investigationonthenonlinearstrengthpropertiesanddamage...

Research ArticleInvestigation on the Nonlinear Strength Properties and DamageStatistical Constitutive Model for Frozen Sandy Soils

De Zhang12 Enlong Liu 13 Xingyan Liu12 Ge Zhang12 Xiao Yin3 and Bingtang Song12

1Northwest Institute of Eco-Environment and Resources State Key Laboratory of Frozen Soil EngineeringChinese Academy of Sciences Lanzhou Gansu 730000 China2University of Chinese Academy of Sciences Beijing 100049 China3State Key Laboratory of Hydraulics and Natural River Engineering College of Water Resource and HydropowerSichuan University Chengdu Sichuan 610065 China

Correspondence should be addressed to Enlong Liu liuenlonglzbaccn

Received 11 September 2017 Accepted 27 November 2017 Published 11 March 2018

Academic Editor Andres Sotelo

Copyright copy 2018 De Zhang et al (is 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

(ere are many flaws such as fissures cavities and inclusions in geomaterials which make their mechanical properties with greatrandomness and uncertainty Upon loading the soil structure gradually losses the bearing capacity due to the transformation frommicrodefects to macroscopic breakage bands Based upon the experimental data of frozen sandy soils a new nonlinear strengthequation between the first and third principal stresses was proposed and then the nonlinear strength properties for frozen sandy soilsin σ-τ plane were analyzed In addition by assuming that the microstrength of frozen sandy soil obeys the Weibull distributionfunction a statistical damage constitutive model was established based upon the framework of continuum damage mechanics(CDM) with few parameters and a high accuracy Compared with experimental data the new model can well grasp the nonlinearstrength properties and simulate the stress-strain relationships under different confining pressures for frozen sandy soils

1 Introduction

Frozen soils are geotechnical materials which are mainlyformed in cryogenic environment Frozen soils are definedas those containing some ice and having a temperature at orlower than 0degC [1] It is distributed all around the worldregularly such as Russia America China and Canadawhich accounts for 24 of the worldrsquos land area [2] As far asthe components are concerned frozen soils are compositematerials consisting of mineral particles ice inclusionsliquid water and gaseous inclusions [3] which are sensitiveto temperature loading rate external load soil types watercontent confining pressure etc(e typical problems in coldregions can be mainly summarized as the frost-heavingeffect thawing-settlement effect and freeze-thaw cycle ef-fect in these regions It is well known that the frost-heavingeffect causes the structure damage due to the extension ofmicrocracks Simultaneously the thawing-settlement effectinduces irreversible plastic deformation Terribly the in-teractions of the two effects aggravate the damage and

destruction of structures Hence up to present many re-searchers have studied the physical and mechanical char-acteristics of permafrost and seasonal frozen soils such asnonlinear strength properties and damage deformationbehaviors

Currently a great number of research achievements havebeen obtained on this field for unfrozen soils and frozensoils For instance the mechanical properties of unfrozensoils are obtained with great progress Matsuoka and Nakai[4] considered the stress deformation and strength char-acteristics of soil under three different principal stresses Liuet al [5 6] proposed a general strength criterion for geo-materials considering isotropic and anisotropic effectsMeanwhile Liu et al [7] employed a nonlinear DruckerndashPrager and MatsuokandashNakai unified failure criterion forgeomaterials to describe the nonlinear strength propertieswith separated stress invariants Yao et al [8 9] establisheda general nonlinear strength theory which was extensivelyemployed in concrete rock sand clay etc Moreover Yaoet al [10 11] proposed a unified strength criterion (USC) for

HindawiAdvances in Materials Science and EngineeringVolume 2018 Article ID 4620101 15 pageshttpsdoiorg10115520184620101

geomaterials to describe the triangle curved shape whichwas the combination of SMP criterion and extended Misescriterion in the deviatoric plane and then he defined a linearinterpolation function between the two criteria to describethe shape function (e nonlinear strength envelope men-tioned above can be described by hyperbolic parabolicexponential power functions etc For structured soils Liuet al [12] proposed a new strength criterion based on binary-medium constitutive model (BMCM) and its expressions onboth meridian and deviator planes are given Due to thepeculiar and complicated properties of frozen soils most ofthe strength criteria are not completely suitable and shouldbe modified for further applications based upon experi-mental results Fish [13] and Ma et al [14] gave a parabolicfunction to describe the strength properties Qi and Ma [15]modified the classical MohrndashCoulomb strength criterion toapply for frozen sandy soils considering the influence ofconfining pressures Lai et al [16] established a paraboliccurve for frozen silt soils in the meridian plane andLadendashDuncan model in the deviatoric plane to describe theshape function Lai et al [17] and Liao et al [18] modified thehydrostatic pressure to simulate the strength in the q-p planeand employed a combination of LD and SMP models todescribe the deviatoric plane Lai et al [19] proposed a re-lationship between the first principal σ1 and the thirdprincipal stress σ3 and then they established a nonlinearstrength formula for frozen sandy soils (e researchesmentioned above mainly concentrate on the nonlinearproperties of unfrozen soils and frozen soils (e classicaland modified strength criterion cannot completely reflectthe nonlinear characteristics for frozen soils so it is nec-essary to establish an appropriate strength criterion to de-scribe the nonlinear properties of frozen soils

At the same time many research results of the stress-strain curves were obtained especially in rock damagemechanism mechanical properties and strength and de-formation characteristics [20ndash29] On the theoreticalframework of continuum damage mechanics (CDM)probability and statistics theory the damage constitutivemodels were established and could well simulate the stress-strain process for geological materials For instance Renet al [30] studied microdamage mechanism and damageconstitutive model on the basis of uniaxial compression testsand found that the damage mechanism could appropriatelydescribe the variation of stress-strain relationship Latera series of experimental study on rock were conducted byZhang et al [31] to analyze the weakened mechanism anddamage properties under the freezing-thawing conditionsRen [32] carried out computed tomography (CT) tests tostudy breakage mechanism for frozen cracked rock froma mesoscopic point of view Lai et al [33] and Li et al [34]proposed a new and an improved statistical damage con-stitutive model for warm frozen clay and warm ice-richfrozen clay based upon the experimental results and appliedMohrndashCoulomb criterion to judge whether the frozen soilelement is damaged or not Xu et al [35] made an in-vestigation on strength and deformation characteristics ofice-saturated frozen sandy soil and then proposed a non-linear strength criterion and simultaneously employed an

improved DuncanndashChang model to simulate the de-formation properties (e researches mentioned above arethe latest research achievements

Due to the peculiar and complicated characteristics offrozen soils some strength criteria such as MohrndashCoulombDruckerndashPrager and HoekndashBrown strength criterion arenot suitable for frozen soils So the classical strength cri-terion should be modified or a new strength criterionmay beestablished to investigate the nonlinear strength propertiesand breakage mechanism for frozen soils (erefore in thispaper a relationship between the first and third principalstresses is proposed and then the nonlinear strengthproperties of frozen sandy soils are investigated Further-more a new statistical damage constitutive model is alsoproposed and the model parameters are determined bycryogenic triaxial compression tests data Finally the ap-plicability of the new constitutive model is validated bycomparisons between predicted and experimental data

2 Test Result Analysis

As is illustrated in Figure 1 from Lai et al [17] it can be foundthat the stress-strain curves go through the linear stage underrelatively small axial strain to the elastoplastic stage In ad-dition the curves present strain-softening phenomenonunder low confining pressures and strain-hardening phe-nomenon under relatively high confining pressures From thevolumetric strain-axial strain curves it is indicated that thevolume of specimen is compressed at first and then dilatedunder low confining pressures while it is merely compressedunder high confining pressures(e determinationmethod ofpeak values of stress-strain curves can be obtained as follows(a) when the stress-strain curves present strain softening themaximum value of σ1 minus σ3 is taken as the strength of frozensoil in this paper and (b) when the stress-strain curves presentstrain hardening the value of σ1 minus σ3 at axial strain εa 20is taken as the strength of frozen soil [33] (e relationshipbetween deviatoric stress q and mean stress p is depicted inFigure 2 which indicates that the strength increases first andthen decreases with increasing mean stress p

3 Nonlinear Strength Theory of FrozenSandy Soils

31 Existing Classical Strength Criterion

311 MohrndashCoulomb Strength Criterion In geomaterialengineering the MohrndashCoulomb criterion is widely used topredict the strength and deformation properties hence oneof the classical strength criteria can be described as follows

σ1 minus σ3 tan2 45deg +φ2

1113874 1113875minus 2c tan 45deg +φ2

1113874 1113875 0 (1)

where c and φ denote the cohesive force and internalfrictional angle respectively In this paper for the test datac 4332 and φ 03792

312 HoekndashBrown Nonlinear Criterion In the early yearsHoek [36] proposed the HoekndashBrown criterion based upon

2 Advances in Materials Science and Engineering

hundreds of experimental results and numerous eld testndings for rock materials It is an empirical strength cri-terion and the expression can be presented in the followingform

σ1 minus σ3 minusmσ3σc + sσ2cradic

0 (2)

where σcm and s are material parameters In this paper forthe test data σc 363 m 1517 and s 6782

32 e Evolution of the Proposed Strength Criterion Basedupon the experimental data from Figure 1 we can obtain therelationship between σ1 and σ3 as illustrated in Figure 3 Inorder to investigate the nonlinear strength characteristicswe propose an empirical equation as follows

σ1 Kσ3 + s 1minusσ3σc( )

n

[ ]expσ3σc( )

n

[ ] (3)

where K s n and σc are material parameters respectivelywhich can be determined based on experimental results Inthis paper for frozen sandy soils K 6449 s 8252σc 282 and n 0394 and the tting coecientR2 0997

For convenience in the σ-τ plane (3) can be rewritten as

f σ1 minusKσ3 minus s 1minusσ3σc( )

n

[ ]expσ3σc( )

n

[ ] (4)

0 4 8 12 16 200

4

8

12

16

20

24

28

32

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa

σ3 = 80 MPaσ3 = 100 MPaσ3 = 140 MPaσ3 = 160 MPa

σ3 = 20 MPa

Axial strain ε1 ()

Dev

iato

ric st

ress

σ1minusσ 3

(M

Pa)

24 28

T = ndash6degC

(a)

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa

σ3 = 20 MPaσ3 = 80 MPaσ3 = 100 MPaσ3 = 140 MPaσ3 = 160 MPa

Axial strain ε1 () 0 4 8

ndash3

ndash2

ndash1

0

1

2

3

4

5

Vol

umet

ric st

rain

ε v (

)

T = ndash6degC

16 20 24 2812

(b)

Figure 1 e curves of deviatoric stress-axial strain (a) and vol-umetric strain (b) of frozen sandy soils (Lai et al [17])

0

4

8

12

16

20

24

28

32

Mean stress p (MPa)

Dev

iato

ric st

ress

q=σ 1

minusσ 3

(MPa

)

Experimental results

0 5 10 15 20 25 30

Figure 2 e strength envelope in the meridian plane

0 4 80

10

20

30

40

50

n

σcσ3σ1 = Kσ3 + s exp1 minus

ird principal stress σ3 (MPa)

Firs

t prin

cipa

l str

ess σ

1 (M

Pa)

Experimental resultsPredicted results

12 16 20

n

σcσ3

Figure 3e relationship between the rst principal stress and thethird principal stress of frozen sandy soils

Advances in Materials Science and Engineering 3

According to experimental results compared with theclassical strength criteria such as MohrndashCoulomb strengthcriterion and HoekndashBrown nonlinear criterion the pro-posed nonlinear strength results are more close to the ex-perimental data as is depicted in Figure 4 It demonstratesthat the new strength criterion can simulate the experi-mental results of frozen sandy soils well

33NonlinearStrengtheoryof FrozenSandySoils Figure 5shows the graphical illustration of the relationship betweenstress state A(σ τ) and failure envelop f at is when thestress state A(σ τ) is at the top of Mohrrsquos circle the failureenvelope is represented by τ f(σ) It should be noted thatthe failure loci presents a nonlinear changing tendencywhich is also called a failure state line e experimentalmaximum stress in the failure envelope is tangent to Mohrrsquoscircle which is the intersection point ofA(σ τ) From Figure 5it is known that the failure stress state is represented by thepoint A which cannot transcend the failure loci τ f(σ)e mathematical relationship of Mohrrsquos circle can be de-scribed as

g σ1 σ3( ) σ minusσ1 + σ3

2( )

2+ τ2 minus

σ1 minus σ32

( )2 0 (5)

By dierentiating (4) and (5) we can obtain the fol-lowing equations

df σ1 σ3( ) zf

zσ1dσ1 +

zf

zσ3dσ3 0 (6)

dg σ1 σ3( ) zg

zσ1dσ1 +

zg

zσ3dσ3 0 (7)

Due to the independence of σ1 and σ3 the relationship ofdσ1dσ3 can be calculated as

dσ1dσ3

zσ1zσ3

minuszgzσ3zgzσ1

minuszfzσ3zfzσ1

(8)

Based on the framework of the envelope theorem by Laiet al [19] for simplication (8) can be rewritten as

zf

zσ1middotzg

zσ3minus

zf

zσ3middotzg

zσ1 0 (9)

Dierentiating (4) for σ1 and σ3 respectively we canobtain the following

zf

zσ1 1 +

zσ3zσ1

sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ]minusK

(10)

zf

zσ3zσ1zσ3minusK +

sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ] (11)

zg

zσ1 minus σ + σ3 +

zσ3zσ1

σ1 minus σ( ) (12)

zg

zσ3 minus σ + σ1 minus

zσ1zσ3

σ minus σ3( ) (13)

Substituting (12) and (13) into (8) the normal stress isexpressed as

σ σ3 +σ1 minus σ3

1 + zσ1zσ3( ) (14)

Substituting (14) into (5) the shear stress is expressed as

τ σ1 minus σ3

1 + zσ1zσ3( )

zσ1zσ3

radic

(15)

Furthermore we can easily get the expression of internalfrictional angle based upon the triangle relationship shownin Figure 5 as follows

0 4 8 12 16 200

10

20

30

40

50

Test resultsMohrndashCoulomb criterion (predicted results)HoekndashBrown criterion (predicted results)is proposed model (predicted results)


Firs

t prin

cipa

l str

ess σ

1 (M

Pa)

Figure 4 Comparisons between predicted results and experi-mental data

τ

σ

Shear stress τ (MPa)Effective normal stress σ (MPa)

0

Mohrrsquos circle

2σ1 + σ3 σ1σ3

φ

A(στ)

τ = f (σ)

0

Figure 5 Schematic description of Mohrrsquos circle


tanφ σ1 minus σ3( 111385711138702( 1113857minus σ

τ (16)

Substituting (14) and (15) into (16) we can obtain thefollowing

φ tanminus1zσ11113870zσ3( 1113857minus 1

2zσ11113870zσ3

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)

Differentiating (4) for σ3 the result is presented as follows

zσ1zσ3

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891 (18)

Finally substituting (18) into (14) (15) and (16) in theσ-τ plane the expressions of normal stress σ shear stress τand internal frictional angle φ of the nonlinear strengthcriterion can be rewritten as

σ Kσ3 minus sn σ3σc( 1113857

2nexp σ3σc( 1113857n

1113858 1113859 + σ1K + 1minus snσ3 σ3σc( 1113857


1113858 1113859 (19)

τ σ1 minus σ3

K + 1minus sn1113870σ3 σ3σc( 11138572nexp σ3σc( 1113857

n1113858 1113859

times

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891

11139741113972

(20)

φ tanminus1Kminus 1minus snσ3 σ3σc( 1113857


1113858 1113859

2

Kminus sn1113870σ3 σ3σc( 11138572nexp σ3σc( 1113857

n1113858 1113859

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)

where the expressions of (19) and (20) can be known asfailure envelope in the σ-τ plane for frozen sandy soils

34 Verification Based upon experimental data the non-linear strength envelope can be obtained from (19) and(20) as is illustrated in Figure 6 in which the same pa-rameters in Section 32 are used (e schematic diagramdistinctively presents the nonlinear strength propertiesCompared with experimental results the proposed ex-pression of (4) about σ1 and σ3 can well grasp and simulatethe nonlinear strength characteristics of frozen sandy soilsIt can be found that the proposed relationship is closer tothe experimental results than MohrndashCoulomb strengthcriterion or HoekndashBrown strength criterion It is convincedthat the proposed equation can appropriately reflect thenonlinear strength properties with the increase of confiningpressures

(e influence of confining pressure on the internalfrictional angle is investigated as well as illustrated in httpwwwnrcresearchpresscomdoifull101139javascriptvoid(0)(Figure 7) It can be found that the frictional angle decreaseswith increasing confining pressure due to pressure meltingand crushing phenomenon It should be noted that the

internal frictional angle is negative when the confiningpressure is nearly 140MPa which is coincident with theresearch results by Ma et al [37] It is indicated that thetemperature of ice increases by 1degC when the applied pressureincreases by 135MPa in the theoretical study on the basis ofthe Clapeyron equation And thus the predicted results canappropriately account for the phenomenon of pressuremeltingand crushing

4 Damage Statistical Constitutive Model forFrozen Sandy Soils

41 Formulation of the Damage Statistical ConstitutiveModel Similar to unfrozen geomaterials such as rockconcrete rockfill material sand and clay the damageprocess can be mainly accounted for by the same token(atis the degradation of the material makes the effective areasdecrease and the effective stress increase Accordingly theintact specimens are easy to damage due to the appearance offissures cracks defects and shear bands

Based on the strain equivalent principle [38] the straincaused by apparent stress σij applied to a damaged materialis equal to the strain caused by equivalent stress σlowastij acting onthe undamaged material Hence the tensor expression canbe described as follows

εij Mijklσlowastkl M

lowastijklσkl Iijkl minusDijkl1113872 1113873Mklmnσmn (22)

where Mlowastijkl and Mijkl denote the elastic flexibility matrixtensors of damaged material and undamaged materialrespectively σlowastij and σij represent the equivalent stressmatrix tensor and apparent stress matrix tensor re-spectively εij represents the strain matrix tensor Iijkl isan identity matrix tensor and Dijkl is a damage matrixtensor

Based on phenomenological method basic assump-tions by Lai et al [33 39] are carried out for frozen soils Atfirst on a macroscale the soil specimen regarded asa representative volume element (abbreviated as RVE) isisotropic and contains the basic information of damagewhile it is a microheterogeneous material in the mesolevelIn addition the linear elastic law is applied when the frozensoil element remains undamaged and the nonlinearity ofthe stress-strain relationship is derived from the damage ofthe material So we can take the initial tangent modulus asthe elastic modulus of the undamaged material (edamage variable is defined as the ratio of damaged sectionarea to total section area in the mesolevel and in mac-roscopic view it is defined as the ratio of the number ofdamaged frozen soil elements to the number of all frozensoil elements So the damage expression defined is given asfollows

D Ndamage

Ntotal (23)

where Ndamage denotes the number of damaged frozen soilelements and Ntotal stands for the number of all frozen soilelements respectively


0 5 10 15 20 25 30 35 40 45 500

7

14

21

28

Normal stress σ (MPa)

Shea

r str

ess τ

(MPa

)

Confining pressure σ3 = 100 MPaConfining pressure σ3 = 20 MPaConfining pressure σ3 = 140 MPaConfining pressure σ3 = 40 MPaConfining pressure σ3 = 160 MPaConfining pressure σ3 = 50 MPaMohrndashCoulomb criterion (predicted results)Confining pressure σ3 = 60 MPa

Confining pressure σ3 = 10 MPa Confining pressure σ3 = 80 MPa

(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28

Confining pressure σ3 = 100 MPaConfining pressure σ3 = 20 MPaConfining pressure σ3 = 140 MPaConfining pressure σ3 = 40 MPaConfining pressure σ3 = 160 MPaConfining pressure σ3 = 50 MPa

Confining pressure σ3 = 60 MPa


HoekndashBrown strength criterion (predicted results)


Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28





The proposed model (predicted results)

Shea

r str

ess τ

(MPa

)

(c)

Figure 6 Nonlinear strength envelope for frozen sandy soils under various conning pressures Comparisons between (a) MohrndashCoulombstrength criterion and experimental results (b) HoekndashBrown strength criterion and experimental results and (c) the proposed model andexperimental results


42 Weibull Distribution Function Compared with theother probability distribution function [40] it is found thatthe Weibull distribution can well describe the breakageprocess of rock [20ndash22] and warm frozen silt clay [33 39] Inthe paper it is assumed that the microstrength of frozensandy soil obeys the Weibull distribution as well When thestress level reaches the value of F as is illustrated in the latersection the internal aws and ssures gradually increase andtransform to macrocracks or shear bands So the expressionof probability density for damaged frozen sandy soil ele-ments can be obtained as follows

f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)

where f(F) represents the probability density function ofmicrostrength for frozen sandy soil F denotes the yieldcriterion and m and F0 are material parameters of theWeibull distribution function

e yield criterion for frozen sandy soils can be written asF σlowastij( ) ki (25)

where ki denotes the failure strength which varies withinternal state parameters such as stress level stress historystress path cohesive force and internal frictional angle

By integrating the microstrength function of (24) we canget the number of damaged frozen sandy soil elements asfollows

Ndamage intF

0Ntotalf(x)dx (26)

Substituting (24) and (26) into (23) the damage variableD can be obtained as follows

D 1minus exp minusF

F0( )

m

[ ] (27)

From (25) and (27) it can be found that the independentvariable F changes with the varying stress state

43 A New Strength Criterion for Frozen Sandy Soil Basedupon the framework of critical state soil mechanics [41]for unfrozen soils as noncohesive materials the strengthenvelope is a straight line and is passing through theoriginal point of the coordinates in the meridian planeWith respect to frozen soils the bonding eect of icecrystals and soil particles possesses tensile strength whichwill be weakened due to the pressure melting under rel-atively high conning pressures similar to the cohesivebehaviors of cemented clay In the previous study[13 14 17 18 37] it is indicated that the strength of frozensoils increases rst and then decreases with the increase ofconning pressures e maximum value qmax of the q-pcurve corresponds to failure mean stress pcr When thestress p is lower than pcr the q-p curve can be replaced bylinear strength criterion in the meridian plane such asMohrndashCoulomb criterion and DruckndashPrager criterionWhile the stress p is higher than pcr the q-p curve starts tobend downward due to pressure melting In order tosimulate evolution laws of the failure state for frozen sandysoils of the phenomenon Fish [13] Ma et al [14 37] andLai et al [16] suggested a parabolic formula for frozen soilsin the meridian plane and Lai et al [16] meanwhile pro-posed the LadendashDuncan model to simulate the strengthcharacteristics in the deviatoric plane Later Nguyen et al[42] proposed that the CSL of the cemented clay eventuallyreduced to an asymptote coinciding with the CSL of naturalclay Lai et al [17] and Liao et al [18] employed a modiedmean eective stress expression to describe the strengthcriterion for frozen soils

Hence in this paper based on the research results ofNguyen et al [42] Lai et al [17] and Liao et al [18] wepropose a modied strength criterion to simulate the var-iation tendency of the rst increasing and then decreasingphenomena for frozen sandy soils Here the graphic illus-tration and the proposed modied strength criterion aregiven in Figure 8 and (28) to (30)

From Figure 8 the modied failure state line can bewritten as follows

ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)

Confining pressure σ3 (MPa)0 3 6 9 12 15 18

Figure 7 e evolution law of internal frictional angle

c0

q = Mlowastp

qcr

pcr

M0mdashe slope of CSLMlowast

mdashe modified slope of failure stress statec0mdashInitial cemented force

q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)

Mean effective stress p (MPa)

t t

Figure 8 Relationships of stress-strain curves for frozen soils in themeridian plane


f(p q) minusq +Mlowast p (28)

Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)

Substituting (29) into (28) the expression can be re-written as follows

f(p q) minusq +M0p + c0 1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (30)

where q is the deviatoric stress p is the mean stress andc0 is the intercept of the strength curve in the meridian

Table 1 Basic physical parameters determined

Conning pressuresσ3 (MPa)

Bulk modulusK (MPa)

Shear modulusG (MPa)

10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)

Confining pressure σ3 (MPa)


0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)

Figure 9e predicted results of the bulk modulus K (a) and shearmodulus G (b) with experimental data

Table 2 Parameter determination of the strength criterion

Parameter values of frozen sandy soilR2

M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32

Mean stress p (MPa)D

evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1

pq = M0 p + c0 exp1 minust

C1

p

Figure 10 Comparisons between experimental data and predictedresults under dierent conning pressures

Table 3 Parameter determination of the damage constitutivemodel of (46)


Parameter determinationm F0

10 11948Eminus01 24424Eminus0120 73400Eminus02 24753Eminus0240 48047Eminus02 56572Eminus0550 35515Eminus02 46744Eminus0760 27221Eminus02 16468Eminus1080 25790Eminus02 13385Eminus11100 20809Eminus02 40019Eminus15140 17408Eminus02 21848Eminus18160 16381Eminus02 44111Eminus19


planeM0 is the initial failure stress ratio equal to the slopeof the curve e parameter C1 is related to pressuremelting which reects the changing rate about q with theincrease of hydrostatic pressure and t denotes the materialparameter

As is depicted in Figure 8 when the mean stress p islower than pcr the modied failure state line can describe thestrengthened eect when the mean stress p is higher thanpcr it denotes the weakened eect for frozen sandy soils

e mean stress p and the deviatoric stress q can bewritten in the following form

p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24

e fitting results of (46)

e value of X (minus)

e v

alue

of Y

(minus)

σ3 = 10 MPa (experimental results)σ3 = 10 MPa (predicted results)σ3 = 20 MPa (experimental results)σ3 = 20 MPa (predicted results)σ3 = 40 MPa (experimental results)

σ3 = 40 MPa (predicted results)σ3 = 50 MPa (experimental results)σ3 = 50 MPa (predicted results)σ3 = 60 MPa (experimental results)σ3 = 60 MPa (predicted results)

(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)

σ3 = 140 MPa (experimental results)σ3 = 140 MPa (predicted results)σ3 = 160 MPa (experimental results)σ3 = 160 MPa (predicted results)


(b)

Figure 11 e relationship between the parameters Y and X of (46) under dierent conning pressures (a) 10ndash60MPa and(b) 80ndash160MPa


sij σij minus13δijσkk (33)

where σ1 σ2 and σ3 are the rst second and third principalstresses respectively J2 is the second deviatoric stress var-iant σij is the stress tensor sij is the deviatoric stress tensorand δij is the Kronecker function

Based upon the assumptions it is easily accepted that themicrostrength of frozen soils satises the modied failurestate line so the strength criterion expression of the eectivestress can be obtained as followsF plowast qlowast( ) f plowast qlowast( )

minusqlowast +M0plowast + c0 1minus

plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)

where plowast and qlowast denote the eective mean stress anddeviatoric stress respectively

According to (22) the eective stress and apparent stresscan be rewritten as follows

σij Iijkl minusDijkl( )σlowastkl (35)

σlowastij λδijεkk + 2Gεij (36)

Substituting (27) and (36) into (35) the stress expressioncan be rewritten as

σij λδijεkk + 2Gεij( )exp minusF

F0( )

m

[ ] (37)

e elastic constants λ and G can be obtained as follows

λ Eμ

(1 + μ)(1minus 2μ) Kminus

23G (38)

G E

2(1 + μ) (39)

where λ denotes the Lame constant and K and G are the bulkmodulus and shear modulus respectively

Based upon the generalized Hookersquos law the expressionof axial strain ε1 can be expressed by

ε1 1

1minusDmiddotσ1 minus μ σ2 + σ3( )

E (40)

Hence the eective mean stress and deviatoric stress canbe obtained as follows

plowast p

1minusD

Eε1pσ1 minus μ σ2 + σ3( )

(41)

qlowast q

1minusD

Eε1qσ1 minus μ σ2 + σ3( )

(42)

Substituting (38) to (42) into (37) the formula can beobtained by a series of mathematical calculation andtransformation

ln minuslnσij

λδijεkk + 2Gεij[ ] mlnFminusmlnF0 (43)

In order to obtain the material parameters (43) can beconverted to the following form

X ln minuslnσij

λδijεkk + 2Gεij[ ] (44)

Y lnF plowast qlowast( )

ln minusqlowast +M0plowast + c0 1minus

plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)

Combining (44) with (45) we can get

Y 1mX + ln F0 (46)

Based upon triaxial compression test data from Figure 1(46) can be used to obtain the parameters m and F0

44 Parameter Determination

441 Determination of BulkModulus K and ShearModulus GBased upon test data from Figure 1 the basic physical pa-rameters can be determined such as the bulk modulusK andshear modulus G in the initial elastic stage as shown inTable 1

In order to describe the changing evolution of the bulkmodulus K and shear modulus G the mathematical re-lationship can be expressed by

K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583

Figure 12 e relationship between the parameter m and con-ning pressures


where Pa is the atmospheric pressure (Pa 010133MPa) andKp ap np and ag bg cg are material parameters In thispaper the tting results are illustrated in Figure 9Kp 13350 ap minus205 np 03637 and ag minus03757bg 8586 cg 3534

442 Parameter Determination of the Strength CriterionAccording to the cryogenic triaxial compression testdata as is illustrated in Figure 1 the values of tting

parameters are shown in Table 2 and the correspondingtting curve in Figure 10 It can be found that the pro-posed criterion has a good application for frozen sandysoils especially the correlation coecient R2 reaching thevalue of 09971

443 Parameter Determination of the Damage StatisticalConstitutive Model e values of tting parameters of (46)can be obtained from triaxial compression test data Hencem

0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

σ3 = 10 MPa (predicted results)σ3 = 10 MPa (experimental results)σ3 = 20 MPa (predicted results)σ3 = 20 MPa (experimental results)σ3 = 40 MPa (predicted results)σ3 = 40 MPa (experimental results)

(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)

Figure 13 Comparisons between test data and predicted results of deviatoric stress-axial strain under dierent conning pressures(a) 10ndash40MPa (b) 50ndash80MPa and (c) 100ndash160MPa


and F0 can be found in Table 3 (e fitting results of (46) aredepicted in Figure 11 It can be found that the relationshipbetween Y and X presents a good linear relationship

As presented in Figure 12 the relationship between theparameter m and confining pressures can be expressed asfollows

m 01544 exp minus04583σ3σ

1113874 11138751113876 1113877 + 001874 (49)

where σ means 10MPa to keep the σ3σ dimensionless

45 Verification

451 Verification of Deviatoric Stress-Axial Strain Afterdetermining themodel parameters the stress under differentconfining pressures can be calculated by (37) Comparisonsof the curves between experimental data and predicted re-sults are depicted in Figure 13 It can be found that thisproposed statistical damage constitutive model can simulatethe softening phenomenon under low confining pressuresand hardening phenomenon under high confining pres-sures As a whole the proposed model has a relatively highaccuracy so it is quite convenient for practical engineering

452 Verification of Volumetric Strain-Axial Strain Inorder to simulate the volumetric strain of frozen sandy soilswe give a damage relationship from the mesolevel and theexpression is shown as follows

εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)

where Vinitial denotes the initial volume of the specimenVdamage denotes the damage volume of the specimen and V

denotes the total volume of the specimenFrom (30) we know the invariant relationship of

V Vinitial + Vdamage hence (50) can be rewritten as

εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)

Based on the theory framework of continuum damagemechanics (CDM) the damage variable can also be rede-fined as

D Vdamage

V (52)

So the strain tensor of (51) can be rewritten as follows

εij Iijkl minusDijkl1113872 1113873εinitialkl + Dijklεdamagekl (53)

Under the triaxial symmetric compression condition thevolumetric strain of (53) can be expressed as

εv (1minusD)εinitialv + Dεdamagev (54)

We make an assumption of εinitialv pK for the initialelastic part and εdamage

v mdεnd

1 for the damage part hencethe following expression can be obtained

εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)

where md and nd are the material parameters which can bedetermined by experimental data

From Tables 3 and 4 the curves of volumetric strain-axial strain can be obtained as are illustrated in Figure 14 Itcan be found from Figure 14 that (55) can well simulate theexperimental data which present firstly compression andthen dilation phenomenon under relatively low confiningpressures and merely compression under high confiningpressures

5 Conclusions

In this paper a new nonlinear strength criterion is proposedand a damage statistical constitutive model is also estab-lished (e following conclusions can be reached

(1) (e relationship between σ1 and σ3 is proposed basedupon experimental results of frozen sandy soils It isillustrated that the proposed equation is more close toexperimental results than the classical strength cri-teria such as MohrndashCoulomb and HoekndashBrownstrength criteria And then the equation is introducedinto the σ-τ plane to investigate the nonlinear strengthproperties based on failure envelope theoremCompared with test data the predicted results havea good accuracy and appropriateness

Table 4 Parameter determination of (55)

Confining pressuresσ3 (MPa)

Parameter determinationmd nd

10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059


(2) e microstrength of frozen sandy soils obeysthe Weibull distribution function e strength cri-terion which contains the damage propertiesofmicrostrength is chosen as an independent variablein damage variable D e parameters related to thedamage statistical constitutive model are determinedby experimental data and they can well simulate boththe curves of deviatoric stress-axial strain and volu-metric strain-axial strain

Additional Points

Highlights (1) A new strength criterion is proposed based onexperimental results and the nonlinear strength propertiesare considered (2)e stress andvolumetric strain-axial strainat dierent conning pressures are validated by damage sta-tistical constitutive model (3)e damage constitutive modelcan simulate the strain softening and hardening phenomenonfor frozen sandy soils

minus6

minus4

minus2

0

2

4

Axial strain εa ()

σ3 = 10 MPa (experimental results)σ3 = 10 MPa (predicted results)σ3 = 20 MPa (experimental results)σ3 = 20 MPa (predicted results)σ3 = 40 MPa (experimental results)σ3 = 40 MPa (predicted results)

0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()

σ3 = 50 MPa (experimental results)σ3 = 50 MPa (predicted results)σ3 = 60 MPa (experimental results)

σ3 = 80 MPa (predicted results)

σ3 = 60 MPa (predicted results)σ3 = 80 MPa (experimental results)

0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5

σ3 = 140 MPa (experimental results)σ3 = 140 MPa (predicted results)

0 3 6 9 12 15 18 21

(d)

Figure 14 Comparisons between test data and predicted results of volumetric strain-axial strain under dierent conning pressures(a) 10ndash40MPa (b) 50ndash80MPa (c) 100 and 160MPa and (d) 140MPa


Conflicts of Interest

(e authors declare that there are no conflicts of interest

Acknowledgments

(e authors appreciate the funding provided by the CASPioneer Hundred Talents Program (Dr Liu Enlong) andNational Science Foundation of China (41771066)

References

[1] X Z Xu J C Wang and L X Zhang Physics of Frozen SoilsScience press in Chinese Beijing China 2009

[2] H M French lte Periglacial Environment Essex LondonUK 2nd edition 1996

[3] N A Tsytovich lte Mechanics of Frozen Ground SciencePress Beijing China 1985

[4] H Matsuoka and T Nakai ldquoStress-deformation and strengthcharacteristics of soil under three different principal stressesrdquoJapan Society of Civil Engineers vol 1974 no 232 pp 59ndash701974

[5] M D Liu and J P Carter ldquoGeneral strength criterion forgeomaterialsrdquo International Journal of Geomechanics vol 3no 2 pp 253ndash259 2003

[6] M D Liu and B N Indraratna ldquoGeneral strength criterion forgeomaterials including anisotropic effectrdquo InternationalJournal of Geomechanics vol 11 no 3 pp 251ndash262 2010

[7] M C Liu Y F Gao and H L Liu ldquoA nonlinear Drucker-Prager and Matsuoka-Nakai unified failure criterion forgeomaterials with separated stress invariantsrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 50pp 1ndash10 2012

[8] Y P Yao and D A Sun ldquoApplication of Ladersquos criterion toCam-clay modelrdquo Journal of Engineering Mechanics vol 126no 1 pp 112ndash119 2000

[9] Y P Yao and A N Zhou ldquoNon-isothermal unified hardeningmodel a thermo-elasto-plastic model for claysrdquo Geotechniquevol 63 no 15 pp 1328ndash1345 2013

[10] Y P Yao D A Sun and T Luo ldquoA critical state model forsands dependent on stress and densityrdquo International Journalfor Numerical and Analytical Methods in Geomechanicsvol 28 no 4 pp 323ndash337 2004

[11] Y P Yao W Hou and A N Zhou ldquoUH model three-dimensional unified hardening model for overconsolidatedclaysrdquo Geotechnique vol 59 no 5 pp 451ndash469 2009

[12] E L Liu Q Nie and J H Zhang ldquoA new strength criterionfor structured soilsrdquo Journal of Rock Mechanics andGeotechnical Engineering vol 5 pp 156ndash161 2013

[13] A M Fish ldquoStrength of frozen soil under a combined stressstaterdquo Sixth International Symposium on Ground Freezingvol 1 pp 135ndash145 1991

[14] W Ma Z W Wu and Y Sheng ldquoEffect of confining pressureon strength behaviour of frozen soilrdquo Chinese JournalGeotechnical Engineering vol 17 no 5 pp 7ndash11 1995

[15] J L Qi and W Ma ldquoA new criterion for strength of frozensand under quick triaxial compression considering effect ofcofining pressurerdquoActa Geotechnica vol 2 pp 221ndash226 2007

[16] Y M Lai Y G Yang X X Chang and S Y Li ldquoStrengthcriterion and elastoplastic constitutive model of frozen silt ingeneralized plastic-mechanicsrdquo International Journal ofPlasticity vol 26 no 10 pp 1461ndash1484 2010

[17] Y M Lai M K Liao and K Hu ldquoA constitutive model offrozen saline sandy soil based on energy dissipation theoryrdquo

International Journal of Plasticity vol 78 no 3 pp 84ndash1132016

[18] M K Liao Y M Lai and C Wang ldquoA strength criterion forfrozen sodium sulfate saline soilrdquo Canadian GeotechnicalJournal vol 53 no 7 pp 1176ndash1185 2016

[19] Y M Lai Z H Gao and S J Zhang ldquoStress-strain re-lationships and nonlinear Mohr strength criteria of frozensandy clayrdquo Soils and Foundations vol 50 no 1 pp 45ndash532010

[20] W G Cao Z L Fang and X J Tang ldquoA study of statisticalconstitutive model for softening and damage rocksrdquo ChineseJournal of Rock Mechanics and Engineering vol 17 no 6pp 628ndash633 1998

[21] W G Cao M H Zhao and C X Liu ldquoA study on damagestatistical strength theory for rockrdquo Chinese Journal ofGeotechnical Engineering vol 26 no 6 pp 820ndash823 2004

[22] W G Cao P Li and M H Zhao ldquoOn statistical damageconstitutive model and its parameters for rock based onnormal distributionrdquo Chinese Hydrogeology amp EngineeringGeology vol 32 no 3 pp 11ndash14 2008

[23] J Deng and D Gu ldquoOn a statistical damage constitutivemodel for rock materialsrdquo Computers amp Geosciences vol 37no 2 pp 122ndash128 2011

[24] X Li W G Cao and Y H Su ldquoA statistical damage con-stitutive model for softening behavior of rocksrdquo EngineeringGeology vol 143 pp 1ndash17 2012

[25] H Li H Liao and G Xiong ldquoA three-dimensional statisticaldamage constitutive model for geomaterialsrdquo Journal ofMechanical Science and Technology vol 29 no 1 pp 71ndash772015

[26] Z L Wang Y C Li and J G Wang ldquoA damage-softeningstatistical constitutive model considering rock residualstrengthrdquo Computers amp Geosciences vol 33 no 1 pp 1ndash92007

[27] H Xie and F Gao ldquo(e mechanics of cracks and a statisticalstrength theory for rocksrdquo International Journal of RockMechanics amp Mining Sciences vol 37 no 3 pp 477ndash4882000

[28] W Y Xu and L D Wei ldquoStudy on statistical damage con-stitutive model of rockrdquo Chinese Journal of Rock Mechanicsand Engineering vol 21 no 6 pp 787ndash791 2002

[29] A Shojaei A D Taleghani and G Li ldquoA continuum damagefailure model for hydraulic fracturing of porous rocksrdquoInternational Journal of Plasticity vol 59 pp 199ndash212 2014

[30] J X Ren and X R Ge ldquoStudy of rock meso-damage evolutionlaw and its constitutive model under uniaxial compressionloadingrdquo Chinese Journal of Rock Mechanics and Engineeringvol 20 no 4 pp 425ndash431 2001

[31] S J Zhang YM Lai and XM Su ldquoA laboratory study on thedamage propagation of rocks under freeze-thaw cycle con-ditionrdquo Chinese Journal of Rock Mechanics and Engineeringvol 23 no 24 pp 4105ndash4111 2004

[32] J X Ren ldquoReal-time CT test of damage failure mechanism offrozen cracked rock in loading and unloading conditionrdquoChinese Journal of Geotechnical Engineering vol 26 no 5pp 641ndash644 2004

[33] Y M Lai S Y Li J L Qi Z H Gao and X X ChangldquoStrength distributions of warm frozen clay and its stochasticdamage constitutive modelrdquo Cold Regions Science andTechnology vol 53 pp 200ndash215 2008

[34] S Y Li Y M Lai S J Zhang and D Liu ldquoAn improvedstatistical damage constitutive model for warm frozen claybased on Mohr-Coulomb criterionrdquo Cold Regions Science andTechnology vol 57 no 2-3 pp 154ndash159 2009


[35] X T Xu Y M Lai Y H Dong and J L Qi ldquoLaboratoryinvestigation on strength and deformation characteristics ofice-saturated frozen sandy soilrdquo Cold Regions Science andTechnology vol 68 no 1 pp 98ndash104 2011

[36] E Hoek ldquo(e Hoek-Brown failure criterionrdquo in Proceedingsof 15th Canadian Rock Mechanics Symposium pp 31ndash38Department of Civil Engineering University of TorontoToronto Canada 1988

[37] W Ma Z W Wu L X Zhang and X Chang ldquoAnalyses ofprocess on the strength decrease in frozen soils under highconfining pressuresrdquo Cold Regions Science and Technologyvol 29 no 1 pp 1ndash7 1999

[38] J A Lemaitre ldquoHow to use damage mechanicsrdquo NuclearEngineering and Design vol 80 no 2 pp 233ndash245 1984

[39] Y M Lai J B Li and Q Z Li ldquoStudy on damage statisticalconstitutive model and stochastic simulation for warm ice-rich frozen siltrdquo Cold Regions Science and Technology vol 71no 2 pp 102ndash110 2012

[40] W Weibull ldquoA statistical distribution function of wideapplicabilityrdquo Journal of Applied Mechanics vol 18pp 293ndash297 1951

[41] K H Roscoe and J B Burland ldquoOn the generalized stress-strain behaviour of wet clayrdquo Engineering Plasticity Cam-bridge University Press Cambridge UK 1968

[42] L D Nguyen B Fatahi and H Khabbaz ldquoA constitutivemodel for cemented clays capturing cementation degrada-tionrdquo International Journal of Plasticity vol 56 pp 1ndash18 2014


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Journal of


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ate

ria

ls


Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

geomaterials to describe the triangle curved shape whichwas the combination of SMP criterion and extended Misescriterion in the deviatoric plane and then he defined a linearinterpolation function between the two criteria to describethe shape function (e nonlinear strength envelope men-tioned above can be described by hyperbolic parabolicexponential power functions etc For structured soils Liuet al [12] proposed a new strength criterion based on binary-medium constitutive model (BMCM) and its expressions onboth meridian and deviator planes are given Due to thepeculiar and complicated properties of frozen soils most ofthe strength criteria are not completely suitable and shouldbe modified for further applications based upon experi-mental results Fish [13] and Ma et al [14] gave a parabolicfunction to describe the strength properties Qi and Ma [15]modified the classical MohrndashCoulomb strength criterion toapply for frozen sandy soils considering the influence ofconfining pressures Lai et al [16] established a paraboliccurve for frozen silt soils in the meridian plane andLadendashDuncan model in the deviatoric plane to describe theshape function Lai et al [17] and Liao et al [18] modified thehydrostatic pressure to simulate the strength in the q-p planeand employed a combination of LD and SMP models todescribe the deviatoric plane Lai et al [19] proposed a re-lationship between the first principal σ1 and the thirdprincipal stress σ3 and then they established a nonlinearstrength formula for frozen sandy soils (e researchesmentioned above mainly concentrate on the nonlinearproperties of unfrozen soils and frozen soils (e classicaland modified strength criterion cannot completely reflectthe nonlinear characteristics for frozen soils so it is nec-essary to establish an appropriate strength criterion to de-scribe the nonlinear properties of frozen soils

At the same time many research results of the stress-strain curves were obtained especially in rock damagemechanism mechanical properties and strength and de-formation characteristics [20ndash29] On the theoreticalframework of continuum damage mechanics (CDM)probability and statistics theory the damage constitutivemodels were established and could well simulate the stress-strain process for geological materials For instance Renet al [30] studied microdamage mechanism and damageconstitutive model on the basis of uniaxial compression testsand found that the damage mechanism could appropriatelydescribe the variation of stress-strain relationship Latera series of experimental study on rock were conducted byZhang et al [31] to analyze the weakened mechanism anddamage properties under the freezing-thawing conditionsRen [32] carried out computed tomography (CT) tests tostudy breakage mechanism for frozen cracked rock froma mesoscopic point of view Lai et al [33] and Li et al [34]proposed a new and an improved statistical damage con-stitutive model for warm frozen clay and warm ice-richfrozen clay based upon the experimental results and appliedMohrndashCoulomb criterion to judge whether the frozen soilelement is damaged or not Xu et al [35] made an in-vestigation on strength and deformation characteristics ofice-saturated frozen sandy soil and then proposed a non-linear strength criterion and simultaneously employed an

improved DuncanndashChang model to simulate the de-formation properties (e researches mentioned above arethe latest research achievements

Due to the peculiar and complicated characteristics offrozen soils some strength criteria such as MohrndashCoulombDruckerndashPrager and HoekndashBrown strength criterion arenot suitable for frozen soils So the classical strength cri-terion should be modified or a new strength criterionmay beestablished to investigate the nonlinear strength propertiesand breakage mechanism for frozen soils (erefore in thispaper a relationship between the first and third principalstresses is proposed and then the nonlinear strengthproperties of frozen sandy soils are investigated Further-more a new statistical damage constitutive model is alsoproposed and the model parameters are determined bycryogenic triaxial compression tests data Finally the ap-plicability of the new constitutive model is validated bycomparisons between predicted and experimental data

2 Test Result Analysis

As is illustrated in Figure 1 from Lai et al [17] it can be foundthat the stress-strain curves go through the linear stage underrelatively small axial strain to the elastoplastic stage In ad-dition the curves present strain-softening phenomenonunder low confining pressures and strain-hardening phe-nomenon under relatively high confining pressures From thevolumetric strain-axial strain curves it is indicated that thevolume of specimen is compressed at first and then dilatedunder low confining pressures while it is merely compressedunder high confining pressures(e determinationmethod ofpeak values of stress-strain curves can be obtained as follows(a) when the stress-strain curves present strain softening themaximum value of σ1 minus σ3 is taken as the strength of frozensoil in this paper and (b) when the stress-strain curves presentstrain hardening the value of σ1 minus σ3 at axial strain εa 20is taken as the strength of frozen soil [33] (e relationshipbetween deviatoric stress q and mean stress p is depicted inFigure 2 which indicates that the strength increases first andthen decreases with increasing mean stress p

3 Nonlinear Strength Theory of FrozenSandy Soils

31 Existing Classical Strength Criterion

311 MohrndashCoulomb Strength Criterion In geomaterialengineering the MohrndashCoulomb criterion is widely used topredict the strength and deformation properties hence oneof the classical strength criteria can be described as follows

σ1 minus σ3 tan2 45deg +φ2

1113874 1113875minus 2c tan 45deg +φ2

1113874 1113875 0 (1)

where c and φ denote the cohesive force and internalfrictional angle respectively In this paper for the test datac 4332 and φ 03792

312 HoekndashBrown Nonlinear Criterion In the early yearsHoek [36] proposed the HoekndashBrown criterion based upon




0 (2)




n

[ ]expσ3σc( )

n

[ ] (3)




n

[ ]expσ3σc( )

n

[ ] (4)

0 4 8 12 16 200

4

8

12

16

20

24

28

32

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa


σ3 = 20 MPa

Axial strain ε1 ()

Dev

iato

ric st

ress

σ1minusσ 3

(M

Pa)

24 28

T = ndash6degC

(a)

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa



ndash3

ndash2

ndash1

0

1

2

3

4

5

Vol

umet

ric st

rain

ε v (

)

T = ndash6degC

16 20 24 2812

(b)


0

4

8

12

16

20

24

28

32

Mean stress p (MPa)

Dev

iato

ric st

ress

q=σ 1

minusσ 3

(MPa

)


0 5 10 15 20 25 30


0 4 80

10

20

30

40

50

n



Firs

t prin

cipa

l str

ess σ

1 (M

Pa)


12 16 20

n

σcσ3






2( )

2+ τ2 minus

σ1 minus σ32

( )2 0 (5)


df σ1 σ3( ) zf

zσ1dσ1 +

zf

zσ3dσ3 0 (6)

dg σ1 σ3( ) zg

zσ1dσ1 +

zg

zσ3dσ3 0 (7)


dσ1dσ3

zσ1zσ3

minuszgzσ3zgzσ1

minuszfzσ3zfzσ1

(8)


zf

zσ1middotzg

zσ3minus

zf

zσ3middotzg

zσ1 0 (9)


zf

zσ1 1 +

zσ3zσ1

sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ]minusK

(10)

zf


sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ] (11)

zg


zσ3zσ1

σ1 minus σ( ) (12)

zg


zσ1zσ3

σ minus σ3( ) (13)



1 + zσ1zσ3( ) (14)


τ σ1 minus σ3

1 + zσ1zσ3( )

zσ1zσ3

radic

(15)


0 4 8 12 16 200

10

20

30

40

50



Firs

t prin

cipa

l str

ess σ

1 (M

Pa)


τ

σ


0

Mohrrsquos circle

2σ1 + σ3 σ1σ3

φ

A(στ)

τ = f (σ)

0




τ (16)



2zσ11113870zσ3

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)


zσ1zσ3

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891 (18)




1113858 1113859 + σ1K + 1minus snσ3 σ3σc( 1113857


1113858 1113859 (19)

τ σ1 minus σ3


n1113858 1113859

times

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891

11139741113972

(20)



1113858 1113859

2


n1113858 1113859

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)












D Ndamage

Ntotal (23)



0 5 10 15 20 25 30 35 40 45 500

7

14

21

28


Shea

r str

ess τ

(MPa

)



(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r str

ess τ

(MPa

)

(c)




f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






0 (2)




n

[ ]expσ3σc( )

n

[ ] (3)




n

[ ]expσ3σc( )

n

[ ] (4)

0 4 8 12 16 200

4

8

12

16

20

24

28

32

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa


σ3 = 20 MPa

Axial strain ε1 ()

Dev

iato

ric st

ress

σ1minusσ 3

(M

Pa)

24 28

T = ndash6degC

(a)

σ3 = 10 MPa

σ3 = 40 MPaσ3 = 50 MPaσ3 = 60 MPa



ndash3

ndash2

ndash1

0

1

2

3

4

5

Vol

umet

ric st

rain

ε v (

)

T = ndash6degC

16 20 24 2812

(b)


0

4

8

12

16

20

24

28

32

Mean stress p (MPa)

Dev

iato

ric st

ress

q=σ 1

minusσ 3

(MPa

)


0 5 10 15 20 25 30


0 4 80

10

20

30

40

50

n



Firs

t prin

cipa

l str

ess σ

1 (M

Pa)


12 16 20

n

σcσ3






2( )

2+ τ2 minus

σ1 minus σ32

( )2 0 (5)


df σ1 σ3( ) zf

zσ1dσ1 +

zf

zσ3dσ3 0 (6)

dg σ1 σ3( ) zg

zσ1dσ1 +

zg

zσ3dσ3 0 (7)


dσ1dσ3

zσ1zσ3

minuszgzσ3zgzσ1

minuszfzσ3zfzσ1

(8)


zf

zσ1middotzg

zσ3minus

zf

zσ3middotzg

zσ1 0 (9)


zf

zσ1 1 +

zσ3zσ1

sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ]minusK

(10)

zf


sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ] (11)

zg


zσ3zσ1

σ1 minus σ( ) (12)

zg


zσ1zσ3

σ minus σ3( ) (13)



1 + zσ1zσ3( ) (14)


τ σ1 minus σ3

1 + zσ1zσ3( )

zσ1zσ3

radic

(15)


0 4 8 12 16 200

10

20

30

40

50



Firs

t prin

cipa

l str

ess σ

1 (M

Pa)


τ

σ


0

Mohrrsquos circle

2σ1 + σ3 σ1σ3

φ

A(στ)

τ = f (σ)

0




τ (16)



2zσ11113870zσ3

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)


zσ1zσ3

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891 (18)




1113858 1113859 + σ1K + 1minus snσ3 σ3σc( 1113857


1113858 1113859 (19)

τ σ1 minus σ3


n1113858 1113859

times

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891

11139741113972

(20)



1113858 1113859

2


n1113858 1113859

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)












D Ndamage

Ntotal (23)



0 5 10 15 20 25 30 35 40 45 500

7

14

21

28


Shea

r str

ess τ

(MPa

)



(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r str

ess τ

(MPa

)

(c)




f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







2( )

2+ τ2 minus

σ1 minus σ32

( )2 0 (5)


df σ1 σ3( ) zf

zσ1dσ1 +

zf

zσ3dσ3 0 (6)

dg σ1 σ3( ) zg

zσ1dσ1 +

zg

zσ3dσ3 0 (7)


dσ1dσ3

zσ1zσ3

minuszgzσ3zgzσ1

minuszfzσ3zfzσ1

(8)


zf

zσ1middotzg

zσ3minus

zf

zσ3middotzg

zσ1 0 (9)


zf

zσ1 1 +

zσ3zσ1

sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ]minusK

(10)

zf


sn

σ3

σ3σc( )

2n

expσ3σc( )

n

[ ] (11)

zg


zσ3zσ1

σ1 minus σ( ) (12)

zg


zσ1zσ3

σ minus σ3( ) (13)



1 + zσ1zσ3( ) (14)


τ σ1 minus σ3

1 + zσ1zσ3( )

zσ1zσ3

radic

(15)


0 4 8 12 16 200

10

20

30

40

50



Firs

t prin

cipa

l str

ess σ

1 (M

Pa)


τ

σ


0

Mohrrsquos circle

2σ1 + σ3 σ1σ3

φ

A(στ)

τ = f (σ)

0




τ (16)



2zσ11113870zσ3

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)


zσ1zσ3

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891 (18)




1113858 1113859 + σ1K + 1minus snσ3 σ3σc( 1113857


1113858 1113859 (19)

τ σ1 minus σ3


n1113858 1113859

times

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891

11139741113972

(20)



1113858 1113859

2


n1113858 1113859

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)












D Ndamage

Ntotal (23)



0 5 10 15 20 25 30 35 40 45 500

7

14

21

28


Shea

r str

ess τ

(MPa

)



(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r str

ess τ

(MPa

)

(c)




f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





τ (16)



2zσ11113870zσ3

1113969⎡⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎦ (17)


zσ1zσ3

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891 (18)




1113858 1113859 + σ1K + 1minus snσ3 σ3σc( 1113857


1113858 1113859 (19)

τ σ1 minus σ3


n1113858 1113859

times

Kminussn

σ3

σ3σc

1113888 1113889

2n

expσ3σc

1113888 1113889

n

1113890 1113891

11139741113972

(20)



1113858 1113859

2


n1113858 1113859

1113969⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ (21)












D Ndamage

Ntotal (23)



0 5 10 15 20 25 30 35 40 45 500

7

14

21

28


Shea

r str

ess τ

(MPa

)



(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r str

ess τ

(MPa

)

(c)




f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




0 5 10 15 20 25 30 35 40 45 500

7

14

21

28


Shea

r str

ess τ

(MPa

)



(a)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r stre

ss τ

(MPa

)

(b)

0 5 10 15 20 25 30 35 40 45 500

7

14

21

28






Shea

r str

ess τ

(MPa

)

(c)




f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





f(F) m

F0

F

F0( )

mminus1

exp minusF

F0( )

m

[ ] (24)





Ndamage intF

0Ntotalf(x)dx (26)


D 1minus exp minusF

F0( )

m

[ ] (27)





ndash15

0

15

30

45

Predicted results

Inte

rnal

fric

tiona

l ang

le φ

(deg)



c0

q = Mlowastp

qcr

pcr



q = Mp + c0

1

M

minus+= M0 C1

pC1

ppc0Mlowast exp1

Dev

iato

ric st

ress

q=

σ 1 minus

σ3 (

MPa

)


t t




Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





Mlowast M0 +c0p

1minusp

C1( )

t

[ ]expp

C1( )

t

[ ] (29)



C1( )

t

[ ]expp

C1( )

t

[ ] (30)




Bulk modulusK (MPa)


10 5051 440220 11331 514940 21047 642250 25311 694760 32442 739780 41162 8076100 48981 8459140 53292 8334160 54555 7827

0

1000

2000

3000

4000

5000

6000

Bulk

mod

ulus

K (M

Pa)



0 3 6 9 12 15 18

(a)


0

200

400

600

800

1000

Shea

r mod

ulus

G (M

Pa)


0 3 6 9 12 15 18

(b)




M0 c0 C1 t1503 1494 1668 09479 09971

0 5 10 15 20 25 300

8

16

24

32


evia

toric

stre

ss q

=σ 1

minusσ 3

(MPa

)


t

C1


C1

p










p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







p σ1 + σ2 + σ3

313σii (31)

q 3J2radic

32sijsij

radic (32)

04 06 08 10 12 14 160

4

8

12

16

20

24



e v

alue

of Y

(minus)



(a)

e value of X (minus)06 07 08 09 10 11 12

5

10

15

20

25

30


e v

alue

of Y

(minus)



(b)







plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

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ate

ria

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plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(34)







F0( )

m

[ ] (37)


λ Eμ


23G (38)

G E

2(1 + μ) (39)



ε1 1


E (40)


plowast p

1minusD


(41)

qlowast q

1minusD


(42)


ln minuslnσij



X ln minuslnσij




plowast

C1( )

t

[ ]expplowast

C1( )

t

[ ]

(45)


Y 1mX + ln F0 (46)





K KpPa ap +σ3Pa( )

np

[ ] (47)

G Pa agσ3Pa( )

2

+ bgσ3Pa+ cg (48)

0 2 4 6 8 10 12 14 16 18000

003

006

009

012

015

018

021

Para

met

ers o

f m (minus

)



+ 001874σσ3m = 01544 exp minus04583







0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls








0 12 16 20 24840

7

14

21

28

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)


(a)


Axial strain εa ()

Dev

iato

ric st

ress

σ1ndashσ 3

(MPa

)

0

6

12

18

24

30

0 12 16 20 2484

(b)

Axial strain εa ()

Dev

iato

ric st

ress

σ1minusσ 3

(MPa

)

0 12 16 20 24840

8

16

24

32


(c)






1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls







1113874 11138751113876 1113877 + 001874 (49)


45 Verification



εij Vinitial

Vεinitialij +

Vdamage

Vεdamage

ij (50)




εij 1minusVdamage

V1113888 1113889εinitialij +

Vdamage

Vεdamage

ij (51)


D Vdamage

V (52)






v mdεnd


εv exp minusF

F01113888 1113889

m

1113890 1113891p

K+ 1minus exp minus

F

F01113888 1113889

m

1113890 11138911113896 1113897mdεnd

1

(55)



5 Conclusions






10 minus1051 0425320 00822 0848740 05258 0586250 04875 0757060 05064 0793080 05189 08119100 05487 08896140 05201 09199160 04083 1059



Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls





Additional Points


minus6

minus4

minus2

0

2

4

Axial strain εa ()


0 3 6 9 12 15 18 21

Volu

met

ric st

rain

ε v (

)

(a)

0

1

2

3

4

5

Volu

met

ric st

rain

ε v (

)

Axial strain εa ()




0 3 6 9 12 15 18 21

(b)

Volu

met

ric st

rain

ε v (

)


Axial strain εa ()

0

1

2

3

4

5

0 3 6 9 12 15 18 21

(c)

Axial strain εa ()

Volu

met

ric st

rain

ε v (

)

0

1

2

3

4

5


0 3 6 9 12 15 18 21

(d)





Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls






Acknowledgments


References
















































Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls















Advances in



Journal of

Chemistry















Journal of





Volume 2018









Journal of


Na

nom

ate

ria

ls




investigationonthenonlinearstrengthpropertiesanddamage...

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