sample to back facet is apparent

44 Impact Failure Mode of Frozen Soil The failure processof frozen soil in the SHPB experiment is on the level of

microseconds level and cannot usually be observed Evenwith the use of high-speed camera only the damage of theouter surface on frozen soil can be roughly observed Thedestruction of internal frozen soil cannot be observed WithDYNA numerical simulations the entire failure process andfailure mode of frozen soil can be observed in detail in theform of slices Based on the numerical simulation results thefailure process can be divided into three stagesThe first stageoccurs before the failure of the sample in this stage uniformstress is achieved through reflection of the shock waves in thesample The second stage is called the crack formation stageThe third stage is called the crushed sample stage

The first stage is illustrated in Figure 19 The uniformstress in the sample is achieved before the failure of thesample To see the internal stress distribution the samplemust be slicedThe stress clouds from left to right are the slices

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 Shock and Vibration

0000

0002

0004

0006

0008St

ress

(MPa

)

Element

E1

E2 E3 E4 E5 E6

76543210

(a) 191120583119904

00

07

14

21

28

35

E6E5

Stre

ss (M

Pa)

E1

E2

E3

E4

Element76543210

(b) 201 120583119904

0

4

8

12

16

20

E6E5E4

E3

E2

Stre

ss (M

Pa)

E1

Element76543210

(c) 209120583119904

E6E5

E4

E3

E2E1

Element76543210

0

4

8

12

16

20

24

Stre

ss (M

Pa)

(d) 213 120583119904

Figure 20 Comparison of stress values at different locations

from the front facet to the back facet of the sample at somemoment

Figure 19(a) shows that the sample is not subjected toforce before the shock wave and remains at an equilibriumstress state In Figure 19(b) the stress wave has just come intocontact with the sample and the front facet of the sample issubjected to the forceThen the pressure is transferred to theback facet In Figure 19(c) the stress wave is just reflected onthe back facet and the back facet is under tension During theprocess shown in Figures 19(d) and 19(e) the internal stressof the sample is primarily the same and the sample is thoughtto achieve a uniform stress state

To observe the propagation of the stress wave in thesample more clearly and intuitively six data points wereobtained as shown in Figure 17 E1 is a point on the front facetof the sample and E6 is a point on the back facet The stressanalyses of the six points are shown in Figure 20

In Figure 20(a) the front facet has just been subjectedto a stress wave and the stress value of E1 is significantlygreater than that at the other points In Figure 20(b) thestress value of E6 becomes negative indicating that the stress

wave is reflected and has a tensile function on the back facetIn Figure 20(c) the stress values of the six points exhibit adecreasing trend indicating the transmission of the stresswave from the front facet to the back facet of the sampleAfter a period of reflection the frozen soil sample reaches auniform stress state as observed in Figure 20(d)

After the uniform stress is attained in the sample with thespread of the stress wave the stress of the sample is graduallyincreased Then the second stage occurs as observed inFigure 21

Because of the boundary effect the forces on the frontand back facets of the sample are greater than on the othersurfaces A compression wave forms from the tension waveafter it is reflected on the side surface of the sample thatis free Although the tensile strength is not large becausethe tensile strength of frozen soil is small the exterior ofthe sample would be destroyed first as illustrated in Figures21(a) and 21(b) Afterwards the destruction on the two endfaces is extended along the outside and central surfacesGradually the larger pieces shown in Figures 21(c)ndash21(f) areformed

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 15

minus9685e minus 05

minus2504e minus 05

minus6094e minus 05

minus1327e minus 04

minus1687e minus 04

minus2046e minus 04

minus2405e minus 04

Fringe levels

(a) 215 120583119904

minus1583e minus 04

minus6179e minus 06

minus8222e minus 05

minus2343e minus 04

minus3104e minus 04

minus3864e minus 04

minus4624e minus 04

Fringe levels

(b) 225120583119904

minus5088e minus 05

3118e minus 05

minus9849e minus 06

minus9191e minus 05

minus1329e minus 04

minus1740e minus 04

minus2150e minus 04

Fringe levels

(c) 237120583119904

minus2430e minus 05

3691e minus 05

6308e minus 06

minus5490e minus 05

minus8550e minus 05

minus1161e minus 04

minus1467e minus 04

Fringe levels

(d) 243120583119904

minus1724e minus 05

3980e minus 05

1128e minus 05

minus4576e minus 05

minus7428e minus 05

minus1028e minus 04

minus1313e minus 04

Fringe levels

(e) 251 120583119904

minus2882e minus 05

4967e minus 05

1042e minus 05

minus6807e minus 05

minus1073e minus 04

minus1466e minus 04

minus1858e minus 04

Fringe levels

(f) 255120583119904

Figure 21 Stress clouds of damage stage

3821e minus 051702e minus 05

minus4168e minus 06minus2536e minus 05minus4655e minus 05minus6773e minus 05minus8892e minus 05

Fringe levels2629e minus 051042e minus 05

minus5451e minus 06minus2132e minus 05minus3719e minus 05minus5306e minus 05minus6892e minus 05

Fringe levels1889e minus 051067e minus 052447e minus 06

minus5775e minus 06minus1400e minus 05minus2222e minus 05minus3044e minus 05

Fringe levels

315120583s267120583s261120583s

Figure 22 Stress clouds of crushed sample stage

If the strain rate is sufficiently high the sample isdestroyed sequentially The fragments are smaller and theirquantity is greaterThis stage is called the third stage (crushedsample stage) and is illustrated in Figure 22

A higher impact velocity results in a greater loading strainrate and a smaller broken sample The numerical simulationand experimental results were identical

5 Conclusions

SHPBs with diameters of 30mm were used to performimpact experiments of frozen soil under various impactvelocities and temperatures In addition using the finiteelement analysis software LS-DYNA SHPB experiments offrozen soil were simulated

(1) The strain rate effect and temperature effect of frozensoil under impact loadings were investigated in theexperiments For a given frozen soil temperature thepeak stress and final strain increased with increasingstrain rate For a given strain rate the peak stressincreased with decreasing temperature and the finalstrain converged

(2) Using the HJC model the dynamic mechanicalbehavior of frozen soil under impact loadings wasnumerically simulated The strain rate effect andtemperature effect of frozen soil under impact load-ings were verified In addition to determine morereasonable parameters for the model the effects ofthe sensitive parameters in the HJC model on thecalculation results were evaluated

(3) Using numerical simulations the stress-strain curvesof frozen soil under impact loadings were obtainedand compared with the corresponding experimentalcurves The curve fitting was good and the stressuniformity of the frozen soil sample was verifiedThe stress-time curves of selected points on a verticalaxis in sample were obtained The stress value ofeach section reached a uniform stress state before itsdestruction In addition the propagation of the stresswave was reflected inside the sample

(4) Based on the numerical simulation the destructionprocess of frozen soil under impact loadings canbe divided into three stages a uniform stress stagecrack formation stage and crushed sample stage

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 Shock and Vibration

In addition for a higher impact velocity the loadingstrain rate was greater and the broken sample wassmaller The numerical simulation and experimentalresults were identical

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (11172251) and the Project of SichuanProvincial Youth Science and Technology Innovation TeamChina (2013TD0004)

References

[1] B S Chen S S Hu Q Y Ma and Z Y Tu ldquoExperimentalresearch of dynamic mechanical behaviors of frozen soilrdquoChinese Journal ofTheoretical andAppliedMechanics vol 37 no6 pp 724ndash728 2005

[2] H-D Zhang Z-W Zhu S-C SongG-Z Kang and J-GNingldquoDynamic behavior of frozen soil under uniaxial strain andstress conditionsrdquo Applied Mathematics and Mechanics vol 34no 2 pp 229ndash238 2013

[3] Q YMa J S ZhangW F Chen and P Yuan ldquoAnalysis of SHPBtest and impact compression in confining pressure for artificialfrozen soilrdquo Rock and Soil Mechanics vol 35 no 3 pp 637ndash6402014

[4] Q Y Ma P Yuan W F Chen and J S Zhang ldquoComparativeanalysis on dynamic mechanical properties of artificial frozensoil under uniaxial load and confining pressurerdquo ChineseJournal of Underground Space and Engineering vol 10 no 1 pp26ndash29 2014

[5] Z-Q Liu J-K Liu B Wang H-L Zhang and X-F LildquoDynamic characteristics of frozen clay by using SHPB testsrdquoChinese Journal of Geotechnical Engineering vol 36 no 3 pp409ndash416 2014

[6] Y Ma Z-W Zhu W Ma and J-G Ning ldquoCharacteristics ofstress-strain curves and convergence phenomenon of frozensoil under dynamic loadingrdquo EngineeringMechanics vol 32 no10 pp 52ndash59 2015

[7] M D Furnish ldquoMeasuring static and dynamic properties offrozen silty soilsrdquo Tech Rep 98-1497 Office of Scientific ampTechnical Information 1998

[8] M Y Les A Fossum and S Laurence ldquoFrozen soil materialtesting and constitutive modelingrdquo Sandia Report SAND 2002-0524 2002

[9] X TWu S F Sun andH P Li ldquoNumerical simulation of SHPBtests for concrete by using HJC modelrdquo Explosion and ShockWaves vol 29 no 2 pp 137ndash142 2009

[10] W C Zhu Y Bai X B Li and L L Niu ldquoNumerical simulationon rock failure under combined static and dynamic loadingduring SHPB testsrdquo International Journal of Impact Engineeringvol 49 pp 142ndash157 2012

[11] C E Anderson Jr P E OrsquoDonoghue J Lankford and JD Walker ldquoNumerical simulations of SHPB experiments forthe dynamic compressive strength and failure of ceramicsrdquoInternational Journal of Fracture vol 55 no 3 pp 193ndash208 1992

[12] T Chakraborty ldquoImpact simulation of rocks under SHPB testrdquoProceedings of the Indian National Science Academy vol 79 no4 pp 605ndash613 2013

[13] T J Holmquist G R Johnson and W H Cook ldquoA computa-tional constitutive model for concrete subjected to large strainshigh strain rates and high pressuresrdquo in Proceedings of the14th International Symposium on Ballistics vol 9 pp 591ndash600Quebec Canada 1993

[14] X Haibin ldquoThe relationship between uniaxial compressivestrength of artificial frozen soil and temperature moisturecontentrdquo Geotechnical Engineering Word vol 11 no 4 pp 60ndash63 2008

[15] Z Jingfeng ldquoAn experimental study on the relationship betweentensile strength and temperature and water ratio of frozen soilrdquoGeology and Prosprcting vol 47 no 6 pp 1158ndash1161 2011

[16] L Wang Q Hu X Ling D Cai and X Xu ldquoExperimentalstudy on dynamic shear modulus of remolded frozen silty clayfor Qinghai-Xizang Railwayrdquo Journal of Earthquake Engineeringand Engineering Vibration vol 27 no 2 pp 177ndash180 2007

[17] Z Fenghua W Lili and H Shisheng ldquoOn the effect of stressnonuniformness in polymer specimen of SHPB testsrdquo Journalof Experimental Mechanics vol 7 no 1 pp 23ndash29 1992

[18] P Feng Q-M Zhang L Chen and W Yao ldquoInfluence ofincident pulse of slope on stress uniformity and constant strainrate in SHPB testrdquo Transaction of Beijing Institute of Technologyvol 30 no 5 pp 513ndash516 2010

[19] L Song and S-S Hu ldquoStress uniformity and constant strain ratein SHPB testrdquoExplosion and ShockWaves vol 25 no 3 pp 207ndash216 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of