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Research ArticleResearch on Characteristic Stress and Constitutive Equation ofConfined Sandstone during Damage Evolution Based on EnergyEvolution Analysis

Xiaobin Yang ,1 Hongming Cheng ,1,2 Xin Hou,1 Chaogang Nie,1 and Jiaqi Lv1

1School of Emergency Management and Safety Engineering, China University of Mining and Technology, Beijing 100083, China2School of Coal Engineering, Shanxi Datong University, Datong 037003, China

Correspondence should be addressed to Hongming Cheng; [email protected]

Received 3 April 2019; Revised 3 June 2019; Accepted 24 June 2019; Published 7 July 2019

Guest Editor: Dariusz Rozumek

Copyright © 2019 Xiaobin Yang et al.,is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

,is study implements the cyclic loading-unloading triaxial compression tests with confining pressures of 5, 10, 20, and 30MPafor determining the characteristic stress and constitutive equation of sandstone during damage evolution. ,e energy evolutioncharacteristics and transformations are analyzed, the energy proportion evolution law is defined and analyzed, and its process isdivided into five stages according to the whole stress-strain curve. ,en, the dissipation energy proportion method for de-termining the characteristic stress is proposed and compared with the lateral strain method and the volumetric strain method, andthe differences of characteristic stress determined by the three methods are within the acceptable error range, i.e., the proportionof crack initiation stress and crack damage stress to peak stress are all about 40% and 70%, respectively, and least affected by theconfining pressures. ,e new method has strong maneuverability, clearer response to the damage evolution, better coordinationwith the damage evolution, and less subjective initiative. In addition, the dissipation energy proportion behavior is captured byfitting function and established a relationship with elastic modulus of sandstone, and then the damage evolution constitutiveequation of sandstone is established based on energy proportion. Finally, the theoretical curve and experimental curve of damageevolution are compared according to the five stages of dissipation energy proportion evolution, and the result shows that thedamage evolution constitutive equation is reasonable.

1. Introduction

,e stability of geotechnical engineering depends on thephysical and mechanical environment in which rock ma-terials are located and their mechanical behavior. As aheterogeneous natural geological material with internalmicrostructure (joints, cracks, inclusions, etc.), the study ofthe damage evolution process of rock materials has im-portant guiding significance for the design, construction,and operation stability evaluation of geotechnical engi-neering. At present, laboratory tests of rock materialsgenerally show that the damage evolution process of loadedrock specimens has experienced crack closure, crack initi-ation, crack penetration, and crack coalescence [1–3].According to the damage evolution state of rock samples,many scholars divide the whole stress-strain curve into five

important stages, including the crack closure stage, linear-elastic deformation stage, stable crack growth stage, unstablecrack growth stage, and postpeak damage stage. Corre-sponding to the five stages of damage evolution, the char-acteristic stress and crack evolution laws in each stage aregiven, namely, the crack closure stress (σcc), the crack ini-tiation stress (σci), the crack damage stress (σcd), the peakstress (σp), and the residual stress (σc) [4–8]; in particular,the σci and σcd can guide geotechnical engineering design,construction, and operation stability evaluation. Hoek andBrown [9] found that when the far-field maximum stressmagnitude exceeded 0.15 of the UCS, spalling was observedaround the underground opening of square tunnels in SouthAfrica. Martin and Christiansson [10] and Andersson et al.[11] voted that the σci determined by uniaxial compressivetests could be used as an estimate for the in situ spalling

HindawiAdvances in Materials Science and EngineeringVolume 2019, Article ID 4145981, 13 pageshttps://doi.org/10.1155/2019/4145981

mailto:[email protected]://orcid.org/0000-0002-9911-2269https://orcid.org/0000-0002-1630-8670https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2019/4145981

strength, while the ISRM makes no mention of the methodfor determining characteristic stress in its “Suggestedmethods for determining the uniaxial compressive strengthand deformability of rock materials.” ,us, finding a reliablemethod for determining characteristic stress may be bene-ficial to geotechnical engineering when estimating thedamage evolution state.

To better estimate characteristic stress, various scholarshave proposed different methods based on unconfined andconfined laboratory tests. Brace et al. [4] utilized the vol-umetric strain-axial strain curve to determine the σci; theynoted that the volumetric strain presented linear charac-teristics before the σci, while it presented nonlinear char-acteristics after the σci, so the point deviated from the linearportion was the σci. Considering initial cracks of rockmaterials, Martin and Chandler [5] proposed that using thecrack volumetric strain-axial strain curve determined the σci;they noted that the curve had a horizontal section with avalue of zero, the onset of the horizontal section was the σcc,and the end was the σci. It is well known that the lateral strainis more sensitive than the axial strain, for the lateral di-latancy of rock samples at the stable crack growth stage.Consequently, Lajtai [12] drew a tangent line on the curve ofaxial stress-lateral strain, and the point where the lateralstrain deviated from linearity was the σci. Nicksiar andMartin [13] proposed the lateral strain response method;firstly, using the volumetric strain method determined theσcd and connected the value of lateral strain correspondingthe σcd to the original point as a reference line; then, thedifference between lateral strain and reference line wascaptured and drawn vs axial stress, and the maximum valuepoint on the curve was the σci. Latterly, with the applicationof acoustic emission in monitoring the damage evolution ofloaded rock specimens, it is feasible that acoustic emissionreveals the phenomenon of crack at different stages ofdamage deformation. Eberhardt et al. [14], Zhao et al. [15],and Xue et al. [16] drew a tangent line on the AE count-axialstress curve, and the point where the AE count deviated fromlinearity was the σci. In practical applications, some of thesemethods feebly respond to the crack evolution and more orless have subjectivity, such as determining the linear seg-ment by using the lateral strain method and AE method. Inaddition, the change of index of these methods is not incoordination with the damage evolution, which cannotreflect all characteristic stress in the whole stress-straincurve, such as the volumetric strain method only de-termining the σcd, and the σcd should be determined firstlyby the volumetric strainmethod when using the lateral strainresponse method [17–19]. To divide the process of damageevolution more intuitively and determine characteristicstress more accurately, it is urgent to find new parametersand propose new methods.

In nature, a series of physical changes during loadingrock specimens, such as crack closure, initiation, growth,and coalescence, are energy transformation, i.e., the damageof loaded rock specimens going through energy absorption,accumulation, dissipation, and release. Xie et al. [20, 21]considered that the deformation and damage of rock were aprocess of exchanging material and energy with the outside

and delineated inner relation between deformation damageand energy dissipation/release. Zhang and Gao [22] de-termined the elastic strain energy (ESE) and dissipation energy(DE) evolution and distribution through energy tests for redsandstone under uniaxial compression. Deng et al. [23]designed uniaxial cyclic loading tests and explored the pa-rameter variation and interrelation, such as the total absorptionenergy (TAE), the ESE, and the DE. Meng et al. [24] exploredthe character of energy accumulation and dissipation duringdeformation and damage of rock specimens under variousloading and unloading schemes and revealed the energy ac-cumulation and dissipation evolution and distribution at theprepeak stage. Under the current scope of work, a unifiedunderstanding of energy transformation during loading hasbeen obtained: the TAE transforms into the ESE and the DE,and more absorption energy transforms into the ESE which isstored in rock specimens than the DE at the prepeak stage,while the DE increases and the ESE dissipates rapidly at thepostpeak stage. In contrast, few articles care about energytransformation corresponding to the physical change of crackclosure, initiation, growth, and coalescence during loading rockspecimens. Whether the key points of energy evolution as-sociate with the characteristic stress in damage evolution ornot, whether the energy evolution relates to the material pa-rameter evolution of rock specimens or not, and whether theconstitutive equation of rock is established from the energyevolution point of view or not have not been explored in depth.

Based on this fact, this paper designed the cyclic loading-unloading triaxial compression tests, introduced the energyproportion, proposed a new method to research crackclosure, initiation, growth, and coalescence of loaded rockspecimens, and established and verified the constitutiveequation from the point of view of energy evolution.

2. Experimental

2.1. Rock Specimen Preparation. ,e experimental speci-mens were sandstone, all machined from the same sandstoneblock, exhibiting fine consistency and integrity with noobvious joints and cracks, and were carefully producedstandard specimens with 50mm in diameter and 100mm inheight and were ground to produce flat parallel surfaces with±0.02mm parallelism (see Figure 1) according to the ISRM-suggested method.

2.2. Test Equipment and Procedure. We implemented con-ventional triaxial compression tests and cyclic loading-unloading triaxial compression tests with the RLJW-2000triaxial shear-creep testing system (see Figure 2(a)), with2000 kN maximum axial loading capacity and 50MPamaximum confining pressure. Axial and lateral displace-ment data were collected by extensometers (see Figure 2(b)).

Firstly, the conventional triaxial compression tests wereimplemented with confining pressures of 5, 10, 20, and30MPa, and the triaxial compressive strength (TCS) wasscored. ,en, the cyclic loading-unloading triaxial com-pression tests were implemented with confining pressures of5, 10, 20, and 30MPa: (1) First, the specimens were wrapped

2 Advances in Materials Science and Engineering

up with plastic �lm to prevent pressure �uid entry. (2)Whentesting, specimens were �xed by loading small axial stressand then con�ning pressure was loaded to a speci�ed value;afterwards, axial stress was loaded to a speci�ed value at adisplacement rate of 0.005mm/s; �nally, axial stress wasunloaded to zero at a displacement rate of 0.005mm/s withconstant con�ning pressure; the load and unload process

was done repeatedly until the test ended. (3) At the prepeakstage, the unloading points were chosen which normalizedto TCS (by the conventional triaxial compression tests)varying from 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, to 0.9; at thestrain-softening stage and residual strength stage, one ormore loading-unloading tests were implemented, re-spectively (see Figure 3).

Loading system

Data collection system

(a)

Rock specimen

Extensometers Confiningpressure-loading

system

(b)

Figure 2: Test equipment. (a) Rock triaxial testing system. (b) Extensometers.

0.10.20.30.40.50.60.70.80.9

Testing time (s)

σ1TCS

Axi

al fo

rce (

kN)

t1 t2 t3 t4 t5 t6 t7 t8 t9

9th8th

7th6th

5th4th

3rd2nd

1st

Prepeak stage Strain-softeningstageResidual strength

stage

One or more

One or more

Load

ing

Unloading

Figure 3: Experimental loading-unloading path.

Figure 1: Sandstone specimens.

Advances in Materials Science and Engineering 3

3. Experimental Results and Analysis

e axial stress-strain curves during conventional triaxialcompression tests are shown in Figure 4(a). Two experimentalgroups were performed at each con�ning pressure during thecyclic loading-unloading triaxial compression tests. As thelimitation of this article, we only showed one experimentalgroup at each con�ning pressure in Figure 4(b).

We analyzed energy evolution of rock specimens based ontest data. Considering the heterogeneity of rock specimens anddierences between rock specimens, some simpli�cations weremade when calculating energy for single-rock specimens:(1) neglecting the strain energy stored in rock by hydrostaticpressure working; (2) calculating the elastic modulus of a singlecycle by using secant modulus, i.e., connecting the maximumstress point with the minimum stress point on a single cyclicstress-strain curve [25]; and (3) calculating the TAE and theESE of a single cycle by the integral method (see Figure 5).

3.1. Energy Density Evolution Analysis. Energy is the drivingforce for internal defect evolution and propagation in rockmaterials [20, 21]. At each stage of deformation of loadedrock specimens, the energy transformation is in progress[26, 27]. During the triaxial stress state, the loaded rockspecimens were taken as a closed-loop system with thesupposition that no thermal transmission occurred betweenthe rock specimen and the external environment [28].According to the �rst law of thermodynamics, the TAE perunit volume of the rock specimen could be described as [28]

W � ∫ σ1dε1 + 2∫ σ3dε3, (1)

where σ1 and ε1 are the axial stress and strain, respectively,and σ3 and ε3 are the lateral stress and strain, respectively.

Taking the i-th time loading-unloading stress-strain curveas an example (see Figure 5), the relationship among the TAE,the ESE, and the DE was explained. As shown in Figure 5, theloading curve AB was higher than the unloading curve BC; thetotal deformation (εi) of rock caused during loading containedrecoverable deformation (εei ) which was released at theunloading stage and irrecoverable deformation (εdi ) used forrock damage and plastic strain. From the energy perspective,the TAE was the area under the loading curve AB and the ESEwas the area under the unloading curve BC, and the dierenceof the above two energies was the DE:

Wi � ∫εi

0σdε,

Wei � ∫εei

0σdε,

Wdi �Wi −Wei � ∫

εi

0σdε− ∫

εei

0σdε,

(2)

where Wi, Wei , and Wdi are the TAE, the ESE, and the DE

during the i-th loading-unloading, respectively.On the axial stress-strain curves during the cyclic

loading-unloading triaxial compression tests, the TAE-

0

50

100

150

200

250

0 0.5 1

Stre

ss (M

Pa)

Axial strain (10–2)1.5

εi

εieεid

Loadingcurve

Unloadingcurve

A

B

C

Figure 5: Relationship among the TAE, the ESE, and the DEduring the i-th loading-unloading.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

50

100

150

200

250σ 1

(MPa

)

ε1 (10–2)

5MPa10MPa

20MPa

30MPa

(a)

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

ε1 (10–2)

5MPa

10MPa

20MPa

30MPa

σ 1 (M

Pa)

(b)

Figure 4: e axial stress-strain curves from (a) conventional triaxial compression tests and (b) cyclic loading-unloading triaxial com-pression tests.


strain curves, the ESE-strain curves, and the DE-straincurves were drawn, as shown in Figure 6:

(1) At the prepeak stage of the same rock specimens, thedeformation changed from nonlinearity to linearity andthen to nonlinearity with the increase of the number ofcycles, corresponding to the crack closing �rst and thengrowing stably and �nally growing unstably till mac-roscopic damage. At the postpeak damage stage, due tothe con�ning pressure, the rock specimen still hadcarrying capacity. With the increase of con�ningpressure, the degree of nonlinear deformation at thestage of crack closure slowed down gradually, the TCSof rock specimens increased, and the residual strengthalso increased; that is, the carrying capacity increasedwhen rock specimens were damaged.

(2) At the prepeak stage, the TAE, the ESE, and the DEincreased with the axial strain increase; basically, theTAE and the ESE reached a maximum value. After thepeak point, i.e., the initial postpeak damage stage, theESE which was stored in the rock specimens at theprepeak stage dropped rapidly and converted into theDE for the macroscopic crack growth, coalescence, andcaving damage of rock specimens, and the DE reached

the maximum value. At the residual strength stage, theenergy decreased and �nally leveled o with no newdamage. On the whole, with the con�ning pressureincrease, the TAE, the ESE, and the DE increasedduring a single cyclic loading-unloading.

(3) Taken as a whole, the TAE, the ESE, and the DEdisplayed nonlinearity at the crack closure stage, whilethe degree of nonlinearity gradually slowed down withthe con�ning pressure increase, corresponding to thechange of stress; at the linear-elastic deformation stageand the stable crack growth stage, the three kinds ofenergies represented a continuous linear increase, whiledisplayed nonlinearity at the unstable crack growthstage; the ESE and the DE changed suddenly at theinitial postpeak damage stage.

3.2. Material Parameter Evolution Analysis. We calculatedthe elastic modulus of a single cycle at each con�ningpressure and drew elastic modulus-strain curves duringcyclic loading-unloading, as shown in Figure 7.

Taken as a whole, the elastic modulus rose with thecon�ning pressure increase. Under the same con�ningpressure, the initial elastic modulus rose, whereas the rate

0.0 0.5 1.0 1.5 2.0 2.50

20

40

60

80

100

120

140En

ergy

den

sity

(10–

2 MJm

–3)

σ 1 (M

Pa)

ε1 (10–2)

ESE

DE

TAE

Stress-strain

(a)

Ener

gy d

ensit

y (1

0–2 M

Jm–3

)σ 1

(MPa

)

0.0 0.5 1.0 1.5 2.0 2.50

30

60

90

120

150

180

ε1 (10–2)

DEESE

TAE

Stress-strain

(b)

Ener

gy d

ensit

y (1

0–2 M

Jm–3

)σ 1

(MPa

)

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

ε1 (10–2)

DEESE

TAE

Stress-strain

(c)

Ener

gy d

ensit

y (1

0–2 M

Jm–3

)σ 1

(MPa

)

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

ε1 (10–2)

DE

ESE

TAE

Stress-strain

(d)

Figure 6: Energy evolution curves during the cyclic loading-unloading triaxial compression tests with con�ning pressures of 5MPa(a), 10MPa (b), 20MPa (c), and 30MPa (d).


of rise gradually decreased and then underwent a period ofapproximately linear increase and reached the maximumvalue, corresponding to the crack closure stage, the linear-elastic deformation stage, and the stable crack growthstage, respectively. ,e stable crack closing and growthprompted the ESE to be stored in the rock specimens; thatis, the elasticity of rock specimens was enhanced, and themaximum values of the elastic modulus with confiningpressures of 5, 10, 20, and 30MPa were 16, 22, 24, and28 GPa, respectively. During the unstable crack growthstage, the elastic modulus gradually decreased with theESE release. ,e elastic modulus decreased suddenly at theinitial postpeak damage stage and then gradually de-creased with no new damage.

4. Proposed Method for DeterminingCharacteristic Stress

As previously mentioned, the loaded rock specimens wentthrough energy absorption, accumulation, dissipation, andrelease. ,e ESE was accumulated firstly and then releasedwhen the rock specimens damaged, while the DEwhich existedin the whole loading process was used for original crack closureand new crack initiation, growth, and coalescence. So, theproportion of ESE orDE inTAE reflected the relation of energytransformation during loading. Especially under some stress orstrain levels, the proportion of DE in TAE could explain thecondition of damage development and plastic deformation ofrock specimens which directly affected rock material param-eters [29, 30]. ,erefore, we defined the parameter of energyproportion, i.e., the ESE proportion (c) and the DE proportion(η), and analyzed their evolution law; then, a new method fordetermining the characteristic stress in damage evolution wasproposed based on DE proportion, and it was applied andverified finally.

4.1. Definition of Energy Proportion. We defined the pa-rameter of energy proportion, which was taken as the

proportion of ESE and DE in TAE. ,e ESE proportion ofthe i-th loading-unloading could be described as

ci �WeiWi

, (3)

and the DE proportion of the i-th loading-unloading couldbe described as

ηi � 1− ci � 1−WeiWi

. (4)

On the axial stress-strain curves during the cyclicloading-unloading triaxial compression tests, we drew en-ergy proportion-strain curves (see Figure 8).

4.2. Energy Proportion Evolution Analysis. As shown inFigure 8, under the confining pressures of 5, 10, 20, and30MPa, the ESE proportion increased, while the DEproportion deceased at the initial stage, but most of theTAE transformed into the DE for original crack closing,corresponding to the crack closure stage on the axial stress-strain curves; at the later crack closure stage, two kinds ofenergy proportion were equal. When the loaded rockspecimens came into the stage of linear-elastic de-formation, the ESE played an important role; contrary tothe DE proportion evolution, the ESE proportion in-creased, while the evolution rate of two kinds of energyproportion gradually decreased. With loading, the evolu-tion rate was gradually steady, and the ESE proportion andthe DE proportion approximately linearly developed; thatis, the energy transformation developed steadily, and theloaded rock specimens stayed at the stable crack growthstage. When the evolution rate was zero, the ESE pro-portion reached the maximum value and the DE pro-portion reached the minimum value in the meanwhile.When the loading stress exceeded the crack damage stressof rock specimens, the rock specimens came into theunstable crack growth stage, and the crack growth con-sumed energy and caused the DE proportion increase andthe ESE proportion decrease, while the evolution rategradually increased. When the loading stress exceeded thepeak stress, the evolution rate suddenly increased, the DEproportion reached the peak value and finally leveled off,and the ESE proportion evolution was opposite.

Taken as a whole, under the confining pressures of 5, 10,20, and 30MPa, the energy proportion (for example, the DEproportion) evolution was similar to the whole axial stress-strain curves, which could be divided into five stages (seeFigure 8):

(i) Decreased suddenly (stage I)(ii) Decreased nonlinearly (stage II)(iii) Decreased approximately linearly (stage III)(iv) Reached the minimum value and increased grad-

ually (stage IV)(v) Increased suddenly and leveled off (stage V)

In contrast, the evolution of the ESE proportion was inthe opposite direction (see Figure 8).

5MPa10MPa

20MPa

30MPa

0

5

10

15

20

25

30

Elas

tic m

odul

us (G

Pa)

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

Figure 7: Elastic modulus-strain curves during cyclic loading-unloading.


4.3. Proposed New Method. As previously mentioned, cor-responding to the stages of the whole axial stress-straincurves during the cyclic loading-unloading triaxial com-pression tests, the energy proportion evolved obviously ateach stage. Based on this, considering geotechnical engi-neering focused on the rock damage, we proposed a newmethod, energy proportion method (EPM), for determiningcharacteristic stress during rock damage evolution using theDE proportion curves:

(i) Specimen preparation: rock samples are selectedon-site, and standard specimens are produced bythe ISRM-suggested method.

(ii) Test procedure: conventional triaxial compressiontests are performed to determine the TCS; cyclicloading-unloading triaxial compression tests areperformed with the unloading points which nor-malized to TCS varying from 0.1, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, to 0.9 (or the hierarchical interval isreduced moderately to improve the accuracy).

(iii) Calculation: energy density is calculated (the TAEand the ESE) during a single cyclic loading using thearea integral method by MATLAB, and then the DEproportion is calculated.

(iv) Plot: the DE proportion-strain curves are drawn onthe axial stress-strain curves during the cyclicloading-unloading triaxial compression tests.

(v) Determination (taking the confining pressures of 5and 20MPa as an example (Figure 9)): firstly, theminimum value of the DE proportion is found (thegreen point), the vertical line is drawn (the purpleline) starting with the green point, and the σcdcontact with the axial stress-strain curve is de-termined; then, the tangent line is drawn startingwith the green point (the blue line) and the σcicontact with the axial stress-strain curve is de-termined where the DE proportion deviated fromlinearity (the yellow point); and finally, the tangentis drawn starting with the yellow point (the blackline) and the σcc where the axial stress deviated fromlinearity is determined. ,is could be concluded as“two points and three lines.”

4.4. EPM: Application and Verification. We determinedcharacteristic stress of sandstone specimens during damageevolution under the confining pressures of 5, 10, 20, and30MPa. ,e determination process is shown in Figure 9

Stress-strain

DE proportion

σ 1 (M

Pa)

ESE proportion

VIVIIIIII

0

20

40

60

80

100

120

140

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

(a)

σ 1 (M

Pa)

VIVIIIII

Stress-strainESE proportion

DE proportion

I

0

30

60

90

120

150

180

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

(b)

σ 1 (M

Pa) Stress-strain

DE proportion

III VIVIII

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0

50

100

150

200

ESE proportion

(c)

σ 1 (M

Pa) Stress-strain

ESE proportion

DE proportionVIVIIIIII

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0

50

100

150

200

250

(d)

Figure 8: Energy proportion-strain curves with confining pressures of 5MPa (a), 10MPa (b), 20MPa (c), and 30MPa (d).


(only the con�ning pressures of 5 and 20MPa are shown),and the results are shown in Table 1.

In Table 1, the characteristic stress was determined byEPM: with the con�ning pressure increase, the σcc decreasedand the σci and the σcd increased, whereas the ratio ofcharacteristic stress to peak stress was dierent; the con-�ning pressure had a signi�cant eect on σcc/σp whichdecreased with the con�ning pressure increase, while the σci/σp and the σcd/σp barely changed. Under the con�ningpressures of 5, 10, 20, and 30MPa, the σci/σp was 41.39%,39.19%, 39.69%, and 38.46%, with an average of 39.68%, andthe σcd/σp was 69.68%, 62.50%, 69.85%, and 70.73%, with anaverage of 68.19%; that is, under dierent con�ning pres-sures, the σcd/σp was all about 40% and the σcd/σp was allabout 70%. From an energetic standpoint, the con�ningpressure ampli�ed the ability of rock specimens to store anddissipate energy, while the distribution proportion of energyalmost unchanged.

For the memory function of the rock specimen, we drewthe outer envelopes of the axial stress-lateral strain curvesand the volumetric strain-axial strain curves which wereobtained from the cyclic loading-unloading triaxial com-pression tests (see Figure 10). e lateral strain method(LSM) and the volumetric strainmethod (VSM) were chosento verify the EPM. e determination process is shown inFigure 10 (taking the con�ning pressures of 5 and 20MPafor example): the start and the end of the linear region on theouter envelope of the lateral strain corresponded to the σccand the σci, respectively, and the peak of volumetric strain

corresponded to the σcd (see Table 1). For the changing trendof characteristic stress, the results determined by LSM andVSM were similar to those by EPM. For the dierence ofcharacteristic stress, the results determined by LSM andVSM were larger than those by EPM, with an average of2.61MPa (σcc), 1.05MPa (σci), and 2.25MPa (σcd). For theratio of characteristic stress to peak stress, the trend of σcc/σpdetermined by LSM was similar to that by EPM, i.e., de-creasing trend with the con�ning pressure increase; the σci/σp was still about 40% determined by LSM, and the σcd/σpwas still about 70% determined by VSM.

Comparisons showed that the EPM was feasible. Inpractical applications, the DE proportion had clearer re-sponse to damage evolution, better coordination withdamage evolution, and less subjective initiative, and the σcd,σci, and σcc could be determined in turn by “two points andthree lines” using the EPM.

5. Equation of Damage Evolution

e mechanical evolution of loaded rock specimens is aprogressive damage. Research on damage evolution from anenergetic standpoint contains mechanical processes, such asstrength and deformation characteristics, so it can reveal thedamage and failure mechanism of rock more essentially. Jinet al. [31] de�ned and calculated the damage variable froman energetic standpoint and researched the damage prop-agation law of rock materials. Zhao et al. [32] revised thedamage variable, derived the relationship between dissipated

σ 1 (M

Pa) Stress-strain

DE proportion

σcc

σci

σcd

0

20

40

60

80

100

120

140

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

(a)

σ 1 (M

Pa)

σcc

σci

σcdStress-strain

DE proportion

0

50

100

150

200

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0.0

0.2

0.4

0.6

0.8

1.0

Ener

gy p

ropo

rtio

n

(b)

Figure 9: Determination process of characteristic stress by EPM with the con�ning pressures of 5MPa (a) and 20MPa (b).

Table 1: Comparison of characteristic stress determined by the three methods.

Con�ningstress (MPa)

Peak stressσp (MPa)

Characteristic stress (MPa) Dierence(MPa)

Ratio of characteristic stress to σci (%)EPM LSM VSM EPM LSM VSM

σcc σci σcd σcc σci σcd σcc σci σcd σcc/σp σci/σp σcd/σp σcc/σp σci/σp σcd/σp5 126.00 26.80 52.15 87.80 28.40 52.25 85.80 1.60 0.10 −2.0 21.27 41.39 69.68 22.54 41.47 68.1010 160.00 20.35 62.70 100.00 22.40 65.30 105.00 2.05 2.60 5.00 12.72 39.19 62.50 14.00 40.81 65.6320 194.00 18.70 77.00 135.50 21.82 77.50 137.00 3.12 0.50 1.50 9.64 39.69 69.85 11.25 39.95 70.6230 234.00 12.15 90.00 165.50 15.80 91.00 170.00 3.65 1.00 4.50 5.19 38.46 70.73 6.75 38.89 72.65Mean — — — — — — — 2.61 1.05 2.25 — 39.68 68.19 — 40.28 69.25


Outer envelope

Outer envelope

–0.6 –0.4 –0.2 0.0 0.2–0.8ε3 (10–2)

0

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120

140

0

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0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0.0

0.1

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0.5

0.6

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

σ 1 (M

Pa)

ε v (1

0–2 )

σcd

σci

σcc

(a)

Outer envelope

Outer envelope

0

50

100

150

200

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

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–0.4 –0.2 0.0–0.6ε3 (10–2)

0.0

0.1

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0.6

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

σ 1 (M

Pa)

σcd

σci

σcc

(b)

Figure 10: Determination process of characteristic stress by LSM and VSM with the con�ning pressures of 5MPa (a) and 20MPa (b).


energy and damage variable, and then established thedamage equation. Liu et al. [33] introduced the damagevariable and compaction factor based on energy and revisedthe equation of damage evolution and veri�ed it using ex-perimental data. Li et al. [34] de�ned the damage variableand calculated damage modulus from an energetic stand-point and then analyzed damage modulus evolution withcon�ning pressures. Moreover, the above research studiesalso showed that the energy proportion evolution couldexplain the deformation of loaded rock specimens. Based onthis, we established the relationship between elastic modulusand DE proportion, �tted the mathematical expression ofthe DE proportion, and then derived the equation of damageevolution of loaded rock specimens.

eevolution laws of elastic modulus which were illus-trated in Section 3.2 were similar to the evolution laws ofenergy proportion illustrated in Section 4.2, so we de�nedthe relationship between elastic modulus and DEproportion:

E � a(1− η)E0, (5)

where a is a scaling factor, greater than 1, relating to thecon�ning pressures, and E0 is the maximum value of theelastic modulus at the linear-elastic deformation stage.

5.1. Equation of Energy Proportion. We �tted the DE pro-portion under con�ning pressures of 5, 10, 20, and 30MPa(see Figure 11), and the �tting function and parameters areshown in Table 2. From the �tting curves, at the initial stageof DE proportion, i.e., the I, II, and III stages, the dierenceof DE proportion gradually decreased to be aligned at thesame strain level, with the con�ning pressures of 5, 10, 20,and 30MPa, while the dierence of DE proportion at thesame strain level gradually increased at the IV stage and theprevious V stage and decreased to be stable at the later stageof V.

emeaning of �tting function and parameters is shownin Figure 12, and combined with the �tting curves inFigure 11, we concluded the following:

(i) η0 is the maximum value of DE proportion at the Vstage

(ii) A is the dierence of DE proportion between theminimum value at the III stage and the maximumvalue at the V stage

(iii) εc is the axial strain corresponding to the minimumvalue of the DE proportion at the III stage

(iv) w is the total axial strain corresponding to the IIIstage and IV stage of the DE proportion

From Table 2, it is seen that the �tting parameters, like η0and A, almost did not change with the con�ning pressurechange, as explained in Section 4.4 (the distribution pro-portion of energy in rock almost unchanged with thecon�ning pressure); εc and w increased with the con�ningpressure increase, which also could be obtained in Figure 8.

5.2. Constitutive Equation of Damage Evolution Based onEnergy Proportion. In Figure 5, during the i-th cyclicloading, the total axial strain εi contained the recoverablestrain εei and the residual strain ε

di :

εi � εei + ε

di . (6)

Fitting curve (5MPa)Fitting curve (10MPa)Fitting curve (20MPa)Fitting curve (30MPa)

Test data (5MPa)Test data (10MPa)Test data (20MPa)Test data (30MPa)

0.0

0.2

0.4

0.6

0.8

1.0

DE

prop

ortio

n

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

Figure 11: Experimental and �tting curves of DE proportion.

A

A/2w

η0

(εc, η0 – A)

Diss

ipat

ed en

ergy

ratio

(%)

Axial strain (10–2)

η0 – Ae–((ε–εc)2/2w2)

Figure 12: Meaning of the function image and the parameters ofthe DE proportion.

Table 2: Fitting function and parameters of DE proportion.

Con�ning stress (MPa)Fitting function and parameters

η � η0 −Ae−((ε− εc)2/2w2)

η0 A εc w

5 0.7 0.42 0.75 0.3510 0.63 0.43 0.75 0.3720 0.64 0.43 0.8 0.4630 0.62 0.42 0.9 0.5


By using formulas (2) and (3), the following relationswere obtained approximately:

εei � 1− ηi( )εi,

εdi � ηiεi.(7)

e stress of loaded rock specimens contains the elasticpart and damage part, which can be described by using theelastic part [35]:

σ � Eεe. (8)

Taking E and εe which contain damage parameter (theDE proportion), the stress is

σ � a(1− η)2E0ε � a 1− η0 −Ae− ε−εc( )2/2w2( )( )

2E0ε. (9)

is equation could describe the axial stress-straincurves, and the values of a were 1.65, 1.48, 1.38, and 1.28.en, we obtained the theoretical axial stress-strain curves

under the con�ning pressures of 5, 10, 20, and 30MPa,respectively (see Figure 13).

As shown in Figure 13, �tting function for the theo-retical curves described the experimental curves. But someof the shortcomings existed corresponded to the stages ofthe experimental curves which were divided by the DEproportion (see Figures 8 and 13): At the I stage, the the-oretical curves coincided with the experimental curvesunder the con�ning pressures of 5 and 10MPa, while theygradually deviated and depressed with the con�ning pres-sure increase. ey had an upward departure with lowcon�ning pressures at the II stage, while downward de-viation with the con�ning pressure increase, and the de-viation degree was minimum with the con�ning pressure of10MPa. ey also had an upward departure with lowcon�ning pressures at the III stage, but downward deviationwith the con�ning pressure increase, and the deviationdegree was minimum with the con�ning pressure of20MPa. ey had an upward departure at the IV stage,while the deviation degree gradually decreased with thecon�ning pressure increase. At the IV stage, the deviation

σ 1 (M

Pa) �eoretical curve

σcc

σci

σcd

Experimental curve

VIVIIIIII

0

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0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

(a)

σ 1 (M

Pa)

σcc

σci

σcd�eoretical curve

Experimental curve

VIVIIII II

0

30

60

90

120

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180

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

(b)

σ 1 (M

Pa)

σcc

σci

σcd�eoretical curve

Experimental curve

VIVIIIIII

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

0

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200

(c)

σ 1 (M

Pa)

σcc

σci

σcd

�eoretical curve

Experimental curve

VIVIIIIII

0

50

100

150

200

250

0.5 1.0 1.5 2.0 2.50.0ε1 (10–2)

(d)

Figure 13: eoretical stress-strain curves based on energy proportion with the con�ning pressures of 5MPa (a), 10MPa (b), 20MPa(c), and 30MPa (d).


degree was minimum with the confining pressure of20MPa.

6. Conclusion

,e objective of this paper is to propose a new method fordetermining characteristic stress and to establish a consti-tutive equation for damage evolution of sandstone from anenergetic standpoint. ,e cyclic loading-unloading triaxialcompression tests which were implemented with confiningpressures of 5, 10, 20, and 30MPa revealed that the energyevolution corresponded to the damage deformation of rockspecimens, and the energy transformation could describecrack closure, initiation, growth, and coalescence under theloading process. Furthermore, the parameter of energyproportion was defined. Taking the DE proportion as anexample, it was divided into five stages corresponding to thefive stages of the whole stress-strain curve. ,e EPM fordetermining characteristic stress was proposed and was usedto determine the characteristic stress of sandstone during thecyclic loading-unloading triaxial compression tests. ,eresults showed that, with the confining pressure increase, theσcc decreased, while the σci and the σcd increased; underdifferent confining pressures, the σcc/σp decreased, the σcd/σp was about 40%, and the σcd/σp was about 70% because thedistribution proportion of energy in rock specimens almostwas unchanged with different confining pressures. Also, theresults were verified by LSM and VSM: for the difference ofcharacteristic stress, the results determined by LSM andVSM were larger than those by ERM, while they were in theacceptable error range; for the ratio of characteristic stress topeak stress, the trend of σcc/σp determined by LSM wassimilar to that by ERM, the σci/σp was still about 40% de-termined by LSM, and the σcd/σp was still about 70% de-termined by VSM.

Based on the analysis of material parameter evolution,the relationship between elastic modulus and DE proportionwas defined. ,e DE proportions fitted under confiningpressures, and the fitting function and parameters hadcertain significance. Furthermore, the equation of damageevolution based on energy proportion was established.According to the five stages of DE proportion evolution, thetheoretical curve and experimental curve of the damageevolution equation were compared, and then the rationalityof the damage constitutive theory established by DE pro-portion evolution was verified.

Data Availability

,e data used to support the findings of this study areavailable from the corresponding author upon request.

Conflicts of Interest

,e authors declare that they have no conflicts of interest.

Acknowledgments

,is study was supported by the National Natural ScienceFoundation of China (nos. 50904071 and 51274207).

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