J. agric. Engng. Res. (2001) 80 (2), 217}222doi:10.1006/jaer.2001.0711, available online at http://www.idealibrary.com onSW*Soil and Water

Mechanical Properties of Soils under Explosive Loading

L. Zhixiong1; W. Yaohua2; P. Junzheng3

1Agricultural Engineering College, Nanjing Agricultural University, Nanjing 210032, People's Republic of China; e-mail of corresponding author:luzx@jlonline.com

2Nanjing Engineering Institute, Nanjing 210007, People's Republic of China; e-mail: wangyh@jlonline.com3Agricultural Engineering College, Nanjing Agricultural University, Nanjing 210032, People's Republic of China; e-mail: panjzh@jlonline.com

(Received 2 June 2000; accepted in revised form 17 February 2001; published online 10 August 2001)

A series of stress histories were detected successfully by plate explosive impact, using piezoelectric crystal gaugespre-positioned at di!erent depths of clayey and sandy soils. The soil dynamic constitutive relation was obtainedfrom the measured stress histories, by "tting a stress}depth}time surface. Results showed that the stress}straincurves had obvious stagnant-return phenomena, strain rate e!ect and waveform dissipation. These propertiesare important for correct evaluation of the dynamic behaviour of soils. Di!erent deformation characteristics ofsoils were found at di!erent depths. The surface layer displayed #uid elastoplastic property, while the middleand deeper layer displayed viscoplastic and elastic properties, respectively.

( 2001 Silsoe Research Institute

1. Introduction

A soil dynamic constitutive relation is fundamental forthe further study of the soil failure mechanism, stresswave transmission and the feasibility of soil tillage byexplosive loading. However, it is di$cult to formulatedynamic constitutive relations for soils under high velo-city impact loading by the traditional method in whicha theoretical model was "rstly proposed with adjustablecoe$cients determined afterwards through experiments.In order to determine the constitutive relation of a mater-ial under uniaxial strain conditions, Lagrangian analysiswas applied. The Lagrangian analysis is a procedure forderiving the stress}strain relations for a material froma series of stress or velocity histories measured duringone-dimensional stress wave travels through thematerial.

Fowles and Williams (1970) introduced the Lagran-gian analysis for processing data from a series of stress orvelocity gauges in material through which one-dimen-sional planar waves were passing. Later, Cowperthwaiteand Williams (1971) generalized the method to accountfor attenuation of the wave peaks. Grady (1973) intro-duced the concept of a path line to aid in computing

0021-8634/01/100217#06 $35.00/0 217

derivatives needs for attenuating #ow. Seaman (1974)constructed a computer program for routine applicationof the path line method. Vantine and Curtis (1981) ap-plied the Lagrangian method to waves under explosivesloading and compared the results of integrations from&directly' computed derivatives with those from deriva-tives based on path lines. They recommended use of thepath line method. The Lagrangian method has also beenapplied to analyse large-scale "eld experiments in spheri-cal and cylindrical geometries by Fowles (1970). Seaman(1987) and Forest (1991) improved Lagrangian analysiswith a curve "tting method.

However, these studies are only on small strain-ratematerials. The paper studies the dynamic constitutiverelation for large strain-rate soils to determine the feasi-bility of soil tillage by explosive loading.

2. Measurement of soil stress

Clayey and sandy soils were selected for measurementand analysis of soil stress}strain. The bulk density, speci-"c gravity, water content of the sample clayey soil was1)904}1)927g cm~3, 2)75, 15)3% respectively. Particle

( 2001 Silsoe Research Institute

Fig. 1. Scheme of planar impact experiment

L. ZHIXIONG E¹ A¸ .218

Notation

Ai"tting values at loading

Bi"tting values at unloading

E internal energyh Lagrangian position (depth), mj path linet time, su particle velocity, m s~1

e straine0

strain at the beginning of unloadingl speci"c volume

o1

initial density, g cm~3

p stress in the direction of propagation, Pap0

stress at the beginning of unloading, Pa

sizes of the sample clayey soil collected from the explos-ive site are shown in Table 1.

Sandy soil is a natural "ne sand, with a moisturecontent of 7)6% and bulk density of 1)51}1)65 g cm~3.

The explosive site is in the vicinity of Nanjing, China.A soil pit (300 cm by 300 cm by 200 cm) was dug and"lled with clayey or sandy soil. Three piezoelectric crystalgauges were embedded at depths of 20, 40, and 60 cm inthe soil (Fig. 1) to sense the soil stress.

In order to generate only one-dimensional waves,a planar explosive was applied. A steel plate, covered byexplosives of ammonium nitrate fuel oil (ANFO) No. 2 ata rate of 1)3}1)5 g cm~2, was placed on the surface of thepit. Experiments were conducted for 8 and 4 times withclayey and sandy soils, respectively. The plan of theexperiment is shown in Table 2.

The explosive detonation occurred immediatelyafter igniting the detonator. A one-dimensional stresswave was formed and travelled through the soil. Thegauges are placed at each of the three depths, measuringthe progress of the wave. The wave histories for clayeysoil and for sandy soil are shown in Figs 2 and 3, respec-tively.

Table 1Particle size of sample clayey soil

Soil fraction, mm Sample analysis, %

(0)001 36)080)005}0)001 28)940)01}0)005 13)930)05}0)01 11)790)25}0)05 8)941)00}0)25 0)32

3. Calculation method

The Lagrangian analysis method was used to formu-late a stress}depth}time surface from measurements ofa series of stress histories during the one-dimensionalstress wave travelling through the soil. From the "ttedsurface, derivatives of #ow variables were obtained. Thenan integration procedure was developed for computingthe unknown #ow quantities from the conservationrelations.

In Lagrangian coordinates, the conservation laws formass, momentum, and energy are, respectively:

ALlLtB

h

!

1

o1ALu

LhBt

"0 (1)

ALu

LtBh

!

1

o1ALpLhB

t

"0 (2)

ALE

LtBh

#

po1ALu

LhBt

"0 (3)

where: l is the speci"c volume; t is the time in s; o1

is theinitial density in g cm~3; u is the particle velocity inm s~1; h is the Lagrangian position (depth) in m; p is thestress in the direction of propagation in Pa; and E is theinternal energy.

To determine the stress, particle velocity, volume, andenergy histories at each gauge plane, the preceding equa-tions were integrated along lines of constant values forh (the gauge path). For the numerical calculations, theintegrations were performed over a short time intervalfrom t

1to t

2. The integrated forms of Eqns (1}3) are:

l2"l

1!

1

o1P

t2

t1ALu

LhBt

dt (4)

u2"u

1!

1

o1P

t2

t1ALpLhB

t

dt (5)

E2"E

1!

1

o1P

t2

t1

p ALu

LhBt

dt (6)

Table 2The plan of stress measurement on a steel plate 200 mm long by 16 mm thick of variable width, with the detonator placed centrally

No. Plate width, mm Mass of explosive, g Thickness of explosive, mm Soil

1 100 300 20 clayey2 100 300 20 clayey3 100 300 20 clayey4 100 300 20 clayey5 150 400 18 clayey6 150 400 18 clayey7 150 400 18 clayey8 150 400 18 clayey9 100 300 20 sandy

10 100 300 20 sandy11 150 400 18 sandy12 150 400 18 sandy

219SOILS UNDER EXPLOSIVE LOADING

where subscript 1 and 2 denote the initial and "nalvalues, respectively. Equation (2) can be integrated alonga line of constant time to obtain:

p2"p

1!o

1 Ph2

h1ALu

LtBh

dh (7)

For integrated Eqns (4}7), a stress}depth}time surfacewas "tted with the measuring a series of stress historydata. The surface was divided into a series of regions asshown in Fig. 4. Within each region, the stress is mono-tonic on all records. The path lines dividing regionsrepresent peak stress or other distinguished features suchas the top of the precursor wave.

For "tting the surface, a function that represents, in theleast-squares sense, a region of the surface between twopath lines was constructed. This surface function has thegeneral form:

p"f (h, t ) (8)

From the surface, numerical approximations to thedepth and time derivatives of stress (Lp/Lh), (Lp/Lt), and

Fig. 2. The stress histories for clayey soil (explosive no. 990

(L2p/Lh2) were obtained. These values of the derivationswere used in the Eqns (4}7) to compute the particlevelocity and strain at the gauge locations. Thusstress}strain paths were obtained in the soil throughwhich the stress wave passed.

4. Results and discussion

According to the Lagrangian analysis method, a pro-gram for calculating the dynamic constitutive relationwas made with FORTRAN. The stress data were used forcalculation. The resulting stress}strain curves for clayeyand sandy soils are shown in Figs 5 and 6.

For "tting these constitutive relation curves, stressp was "tted to quarter function of strain e, the "ttedfunctions are then:

p"3+i/1

Aiei dp'0 (9)

p"p0#

3+i/0

Bi(e!e

0)i dp(0 (10)

101) at three depths: (a) 20 cm; (b) 40 cm; and (c) 60 cm

Fig. 3. The stress histories for sandy soil (explosive no. 990109) at three depths: (a) 20 cm; (b) 40 cm; and (c) 60 cm

L. ZHIXIONG E¹ A¸ .220

where A and B are coe$cients for the loading and un-loading stress}strain curves; p

0is the stress at the begin-

ning of unloading; and e0

is the strain at the beginning ofunloading.

All coe$cients of stress}strain "tting curves for clayeyand sandy soils are shown in Table 3.

From Figs 5 and 6 and Table 3 it was observed that:

(a) the stress}strain curves were concave downwardwhile loading but upward unloading;

(b) the stagnant-return circle for clayey soil was biggerthan that for sandy soil, indicating that clayey soilabsorbed more energy than sandy soil; and

(c) the stress}strain curves in the loading process werenot superposed with the curves in the unloading pro-cess, indicating that a volume strain after unloadingcannot completely come back.

A non-linear relationship between the strain and stressmodules existed. In the loading process, the stress}straincurves of clayey and sandy soils had an obvious concaveshape in the downward direction. When loading reachesits peak value, the gradient of the curves clearly declines.

Fig. 4. Plane for constructing a stress}depth}time surface fromstress histories; h1}h3, depths; j0}j3, path lines

A smaller stress increment will bring a bigger strainincrement.

At the beginning of unloading, the gradient ofstress}strain curves is bigger. With stress decreases, thegradient becomes #attened. At the end of unloading,a bigger remnant strain still exists. It is obvious that thevolume strain of soils under explosive loading includestwo parts: resumptive and non-resumptive strain. Thenon-resumptive strain is 60}80% of the total.

The stress}strain curves demonstrate a strain-rate ef-fect. As the loading varies in each measurement depth forone test, the unloading paths of each gauge have obviousdi!erences and have no superposition. The reason maybe that an inner clayey dissipation force exists in soils.The dynamic yield intensity is related to strain rate. Thedynamic yield intensity increases with strain rate.

Under impact loading at rate of 1)3}1)5 g cm~2 explos-ive, clayey, and sandy soils at depths of 20}60 cm hadintense visco-plasticity with the shapes of thestress}strain curves in Figs 5 and 6.

At soil depths of 60 cm and greater, the gradient ofstress}strain curves in the whole loading or unloadingprocess was about linear. The stress}strain curves werenot a!ected by loading rate, and soil viscosity can beneglected. Thus, a linear elastic model would take placeat soil depths of 60 cm and greater.

5. Conclusion

(1) Dynamic constitutive relations of clayey and sandysoils were obtained by "tting the experimental datawith Lagrangian analysis. Stress}strain curvesshowed visco-plasticity. Maximum strain lags behindmaximum stress, forming an obvious stagnant-returncircle. The circle for clayey soil was bigger than thatfor sandy soil, indicating that clayey soil absorbedmore energy than sandy soil. Relatively big remnantstrain existed by the end of unloading.

Fig. 5. Stress}strain relations of clayey soil (no. 1) at three depths: (a) 20 cm; (b) 40 cm; and (c) 60 cm

Fig. 6. Stress}strain relations of sandy soil (no. 9) at three depths: (a) 20 cm; (b) 40 cm; and (c) 60 cm; stagnant-return circle shownhatched

Table 3The coe7cients A and B of stress+strain 5tting curves under explosive loading

Depth Clayey soil Sandy soil

20 cm 40 cm 20 cm 40 cm

Initial density, g cm~3 1)908 1)915 1)51 1)56A

167)42 16)18 31)02 12)19

LoadingA

2!193)65 !226)64 3)86 !48)85

A3

200)29 1188)4 47)87 80)38

B0

!3)61 !0)09 !6)03 !0)33B1

!31)76 !10)48 !42)46 !15)49Unloading

B2

50)63 !186)65 161)74 64)29B3

!8)36 5508)8 !276)89 !208)33

Maximum stress, MPa 8)78 0)45 10)21 1)08

221SOILS UNDER EXPLOSIVE LOADING

L. ZHIXIONG E¹ A¸ .222

(2) The stress}strain curves of clayey and sandysoils demonstrate the stagnant-return phenomena,strain-rate e!ect and waveform dissipation.These properties are important to comprehend cor-rectly the dynamic behaviour of clayey and sandysoils.

(3) Di!erent deformation characteristics werefound at di!erent depths, i.e. #uid elastoplastic insurface layer (0}20 cm) and viscoplastic in middle(20}60 cm).

References

Cowperthwaite M; Williams R F (1971). Determination of con-stitutive relationship with multiple gauges in nondivergentwaves. Journal of Applied Physics, 42, 456}462

Forest C A (1991). Lagrangian analysis with variance estimatedusing the impulse time integral. Bulletin of American PhysicsSociety, 36, 1825}1834

Fowles R (1970). Conservation relations for spherical and cylin-drical stress wave. Journal of Applied Physics, 41, 2740}2748

Fowles R; Williams R F (1970). Plane stress wave propagationin solids. Journal of Applied Physics, 41, 360}362

Grady D E (1973). Experimental analysis of spherical wavepropagation. Journal Geophysics Research, 78, 1299}1307

Seaman L (1974). Lagrangian analysis for multiple stress orparticle velocity gauges in attenuating wave. Journal ofApplied Physics, 45, 4303}4314

Seaman L (1987). Analysis of dynamic in situ back"ll propertytests. Part 2, An improved Lagrangian analysis for stress andparticle velocity gauge arrays. Technical report SL-87-11,34}58

Vantine H C; Curtis W D (1981). A comparison of stress andvelocity measurements in PBX-9404 explosive. Proceedingsof the 18th Symposium (International) on Combustion, TheCombustion Institute, pp 135}146