Research ArticleExperimental Analysis and Discussion onthe Damage Variable of Frozen Loess

Cong Cai12 Wei Ma1 Shuping Zhao3 and YanhuMu1

1State Key Laboratory of Frozen Soil Engineering Cold and Arid Regions Environmental and Engineering Research InstituteChinese Academy of Sciences Lanzhou 730000 China2University of Chinese Academy of Sciences Beijing 100049 China3Key Lab of Visual Geographic Environment Ministry of Education Nanjing Normal University Nanjing 210023 China

Correspondence should be addressed to Wei Ma cc20109163163com

Received 25 October 2016 Revised 2 January 2017 Accepted 8 February 2017 Published 13 March 2017

Academic Editor Paulo M S T De Castro

Copyright copy 2017 Cong Cai et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The damage variable is very important to study damage evolution of material Taking frozen loess as an example a series oftriaxial compression and triaxial loading-unloading tests are performed under five strain rates of 50 times 10minus6ndash13 times 10minus2s at atemperature of minus6∘C A damage criterion of frozen loess is defined and a damage factor119863119888 is introduced to satisfy the requirementsof the engineering application The damage variable of frozen loess is investigated using the following four methods the stiffnessdegradationmethod the deformation increasemethod the dissipated energy increasemethod and the constitutivemodel deducingmethod during deformation process In addition the advantages and disadvantages of the fourmethods are discussedwhen they areused for frozen loess material According to the discussion the plastic strain may be the most appropriate variable to characterizethe damage evolution of frozen loess during the deformation process based on thematerial properties and the nature of thematerialservice

1 Introduction

There are many defects in a material such as voids andcracks which may constantly grow and accumulate underexternal loadsThese inner defects and their change processesunder external loads are defined as the damage and thedamage evolution of material [1] The damage evolution canalso be considered the degradation of material propertiesXu introduced a continuous variable and an evaluation lawto describe the rupture process of metal materials underthe creep process [1] Since then many achievements in thedamage evaluation of materials were obtained [2ndash6] and anew discipline in solid mechanics has been developed whichis named continuum damage mechanics (CDM)

In general the main methods of characterizing the dam-age and damage evolution of a material are the mesomethodbased on material science (metal physics) the macromethodbased on phenomenology and the micromethod based onstatistics The study covers the following three major com-ponents selection of the damage variable definition of the

damage threshold value and establishment of the damageevolution law However because of the randomness of thedefects it is too difficult to consider the effects of all defectson the degradation of material properties under externalloads Therefore the most common investigation methodof damage mechanics is to introduce a continuous variablewhich can describe the changing process of microstructuredefects during deformation process Then the changes ineffective loading area or macromechanical properties of thematerial under loads are characterized thus this continuousvariable is also called the damage factor [7] The damagevariable is introduced to describe the change of the damagefactor An appropriate damage variable should include threefeatures First it must have a direct response to the meso-scopic changes of the material Second the damage variableof different materials should select this physical index whichcan directly or indirectly characterize the main property ofthematerialThird the selected variable better can be directlyobtained from the experiment Before selecting the damage

HindawiAdvances in Materials Science and EngineeringVolume 2017 Article ID 1689251 13 pageshttpsdoiorg10115520171689251

2 Advances in Materials Science and Engineering

Table 1 Grain size distribution and physical properties of frozen loess

Grain size distribution of loess in Lanzhou Initial moisture

Liquid limit

Plastic limit005ndash0075mm 0005ndash005mm lt0005mm

2388 6657 955 640 1570 2430

variable and establishing the damage evolution law the initialcriterion of damage (entering the damage status) and failurecriterion of damage (damage thresholds of material) shouldbe definedThe theory was first introduced to study themate-rial uniaxial tension 119863 = 0 undamaged state 119863 = 1 failurestate and 0 lt 119863 lt 1 damaged intermediate state [8] Voyi-adjis also raised a key issue concerning damage of differentmaterials ldquoWhat is failurerdquo [9] According to Voyiadjis thefailure criteria of different materials (119863 = 1) should be clearlydefined by considering the material properties and purposesof engineering applications otherwise the definition of thedamage variable will lose its meaning According to theideal condition the failure criterion of material is that theeffective stress of the material reaches the residual strengthvalue However for engineering applications considering theunpredictable loads and structural instability the situationthat cannot satisfy safety requirements will appear even ifthe stress conditions do not reach failure conditions (theresidual strength) [1] For engineering requirements CDMoften provides a critical damage 119863119888 to represent failureconditions Based on the abovementioned developmentsresearchers have proposed many different methods such asthe stiffness degradation [10] dissipated energy increase [11]and development of plastic strain [12] Lapovok [12] and Shi etal [13] studied the damage evolution of metal by deformationand density variation Cao et al studied the damage evolutionof rock using the digital speckle correlation method (DSCM)and compared with the results from computed tomography(CT) [14] Swoboda and Yang considered that the dissipatedenergy could be turned as the driving force of crack growthbased on the view of Griffith crack in fracture mechanicsunder the isothermal and insulate condition [11] In additionthe constitutive model of some materials has been obtainedbut it cannot clearly observe the damage evolution lawThenthe law can be derived by the theory of effective stress[7 15]

However few studies compared the advantages and dis-advantages among different damage variables or discussedthe selection criteria of the damage variable to the samematerial In this paper the frozen loess is a typical com-plex material with a four-phase medium [16] which isnotably sensitive to environmental conditions In additionthe microdefects are difficult to describe because of thecomplexity of the material structure Thus many microtestmethods including CT and scanning electron microscope(SEM) cannot be used for their damage testing [5] Atpresent the common method of damage research of frozenloess depends on the macroexperimental results This paperstudies the damage evolution of frozen loess under externalloads based on a series of triaxial test results on frozenloess the evolution characteristics of different damage vari-ables during deformation are investigated and compared

The comparison analysis is also conducted by using differentdamage factors to evaluate the mechanical properties offrozen loess based on experimental results and previousstudies

2 Specimen Preparation and Test Conditions

Loess soils from Jiuzhoutai town (a seasonal frozen region)of Lanzhou City northwest China are used in this studyTheparticle size distribution and associated physical parametersof the soils are shown in Table 1 To ensure the uniformitythe soil specimens are prepared according to the followingsteps air drying of the undisturbed soils crushing and 2mmscreening measuring the initial moisture content mixingto obtain a 165 moisture content and maintaining thatcontent for 6 h without evaporation The specimens arecompacted using variable loading rates in a cylinder Thedimensions of the specimens are 618mm in diameter and1250mm in height After the compaction the specimensare completely saturated using distilled water in vacuumconditions for 10 h Then they are put into rigid molds andquickly frozen at a low temperature of minus25∘C for over 48 hto limit water loss and migration After the freezing thespecimens are taken from themolds andmountedwith epoxyresin platens on both ends and covered with rubber sleevesFinally the specimens are kept in an incubator for over 12 huntil the specimens reached the target testing temperature ofminus6∘C [17]

Two types of mechanical tests are conducted using amaterial testing machine (MTS-810) constant strain ratetriaxial compression (119862) and loading-unloading compression(119871-119880-119862) All tests are performed at a constant confiningpressure (1205903 = 01MPa) and a constant temperature (119879 =minus6∘C)The strain rates of the tests are 13times 10minus2s 50times 10minus3s50 times 10minus4s 50 times 10minus5s and 50 times 10minus6s The loading andunloading tests are performed at the identical strain rate toavoid the effect of the strain rate on the same trial

3 Results and Analysis

The 119862 test results under five different strain rates are shownin Figure 1 The deviator stress (1205901 minus 1205903) and axial strain(1205761) relation of frozen loess strongly depends on the strainrate At small strain the stress-strain behavior performsas hardening With the continuous loading the specimendamage gradually develops and the stress-strain behaviorperforms as strain softening With the increase in strain ratethe peak axial strain 1205761199061 sharply increases and approaches themaximum at a strain rate of 50 times 10minus3s Then the peak axialstrain decreases with the strain rate increase as shown inFigure 2 This result suggests that the damage of frozen loessis strain-dependent and strongly rate-dependent Variations

Advances in Materials Science and Engineering 3

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6

휀1 ()

3024181260

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

T = minus6∘C휎3 = 01 MPa

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 1 Deviator stress-axial strain curves for frozen loess under119862 tests

휀1 (10minus4s)

1501209060300minus30

12

15

18

21

휀u 1(

)

Figure 2 Peak strain for frozen loess under 119862 tests

of the stress-strain relations and the peak axial strains withthe strain rates are related to the rheological characteristicsof frozen loess

The 119871-119880-119862 test results under the five different strain ratesare shown in Figure 3 and the comparison of 119862 and 119871-119880-119862test results is shown in Figure 4 It can be seen that the loadingand unloading cycles test do not affect the deformationbehavior of frozen loess To verify the accuracy of the testdata some repeated tests are carried out as shown in Figure 5

4 Damage Features of Frozen Loess

In this paper the following studies are conducted on thepremise of the phenomenalism and isotropic continuummedium The changes of the mesofabric are characterized by

T = minus6∘C휎3 = 01 MPa

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

휀1 ()

3024181260

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 3 Deviator stress-axial strain curves for frozen loess under119871-119880-119862 tests

introducing a continuous variable Then the macro descrip-tion of damage accumulation can be expressed by a scalarequation as follows [12 18]

119863 = 119863(120590119894119895 120576119894119895 120576119894119895) (1)

There were different damage variables with differentdamage evolution lawsiumlijŇ even when the identical physicalprocess is describedThe concept of damage variable was firstintroduced from the concept of effective stress which wasproposed by Kachanov [19]

120590eff = 12059001 minus 119863 (2)

Then the damage evolution law can be written as

119889119863119889119905 = 119891 (119863 120590119894119895) (3)

where119863 is the damage variable 120590eff is the effective stress and1205900 is the nominal stress If a reasonable damage evolution lawcan be established then the damage state residual strengthand longevity of material can be easily and accurately pre-dicted

41 Damage Criterion of Frozen Loess According to thestudies of Zhao et al [5] Xu et al [6] and Huang and Li[20] and the test results of this study the material is softeningwhen the deformation reaches a certain level Then thecracks in the material expand rapidly the strength decreasesrapidly and the structure becomes unstable Therefore someresearchers proposed that the failure condition of materialcould be defined as the status that the stress reaches theextreme value point [1] The status can be expressed asfollows

119889120590119889120576

10038161003816100381610038161003816100381610038161003816119863119888

= 0 (4)

4 Advances in Materials Science and Engineering

휀1 ()2520 30151050

L-U-C

C

0

1

2

3

4

5

6휎1minus휎3(M

Pa)

휀1 = 13 times 10minus2 (sminus1)

(a)

휀1 ()2520 30151050

휀1 = 50 times 10minus3

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(b)

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

00

07

14

21

28

35

휎1minus휎3(M

Pa)

(sminus1)

(c)

휀1 ()2520 30151050

휀1 = 50 times 10minus5

00

05

10

15

20

25

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(d)

휀1 ()2520 30151050

휀1 = 50 times 10minus6

L-U-C

C

00

05

10

15

20

휎1minus휎3(M

Pa)

(sminus1)

(e)

Figure 4 The comparison of 119862 and 119871-119880-119862 tests under different strain rates

Advances in Materials Science and Engineering 5

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus5

L-U-C

C

0

1

2

3휎1minus휎3(M

Pa)

(sminus1)

(a)

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

0

1

2

3

4

휎1minus휎3(M

Pa)

(sminus1)

(b)

Figure 5 The accuracy of the test data

Rapid development of defects

Slow development of defects

Peak strain 182

3024181260

휀1 ()

휀1 = 50 times 10minus4

00

07

14

21

28

35

휎1minus휎3(M

Pa)

d휎

d휀

D푐

= 0(sminus1)

Figure 6 The selected criterion of critical damage for frozen loess

Based on a series of repeated tests the peak strain offrozen loess is obtained under different strain rates accordingto the statistics principle The new test is accomplished whenthe strain reaches the peak strain The typical physical statusof frozen loess around the peak strain is shown in Figure 6The cracks on the frozen loess surface are large and rapidlydevelop after the peak strain Thus this status is unsuitedfor continuous loading and can be referred to as the criticaldamage status of frozen loess material

In addition the selection of the damage factor andquantitative relationship between the damage factor andthe damage variable should be considered Based on pre-vious studies and the test results of this study this paperdiscusses four common damage factors and damage varia-bles

42 Damage Variables of Frozen Loess

421 Stiffness DegradationMethod The stiffness of themate-rial continuously degenerates with the increase of defor-mation The degradation of mechanical properties of thematerial is due to the nucleation and development of thematerial damage at the microscale level such as voids andcracks According to the above discussions and previousstudies [8 10 21] the unloading modulus is selected tocharacterize the stiffness of the material under differentdamage states Then the damage variable can be written asfollows

119863 = 1 minus 1198641198640 (5)

where 1198640 is the initial elastic modulus of the material whichis also known as the undamaged modulus 119864 is the elasticmodulus under different damage state

To obtain the component of the elastic strain 1205761198901 plasticstrain 1205761199011 and the change law of unloading modulus duringthe deformation the hysteretic loop is addressed using theapproximation-linearizedmethod [22] as shown in Figure 7

There are two selections for the unloading modulusmodulus of hysteretic loop 119864ℎ and the secant modulus 119864119904The variations of 119864ℎ and 119864119904 with the axial strain are shownin Figures 8(a) and 8(b) respectively

In Figure 8 the unloading modulus of frozen loessdecreases with the increase in axial strain when the strainrate is below 50 times 10minus4s but the change trend is not obviouswhen the strain rate is above 50 times 10minus4s The phenomenonis verified by repeated tests According to Xie [23] and Yu[24] the unloading modulus was selected as a damage factorbased on the equivalence hypothesis of strain and themethodof elastic modulus However this hypothesis can only be

6 Advances in Materials Science and Engineering

Loading

Reloading

Unloading

휀1 ()

휎1minus휎3

Es

Eℎ

휀p1 휀e1 휀i1

Figure 7 The decoupling principle on the hysteretic loop

applied for elastic damage The frozen loess is an elastic-plastic coupling material which can be verified from the 119871-119880-119862 tests that the plastic deformation is dominant even if theaxial strain is 1 Thus the actual damage feature of frozenloess is significantly simplified or hidden when the unloadingmodulus is selected as the damage factor

Many experimental results show that the unloadingmodulus can only decrease when the deformation reaches acertain status [23] This phenomenon suggests that there isa damage threshold value for elastic-plastic materials whichis consistent with the law that the granular material is firstcompacted and subsequently shear failure under externalloadsTheunloadingmodulusmay be greater than or equal tothe initial elastic modulus of the material before this certainstatus Then using the unloading modulus (119864ℎ or 119864119904) tocalculate thematerial damagewill lead to awrong conclusionfor example no damage or even negative damage occursHowever for the unloading modulus 119864119904 may be better than119864ℎ because 119864119904 is only a part of the unloading curve

To study the stiffness degradation of the material dur-ing the deformation some researchers proposed that thedeformation modulus should be adopted which is definedas the slope of the secant line of the stress-strain curve fromthe loading point to the instantaneous damage point (seeFigure 9) [24ndash26] The deformation modulus 119864119889 is used todescribe the damage evolution of the material

The evolution law of 119864119889 with axial strain under differentstrain rates is shown in Figure 10 The damage variable isexpressed as follows

119863 = 1 minus 1198641198891198640 (6)

In addition there is a quantitative relationship betweenthe unloading modulus 119864119904 and the deformation modulus 119864119889under the same strain and stress [23]

119864119889 = 1205761 minus 12057611990111205761 119864119904 (7)

Eℎ(M

Pa)

700

600

500

400

300

휀1 ()0 302418126

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(a)

휀1 ()50 3025201510

300

400

500

600

700

Es(M

Pa)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(b)

Figure 8 Variation of 119864ℎ and 119864119904 with axial strain under differentstrain rates

where 1205761199011 is the axial plastic strain Then the damage variablecan be described with the unloading modulus (119864119904)

119863 = 1 minus 1205761 minus 12057611990111205761 (1198641199041198640) (8)

Then the unloading modulus (119864119904) can describe theincrease in damage and the stiffness degradation of materiallike the deformation modulus (119864119889) Based on the test resultsof this study the relationship between axial strain anddamagevariable is established as follows

119863 = 120576111988612057621 + 1198871205761 + 119888 (9)

where 119886 119887 and 119888 are material parameters corresponding tothe strain rateThe test results and fitting curves are shown in

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 3

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6

휀1 ()

3024181260

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

T = minus6∘C휎3 = 01 MPa

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 1 Deviator stress-axial strain curves for frozen loess under119862 tests

휀1 (10minus4s)

1501209060300minus30

12

15

18

21

휀u 1(

)

Figure 2 Peak strain for frozen loess under 119862 tests

of the stress-strain relations and the peak axial strains withthe strain rates are related to the rheological characteristicsof frozen loess

The 119871-119880-119862 test results under the five different strain ratesare shown in Figure 3 and the comparison of 119862 and 119871-119880-119862test results is shown in Figure 4 It can be seen that the loadingand unloading cycles test do not affect the deformationbehavior of frozen loess To verify the accuracy of the testdata some repeated tests are carried out as shown in Figure 5

4 Damage Features of Frozen Loess

In this paper the following studies are conducted on thepremise of the phenomenalism and isotropic continuummedium The changes of the mesofabric are characterized by

T = minus6∘C휎3 = 01 MPa

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

휀1 ()

3024181260

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 3 Deviator stress-axial strain curves for frozen loess under119871-119880-119862 tests

introducing a continuous variable Then the macro descrip-tion of damage accumulation can be expressed by a scalarequation as follows [12 18]

119863 = 119863(120590119894119895 120576119894119895 120576119894119895) (1)

There were different damage variables with differentdamage evolution lawsiumlijŇ even when the identical physicalprocess is describedThe concept of damage variable was firstintroduced from the concept of effective stress which wasproposed by Kachanov [19]

120590eff = 12059001 minus 119863 (2)

Then the damage evolution law can be written as

119889119863119889119905 = 119891 (119863 120590119894119895) (3)

where119863 is the damage variable 120590eff is the effective stress and1205900 is the nominal stress If a reasonable damage evolution lawcan be established then the damage state residual strengthand longevity of material can be easily and accurately pre-dicted

41 Damage Criterion of Frozen Loess According to thestudies of Zhao et al [5] Xu et al [6] and Huang and Li[20] and the test results of this study the material is softeningwhen the deformation reaches a certain level Then thecracks in the material expand rapidly the strength decreasesrapidly and the structure becomes unstable Therefore someresearchers proposed that the failure condition of materialcould be defined as the status that the stress reaches theextreme value point [1] The status can be expressed asfollows

119889120590119889120576

10038161003816100381610038161003816100381610038161003816119863119888

= 0 (4)

4 Advances in Materials Science and Engineering

휀1 ()2520 30151050

L-U-C

C

0

1

2

3

4

5

6휎1minus휎3(M

Pa)

휀1 = 13 times 10minus2 (sminus1)

(a)

휀1 ()2520 30151050

휀1 = 50 times 10minus3

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(b)

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

00

07

14

21

28

35

휎1minus휎3(M

Pa)

(sminus1)

(c)

휀1 ()2520 30151050

휀1 = 50 times 10minus5

00

05

10

15

20

25

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(d)

휀1 ()2520 30151050

휀1 = 50 times 10minus6

L-U-C

C

00

05

10

15

20

휎1minus휎3(M

Pa)

(sminus1)

(e)

Figure 4 The comparison of 119862 and 119871-119880-119862 tests under different strain rates

Advances in Materials Science and Engineering 5

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus5

L-U-C

C

0

1

2

3휎1minus휎3(M

Pa)

(sminus1)

(a)

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

0

1

2

3

4

휎1minus휎3(M

Pa)

(sminus1)

(b)

Figure 5 The accuracy of the test data

Rapid development of defects

Slow development of defects

Peak strain 182

3024181260

휀1 ()

휀1 = 50 times 10minus4

00

07

14

21

28

35

휎1minus휎3(M

Pa)

d휎

d휀

D푐

= 0(sminus1)

Figure 6 The selected criterion of critical damage for frozen loess

Based on a series of repeated tests the peak strain offrozen loess is obtained under different strain rates accordingto the statistics principle The new test is accomplished whenthe strain reaches the peak strain The typical physical statusof frozen loess around the peak strain is shown in Figure 6The cracks on the frozen loess surface are large and rapidlydevelop after the peak strain Thus this status is unsuitedfor continuous loading and can be referred to as the criticaldamage status of frozen loess material

In addition the selection of the damage factor andquantitative relationship between the damage factor andthe damage variable should be considered Based on pre-vious studies and the test results of this study this paperdiscusses four common damage factors and damage varia-bles

42 Damage Variables of Frozen Loess

421 Stiffness DegradationMethod The stiffness of themate-rial continuously degenerates with the increase of defor-mation The degradation of mechanical properties of thematerial is due to the nucleation and development of thematerial damage at the microscale level such as voids andcracks According to the above discussions and previousstudies [8 10 21] the unloading modulus is selected tocharacterize the stiffness of the material under differentdamage states Then the damage variable can be written asfollows

119863 = 1 minus 1198641198640 (5)

where 1198640 is the initial elastic modulus of the material whichis also known as the undamaged modulus 119864 is the elasticmodulus under different damage state

To obtain the component of the elastic strain 1205761198901 plasticstrain 1205761199011 and the change law of unloading modulus duringthe deformation the hysteretic loop is addressed using theapproximation-linearizedmethod [22] as shown in Figure 7

There are two selections for the unloading modulusmodulus of hysteretic loop 119864ℎ and the secant modulus 119864119904The variations of 119864ℎ and 119864119904 with the axial strain are shownin Figures 8(a) and 8(b) respectively

In Figure 8 the unloading modulus of frozen loessdecreases with the increase in axial strain when the strainrate is below 50 times 10minus4s but the change trend is not obviouswhen the strain rate is above 50 times 10minus4s The phenomenonis verified by repeated tests According to Xie [23] and Yu[24] the unloading modulus was selected as a damage factorbased on the equivalence hypothesis of strain and themethodof elastic modulus However this hypothesis can only be

6 Advances in Materials Science and Engineering

Loading

Reloading

Unloading

휀1 ()

휎1minus휎3

Es

Eℎ

휀p1 휀e1 휀i1

Figure 7 The decoupling principle on the hysteretic loop

applied for elastic damage The frozen loess is an elastic-plastic coupling material which can be verified from the 119871-119880-119862 tests that the plastic deformation is dominant even if theaxial strain is 1 Thus the actual damage feature of frozenloess is significantly simplified or hidden when the unloadingmodulus is selected as the damage factor

Many experimental results show that the unloadingmodulus can only decrease when the deformation reaches acertain status [23] This phenomenon suggests that there isa damage threshold value for elastic-plastic materials whichis consistent with the law that the granular material is firstcompacted and subsequently shear failure under externalloadsTheunloadingmodulusmay be greater than or equal tothe initial elastic modulus of the material before this certainstatus Then using the unloading modulus (119864ℎ or 119864119904) tocalculate thematerial damagewill lead to awrong conclusionfor example no damage or even negative damage occursHowever for the unloading modulus 119864119904 may be better than119864ℎ because 119864119904 is only a part of the unloading curve

To study the stiffness degradation of the material dur-ing the deformation some researchers proposed that thedeformation modulus should be adopted which is definedas the slope of the secant line of the stress-strain curve fromthe loading point to the instantaneous damage point (seeFigure 9) [24ndash26] The deformation modulus 119864119889 is used todescribe the damage evolution of the material

The evolution law of 119864119889 with axial strain under differentstrain rates is shown in Figure 10 The damage variable isexpressed as follows

119863 = 1 minus 1198641198891198640 (6)

In addition there is a quantitative relationship betweenthe unloading modulus 119864119904 and the deformation modulus 119864119889under the same strain and stress [23]

119864119889 = 1205761 minus 12057611990111205761 119864119904 (7)

Eℎ(M

Pa)

700

600

500

400

300

휀1 ()0 302418126

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(a)

휀1 ()50 3025201510

300

400

500

600

700

Es(M

Pa)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(b)

Figure 8 Variation of 119864ℎ and 119864119904 with axial strain under differentstrain rates

where 1205761199011 is the axial plastic strain Then the damage variablecan be described with the unloading modulus (119864119904)

119863 = 1 minus 1205761 minus 12057611990111205761 (1198641199041198640) (8)

Then the unloading modulus (119864119904) can describe theincrease in damage and the stiffness degradation of materiallike the deformation modulus (119864119889) Based on the test resultsof this study the relationship between axial strain anddamagevariable is established as follows

119863 = 120576111988612057621 + 1198871205761 + 119888 (9)

where 119886 119887 and 119888 are material parameters corresponding tothe strain rateThe test results and fitting curves are shown in

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Advances in Materials Science and Engineering

휀1 ()2520 30151050

L-U-C

C

0

1

2

3

4

5

6휎1minus휎3(M

Pa)

휀1 = 13 times 10minus2 (sminus1)

(a)

휀1 ()2520 30151050

휀1 = 50 times 10minus3

0

1

2

3

4

5

6

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(b)

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

00

07

14

21

28

35

휎1minus휎3(M

Pa)

(sminus1)

(c)

휀1 ()2520 30151050

휀1 = 50 times 10minus5

00

05

10

15

20

25

휎1minus휎3(M

Pa)

L-U-C

C

(sminus1)

(d)

휀1 ()2520 30151050

휀1 = 50 times 10minus6

L-U-C

C

00

05

10

15

20

휎1minus휎3(M

Pa)

(sminus1)

(e)

Figure 4 The comparison of 119862 and 119871-119880-119862 tests under different strain rates

Advances in Materials Science and Engineering 5

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus5

L-U-C

C

0

1

2

3휎1minus휎3(M

Pa)

(sminus1)

(a)

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

0

1

2

3

4

휎1minus휎3(M

Pa)

(sminus1)

(b)

Figure 5 The accuracy of the test data

Rapid development of defects

Slow development of defects

Peak strain 182

3024181260

휀1 ()

휀1 = 50 times 10minus4

00

07

14

21

28

35

휎1minus휎3(M

Pa)

d휎

d휀

D푐

= 0(sminus1)

Figure 6 The selected criterion of critical damage for frozen loess

Based on a series of repeated tests the peak strain offrozen loess is obtained under different strain rates accordingto the statistics principle The new test is accomplished whenthe strain reaches the peak strain The typical physical statusof frozen loess around the peak strain is shown in Figure 6The cracks on the frozen loess surface are large and rapidlydevelop after the peak strain Thus this status is unsuitedfor continuous loading and can be referred to as the criticaldamage status of frozen loess material

In addition the selection of the damage factor andquantitative relationship between the damage factor andthe damage variable should be considered Based on pre-vious studies and the test results of this study this paperdiscusses four common damage factors and damage varia-bles

42 Damage Variables of Frozen Loess

421 Stiffness DegradationMethod The stiffness of themate-rial continuously degenerates with the increase of defor-mation The degradation of mechanical properties of thematerial is due to the nucleation and development of thematerial damage at the microscale level such as voids andcracks According to the above discussions and previousstudies [8 10 21] the unloading modulus is selected tocharacterize the stiffness of the material under differentdamage states Then the damage variable can be written asfollows

119863 = 1 minus 1198641198640 (5)

where 1198640 is the initial elastic modulus of the material whichis also known as the undamaged modulus 119864 is the elasticmodulus under different damage state

To obtain the component of the elastic strain 1205761198901 plasticstrain 1205761199011 and the change law of unloading modulus duringthe deformation the hysteretic loop is addressed using theapproximation-linearizedmethod [22] as shown in Figure 7

There are two selections for the unloading modulusmodulus of hysteretic loop 119864ℎ and the secant modulus 119864119904The variations of 119864ℎ and 119864119904 with the axial strain are shownin Figures 8(a) and 8(b) respectively

In Figure 8 the unloading modulus of frozen loessdecreases with the increase in axial strain when the strainrate is below 50 times 10minus4s but the change trend is not obviouswhen the strain rate is above 50 times 10minus4s The phenomenonis verified by repeated tests According to Xie [23] and Yu[24] the unloading modulus was selected as a damage factorbased on the equivalence hypothesis of strain and themethodof elastic modulus However this hypothesis can only be

6 Advances in Materials Science and Engineering

Loading

Reloading

Unloading

휀1 ()

휎1minus휎3

Es

Eℎ

휀p1 휀e1 휀i1

Figure 7 The decoupling principle on the hysteretic loop

applied for elastic damage The frozen loess is an elastic-plastic coupling material which can be verified from the 119871-119880-119862 tests that the plastic deformation is dominant even if theaxial strain is 1 Thus the actual damage feature of frozenloess is significantly simplified or hidden when the unloadingmodulus is selected as the damage factor

Many experimental results show that the unloadingmodulus can only decrease when the deformation reaches acertain status [23] This phenomenon suggests that there isa damage threshold value for elastic-plastic materials whichis consistent with the law that the granular material is firstcompacted and subsequently shear failure under externalloadsTheunloadingmodulusmay be greater than or equal tothe initial elastic modulus of the material before this certainstatus Then using the unloading modulus (119864ℎ or 119864119904) tocalculate thematerial damagewill lead to awrong conclusionfor example no damage or even negative damage occursHowever for the unloading modulus 119864119904 may be better than119864ℎ because 119864119904 is only a part of the unloading curve

To study the stiffness degradation of the material dur-ing the deformation some researchers proposed that thedeformation modulus should be adopted which is definedas the slope of the secant line of the stress-strain curve fromthe loading point to the instantaneous damage point (seeFigure 9) [24ndash26] The deformation modulus 119864119889 is used todescribe the damage evolution of the material

The evolution law of 119864119889 with axial strain under differentstrain rates is shown in Figure 10 The damage variable isexpressed as follows

119863 = 1 minus 1198641198891198640 (6)

In addition there is a quantitative relationship betweenthe unloading modulus 119864119904 and the deformation modulus 119864119889under the same strain and stress [23]

119864119889 = 1205761 minus 12057611990111205761 119864119904 (7)

Eℎ(M

Pa)

700

600

500

400

300

휀1 ()0 302418126

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(a)

휀1 ()50 3025201510

300

400

500

600

700

Es(M

Pa)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(b)

Figure 8 Variation of 119864ℎ and 119864119904 with axial strain under differentstrain rates

where 1205761199011 is the axial plastic strain Then the damage variablecan be described with the unloading modulus (119864119904)

119863 = 1 minus 1205761 minus 12057611990111205761 (1198641199041198640) (8)

Then the unloading modulus (119864119904) can describe theincrease in damage and the stiffness degradation of materiallike the deformation modulus (119864119889) Based on the test resultsof this study the relationship between axial strain anddamagevariable is established as follows

119863 = 120576111988612057621 + 1198871205761 + 119888 (9)

where 119886 119887 and 119888 are material parameters corresponding tothe strain rateThe test results and fitting curves are shown in

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 5

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus5

L-U-C

C

0

1

2

3휎1minus휎3(M

Pa)

(sminus1)

(a)

Repeated tests

휀1 ()2520 30151050

휀1 = 50 times 10minus4

L-U-C

C

0

1

2

3

4

휎1minus휎3(M

Pa)

(sminus1)

(b)

Figure 5 The accuracy of the test data

Rapid development of defects

Slow development of defects

Peak strain 182

3024181260

휀1 ()

휀1 = 50 times 10minus4

00

07

14

21

28

35

휎1minus휎3(M

Pa)

d휎

d휀

D푐

= 0(sminus1)

Figure 6 The selected criterion of critical damage for frozen loess

Based on a series of repeated tests the peak strain offrozen loess is obtained under different strain rates accordingto the statistics principle The new test is accomplished whenthe strain reaches the peak strain The typical physical statusof frozen loess around the peak strain is shown in Figure 6The cracks on the frozen loess surface are large and rapidlydevelop after the peak strain Thus this status is unsuitedfor continuous loading and can be referred to as the criticaldamage status of frozen loess material

In addition the selection of the damage factor andquantitative relationship between the damage factor andthe damage variable should be considered Based on pre-vious studies and the test results of this study this paperdiscusses four common damage factors and damage varia-bles

42 Damage Variables of Frozen Loess

421 Stiffness DegradationMethod The stiffness of themate-rial continuously degenerates with the increase of defor-mation The degradation of mechanical properties of thematerial is due to the nucleation and development of thematerial damage at the microscale level such as voids andcracks According to the above discussions and previousstudies [8 10 21] the unloading modulus is selected tocharacterize the stiffness of the material under differentdamage states Then the damage variable can be written asfollows

119863 = 1 minus 1198641198640 (5)

where 1198640 is the initial elastic modulus of the material whichis also known as the undamaged modulus 119864 is the elasticmodulus under different damage state

To obtain the component of the elastic strain 1205761198901 plasticstrain 1205761199011 and the change law of unloading modulus duringthe deformation the hysteretic loop is addressed using theapproximation-linearizedmethod [22] as shown in Figure 7

There are two selections for the unloading modulusmodulus of hysteretic loop 119864ℎ and the secant modulus 119864119904The variations of 119864ℎ and 119864119904 with the axial strain are shownin Figures 8(a) and 8(b) respectively

In Figure 8 the unloading modulus of frozen loessdecreases with the increase in axial strain when the strainrate is below 50 times 10minus4s but the change trend is not obviouswhen the strain rate is above 50 times 10minus4s The phenomenonis verified by repeated tests According to Xie [23] and Yu[24] the unloading modulus was selected as a damage factorbased on the equivalence hypothesis of strain and themethodof elastic modulus However this hypothesis can only be

6 Advances in Materials Science and Engineering

Loading

Reloading

Unloading

휀1 ()

휎1minus휎3

Es

Eℎ

휀p1 휀e1 휀i1

Figure 7 The decoupling principle on the hysteretic loop

applied for elastic damage The frozen loess is an elastic-plastic coupling material which can be verified from the 119871-119880-119862 tests that the plastic deformation is dominant even if theaxial strain is 1 Thus the actual damage feature of frozenloess is significantly simplified or hidden when the unloadingmodulus is selected as the damage factor

Many experimental results show that the unloadingmodulus can only decrease when the deformation reaches acertain status [23] This phenomenon suggests that there isa damage threshold value for elastic-plastic materials whichis consistent with the law that the granular material is firstcompacted and subsequently shear failure under externalloadsTheunloadingmodulusmay be greater than or equal tothe initial elastic modulus of the material before this certainstatus Then using the unloading modulus (119864ℎ or 119864119904) tocalculate thematerial damagewill lead to awrong conclusionfor example no damage or even negative damage occursHowever for the unloading modulus 119864119904 may be better than119864ℎ because 119864119904 is only a part of the unloading curve

To study the stiffness degradation of the material dur-ing the deformation some researchers proposed that thedeformation modulus should be adopted which is definedas the slope of the secant line of the stress-strain curve fromthe loading point to the instantaneous damage point (seeFigure 9) [24ndash26] The deformation modulus 119864119889 is used todescribe the damage evolution of the material

The evolution law of 119864119889 with axial strain under differentstrain rates is shown in Figure 10 The damage variable isexpressed as follows

119863 = 1 minus 1198641198891198640 (6)

In addition there is a quantitative relationship betweenthe unloading modulus 119864119904 and the deformation modulus 119864119889under the same strain and stress [23]

119864119889 = 1205761 minus 12057611990111205761 119864119904 (7)

Eℎ(M

Pa)

700

600

500

400

300

휀1 ()0 302418126

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(a)

휀1 ()50 3025201510

300

400

500

600

700

Es(M

Pa)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(b)

Figure 8 Variation of 119864ℎ and 119864119904 with axial strain under differentstrain rates

where 1205761199011 is the axial plastic strain Then the damage variablecan be described with the unloading modulus (119864119904)

119863 = 1 minus 1205761 minus 12057611990111205761 (1198641199041198640) (8)

Then the unloading modulus (119864119904) can describe theincrease in damage and the stiffness degradation of materiallike the deformation modulus (119864119889) Based on the test resultsof this study the relationship between axial strain anddamagevariable is established as follows

119863 = 120576111988612057621 + 1198871205761 + 119888 (9)

where 119886 119887 and 119888 are material parameters corresponding tothe strain rateThe test results and fitting curves are shown in

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Advances in Materials Science and Engineering

Loading

Reloading

Unloading

휀1 ()

휎1minus휎3

Es

Eℎ

휀p1 휀e1 휀i1

Figure 7 The decoupling principle on the hysteretic loop

applied for elastic damage The frozen loess is an elastic-plastic coupling material which can be verified from the 119871-119880-119862 tests that the plastic deformation is dominant even if theaxial strain is 1 Thus the actual damage feature of frozenloess is significantly simplified or hidden when the unloadingmodulus is selected as the damage factor

Many experimental results show that the unloadingmodulus can only decrease when the deformation reaches acertain status [23] This phenomenon suggests that there isa damage threshold value for elastic-plastic materials whichis consistent with the law that the granular material is firstcompacted and subsequently shear failure under externalloadsTheunloadingmodulusmay be greater than or equal tothe initial elastic modulus of the material before this certainstatus Then using the unloading modulus (119864ℎ or 119864119904) tocalculate thematerial damagewill lead to awrong conclusionfor example no damage or even negative damage occursHowever for the unloading modulus 119864119904 may be better than119864ℎ because 119864119904 is only a part of the unloading curve

To study the stiffness degradation of the material dur-ing the deformation some researchers proposed that thedeformation modulus should be adopted which is definedas the slope of the secant line of the stress-strain curve fromthe loading point to the instantaneous damage point (seeFigure 9) [24ndash26] The deformation modulus 119864119889 is used todescribe the damage evolution of the material

The evolution law of 119864119889 with axial strain under differentstrain rates is shown in Figure 10 The damage variable isexpressed as follows

119863 = 1 minus 1198641198891198640 (6)

In addition there is a quantitative relationship betweenthe unloading modulus 119864119904 and the deformation modulus 119864119889under the same strain and stress [23]

119864119889 = 1205761 minus 12057611990111205761 119864119904 (7)

Eℎ(M

Pa)

700

600

500

400

300

휀1 ()0 302418126

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(a)

휀1 ()50 3025201510

300

400

500

600

700

Es(M

Pa)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

(b)

Figure 8 Variation of 119864ℎ and 119864119904 with axial strain under differentstrain rates

where 1205761199011 is the axial plastic strain Then the damage variablecan be described with the unloading modulus (119864119904)

119863 = 1 minus 1205761 minus 12057611990111205761 (1198641199041198640) (8)

Then the unloading modulus (119864119904) can describe theincrease in damage and the stiffness degradation of materiallike the deformation modulus (119864119889) Based on the test resultsof this study the relationship between axial strain anddamagevariable is established as follows

119863 = 120576111988612057621 + 1198871205761 + 119888 (9)

where 119886 119887 and 119888 are material parameters corresponding tothe strain rateThe test results and fitting curves are shown in

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 7

Table 2 Material parameters 119886 119887 and 119888 under different strain rates (stiffness)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119886 minus0381 minus0377 minus0544 minus0463 minus0532119887 1142 1150 1189 1134 1127119888 0014 0009 0003 0005 0004

Table 3 Material parameter 120579 under different strain rates (plastic strain)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120579 5520 4710 5448 7241 9075

35

28

21

14

07

00

E3 = 3404

E0 = 8567

E5 = 2558E7 = 1834

E9 = 1291

휎1minus휎3(M

Pa)

휀1 ()2520 30151050

휀1 = 50 times 10minus4 (sminus1)

Figure 9 The definition of deformation modulus 119864119889

Figure 11 and the parameters 119886 119887 and 119888 under different strainrates are listed in Table 2

422 Plastic Strain Method Considering the process offatigue damage Lemaitre [27] and Lapovok [12] proposedthat plastic strain or permanent strain was an irreversibilitymacroscopic behavior as a result of microcrack growth anddevelopment (damage-fatigue equivalent analogy method)Therefore the plastic deformation can be selected as thedamage factor according to the ability of the material to resistpermanent deformation Then according to Liao et al [28]the damage variable can be written as follows

119863 = 1 minus exp(minus12057611990111205761199061 ) (10)

where 1205761199061 is the peak strain (see Figure 2) and 1205761199011 is plasticstrain under different deformation statuses

The variation of 1205761199011 with axial strain under different strainrates is shown in Figure 12 Within the total deformationthe plastic strain is dominant and the elastic deformationcontributes very little In addition the development of plasticstrain is almost independent of the strain rate Based on thetest results of this study the relationship between the damagevariable and the axial strain is shown in Figure 13

280

210

140

70

0

Ed

(MPa

)

2520151050

휀1()

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 10 Variation of 119864119889 with axial strain under different strainrates

Liao et al introduced the following negative exponentialfunction in order to establish the relationship of the damagevariable and the axial strain [28]

119863 = 1 minus exp (minus1205791205761) (11)

where 120579 is amaterial parameter corresponding to time whichcontrols the damage degree and degradation of materialproperties Parameter 120579 under different strain rates is listedin Table 3 according to the test results of this study Differentstrain rates that correspond to the critical damage (119863119888) areshown in Figure 13 The strains considerably vary when thecritical damage approaches under different strain rates butthe critical damage value (119863119888) differs a little This resultsuggests that the plastic strain elastic strain and theircontributions to the total deformation are independent of thedeformation degree and strain rate

423 Energy Dissipation Method According to thermody-namics and fracture mechanics the work performed byexternal loads is not entirely stored as elastic strain energyunder the isothermal and adiabatic condition and a partof it is consumed in damage nucleation and evaluation For

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

휀1 ()2520151050

10

08

06

04

02

00

D

Dc = 0900Dc = 0899 Dc = 0898

Dc = 0861 Dc = 0897

Equation (9)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 11 Variation of the damage variable with axial strain under different strain rates (stiffness)

휀1 ()

25

20

15

10

5

0

휀p 1(

)

302520151050

4427∘

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 12 Variation of the plastic strain with axial strain under different strain rates

convenience it is a hypothesis that the dissipated energy cancontribute to the development of material damage accordingto the view of 119869 integral in fracture mechanics Thus thedissipated energy can also be treated as a damage factor forthe damage evolution Regarding the dissipated energy thehysteresis loop area 119882ℎ of loading and unloading curves iscommonly considered the dissipated energy of the material[6]

According to the principle of energy conservation anddamage evolution the initial damage of the specimen should

be selected as the reference point of damage evolutionHowever the hysteresis loop area is only the dissipated energyof one loading-unloading cycle (the dissipated energy withinthe rebound strain) [29]

Based on the test results in this study the variation of thehysteresis loop area of frozen loess with axial strain underdifferent strain rates is shown in Figure 14 The area of thehysteresis loop decreases after the peak strain because theshear failure of the frozen loess mesostructure makes thestrength reduction Then the hysteresis loop area cannot

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 9

휀1 ()3024181260

10

08

06

04

02

00

D

Dc = 0624 Dc = 0623Dc = 0618

Dc = 0610Dc = 0616

Equation (11)

50 times 10minus5

50 times 10minus613 times 10minus2

50 times 10minus3

50 times 10minus4

(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 13 Variation of the damage variable with axial strain underdifferent strain rates (plastic strain)

0010

0008

0006

0004

0002

0000

Increase

Decrease

Peak strain

휀1 ()302520151050

Wℎ(1

0minus3

Jm

m3

)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 14 Variation of the hysteresis loop area with axial strainunder different strain rates

accurately reflect the mesofailure process of frozen loess Thetest results again indicate that the hysteresis loop area couldnot be directly considered the damage factor

Based on the view of fracture mechanics and energyconservation the dissipated energy is recalculated as follows

119882 = 119882119904 + 119882119887 + 119882119903 (12)

where 119882119904 is the storage elastic energy of the testing systemduring loading 119882119887 is the dissipated energy of the testingmachine damping and 119882119903 is the absorbed energy by thespecimen 119882119904 119882119887 can be ignored because the stiffness of thetesting machine is far greater than the frozen loess specimen

Loading

Reloading

Unloading

휀pij

휀eij

휀ij

Wd

We

휎ij

Figure 15 The calculation method of dissipated energy (120576119901119894119895 is theplastic strain)

Then the entire work (119882) can be considered absorbed by thespecimen (119882119903) In the isothermal and adiabatic conditionsone part of the work performed by external loads is storedin the form of elastic energy (119882119890) which is released afterunloading Another part of the work is consumed by thedevelopment of plastic deformation and cracks based onHuang and Li [20] Thus this part can also be called thedissipated energy (119882119889)

119882 asymp 119882119903 = 119882119890 + 119882119889 = int1205760120590119894119895 119889120576119894119895

119882119890 = int119904120590119894119895 119889120576119890119894119895

(13)

where 119904 is the unloading path and 120576119890119894119895 is the elastic strainThenthe dissipated energy can be calculated from the 119871-119880-119862 testresults and the calculation method is shown in Figure 15

The variation of dissipated energy with axial strain isshown in Figure 16 The dissipated energy increases withthe increase in axial strain With the increase in dissipatedenergy the cracks and holes in the specimen increase andthe specimen eventually achieves the extreme damage statusConsidering the irreversibility of the damage the damagevariable can be expressed as follows

119863 = sum119899119894=1119882119894119889119882119906119889

(14)

where 119882119906119889 is the dissipated energy of the specimen corre-sponding to the critical damage 119882119894119889 is the dissipated energyunder different damage stages The relationship between thedamage variable and the axial strain is shown in Figure 17Based on the test results of this study and Darabi et al [15]the damage evolution law can be written as follows

119863 = 120572( 12057611205761199061 ) exp(120573 12057611205761199061 ) (15)

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 Advances in Materials Science and Engineering

Table 4 Material parameters 120572 and 120573 under different strain rates (dissipated energy)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s120572 0415 0482 0369 0468 0474120573 0881 0748 1001 0773 0760

휀1 ()

12

10

08

06

04

02

002520151050

Ws(1

0minus3

Jm

m3

(

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 16 Variation of dissipated energy with axial strain underdifferent strain rates

12

09

06

03

00

휀1 ()2520151050

D

Equation (15)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 17 Variation of the damage variable with axial strain underdifferent strain rates (dissipated energy)

where 120572 and 120573 are material parameters corresponding to thestrain rate These parameters under different strain rates arelisted in Table 4

424 Damage Evolution Laws Deduced from the ConstitutiveModel The constitutive model of some materials has been

3024181260

휀1 ()

0

1

2

3

5

4

6

휎1minus휎3(M

Pa)

Equation (16)

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 18 The model matches effect

known but the damage evolution law cannot be clearlyobserved from the model Then the damage evolution lawcan be deduced from the constitutive model Bazant andMazars [25] and Lemaitre [30] calculated the damage of amaterial by fitting stress-strain curves according to the phe-nomenological theory

Considering the deformation features of frozen loess ageneralized hyperbolic model is used in this study [31] Withthe model the 119862 test results are fitted as shown in Figure 18

1205901 minus 1205903 = 119897 + 1198981205761(119897 + 1198991205761)2 1205761 = (1 minus 119863) 119864lowast0 1205761 (16)

120590(1 minus 119863) = 120590eff (17)

where119864lowast0 is the initial modulus that corresponds to the fittingcurves 119897 119898 and 119899 are material parameters that are relatedto the strain rate These parameters under different strainrates are listed in Table 5 The physical significance of theseparameters can be found in Prevost [31] The methods cancharacterize stiffness degradation ofmaterial bymodulus (16)and the strength change of material by the effective stress(17) in the deformation process The two variables can bededuced from the same method [23] In the paper we usedthe modulus as an example

Then the equation of damage evolution can be obtainedas follows

119863 = 1 minus 119897 + 1198981205761119864lowast0 (119897 + 1198991205761)2 (18)

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 11

Table 5 Material parameters 119897 119898 and 119899 under different strain rates (constitutive model)

1205761 13 times 10minus2s 50 times 10minus3s 50 times 10minus4s 50 times 10minus5s 50 times 10minus6s119897 001159 001660 002520 002200 001800119898 minus001853 minus003000 minus006200 minus006200 minus004540119899 002570 002090 001660 004720 009450

3024181260

휀1 ()

10

08

06

04

02

00

D

Dc = 0705Dc = 0639

Dc = 0624

Dc = 0597

Dc = 0538

13 times 10minus2

50 times 10minus3

50 times 10minus4

50 times 10minus5

50 times 10minus6(sminus1)(sminus1)(sminus1)

(sminus1)(sminus1)

Figure 19 Variation of the damage variable with axial strain underdifferent strain rates (constitutive model)

The variation of the damage variable with axial strain isshown in Figure 19 The development of damage is closelyrelated to the loading timewhich is consistentwith the resultsof the 119862 tests The critical damage (119863119888) is shown in Figure 19

5 Discussion

At present few studies compared the advantages and dis-advantages among different damage factors or discussedthe selection criteria of the damage variable to the samematerial In this paper four damage factors were studiedand compared based on macro tests results of frozen loessBased on the above studies the four methods can describethe damage and damage evolution of frozen loess under idealconditions However considering the special properties andengineering service purposes on frozen loess the advantagesand disadvantages of these damage factors and damagevariables will be assessed as follows

For frozen loess it is difficult to accurately measure thedissipated energy by experiments and it is not for presentingthe damage process considering the special physical propertyof the granular material [32] In the paper the dissipatedenergy of the frozen loess was calculated based on manyhypothesesThus the dissipated energy is hardly investigatedand not recommended in this areaThe fourthmethod is thatthe damage evolution law is obtained from the constitutivemodel based on the theory of effective stress but the premise

is that the constitutive model of the material is knownBecause of the complexity of the geotechnical property suchas large regional characteristics and the complex structureit is difficult to obtain a simple and accurate model There-fore this method is also difficult to apply The theory ofstiffness degradation can be used for most materials andthe stiffness of material is also easy to measure Thus manyresearchers choose the stiffness as the damage factor to studythe mechanical damage of material The main limitation ofthe method is that damage is uniformly distributed in thevolume on which strain is measured Thus it is also difficultregarding frozen loess For frozen loess engineering the keyengineering problem is the deformation issue for examplesubgrade settlement and foundation deformation which isthe safety evaluation index to these engineering Thereforeplastic deformation (permanent deformation) is the mostsuitable for the damage research of frozen loess based on thematerial properties and the nature of the material service

In addition the problem of establishing a quantitativerelationship between the damage factor and the damage vari-able for different materials and different purposes should beconsidered Different relationships lead to different evolutionprocesses such as the comparison results of the exponentialfunction (10) and linear function ((6) and (14))The key pointto understand is how to exactly define the damage criterions

Selecting an appropriate damage factor has been a hottopic for a long time in CDM Based on this discussion itcan be seen that a new technology (microscopic scanning)and method must be developed to more accurately describethe damage evolution of the material under external loadsThemethod is a comprehensivemethod that containsmacro-and micromethods (the combination of material science andcontinuous media mechanics or mechanical damage andphysical damage) Both methods are essential otherwise thedamage process of the material can only be described fromone side In other words the method must be able to verifyand inverse the macromechanical properties of the materialaccording to the change law of mesoscopic fabric statusSelecting a cross-scale damage variable is also the key bridgeto explore the essence of deformation and failure ofmaterialsThis subject is also a main issue of the study of materialproperties in current solid mechanics [33]

6 Conclusions

This paper presented a series of tests including monotonicand load-unloading-loading tests to the frozen loess underfive strain rates in order to investigate the rate-dependentdeformation behavior and the damage evolution law Fourdamage factors were studied and compared based on tests

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

12 Advances in Materials Science and Engineering

results Some conclusions drawn from experimental andtheoretical results are summarized below

(1) The strength and the deformation behavior of frozenloess strongly depend on the strain rate In addition the load-ing and unloading cycles test does not affect the deformationbehavior of frozen loess

(2) 119864ℎ and 119864119904 cannot describe the stiffness degradation ofthe frozen loess in the deformation process So 119864119889 is selectedbased on the test results and previous studies

(3) The hysteresis loop area cannot accurately reflect thedissipated energy of frozen loess The value of dissipatedenergy is recalculated according to the view of fracturemechanics and energy conservation in order to describe thedevelopment of the material damage

(4) According to the discussion the plastic deformationis the most suitable for the damage research of frozen loessbased on the material properties and the nature of thematerial service

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (41630636 41401077 and 11502266) andthe Science and TechnologyMajor Project of Gansu Province(143GKDA007)

References

[1] J Q XuTheory on the Strength of Materials Shanghai JiaotongUniversity Press 2009

[2] R A Schapery ldquoCorrespondence principles and a generalized Jintegral for large deformation and fracture analysis of viscoelas-tic mediardquo International Journal of Fracture vol 25 no 3 pp195ndash223 1984

[3] S Dhar P M Dixit and R Sethuraman ldquoA continuum damagemechanics model for ductile fracturerdquo International Journal ofPressure Vessels and Piping vol 77 no 6 pp 335ndash344 2000

[4] RW Sullivan ldquoDevelopment of a viscoelastic continuum dam-age model for cyclic loadingrdquo Mechanics of Time-DependentMaterials vol 12 no 4 pp 329ndash342 2008

[5] S P Zhao W Ma J F Zheng and G D Jiao ldquoDamage dis-sipation potential of frozen remolded Lanzhou loess basedon CT uniaxial compression test resultsrdquo Chinese Journal ofGeotechnical Engineering vol 34 pp 2019ndash2025 2012

[6] X T Xu Y H Dong and C X Fan ldquoLaboratory investigationon energy dissipation and damage characteristics of frozen loessduring deformation processrdquoCold Regions Science and Technol-ogy vol 109 pp 1ndash8 2015

[7] J Shao A Chiarelli andNHoteit ldquoModeling of coupled elasto-plastic damage in rock materialsrdquo International Journal of RockMechanics and Mining Sciences vol 35 no 4-5 p 444 1998

[8] J Lemaitre ldquoContinuous damage mechanics model for ductilefracturerdquo Journal of Engineering Materials and TechnologyTransactions of the ASME vol 107 no 1 pp 83ndash89 1985

[9] G Z VoyiadjisHandbook of DamageMechanics Springer 2015

[10] M Alves J Yu and N Jones ldquoOn the elastic modulus degrada-tion in continuum damage mechanicsrdquo Computers and Struc-tures vol 76 no 6 pp 703ndash712 2000

[11] G Swoboda and Q Yang ldquoAn energy-based damage model ofgeomaterialsmdashI Formulation and numerical resultsrdquo Interna-tional Journal of Solids and Structures vol 36 no 12 pp 1719ndash1734 1999

[12] R Lapovok ldquoDamage evolution under severe plastic deforma-tionrdquo International Journal of Fracture vol 115 no 2 pp 159ndash172 2002

[13] Z M Shi H L Ma and J B Li ldquoA novel damage variableto characterize evolution of microstructure with plastic defor-mation for ductile metal materials under tensile loadingrdquoEngineering Fracture Mechanics vol 78 no 3 pp 503ndash513 2011

[14] Y Y Cao S P Ma X Wang and Z Y Hong ldquoA new definitionof damage variable for rock material based on the spatial char-acteristics of deformation fieldsrdquo Advanced Materials Researchvol 146-147 pp 865ndash868 2011

[15] M K Darabi R K AbuAl-Rub E AMasad C-WHuang andD N Little ldquoA thermo-viscoelastic-viscoplastic-viscodamageconstitutivemodel for asphalticmaterialsrdquo International Journalof Solids and Structures vol 48 no 1 pp 191ndash207 2011

[16] H Du W Ma S Zhang Z Zhou and E Liu ldquoStrength proper-ties of ice-rich frozen silty sands under uniaxial compression fora wide range of strain rates andmoisture contentsrdquoCold RegionsScience and Technology vol 123 pp 107ndash113 2016

[17] Z Zhou W Ma S Zhang H Du Y Mu and G Li ldquoMultiaxialcreep of frozen loessrdquo Mechanics of Materials vol 95 pp 172ndash191 2016

[18] LM Kachanov Introduction to ContinuumDamageMechanicsM Nijhoff 1986

[19] L M Kachanov ldquoTime of the rupture process under creep con-ditionsrdquo Akad Nauk SSR Otd Tech vol 8 p 6 1958

[20] D Huang and Y Li ldquoConversion of strain energy in TriaxialUnloading Tests on Marblerdquo International Journal of RockMechanics and Mining Sciences vol 66 pp 160ndash168 2014

[21] A S Chiarelli J F Shao and N Hoteit ldquoModeling of elasto-plastic damage behavior of a claystonerdquo International Journal ofPlasticity vol 19 no 1 pp 23ndash45 2003

[22] G E Mann T Sumitomo C H Caceres and J R GriffithsldquoReversible plastic strain during cyclic loading-unloading ofMgand Mg-Zn alloysrdquo Materials Science and Engineering A vol456 no 1-2 pp 138ndash146 2007

[23] H P Xie Rock and Concrete Damage Mechanics China Univer-sity of Mining and Technology Press 1990

[24] S W Yu Damage Mechanics Tsinghua University Press 1997[25] Z P Bazant and J Mazars ldquoFrance-US workshop on strain

localization and size effect due to cracking and damagerdquo Journalof Engineering Mechanics vol 116 no 6 pp 1412ndash1424 1990

[26] I Carol E Rizzi and K Willam ldquoA unified theory of elasticdegradation and damage based on a loading surfacerdquo Interna-tional Journal of Solids and Structures vol 31 no 20 pp 2835ndash2865 1994

[27] J Lemaitre A Course on Damage Mechanics Springer 1998[28] M Liao Y Lai E Liu and X Wan ldquoA fractional order creep

constitutive model of warm frozen siltrdquo Acta Geotechnica pp1ndash13 2016

[29] Y Yugui G Feng C Hongmei and H Peng ldquoEnergy dissipa-tion and failure criterion of artificial frozen soilrdquo Cold RegionsScience and Technology vol 129 pp 137ndash144 2016

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 13

[30] J Lemaitre ldquoFormulation and identification of damage kineticconstitutive equationsrdquo in Continuum Damage Mechanics The-ory andApplication vol 295 of International Centre forMechan-ical Sciences pp 37ndash89 Springer Vienna Austria 1987

[31] J Prevost ldquoMathematical modelling of monotonic and cyclicundrained clay behaviourrdquo International Journal for Numericaland Analytical Methods in Geomechanics vol 1 no 2 pp 195ndash216 1977

[32] J Lemaitre and J Dufailly ldquoDamage measurementsrdquo Engineer-ing Fracture Mechanics vol 28 no 5-6 pp 643ndash661 1987

[33] F Hild A Bouterf and S Roux ldquoDamage measurements viaDICrdquo International Journal of Fracture vol 191 no 1-2 pp 77ndash105 2015

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpswwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials