Study on the Elastoplastic Damage-Healing Coupled

Research ArticleStudy on the Elastoplastic Damage-Healing CoupledConstitutive Model of Mudstone

Jie Xu12 Jiawang Qu3 Yufeng Gao12 and Ning Xu4

1Geotechnical Research Institute Hohai University Nanjing 210098 China2Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering Hohai University Nanjing 210098 China3Wuhan Surveying-Geotechnical Research Institute Co Ltd MCC Wuhan 430080 China4Laixi Water Conservancy Bureau Laixi 266600 China

Correspondence should be addressed to Jie Xu besurexj163com

Received 9 November 2016 Revised 11 April 2017 Accepted 14 May 2017 Published 12 June 2017

Academic Editor RomanWendner

Copyright copy 2017 Jie Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Under the effect of high ground stress and water-rock chemical interaction the fractures in the damagedmudstone wound undergoa self-healing process and recover the physical andmechanical properties which has a significant impact on the wall-rockrsquos stabilityof high level radioactive waste repository and the migration of radioactive nuclide According to the general thermodynamicsand continuum damage mechanics an internal variable describing mudstone healing properties is introduced and an elastoplasticdamage-healing model reflecting mudstone deformation damage and self-healing evolution is put forward This model is used tosimulate the triaxial compression test of mudstone under different confining pressures whose simulated results are compared withthe test data It is indicated that the model could embody the main mechanical properties of mudstone with the healing effect in aneffective way and the healing part of the model has a great influence on the simulated results

1 Introduction

The reasonable disposal of nuclear waste relates to thesustainable development of society and economy and thesurvival of mankind At present the worldwide recognizednuclear waste disposal method is to bury nuclear waste deepunderground for storage [1ndash4] Among the many geologicbodies mudstone is being paid extensive attention due to itscharacteristics of low permeability and good rheological andself-healing performance which has been the main parentrock for nuclear waste disposal Therefore comprehendingthe mechanical behaviors of mudstone materials is beneficialto assessing designing and building the underground repos-itory for nuclear waste

The self-healing property of mudstone can improve theisolation performance and wall-rockrsquos stability of nuclearwaste repository Zhang et al carried out a series of tests suchas porosity measurement permeability determination andAEmonitoring which validated the existence of damage self-healing process ofmudstone [5 6] Bastiaens et al pointed outthat the main mechanism for mudstone damage self-healing

effect was caused by the inflation consolidation and creepof mudstone materials in response to mechanics and water-power Many domestic experts and scholars have also con-ducted a lot of researches on the damage self-healing effectof materials [7] Jia et al introduced the concept of healingstress and hydrochemistry healing factor to establish theself-healing model which was used to describe the self-healing performance of fractured mudstone [8] Based onthe isotropic hypothesis Miao et al proposed a continuoushealing mechanics model of crushed rock [9] On the ther-modynamic framework Barbero built a continuous damage-healing constitutive model for the polymer composites [10]Darabi et al extended the concepts of the Kachanov effectiveconstruction and effective stress space and put forwardmicrodamage-healing model based on continuum damagemechanics [11]

The current studies on mudstone self-healing perfor-mance mostly focus on the field and laboratory tests [12ndash15]The mechanical analysis on mudstone self-healing perfor-mance is little In this paper an exploration is made to model

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 6431607 7 pageshttpsdoiorg10115520176431607

2 Mathematical Problems in Engineering

the continuous damage self-healing processes based on max-imum energy dissipation principle which is established byintroducing the internal variable of mudstone self-healingperformance

2 Basic Framework of the Model

According to the generalized thermodynamics an internalvariable is introduced to the thermodynamic constitutiverelation to describe the different mechanisms of materialdeformation such as elastoplastic deformation and damage-healing evolution The mudstone deformation is regarded asan isothermal process The second law of thermodynamicscan be expressed as

120590 minus 120588 ge 0 (1)

where 120590 and are the stress and strain tensors 120588 is the massdensity 120595 is the Helmholtz free energy and sdot is the derivativeof variable with respect to time

TheHelmholtz free energy120595 is set to the thermodynamicpotential of mudstone and elastic strain 120576119890 plastic strain120576119901 isotropic damage tensor d and tensor h describing theself-healing performance of materials are set to the internalvariablesso the Helmholtz free energy is expressed as

120595 = 120595 (120576119890 120576119901 d h) (2)

It is assumed that the Helmholtz free energy can bedivided into two parts

120595 (120576119890 120576119901 d h) = 12120576119890119864 (d h) 120576119890 + 120601 (120576119901 d h) (3)

where the first part of the formula is the strain energy thesecond part is the dissipated energy and119864(d h) is the fourth-order damage-healing stiffness tensor

In order to simplify the model it is assumed that the evo-lution of damage and healing surface is isotropic Accordingto the study of [15] the dissipated energy is expressed as

120593 (120576119901 d h) = 120593 (120574 120578 120596)= 1

211988601205742 + 1198861 (120578 minus 11988621198901205781198862)

+ 1198863 (11988641198901205961198864 minus 120596) (4)

where 1198860 1198861 1198862 and 1198863 are material constant and 120574 120578 and 120596are the isotropic hardening parameters which are designed asfollows [16]

120574 = int1199050

radic23 120576119901 120576119901119889119905

120578 = int1199050

radic23 d d 119889119905

120596 = int1199050

radic23 h h 119889119905

(5)

According to formulas (1) (3) and (4) the thermo-dynamic driving forces associated with the correspondinginternal variables are shown as follows

120590 = 120597120595120597120576119890 = 119864 (d h) 120576119890 (6a)

120590119901 = 120590 minus 120597120595120597120576119901 = 120590 minus 1198860120574 120597120574

120597120576119901 (6b)

120590119889 = minus120597120595120597d = minus12120576119890 120597119864120597d 120576119890 minus 1198861 (1 minus 1198901205781198862) 120597120578

120597d (6c)

120590ℎ = 120597120595120597h = 1

2120576119890 120597119864120597h 120576119890 + 1198863 (1198901205961198864 minus 1) 120597120596120597h (6d)

The mechanical dissipation potential of mudstone defor-mation is

Π = 120590119901 120576119901 + 120590119889 d minus 120590ℎ h ge 0 (7)In addition the other conditions need to be satisfied and

they arePlastic dissipation energy Π119901 = 120590119901 120576119901 ge 0 (8a)

Damage dissipation potential Π119889 = 120590119889 d ge 0 (8b)

Healing dissipation potential energy Πℎ = minus120590ℎ hle 0 (8c)

The maximum dissipation theory says that the true stateof thermodynamic driving force is the state that makesthe maximum of dissipation equation among all admissiblestates It is a constrained optimization problem which canbe solved by Lagrange multiplier method Now define thefunction

Γ = Π minus 119901120581119901 (120590119901) minus 119889120581119889 (120590119889) minus ℎ120581ℎ (120590ℎ) (9)

where 119901 119889 and ℎ are Lagrange multipliers related toplasticity damage and healing respectively 120581119889 120581119901 and 120581ℎ arethe dissipation potentials of plasticity damage and healing

In order to make the maximum of Γ the followingcondition needs to be satisfied

120597Γ120597120590119901 = 0120597Γ120597120590119889 = 0120597Γ120597120590ℎ = 0

(10)

The evolution equation of each internal variable can beexpressed as

120576119901 = 119901 120597120581119901120597120590119901

d = 119889 120597120581119889120597120590119889

h = ℎ 120597120581ℎ120597120590ℎ

(11)

Mathematical Problems in Engineering 3

The process of mudstone plastic deformation damageand healing evolution is a coupling process which needs tosatisfy the consistency condition

119891119901 = 120597119891119901120597120590 + 120597119891119901

120597d d + 120597119891119901120597h h = 0

119891119889 = 120597119891119889120597120590119889

119889 + 120597119891119889120597d d = 0

119891ℎ = 120597119891ℎ120597120590ℎ

ℎ + 120597119891ℎ120597h h = 0

(12)

where119891119901 119891119889 and119891ℎ are yield function damage and healingevolution equations

The derivative of (6a) with respect to time leads to theincrement constitutive model of mudstone

= 119864 (d h) 120576119890 + (d h) 120576119890= 119864 (d h) ( 120576 minus 120576119901)

+ (1198641015840d (d h) d + 1198641015840h (d h) h) (120576 minus 120576119901) (13)

where1198641015840d and1198641015840h are the derivatives of119864with respect to d andh

The plastic yield function damage and healing evolutionequations are plugged into the consistency condition Com-bined with the optimization problem of Lagrange multiplierthe Lagrange multipliers related to plasticity damage andhealing can be solved and then the mudstone constitutivemodel is obtained according to (13)

3 Mudstone Plastic Property

The plastic deformation can be described by yield functionand plastic potential Like many geomaterials mudstone is apressure-sensitive distant material which reflects asymmet-ric response under the compression and tensile loadingsTheyield function is expressed as

119891119901 (120590 d h) = 119902 minus 120572119901119875119888 [119873(119862119904 + 119901119875119888)]12

= 0 (14)

where 119901 = 1205901198941198943 119902 = radic31198692 119862119904 is the cohesion coefficient119873 is the curvature of failure surface 119875119888 is the uniaxialcompressive strength of material and 120572119901 is the functiondescribing mudstone plastic hardening The hardening func-tion is obtained by introducing the healing variable h

120572119901 (120574 d h)= (1 minus 1198871trd) (1 + 1198872trh) [1205720119901 + (120572119898119901 minus 1205720119901) 120574

119861 + 120574] (15)

where 1198871 and 1198872 represent the influence of damage and healingto plasticity whose value range is from 0 to 1 In order tosimplify the calculation process 1198871 = 1198872 = 1 is set 119861 isthe parameter used to control plastic hardening rate 1205720119901 and120572119898119901 are the initial yield threshold and final plastic hardening

Unhealed damageHealed damage

F

F

Figure 1 Schematic diagram of healed sample

valueThe plastic flow rule is determined by plastic potentialand the mudstone plastic potential function proposed by Jiaet al [17] is modified to be

119892119901 (120590119894119895 120574 d h) = 119902 minus (1 minus 1198871trd) (1 + 1198872trh)sdot (119901 + 119862119904119875119888)sdot (1205720119901 + (1 minus 1205720119901) 120574

119861 + 120574 minus 120575) (16)

where 120578 is the turning point between compression anddilatation

4 Damage and Healing of Mudstone

When the loading mudstone reaches the conditions ofdamage nucleation and growth some new microholes andmicrocracks begin to initiate and propagate However somemicrocracks may repair themselves The damage can lead tosome deterioration of mechanical property such as strengthreduction and deformation increase while damage healingoften gives rise to the strength recovery and compressibilityreduction

41 Definition of Damage and Healing Variables The dam-age and healing variables are defined based on the one-dimensional space The cylinder sample under uniaxial ten-sile loading is shown in Figure 1

The cross-sectional area119860 can be divided into three partsundamaged area 119860ud unhealed microcrack area 119860uh andhealed microcrack area 119860ℎ and

119860 = 119860ud + 119860119889119860119889 = 119860uh + 119860ℎ

(17)

where 119860119889 is the damaged area


Since the damaged mudstone can not be healed to intactrock unhealed microcrack area 119860uh is greater than 0 whichmeans the damaged part ofmudstone can not heal completelyunder any circumstance The carrying capacity of healingarea 119860ℎ does not always come back to that of the intactmaterial thus proposing a reduction coefficient to describethe effective healing area In order to simplify the calculationit is assumed in the paper that the mechanical propertyof healed areas is identical to that of intact material Thatmeans when microdamage is fully repaired its mechanicalproperties could recover completely The effective bearingarea is

119860119890 = 119860ud + 119860ℎ (18)

The one-dimensional internal variables of damage andhealing are defined as

119889 = 119860119889119860

ℎ = 119860ℎ119860ud

(19)

It is assumed that cracks and holes have no carryingcapacity and themain loads are carried on by the undamagedarea and healed area Thus the effective bearing area is alsoexpressed as

119860119890 = (1 minus 119889) (1 + ℎ)119860 (20)

The effective stress is

1205901015840 = 120590(1 minus 119889) (1 + ℎ) (21)

According to elastic mechanics theories if one-dimen-sional strain 120576 = Δ119871119871 is extended to three-dimensionalspace strain tensor 120576 can be obtained Analogously thetensors d and h used to describe three-dimensional damageand healing also can be acquired with damage internalvariable d and healing internal variable h extended to three-dimensional spaceThe formation and propagation of micro-cracks and holes have the prior direction thus assuming thatthe damage and healing have the same principle direction dand h can be divided into

d = 3sum119894=1

119889119894n119894 otimes n119894

h = 3sum119894=1

ℎ119894n119894 otimes n119894(22)

where n119894 119889119894 and ℎ119894 are damage (healing) principle directiondamage principle value and healing principle value and otimes istensor product

Based on the elastic energy equivalence principle therelationship between 119864(d h) and elastic modulus 119864 of intactmudstone is

119864 (d h) = 120573 119864 120573119879 (23)

where 120573 is a fourth-order transformational tensor which isconsidered as function of d and h

42 Damage and Healing Evolution The damage develop-ment of mudstone can be described by the damage criterionIt is generally acknowledged that weak tensile strength ofgeotechnical materials is one of the larger contributors todamage Thus the damage criterion has to do with tensilestress or tensile strain Given this the following damagecriterion is used to describe the damage development ofmudstone in this paper

119891119889 (120576 d) = (120576+ 120576+)12 minus (1199030 + 1199031trd) le 0 (24)

where 120576+ is the tensile part of strain and 1199030 1199031 are materialparameters

120576+ = 3sum119896=1

120576119896119867(120576119896)V119896 otimes V119896 (25)

where 119867(120576119896) is the Heaviside step function of the 119896th prin-cipal strain 120576119896 and V119896 is the corresponding principal vector

Like rock damage the healing criterion can be used todescribe the healing development process of rock The heal-ing of cracks is associated with compressive strain and thusthe following healing criterion is adopted

119891ℎ (120590ℎ h) = (120576minus 120576minus)12 minus (1199032 + 1199033trh) le 0 (26)

where 120576minus is the tensile part of strain and 1199032 1199033 are materialparameters

120576minus = 120576 minus 120576+ (27)

According to the nonassociated dissipation criterion119891119889 = 120581119889 is obtained Similarly 119891ℎ = 120581ℎ is also achieved andthus the evolution equations of damage and healing areexpressed as

d = 119889 120597119891119889120597120590119889

h = ℎ 120597119891ℎ120597120590ℎ (28)

5 Model Validation

Theparameters of the abovemodel can be obtained by exper-iment For example the parameters associated with plasticyielding119862119904 and119873 are acquired by drawing out failure curveson the 119901 minus 119902 plane with peak stresses in triaxial compressiontests under different confining pressuresThe elastic constants119864 and ] can be determined by using the linear part of stress-strain curves before the initiation of damage and plasticdeformation [10]The controlling parameter of enhancementrate119861 is got by fitting the evolution of hardening function 120572119901with hardening variable 119861 according to (15) The initial yieldthreshold1205720119901 is determined through initial yield surface while120572119898119901 is acquired by fitting the peak strengthThe turning pointbetween compression and dilatation 120578 is ascertained by thestress point in triaxial test where volumetric strain rate 120576V isequal to 0 [17]


In order to validate the rationality and validity of themodel the triaxial compression tests of mudstone samplesare carried out which are taken from deep rock roadwayin Huainan Coalmine Each sample was cut into cylinders50mm in diameter and 100mm in height and the surfaceof the samples appeared relatively smooth without apparentcracksThe samples were loaded by RMT-301 and the processof tests was as follow load the samples to the prescribed con-fining pressure level (0 10 and 20MPa) at constant loadingrate of 0001mms then keep the confining pressure constantand load axially the samples at loading rate of 0001mmsuntil the complete stress-strain curve was available Based onthe above methods the parameters of model are achieved asfollows

119864 = 112GPa] = 0311198860 = 25 times 10minus7119875119888 = 284MPa119862119904 = 01119873 = 36119861 = 7 times 10minus31205720119901 = 029120572119898119901 = 1120575 = 051198861 = 00211198862 = minus64 times 1041199030 = 64 times 10minus51199031 = 37 times 10minus31198863 = 00111198864 = minus015 times 1041199032 = 93 times 10minus51199033 = 41 times 10minus3

(29)

The triaxial compression tests are simulated by craftingthe finite element program based on the above model whichis denoted by Model I In order to explore the influence ofhealing part on the simulated results Mode II is obtained bytaking out the healing part of Model I and then using it tosimulate the triaxial compression tests againThe incrementalconstitutive equation of Mode II is shown as follows

= 119864 (d) ( 120576 minus 120576119901) + 1198641015840 (d) d (120576 minus 120576119901) (30)

As indicated in Figures 2 3 and 4 the comparisonsbetween simulation values ofModels I and II and experiment

Experimental dataModel IModel II

1205761 () 1205763 ()0

5

10

15

20

25

30 1205901 minus 1205903 (MPa)

minus04 minus02 0 02 04 06 08minus06

Figure 2 Comparison between simulation values of Models I andII and experimental data under 0MPa confining pressure

Experimental dataModel 1Model 2

1205901 minus 1205903 (MPa)

1205761 () 1205763 ()05

101520253035404550

minus2 minus1 0 1 2 3minus3

Figure 3 Comparison between simulation values of Models I andII and experiment data under 10MPa confining pressure

data under different confining pressures are carried out Itis found that the simulation values of Model I are exactlyconsistent with experiment data especially in terms of lateraldeformation which reveals that it is feasible to describethe mechanical properties of deep mudstone with Model Iproposed in this paper

AlthoughModel II without considering healing effect canalso be used to simulate the triaxial compression tests ofmudstone the simulated results of Model II do not matchwell with experiment data compared with Model I Addingthe healing part to the model can improve the accuracy ofthe simulation

6 Conclusion

(1) Based on the generalized thermodynamics contin-uum damage mechanics and specificity of mudstone


0

10

20

30

40

50

60

70


1205901 minus 1205903 (MPa)

1205761 () 1205763 ()minus1 minus05 0 05 1 15minus15


the theoretical model proposed in this paper notonly describes the elastic-plastic deformation anddamage evolution of mudstone under different stressconditions but also reflects its healing properties

(2) Comparison of calculation results with relevant testdata shows that the presented elastoplastic damage-healing model can demonstrate mechanical proper-ties of mudstone more effectively

(3) The model proposed in this paper takes account ofthe healing properties of mudstone at the same timethere are certain shortcomings as well the model isput forward for the mudstone with fixed moisturecontent not considering the influence of watermigra-tion and content variation on self-healing effect andthus the model can not reflect the microscopic mech-anism of self-healing property Given this furtherresearch will be conducted in a follow-up study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper and the funding didnot lead to any conflicts of interest regarding the publicationof this paper

Acknowledgments

This work was supported by China Postdoctoral ScienceFoundation (Grant no 2017M611676) the National NaturalScience Foundation of China (Grant no 41630638) theNational Key Basic Research Program of China (ldquo973rdquo Pro-gram) (Grant no 2015CB057901) the Public Service SectorRampD Project of Ministry ofWater Resources of China (Grantno 201501035-03) ldquo333rdquo Projects of Jiangsu Province (Grantno BRA2015312) and the 111 Projects (Grant no B13024)

References

[1] Q Z Fan Y F Gao X H Cui et al ldquoStudy on nonlinear creepmodel of soft rockrdquo Yantu Gongcheng XuebaoChinese Journalof Geotechnical Engineering vol 29 no 4 pp 505ndash509 2007

[2] W Y Xu J W Zhou S Q Yang et al ldquoStudy on creep damageconstitutive relation of greenschist specimenrdquo Chinese Journalof Rock Mechanics and Engineering vol 25 supplement 1 pp3093ndash3096 2006

[3] D Hoxha A Giraud A Blaisonneau F Homand and CChavant ldquoPoroplastic modelling of the excavation and venti-lation of a deep cavityrdquo International Journal for Numerical andAnalytical Methods in Geomechanics vol 28 no 4 pp 339ndash3642004

[4] K K Muraleetharan C Liu C Wei T C G Kibbey and LChen ldquoAn elastoplatic framework for coupling hydraulic andmechanical behavior of unsaturated soilsrdquo International Journalof Plasticity vol 25 no 3 pp 473ndash490 2009

[5] C L Zhang ldquoExperimental evidence for self-sealing of fracturesin claystonerdquo Physics and Chemistry of the Earth vol 36 no 17pp 1972ndash1980 2011

[6] C L Zhang T Rothfuchs K Su et al ldquoExperimental study ofthe thermo-hydro-mechanical behaviour of indurated claysrdquoPhysics and Chemistry of the Earth vol 32 no 8 pp 957ndash9652007

[7] W Bastiaens F Bernier and X L Li ldquoSelfrac experimentsand conclusions on fracturing self-healing and self-sealingprocesses in claysrdquo Physics and Chemistry of the Earth vol 32no 8 pp 600ndash615 2007

[8] S P Jia W Z Chen Y u HD et al ldquoStudy of hydromechanical-damage coupled creep constitutive model of mudstone Part ITheoretical modelrdquo Rock and Soil Mechanics vol 32 no 9 pp2596ndash2602 2011 (in Chinese)

[9] S K Miao M L Wang H L Schreyer et al ldquoConstitutivemodels for healing of materials with application to compactionof crushed rock saltrdquo Journal of Engineering Mechanics vol 121no 10 pp 1122ndash1129 1995

[10] E J Barbero and ldquoContinuum damage-healing mechanicswith application to self-healing compositesrdquo International Jour-nal of Damage Mechanics vol 14 no 1 pp 51ndash81 2005

[11] M K Darabi R K Abualrub and D N Little ldquoA continuumdamage mechanics framework for modeling micro-damagehealingrdquo International Journal of Solids and Structures vol 49no 3 pp 492ndash513 2012

[12] N Conil I Djeran-Maigre R Cabrillac andK Su ldquoThermody-namicsmodelling of plasticity and damage of argilliterdquoComptesRendus Mecanique vol 332 no 10 pp 841ndash848 2004

[13] A S Chiarelli Experimental investigation and constitutivemodeling of coupled elastoplastic damage in hard argillitestones[Doctoral thesis] University de Lille 1 2000

[14] P Bossart P M Meier A Moeri T Trick and J C MayorldquoGeological and hydraulic characterisation of the excavationdisturbed zone in the opalinus clay of the mont terri rocklaboratoryrdquo Engineering Geology vol 66 pp 19ndash38 2002

[15] E J Barbero and P Lonetti ldquoAn inelastic damagemodel for fiberreinforced laminatesrdquo Journal of Composite Materials vol 36no 8 pp 941ndash962 2002

[16] G Z Voyiadjis and B Deliktas ldquoA coupled anisotropic damagemodel for the inelastic response of composite materialsrdquo Com-puter Methods in Applied Mechanics and Engineering vol 183no 3 pp 159ndash199 2000


[17] Y Jia H B Bian K Su et al ldquoElastoplastic damage modeling ofdesaturation and resaturation in argillitesrdquo International Journalfor Numerical and Analytical Methods in Geomechanics vol 34no 2 pp 187ndash220 2010

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the continuous damage self-healing processes based on max-imum energy dissipation principle which is established byintroducing the internal variable of mudstone self-healingperformance

2 Basic Framework of the Model

According to the generalized thermodynamics an internalvariable is introduced to the thermodynamic constitutiverelation to describe the different mechanisms of materialdeformation such as elastoplastic deformation and damage-healing evolution The mudstone deformation is regarded asan isothermal process The second law of thermodynamicscan be expressed as

120590 minus 120588 ge 0 (1)

where 120590 and are the stress and strain tensors 120588 is the massdensity 120595 is the Helmholtz free energy and sdot is the derivativeof variable with respect to time

TheHelmholtz free energy120595 is set to the thermodynamicpotential of mudstone and elastic strain 120576119890 plastic strain120576119901 isotropic damage tensor d and tensor h describing theself-healing performance of materials are set to the internalvariablesso the Helmholtz free energy is expressed as

120595 = 120595 (120576119890 120576119901 d h) (2)

It is assumed that the Helmholtz free energy can bedivided into two parts

120595 (120576119890 120576119901 d h) = 12120576119890119864 (d h) 120576119890 + 120601 (120576119901 d h) (3)

where the first part of the formula is the strain energy thesecond part is the dissipated energy and119864(d h) is the fourth-order damage-healing stiffness tensor

In order to simplify the model it is assumed that the evo-lution of damage and healing surface is isotropic Accordingto the study of [15] the dissipated energy is expressed as

120593 (120576119901 d h) = 120593 (120574 120578 120596)= 1

211988601205742 + 1198861 (120578 minus 11988621198901205781198862)

+ 1198863 (11988641198901205961198864 minus 120596) (4)

where 1198860 1198861 1198862 and 1198863 are material constant and 120574 120578 and 120596are the isotropic hardening parameters which are designed asfollows [16]

120574 = int1199050

radic23 120576119901 120576119901119889119905

120578 = int1199050

radic23 d d 119889119905

120596 = int1199050

radic23 h h 119889119905

(5)

According to formulas (1) (3) and (4) the thermo-dynamic driving forces associated with the correspondinginternal variables are shown as follows

120590 = 120597120595120597120576119890 = 119864 (d h) 120576119890 (6a)

120590119901 = 120590 minus 120597120595120597120576119901 = 120590 minus 1198860120574 120597120574

120597120576119901 (6b)

120590119889 = minus120597120595120597d = minus12120576119890 120597119864120597d 120576119890 minus 1198861 (1 minus 1198901205781198862) 120597120578

120597d (6c)

120590ℎ = 120597120595120597h = 1

2120576119890 120597119864120597h 120576119890 + 1198863 (1198901205961198864 minus 1) 120597120596120597h (6d)

The mechanical dissipation potential of mudstone defor-mation is

Π = 120590119901 120576119901 + 120590119889 d minus 120590ℎ h ge 0 (7)In addition the other conditions need to be satisfied and

they arePlastic dissipation energy Π119901 = 120590119901 120576119901 ge 0 (8a)

Damage dissipation potential Π119889 = 120590119889 d ge 0 (8b)

Healing dissipation potential energy Πℎ = minus120590ℎ hle 0 (8c)

The maximum dissipation theory says that the true stateof thermodynamic driving force is the state that makesthe maximum of dissipation equation among all admissiblestates It is a constrained optimization problem which canbe solved by Lagrange multiplier method Now define thefunction

Γ = Π minus 119901120581119901 (120590119901) minus 119889120581119889 (120590119889) minus ℎ120581ℎ (120590ℎ) (9)

where 119901 119889 and ℎ are Lagrange multipliers related toplasticity damage and healing respectively 120581119889 120581119901 and 120581ℎ arethe dissipation potentials of plasticity damage and healing

In order to make the maximum of Γ the followingcondition needs to be satisfied

120597Γ120597120590119901 = 0120597Γ120597120590119889 = 0120597Γ120597120590ℎ = 0

(10)

The evolution equation of each internal variable can beexpressed as

120576119901 = 119901 120597120581119901120597120590119901

d = 119889 120597120581119889120597120590119889

h = ℎ 120597120581ℎ120597120590ℎ

(11)



119891119901 = 120597119891119901120597120590 + 120597119891119901

120597d d + 120597119891119901120597h h = 0

119891119889 = 120597119891119889120597120590119889

119889 + 120597119891119889120597d d = 0

119891ℎ = 120597119891ℎ120597120590ℎ

ℎ + 120597119891ℎ120597h h = 0

(12)



= 119864 (d h) 120576119890 + (d h) 120576119890= 119864 (d h) ( 120576 minus 120576119901)

+ (1198641015840d (d h) d + 1198641015840h (d h) h) (120576 minus 120576119901) (13)





119891119901 (120590 d h) = 119902 minus 120572119901119875119888 [119873(119862119904 + 119901119875119888)]12

= 0 (14)



119861 + 120574] (15)



F

F




119861 + 120574 minus 120575) (16)






119860 = 119860ud + 119860119889119860119889 = 119860uh + 119860ℎ

(17)




119860119890 = 119860ud + 119860ℎ (18)


119889 = 119860119889119860

ℎ = 119860ℎ119860ud

(19)


119860119890 = (1 minus 119889) (1 + ℎ)119860 (20)


1205901015840 = 120590(1 minus 119889) (1 + ℎ) (21)


d = 3sum119894=1

119889119894n119894 otimes n119894

h = 3sum119894=1

ℎ119894n119894 otimes n119894(22)



119864 (d h) = 120573 119864 120573119879 (23)



119891119889 (120576 d) = (120576+ 120576+)12 minus (1199030 + 1199031trd) le 0 (24)


120576+ = 3sum119896=1

120576119896119867(120576119896)V119896 otimes V119896 (25)





120576minus = 120576 minus 120576+ (27)


d = 119889 120597119891119889120597120590119889

h = ℎ 120597119891ℎ120597120590ℎ (28)

5 Model Validation





(29)


= 119864 (d) ( 120576 minus 120576119901) + 1198641015840 (d) d (120576 minus 120576119901) (30)



1205761 () 1205763 ()0

5

10

15

20

25

30 1205901 minus 1205903 (MPa)




1205901 minus 1205903 (MPa)

1205761 () 1205763 ()05

101520253035404550





6 Conclusion



0

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70


1205901 minus 1205903 (MPa)








Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119891119901 = 120597119891119901120597120590 + 120597119891119901

120597d d + 120597119891119901120597h h = 0

119891119889 = 120597119891119889120597120590119889

119889 + 120597119891119889120597d d = 0

119891ℎ = 120597119891ℎ120597120590ℎ

ℎ + 120597119891ℎ120597h h = 0

(12)



= 119864 (d h) 120576119890 + (d h) 120576119890= 119864 (d h) ( 120576 minus 120576119901)

+ (1198641015840d (d h) d + 1198641015840h (d h) h) (120576 minus 120576119901) (13)





119891119901 (120590 d h) = 119902 minus 120572119901119875119888 [119873(119862119904 + 119901119875119888)]12

= 0 (14)



119861 + 120574] (15)



F

F




119861 + 120574 minus 120575) (16)






119860 = 119860ud + 119860119889119860119889 = 119860uh + 119860ℎ

(17)




119860119890 = 119860ud + 119860ℎ (18)


119889 = 119860119889119860

ℎ = 119860ℎ119860ud

(19)


119860119890 = (1 minus 119889) (1 + ℎ)119860 (20)


1205901015840 = 120590(1 minus 119889) (1 + ℎ) (21)


d = 3sum119894=1

119889119894n119894 otimes n119894

h = 3sum119894=1

ℎ119894n119894 otimes n119894(22)



119864 (d h) = 120573 119864 120573119879 (23)



119891119889 (120576 d) = (120576+ 120576+)12 minus (1199030 + 1199031trd) le 0 (24)


120576+ = 3sum119896=1

120576119896119867(120576119896)V119896 otimes V119896 (25)





120576minus = 120576 minus 120576+ (27)


d = 119889 120597119891119889120597120590119889

h = ℎ 120597119891ℎ120597120590ℎ (28)

5 Model Validation





(29)


= 119864 (d) ( 120576 minus 120576119901) + 1198641015840 (d) d (120576 minus 120576119901) (30)



1205761 () 1205763 ()0

5

10

15

20

25

30 1205901 minus 1205903 (MPa)




1205901 minus 1205903 (MPa)

1205761 () 1205763 ()05

101520253035404550





6 Conclusion



0

10

20

30

40

50

60

70


1205901 minus 1205903 (MPa)








Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119860119890 = 119860ud + 119860ℎ (18)


119889 = 119860119889119860

ℎ = 119860ℎ119860ud

(19)


119860119890 = (1 minus 119889) (1 + ℎ)119860 (20)


1205901015840 = 120590(1 minus 119889) (1 + ℎ) (21)


d = 3sum119894=1

119889119894n119894 otimes n119894

h = 3sum119894=1

ℎ119894n119894 otimes n119894(22)



119864 (d h) = 120573 119864 120573119879 (23)



119891119889 (120576 d) = (120576+ 120576+)12 minus (1199030 + 1199031trd) le 0 (24)


120576+ = 3sum119896=1

120576119896119867(120576119896)V119896 otimes V119896 (25)





120576minus = 120576 minus 120576+ (27)


d = 119889 120597119891119889120597120590119889

h = ℎ 120597119891ℎ120597120590ℎ (28)

5 Model Validation





(29)


= 119864 (d) ( 120576 minus 120576119901) + 1198641015840 (d) d (120576 minus 120576119901) (30)



1205761 () 1205763 ()0

5

10

15

20

25

30 1205901 minus 1205903 (MPa)




1205901 minus 1205903 (MPa)

1205761 () 1205763 ()05

101520253035404550





6 Conclusion



0

10

20

30

40

50

60

70


1205901 minus 1205903 (MPa)








Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







(29)


= 119864 (d) ( 120576 minus 120576119901) + 1198641015840 (d) d (120576 minus 120576119901) (30)



1205761 () 1205763 ()0

5

10

15

20

25

30 1205901 minus 1205903 (MPa)




1205901 minus 1205903 (MPa)

1205761 () 1205763 ()05

101520253035404550





6 Conclusion



0

10

20

30

40

50

60

70


1205901 minus 1205903 (MPa)








Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





0

10

20

30

40

50

60

70


1205901 minus 1205903 (MPa)








Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of













Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Study on the Elastoplastic Damage-Healing Coupled