research article a coupled plastic damage model for ...several plastic damage models of concrete...

Research ArticleA Coupled Plastic Damage Model for Concrete consideringthe Effect of Damage on Plastic Flow

Feng Zhou and Guangxu Cheng

School of Chemical Engineering and Technology Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Guangxu Cheng gxchengmailxjtueducn

Received 8 April 2014 Revised 5 August 2014 Accepted 11 August 2014

Academic Editor Xi Frank Xu

Copyright copy 2015 F Zhou and G Cheng This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A coupled plastic damagemodel with two damage scalars is proposed to describe the nonlinear features of concreteThe constitutiveformulations are developed by assuming that damage can be represented effectively in the material compliance tensor Damageevolution law and plastic damage coupling are described using the framework of irreversible thermodynamicsThe plasticity part isdeveloped without using the effective stress concept A plastic yield function based on the true stress is adopted with two hardeningfunctions one for tensile loading history and the other for compressive loading history To couple the damage to the plasticitythe damage parameters are introduced into the plastic yield function by considering a reduction of the plastic hardening rateThe specific reduction factor is then deduced from the compliance tensor of the damaged material Finally the proposed modelis applied to plain concrete Comparison between the experimental data and the numerical simulations shows that the proposedmodel is able to describe the main features of the mechanical performances observed in concrete material under uniaxial biaxialand cyclic loadings

1 Introduction

The mechanical behavior of concrete is unique due to theinfluence of micromechanisms involved in the nucleationand growth of microcracks and plastic flow This behavioris characterized by several features as follows the tensilestrength which is different from the compressive strengththe irreversible plastic deformation which is found as thepressure on the concrete exceeds a certain value beyondthe peak stress the stress-strain curve displays an unstableregion accompanied by stiffness degradation and strengthsofteningThese features have to be considered in constitutivemodels of concretematerials Tomodel these features severalmechanics theories have been used In general the damagemechanics theories can be used to model the nucleation andgrowth of microcracks [1ndash3] whereas the plasticity theoriescan be used to model the plastic flow component of thedeformation [4]

Because damage mechanics can be used to describe theprogressive weakening of solids due to the developmentof microcracks and because plasticity theories have been

successfully applied to the modeling of slip-type processesseveral plastic damage models of concrete have been devel-oped through combining damage mechanics and plastic-ity theories [5ndash13] These coupled plastic damage models(CPDMs) could be formulated in the irreversible thermo-dynamics framework and can be easily applied to describethe essential nonlinear performances of concrete includingthe strain softening and the stiffness degradation One typeof CPDMs is based on the concept of effective stress whichwas initially proposed by Kachanov [14] for metal creepfailure In these models the plastic yield function is definedin the effective configuration pertaining to the stresses inthe undamaged material [5 7ndash9] Many authors used thisapproach to couple damage to plasticity Some of thesemodels developed by the use of two damage scalars anddamage energy release rate-based damage criteria showexcellent performance in reproducing the typical nonlinearbehavior of concrete materials under different monotonicand cyclic load conditions The other type of CPDMs isopposite of the above In these models the true stress appearsin the plastic process which clearly couples plasticity to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 867979 13 pageshttpdxdoiorg1011552015867979

2 Mathematical Problems in Engineering

damage [6 10 11 15ndash17] However considering the influenceof damage on plastic flow it is difficult to develop a plasticyield function that can be used to describe the plasticdeformation and damage growth of concrete simultaneouslyTaqieddin et al [17] suggested that a damage-sensitive plasticyield function should be introduced in terms of true stressconfiguration Even if several types of expressions for theplastic yield function written in terms of the effective stresshave been successfully applied to model some of the typicalnonlinearities of concrete (such as the volumetric dilationand strength increase undermultidimensional compression)they cannot be directly used in the true stress space Acommon approach to solve this problem is to introduce areduction factor of plastic hardening rate into the plastic yieldfunction The plastic yield function is usually expressed by afunction of the stress tensor and plastic hardening functionso the damage parameters are included in the plastic yieldfunction with the introduction of the reduction factor inthe plastic hardening function Several authors applied thisapproach to develop the CPDM with a true stress spaceplastic yield function Shao et al [15] proposed a singlescalar damage framework based on the elastoplastic damagerelease rate The influence of damage on the plastic flow iscalculated by considering a reduction of the plastic hardeningrate The reduction factor is introduced into the plastic yieldfunction for the coupling between the damage evolutionand the plastic flow Nguyen and Houlsby [10] proposed adouble scalar damage-plasticity model for concrete based onthermodynamic principles In this model the plasticity partis based on the true stress using a yield function with twohardening functions one for the tensile loading history andthe other for the compressive loading history Two reductionfactors are introduced into the tensile and the compressivehardening functions to consider the influence of the tensileand compressive damage mechanisms on the plastic flowrespectively Taqieddin et al [17] proposed a double scalardamage-plasticitymodel for concrete based on the hypothesisof strain energy equivalenceThemodel incorporates a plasticyield function written in terms of the true stress Damagevariables are introduced all over the plastic yield function

For the models mentioned above the reduction factorof the plastic hardening rate is equal to the damage variabledefined in these models and there is no distinction inthe aspect of choosing a reduction factor between uniaxialand biaxial loadings The plastic hardening rate is generallydefined by the equivalent plastic strain so the sum of theprincipal damage values (in three directions 119909 119910 and 119911)seems reasonable to use as the reduction factor of the plastichardening rate undermultiaxial stresses By introducing sucha reduction factor the widely used plastic yield functionssuch as those applied by Lee and Fenves [7] Wu et al [9]and Lubliner et al [18] can be used in the true stress space

In the present work a coupled plastic damage model isformulated in the framework of thermodynamics The con-stitutive formulations are developed by considering an incre-ment in the concrete compliance due to microcrack propa-gation The plasticity part is based on the true stress using ayield function with tensile and compressive hardening func-tions The plasticity yield function widely used in effective

stress space is modified to be applied in this study byconsidering a reduction in the plastic hardening rate Finallythe proposedmodel is applied to plain concrete Comparisonbetween the experimental data and the numerical simulationsshows that the proposed model is able to represent the mainfeatures of mechanical performances observed in concretematerial under uniaxial biaxial and cyclic loadings

2 General Framework of CoupledPlastic Damage Model

Themodel presented in this work is thermodynamically con-sistent and is developed using internal variables to representthe material damage state The assumption of small strainswill be adopted in this work In the isothermal conditionsthe state variables are composed of the total strain tensor120576 scalar damage variables 119889+ and 119889

minus plastic strain 120576119901 andinternal variables for plastic hardening 120581+ and 120581minus The totalstrain tensor is decomposed into an elastic part 120576119890 and plasticpart 120576119901

120576 = 120576119890

+ 120576119901

(1)

To establish the constitutive law a thermodynamicpotential for the damaged elastoplastic material should beintroduced as a function of the internal state variablesConsidering that the damage process is coupled with plas-tic deformation and plastic hardening the thermodynamicpotential can be expressed by

120595 = 120595119890

(119889plusmn

120576119890

) + 120595119875

(119889plusmn

120581plusmn

)

120595119890

=

1

2

120576119890

S 120576119890(2)

where 120595119890 and 120595119875 are the elastic and plastic energy for plastichardening of damaged material respectively S is the fourth-order stiffness tensor of the damaged material

According to the second principle of thermodynam-ics any arbitrary irreversible process satisfies the Clausius-Duhem inequality as

120590 minus ge 0 (3)

Substituting the time derivative of the thermodynamicpotential into the inequality yields

(120590 minus

120597120595119890

120597120576119890) 119890

+ 120590 119901

minus

120597120595

120597119889+

119889+

minus

120597120595

120597119889minus

119889minus

minus

120597120595119901

120597120581++

minus

120597120595119901

120597120581minusminus

ge 0

(4)

Because the inequality must hold for any value of 119890 119901

119889+ 119889minus + and minus the constitutive equality can be obtained

as follows

120590 =

120597120595119890

120597120576119890= S 120576119890 = S (120576 minus 120576119901) (5)

Mathematical Problems in Engineering 3

The constitutive relation between the stress and straintensors can also be expressed utilizing the materialrsquos compli-ance tensor as

120576 minus 120576119901

= C 120590 S = Cminus1 (6)

where C is the fourth-order compliance tensor of the dam-aged material and (sdot)

minus1 is the matrix inverse function It isassumed in (6) that the damage can be represented effectivelyin the material compliance tensor This assumption is in linewith many published papers Budiansky and Orsquoconnell [19]and Horii and Nemat-Nasser [20] indicated that damageinduces degradation of the elastic properties by affecting thecompliance tensor of the material Based on this consider-ation several researchers proposed an elastic degradationmodel in which the material stiffness or compliance wasadopted as the damage variable The elastic degradationtheory was then extended to the multisurface elastic-damagemodel [21] and to the combined plastic damage model [6 1516 22]

Considering that an added compliance tensor is inducedby the microcrack propagation the fourth-order compliancetensor C is decomposed as follows [23]

C = C0 + C119889 (7)

where C0 defines the compliance tensor of the undamagedmaterial and C119889 defines the total added compliance tensordue to microcracks Microcracks in concrete are inducedin two modes the splitting mode and compressive mode[6 23] In general the splitting mode is dominant in tensionwhereas the compressive mode is dominant in compressionTo incorporate these modes into the formulation the stresstensor is decomposed into a positive part120590+ andnegative part120590minus

120590 = 120590+

+ 120590minus

(8)

where the eigenvalues of 120590+ are nonnegative and the eigen-values of 120590minus are all nonpositive According to Faria et al [8]Wu et al [9] and Pela et al [24] each part of the stress tensor120590 can be expressed as

120590+

= P+ 120590 120590minus

= 120590 minus 120590+

= Pminus 120590 (9)

where the fourth-order projection tensors P+ and Pminus areexpressed as

P+ = sum

119894

119867(119894) (p119894otimes p119894otimes p119894otimes p119894)

Pminus = I minus P+(10)

in which I is the fourth-order identity tensor 119894and p119894are the

119894th eigenvalue and the normalized eigenvector of the stresstensor 120590 respectively and 119867(sdot) is the Heaviside function(119867(119909) = 0 for 119909 lt 0 and119867(119909) = 1 for 119909 ge 0)

By introducing the positivenegative projection operatorsPplusmn the total added compliance tensor C119889 is decomposed as[25]

C119889 = P+ C+119889 P+ + Pminus Cminus

119889 Pminus (11)

where C+119889and Cminus

119889represent the added compliance tensors

due to microcracks induced in tension and compressionrespectively

To progress further evolutionary relations are requiredfor the added compliance tensors C+

119889and Cminus

119889 Based on the

previous work of Yazdani and Karnawat [6] Ortiz [23] andWu and Xu [25] the added compliance tensors are expressedin terms of response tensorsM+ andMminus such that

C+119889=

119889+M+ Cminus

119889=

119889minusMminus (12)

where the response tensors M+ and Mminus determine theevolution directions of the added compliance tensors C+

119889and

Cminus119889 respectivelyThe thermodynamic forces conjugated to the correspond-

ing damage variables are given by

119884+

119889= minus

120597120595

120597119889+= minus

1

2

(120576 minus 120576119901

)

120597S120597119889+ (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889= minus

120597120595

120597119889minus= minus

1

2

(120576 minus 120576119901

)

120597S120597119889minus (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(13)

where the terms 120597S120597119889+ and 120597S120597119889minus are the derivatives of thestiffness tensor of the damaged material with respect to thedamage variables 119889+ and 119889minus respectively These terms can becalculated utilizing the materialrsquos compliance tensor as

120597S120597119889+= minusS 120597C

120597119889+ S

120597S120597119889minus= minusS 120597C

120597119889minus S

(14)

Substituting (14) into (13) and calling for (5) one obtains

119884+

119889=

1

2

120590

120597C120597119889+ 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889=

1

2

120590

120597C120597119889minus 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(15)

Finally the rate form of the constitutive equation can begiven as

= C +

119889+ 120597C120597119889+ 120590 +

119889minus 120597C120597119889minus 120590 +

119901

(16)

The derivation of the detailed rate equation from (16)requires determining the evolution laws of the damagevariables and plastic strains

21 Damage Characterization The damage criteria aredefined as [15]

119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

le 0

119892minus

119889(119884minus

119889 119889minus

) = 119884minus

119889minus 119903minus

le 0

(17)


where 119903+119903minus (tensioncompression) represents the thresholdof the damage energy release rate at a given value of damageOn the basis of the normality structure in the continuummechanics the evolution law for the damage variables can bederived by

119889+

=120582+

119889

120597119892+

119889

120597119884+

119889

=120582+

119889

119889minus

=120582minus

119889

120597119892minus

119889

120597119884minus

119889

=120582minus

119889(18)

in which 120582+

119889ge 0 and

120582minus

119889ge 0 denote the tensile and com-

pressive damage multipliers respectively Under the loading-unloading condition

120582+

119889and

120582minus

119889are expressed according to

the Kuhn-Tucker relations

119892+

119889(119884+

119889 119889+

) le 0120582+

119889ge 0

120582+

119889119892+

119889= 0

119892minus

119889(119884minus

119889 119889minus

) le 0120582minus

119889ge 0

120582minus

119889119892minus

119889= 0

(19)

In the special case of elastic damage loading withoutplastic flow ( 120582119901 = 0) the damagemultipliers

120582+

119889and

120582minus

119889can be

determined by calling for the damage consistency conditionunder static loading

120597119884+

119889

120597120590

Cminus1

+120582+

119889(

120597Cminus1

120597119889+ 120576119890

minus

120597119903+

120597119889+) +

120582minus

119889(

120597Cminus1

120597119889minus 120576119890

) = 0

120597119884minus

119889

120597120590

Cminus1

+120582minus

119889(

120597Cminus1

120597119889minus 120576119890

minus

120597119903minus

120597119889minus) +

120582+

119889(

120597Cminus1

120597119889+ 120576119890

) = 0

(20)

22 Plastic Characterization Irreversible plastic deformationwill be formed during the deformation process of concreteThe plastic strain rate can be determined using a plasticyield function with multiple isotropic hardening criteria anda plasticity flow rule The yield function determines underwhat conditions the concrete begins to yield and how theyielding of the material evolves as the irreversible deforma-tion accumulates [26] According to Shao et al [15] and Salariet al [27] the plastic yield criterion 119891

119901can be expressed by a

function of the stress tensor 120590 and the thermodynamic forces119888+(119889+ 120581+) and 119888

minus(119889minus 120581minus) respectively conjugated with the

internal hardening variables 120581+ and 120581minus

119891119901(120590 119888plusmn

) le 0 (21)

The plastic hardening functions 119888+ and 119888minus can be deducedby the derivative of the thermodynamic potential

119888+

(119889+

120581+

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581+

119888minus

(119889minus

120581minus

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581minus

(22)

The plastic potential 119865119901(120590 119889plusmn 120581plusmn) is also a function of thestress tensor the scalar damage variables and the internal

hardening variables Note that 119865119901 is the yield function inthe associated flow rule At the given function of the plasticpotential the evolution law of the plastic strain is expressedas follows

119901

=120582119901

120597119865119901(120590 119889plusmn 120581plusmn)

120597120590

(23)

119891119901(120590 119888plusmn

) = 0120582119901ge 0 119891

119901(120590 119889plusmn

120581plusmn

)120582119901= 0 (24)

In the special case of plastic loading without damageevolution ( 120582+

119889= 0 and

120582minus

119889= 0) the plastic multiplier

120582119901can

be determined from the plastic consistency condition120597119891119901

120597120590

Cminus1

+120582119901

[

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119891119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119891119901

120597120590

) minus

120597119891119901

120597120590

Cminus1 120597119891119901

120597120590

] = 0

(25)

23 Computational Procedure The numerical algorithm ofthe proposed constitutive model is implemented in a finiteelement code The elastic predictor-damage and plasticcorrector integrating algorithm proposed by Crisfield [28]and then used by Nguyen and Houlsby [10] is used herefor calculating the coupled plastic damage model In thisalgorithm the damage and plastic corrector is along thenormal at the elastic trial point which avoids consideringthe intersection between the predicting increments of elasticstress and the damage surfaces Furthermore this algorithmensures that the plastic and damage consistent conditions arefulfilled at any stage of the loading process For details on thenumerical scheme and the associated algorithmic steps referto the published reports [10 28]

The following notational convention is used in thissection (1) 119909trial

119899represents 119909 at the trial point in time step 119899

(2) For simplicity the increment is defined as Δ119909 = 119909119899+1

minus119909119899

The elastic trial stress is defined as follows

120590trial119899+1

= 120590119899+ S119899 Δ120576119899+1

(26)

By solving (16) (17) and (21) in terms of the trial stressthe increments of the equivalent plastic strains Δ120581+ and Δ120581minusplastic strain Δ120576119901 and damage variables Δ119889+ and Δ119889minus can beobtained

Δ120576 = C119899 Δ120590 + Δ119889

+ 120597C120597119889+ 120590 + Δ119889

minus 120597C120597119889minus 120590 + Δ120576

119901

= 0

(27a)

(119891119901)119899+1

= 119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119891119901

120597120581+)

trial

119899+1

Δ120581+

+ (

120597119891119901

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27b)


(119892+

119889)119899+1

= (119892+

119889)

trial119899+1

+ (

120597119892+

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892+

119889

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119892+

119889

120597120581+)

trial

119899+1

Δ120581+

= 0

(27c)

(119892minus

119889)119899+1

= (119892minus

119889)

trial119899+1

+ (

120597119892minus

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892minus

119889

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119892minus

119889

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27d)

The expression for the increments of the plastic strainΔ120576119901can be obtained by substituting (27b) into (23)

Δ120576119901

= minus ((119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+(

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times (

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

(28)

Substituting (28) into (27a) yields the relationshipbetween the stress and the damage variables written inincremental form as follows

Δ120590 = ((119891trial119899+1

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

minus Δ119889+ 120597C120597119889+ 120590

minusΔ119889minus 120597C120597119889minus 120590 [Cst

]

minus1

(29)

where the fourth-order tensor Cst is the constitutive matrixwhich is the tangent with respect to the plasticity and thesecant with respect to damage

Cst= C119899minus ((

120597119891119901

120597120590

)

trial

119899+1

otimes

120597119865119901

120597120590

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

(30)

Substituting (29) into (27c) and (27d) yields the incre-ments of the damage variables Δ119889+ and Δ119889minus Back substitut-ing the increments of the damage variables into (29) resultsin the stress increment Δ120590

3 Determination of SpecificFunctions For Concrete

Specific functions for concrete are proposed based on thegeneral framework of the coupled plastic damage modelgiven in the previous section

Referring to the definition of the isotropic compliancetensor the evolution directions of added compliance tensorscan be defined as [25]

M+ = 1

1198640

[(1 + V0)

120590+otimes120590+

120590+ 120590+minus ]0

120590+otimes 120590+

120590+ 120590+]

Mminus = 1

1198640

[(1 + V0)

120590minusotimes120590minus

120590minus 120590minusminus ]0

120590minusotimes 120590minus

120590minus 120590minus]

(31)

where 1198640is the initial Youngrsquos modulus and V

0is the initial

Poissonrsquos ratioThe symmetrized outer product ldquootimesrdquo is definedas (120590plusmn otimes120590plusmn)

119894119895119896119897= (12)(120590

plusmn

119894119897120590plusmn

119895119896+ 120590plusmn

119894119896120590plusmn

119895119897)

Generally the parameters 119903+ and 119903minus are the functions ofthe damage scalars respectively Their mathematical expres-sions can be logarithmic power or exponential functionsThe specific expressions of 119903+ and 119903minus are given as

119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

0(1 + 119861

+

119889+

)

minus119860+

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


0[1 + 119861

minus ln (1 + 119889minus)]

times (1 + 119889minus

)

minus119860minus

(32)

where 119903+0and 119903minus0are respectively the initial damage energy

release threshold under tension and compression 119860+ and119861+ are the parameters controlling the damage evolution rate

under tension and 119860minus and 119861minus are the parameters controllingthe damage evolution rate under compression

The plastic yield function adopted in this work was firstintroduced in the Barcelona model by Lubliner et al [18] andwas later modified by Lee and Fenves [7] Wu et al [9] andVoyiadjis et al [11] It has proven to be excellent in modelingthe biaxial strength of concrete However the yield functionis usually used in effective stress To apply this function inthe CPDM written in terms of true stress Taqieddin et al[17]modified the function by introducing damage parametersand achieved better results Taqieddin et al showed that usingthe samemodel andmaterial parameters fully coupled plasticdamage yield criterion showed a more pronounced effect ofdamage on the mechanical behavior of the material thandid the weakly coupled plastic damage model yield criterionConsidering that damage induces a reduction in the plastichardening rate the damage parameters are then introducedinto the yield function by the plastic hardening functions


of damaged material [119888+(119889+ 120581+) and 119888minus(119889minus 120581minus)] The yield

function used in the present work is expressed as follows

119891119901= 1205721198681+ radic3119869

2+ 120573119867 (max) max minus (1 minus 120572) 119888

minus

(33)

where 1198681= 12059011+ 12059022+ 12059033

is the first invariant of the stresstensor 120590 119869

2= 119904 1199042 is the second invariant of the nominal

deviatoric stress 119904119894119895= 120590119894119895minus 1205901198961198961205751198941198953

The parameter 120572 is a dimensionless constant given byLubliner et al [18] as follows

120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus

0are the initial equibiaxial and uniaxial

compressive yield stresses respectively Based on the exper-imental results the ratio 119891

minus

1198870119891minus

0lies between 110 and 120

then as derived by Wu et al [9] the parameter 120572 is between008 and 014 In the present work 120572 is chosen as 012 Theparameter 120573 is a parameter expressed from the tensile andcompressive plastic hardening functions

120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)

Because the associative flow rule is adopted in the presentmodel the plastic yield function 119891

119901is also used as the plastic

potential 119865119875 to obtain the plastic strainAccording to (22) a specific expression of the plastic

energy 120595119875 should be given to obtain the plastic hardeningfunctions Considering as previously a reduction in theplastic hardening rate due to damage the plastic energy isexpressed as follows

120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+

) + [1 minus 120601minus

119867] 120595119875minus

0(120581minus

) (36)

where the reduction factors 120601plusmn119867are functions of the damage

scalars 119889plusmn and 120595

119875

0is the plastic hardening energy for

undamaged material which is defined by

120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)

in which 119891+0and 119891minus

0are the initial uniaxial tensile and com-

pressive yield stresses respectively ℎ+ and ℎminus are the tensileand compressive plastic hardening moduli respectively Thehardening parameters 120581+ and 120581

minus are the equivalent plasticstrains under tension and compression respectively definedas [1 17]

120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)

where + and minus are the tensile and compressive equivalentplastic strain rates respectively

According to Wu et al [9] the evolution laws for + maybe postulated as follows

+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901

min are the maximum and minimum eigen-values of the plastic strain rate tensor 119901 respectively maxand min are the maximum and minimum eigenvalues of thestress tensor 120590 119877 is a weight factor expressed as

119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)

in which the symbol ⟨sdot⟩ is the Macaulay bracket defined as⟨119909⟩ = (119909 + |119909|)2 Note that 119877 is equal to one if all of theeigenstresses

119894are positive and it is equal to zero if all of the

eigenstresses 119894are negative

The plastic tensile and compressive hardening functionsof the damaged material are then specified as

119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)

It can be observed from (33) and (41) that the damagevariables are introduced into the plastic yield function

The last expressions to be specified are the reduction fac-tors 120601plusmn

119867 Defined as the sum of the principal damage values in

three directions (119909 119910 and 119911) the reduction factor is obtainedfrom the definition of a particular thermodynamic energyfunction in the present paper Based on the previous works ofLemaitre andDesmorat [29] the following particular form ofthe elastic thermodynamic energy function is used

120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)

where120590119867is the hydrostatic stress and119863

119867= 1198631+1198632+1198633 and

11986311198632 and119863

3are the damage values in119909119910 and 119911directions

respectively The effective damage tensorH is given by

H =[

[

[

(1 minus 1198631)minus12

0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)

Differentiation of the elastic thermodynamic energyyields the strain-stress relations

120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)

where (sdot)119863 is the component of the deviatoric tensor


Substituting (44) into (6) the relation between thesecond-order damage tensor and the compliance tensor of thedamaged material can be given by

119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)

By solving (45) the reduction factor can be obtainedFor example considering that concrete is subjected to biaxialtension loading (120590

21205901= 120574 and 120590

3= 0) the reduction factor

120601+

119867can be obtained

120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)

Similarly 120601minus119867 corresponding to compression loading can

be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)

In the special case of uniaxial loadings or combinedtension-compression loading (120574 = 0) the reduction factorscan be expressed as

120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus+ (1 minus 2V

0)

(48)

4 Numerical Examples andDiscussion of Results

In this section the procedure for the determination of themodel parameters is first presented Then several numericalexamples are provided to investigate the capability of theproposed model in capturing material behavior in bothtension and compression under uniaxial and biaxial loadingsThe examples are taken from [7] with corresponding exper-imental data provided by Kupfer et al [30] Karson and Jirsa[31] Gopalaratnam and Shah [32] and Taylor [33] To complywith the results of the studies [7 9] these numerical exampleswere analyzed using the single quadrilateral finite elementshown in Figure 1

41 Identification of Model Parameters The proposed modelcontains 12 parameters four elastic constants for the undam-aged material (119864

0 V0 119891+0 and 119891

minus

0) two parameters for

plasticity characterization (ℎ+ and ℎminus) and six parameters

for damage characterization (119903+0 119903minus0 119860plusmn and 119861plusmn) All of the

parameters can be identified from the uniaxial tension andcompression tests

The initial elastic constants (1198640 V0 119891+0 and 119891

minus

0) are

determined from the linear part of the typical stress-straincurve in Figure 2

As mentioned before the damage evolution is coupledwith the plastic flow According to Shao it is reasonable toassume that the damage initiation occurs at the same time as

the plastic deformation They are identified as the end of thelinear part of the stress-strain curves (point 119860 in Figure 2)In this case considering 119889plusmn = 0 and 120581plusmn = 0 in the damagecriterion and the plastic yield function the initial damagethresholds 119903+

0and 119903minus0can be determined as follows

119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)

where the stress ratio is expressed as 120574 = 12059021205901

The two parameters (119860+ and 119861+) characterizing the

failure surface and the plastic hardening parameter (ℎ+) canbe determined by fitting both curves of stress-strain andstress-plastic strain obtained in the uniaxial tension testSimilarly the parameters (119860minus 119861minus and ℎ

minus) are determinedby fitting both curves of stress-strain and stress-plastic strainobtained in the uniaxial compression test

The effects of the model parameters (119860+ 119861+ and ℎ+) onthe stress-strain response the stress-plastic strain responseand the damage evolution under uniaxial tension are shownin Figure 3 Each model parameter in turn is varied whilethe others are kept fixed to show the corresponding effecton the behavior of the concrete material A good agreementexists between the experimental data [33] and the numericalsimulations obtained with the associatedmaterial parametersgiven in Table 1 It can also be observed in Figure 3 thatnot only is the stress-strain curve governed by all of theparameters (119860+ 119861+ and ℎ+) of the model but also the plasticstrain and damage evolution are controlled by all of theparametersThis result is in agreement with that described bythe proposed damage energy release rate 119884+

119889and the plastic

yield criterion 119891119875in which both the damage scalar 119889+ and

the plastic hardening rate 120581+ are introducedThese responsesobserved in Figure 3 show the coupled effect of damage andplasticity on the predicted response

To illustrate the effects of the model parameters (119860minus119861minus and ℎ

minus) on the behavior of the proposed model thestress-strain curve the stress-plastic strain curve and thedamage evolution under uniaxial compression are plottedin Figure 4 in which the results obtained from varying eachparameter while keeping the others fixed are also presentedThe model parameters obtained from the experimental data[31] are listed in Table 1 It can be observed in Figure 4 thatthe numerical predictions including not only the hardeningand softening regimes of the stress-strain curve but also thestress-plastic strain curve are in good agreement with theexperimental data [31] It can also be observed that all ofthe responses of the model including the stress-strain curvethe stress-plastic strain curve and the damage evolutionare controlled by all of the parameters (119860minus 119861minus and ℎ

minus)


826

mm

826mm

(a) (b)

(c) (d)

Figure 1 Quadrilateral finite elements under (a) uniaxial tension (b) biaxial tension (c) uniaxial compression and (d) biaxial compression

1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)

Figure 2 Schematization of typical stress-strain curves of concrete in uniaxial (a) tension and (b) compression tests


00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H

Experimental Experimentalh+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)

Figure 3 Effect of the parameters on the model response in tension (a) effect of 119860+ (b) effect of 119861+ and (c) effect of ℎ+

Table 1 Model parameters for concrete material under uniaxial loadings

Elastic parameters Plastic parameters Damage parameters

Uniaxial tension1198640= 31700MPa

119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16

Uniaxial compression1198640= 31000MPa119891minus

0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10

minus3MPa119860minus

= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain

ExperimentalAminus = 190

Aminus = 198

Aminus = 206


Aminus = 198

Aminus = 206 Aminus = 190

Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain

ExperimentalBminus = 28

Bminus = 32

Bminus = 35


Bminus = 32

Bminus = 35 Bminus = 28

Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain

Experimental Experimental

Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa

hminus = 50000MPahminus = 60000MPa hminus = 40000MPa


hminus = 50000MPahminus = 60000MPa

(c)

Figure 4 Effect of the parameters on the model response in compression (a) effect of 119860minus (b) effect of 119861minus and (c) effect of ℎminus

This finding corresponds to the expressions of the proposeddamage energy release rate 119884minus

119889and the plastic yield criterion

119891119875in which both the damage scalar 119889

minus and the plastichardening rate 120581minus are introduced These responses observedin Figure 4 again indicate the coupled effect of damage andplasticity on the predicted behavior

42 Biaxial Loading In this section the behavior of theproposed model in combined loadings (biaxial compressionand biaxial tension-compression) is investigated The experi-mental data of Kupfer et al [30] are used to compare with the

numerical predictions of the model The model parametersused in this series of tests are determined from the uniaxialtest (120590

21205901= 0minus1)Then these parameters are used tomodel

the behavior of the concrete under different stress ratios(12059021205901= minus052minus1 120590

21205901= minus1minus1 and 120590

21205901= 0052minus1)

The model parameters obtained for the concrete are listedin Table 2 Note that after being given the parameters 119891plusmn

0

and 120574 (120574 = 12059021205901) the initial damage threshold 119903

plusmn

0can

be calculated by (49) In addition as the experimental dataare available only in the prepeak regime under the uniaxialtension test some of the model parameters can be assumed


0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40

ExperimentalNumerical

0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)

Figure 5 Comparison of the model predictions with the experimental results under biaxial compression (a) 12059021205901= 0 minus 1 (b) 120590

21205901=

minus052 minus 1 (c) 12059021205901= minus1 minus 1 and (d) 120590

21205901= 0052 minus 1

to yield good fit in biaxial tension-compression such asℎ+= 31000MPa 119860+ = 06 and 119861

+= 14 By using these

parameters simulations of biaxial compression-compressionand compression-tension tests with different stress ratioshave been performed As Figure 5 shows the numericalresults obtained from the model are observed to be in goodagreement with the experimental data

43 Cyclic Uniaxial Loading The cyclic uniaxial tensile testof Taylor [33] and the cyclic compressive test of Karson andJirsa [31] are also compared with the numerical results Theproperties andmodel parameters forTaylorsrsquos andKarsan andJirsarsquos simulation are listed in Table 1 As shown in Figure 6the strength softening and stiffness degrading as well asthe irreversible strains upon unloading can be clearly seenunder both cyclic uniaxial tension and compression Thisdemonstrates the coupling between damage and plasticity as

Table 2 Model parameters for concrete material under biaxialloadings

Elastic parameters Plastic parameters Damageparameters

1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20

well as the capability of themodel in reproducingmechanicalfeatures of concrete

5 Conclusions

A coupled plastic damage model written in terms of thetrue stress has been proposed in this paper to describe


00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)

Figure 6 Comparison of the model predictions with the experimental results in (a) cyclic uniaxial tension and (b) cyclic uniaxialcompression

the nonlinear features of concrete in uniaxial and biaxialloadings Based on the theoretical development of the modelformulation several conclusions have been summarized asfollows

The constitutive formulations are developed by consid-ering an added flexibility due to microcrack growth It isassumed that damage can be represented effectively in thematerial compliance tensor The framework of irreversiblethermodynamics is adopted to describe the damage evolutionand plasticity damage coupling

The plasticity part is based on the true stress using a yieldfunction with two hardening variables one for the tensileloading history and the other for the compressive loadinghistory To couple the damage to the plasticity the damageparameters are introduced into the plastic yield function byconsidering a reduction in the plastic hardening rate Thespecific reduction factor is defined as the sum of the principalvalues of a second-order damage tensor which is deducedfrom the compliance tensor of the damaged material

Themodel contains 12 parameters which can be obtainedby fitting both curves of the stress-strain and the stress-plasticstrain in the uniaxial tension and compression tests Thenumerical results suggest that the proposed model is able todescribe the main features of the mechanical behavior forconcrete under uniaxial biaxial and cyclic loadings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural Foundationof China (Grant no 51175403)

References

[1] G Z Voyiadjis and Z N Taqieddin ldquoElastic plastic and damagemodel for concrete materials part Imdashtheoretical formulationrdquoThe International Journal of Structural Changes in Solids vol 1no 1 pp 31ndash59 2009

[2] J Mazars and G Pijaudier-Cabot ldquoContinuum damage theoryapplication to concreterdquo Journal of Engineering Mechanics vol115 no 2 pp 345ndash365 1989

[3] V A Lubarda D Krajcinovic and S Mastilovic ldquoDamagemodel for brittle elastic solids with unequal tensile and com-pressive strengthsrdquo Engineering Fracture Mechanics vol 49 no5 pp 681ndash697 1994

[4] P H Feenstra and R De Borst ldquoA composite plasticity modelfor concreterdquo International Journal of Solids and Structures vol33 no 5 pp 707ndash730 1996

[5] J W Ju ldquoOn energy-based coupled elastoplastic damagetheories constitutive modeling and computational aspectsrdquoInternational Journal of Solids and Structures vol 25 no 7 pp803ndash833 1989

[6] S Yazdani and S Karnawat ldquoA constitutive theory for brittlesolids with application to concreterdquo International Journal ofDamage Mechanics vol 5 no 1 pp 93ndash110 1996

[7] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[8] R Faria J Oliver and M Cervera ldquoA strain-based plasticviscous-damage model for massive concrete structuresrdquo Inter-national Journal of Solids and Structures vol 35 no 14 pp 1533ndash1558 1998

[9] J YWu J Li andR Faria ldquoAn energy release rate-based plastic-damage model for concreterdquo International Journal of Solids andStructures vol 43 no 3-4 pp 583ndash612 2006

[10] G D Nguyen and G T Houlsby ldquoA coupled damage-plasticitymodel for concrete based on thermodynamic principles part Imodel formulation and parameter identificationrdquo International


Journal for Numerical and Analytical Methods in Geomechanicsvol 32 no 4 pp 353ndash389 2008

[11] G Z Voyiadjis Z N Taqieddin and P I Kattan ldquoTheoreticalformulation of a coupled elastic-plastic anisotropic damagemodel for concrete using the strain energy equivalence con-ceptrdquo International Journal of Damage Mechanics vol 18 no 7pp 603ndash638 2009

[12] D Chen C Du X G Feng and F Ouyang ldquoAn elastoplasticdamage constitutive model for cementitious materials underwet-dry cyclic sulfate attackrdquo Mathematical Problems in Engi-neering vol 2013 Article ID 562410 7 pages 2013

[13] F Shen Q Zhang and D Huang ldquoDamage and failure processof concrete structure under uniaxial compression based onperi-dynamics modelingrdquo Mathematical Problems in Engineeringvol 2013 Article ID 631074 5 pages 2013

[14] L M Kachanov ldquoTime of the rupture process under creepconditionsrdquo Isv Akad NaukSSROtd Tekh Nauk vol 8 pp 26ndash311958

[15] J F Shao Y Jia D Kondo andA S Chiarelli ldquoA coupled elasto-plastic damage model for semi-brittle materials and extensionto unsaturated conditionsrdquo Mechanics of Materials vol 38 no3 pp 218ndash232 2006

[16] L Chen J F Shao and H W Huang ldquoCoupled elastoplas-tic damage modeling of anisotropic rocksrdquo Computers andGeotechnics vol 37 no 1-2 pp 187ndash194 2010

[17] Z N Taqieddin G Z Voyiadjis and A H Almasri ldquoFormula-tion and verification of a concrete model with strong couplingbetween isotropic damage and elastoplasticity and comparisonto a weak coupling modelrdquo Journal of Engineering Mechanicsvol 138 no 5 pp 530ndash541 2012

[18] J Lubliner J Oliver S Oller and E Onate ldquoA plastic-damagemodel for concreterdquo International Journal of Solids and Struc-tures vol 25 no 3 pp 299ndash326 1989

[19] B Budiansky and R J Orsquoconnell ldquoElastic moduli of a crackedsolidrdquo International Journal of Solids and Structures vol 12 no2 pp 81ndash97 1976

[20] H Horii and S Nemat-Nasser ldquoOverall moduli of solids withmicrocracks load-induced anisotropyrdquo Journal of theMechanicsand Physics of Solids vol 31 no 2 pp 155ndash171 1983

[21] J F Shao and J W Rudnicki ldquoMicrocrack-based continuousdamage model for brittle geomaterialsrdquoMechanics of Materialsvol 32 no 10 pp 607ndash619 2000

[22] G Meschke R Lackner and H A Mang ldquoAn anisotropicelastoplastic-damage model for plain concreterdquo InternationalJournal for Numerical Methods in Engineering vol 42 no 4 pp703ndash727 1998

[23] M Ortiz ldquoA constitutive theory for the inelastic behavior ofconcreterdquoMechanics of Materials vol 4 no 1 pp 67ndash93 1985

[24] L Pela M Cervera and P Roca ldquoAn orthotropic damagemodel for the analysis of masonry structuresrdquoConstruction andBuilding Materials vol 41 pp 957ndash967 2013

[25] J Y Wu and S L Xu ldquoReconsideration on the elastic dam-agedegradation theory for themodeling ofmicrocrack closure-reopening (MCR) effectsrdquo International Journal of Solids andStructures vol 50 no 5 pp 795ndash805 2013

[26] Y Zhang R Luo and R L Lytton ldquoCharacterization ofviscoplastic yielding of asphalt concreterdquo Construction andBuilding Materials vol 47 pp 671ndash679 2013

[27] M R Salari S Saeb K J Willam S J Patchet and R C Car-rasco ldquoA coupled elastoplastic damage model for geomaterialsrdquoComputer Methods in Applied Mechanics and Engineering vol193 no 27ndash29 pp 2625ndash2643 2004

[28] M A CrisfieldNon-Linear Finite Element Analysis of Solids andStructure Wiley New York NY USA 1997

[29] J Lemaitre and R Desmorat Engineering Damage MechanicsDuctile Creep Fatigue and Brittle Failures Springer BerlinGermany 2005

[30] H Kupfer H K Hilsdorf and H Rusch ldquoBehavior of concreteunder biaxial stressesrdquo ACI Journal Proceedings vol 66 no 8pp 656ndash666 1969

[31] I D Karson and J O Jirsa ldquoBehavior of concrete undercompressive loadingsrdquo Journal of the Structural Division vol 95no 12 pp 2535ndash2563 1969

[32] V S Gopalaratnam and S P Shah ldquoSoftening response of plainconcrete in direct tensionrdquo ACI Journal Proceedings vol 82 no3 pp 310ndash323 1985

[33] R L Taylor ldquoFEAP a finite element analysis program forengineering workstationrdquo Tech Rep UCBSEMM-92 (DraftVersion) 1992

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


damage [6 10 11 15ndash17] However considering the influenceof damage on plastic flow it is difficult to develop a plasticyield function that can be used to describe the plasticdeformation and damage growth of concrete simultaneouslyTaqieddin et al [17] suggested that a damage-sensitive plasticyield function should be introduced in terms of true stressconfiguration Even if several types of expressions for theplastic yield function written in terms of the effective stresshave been successfully applied to model some of the typicalnonlinearities of concrete (such as the volumetric dilationand strength increase undermultidimensional compression)they cannot be directly used in the true stress space Acommon approach to solve this problem is to introduce areduction factor of plastic hardening rate into the plastic yieldfunction The plastic yield function is usually expressed by afunction of the stress tensor and plastic hardening functionso the damage parameters are included in the plastic yieldfunction with the introduction of the reduction factor inthe plastic hardening function Several authors applied thisapproach to develop the CPDM with a true stress spaceplastic yield function Shao et al [15] proposed a singlescalar damage framework based on the elastoplastic damagerelease rate The influence of damage on the plastic flow iscalculated by considering a reduction of the plastic hardeningrate The reduction factor is introduced into the plastic yieldfunction for the coupling between the damage evolutionand the plastic flow Nguyen and Houlsby [10] proposed adouble scalar damage-plasticity model for concrete based onthermodynamic principles In this model the plasticity partis based on the true stress using a yield function with twohardening functions one for the tensile loading history andthe other for the compressive loading history Two reductionfactors are introduced into the tensile and the compressivehardening functions to consider the influence of the tensileand compressive damage mechanisms on the plastic flowrespectively Taqieddin et al [17] proposed a double scalardamage-plasticitymodel for concrete based on the hypothesisof strain energy equivalenceThemodel incorporates a plasticyield function written in terms of the true stress Damagevariables are introduced all over the plastic yield function

For the models mentioned above the reduction factorof the plastic hardening rate is equal to the damage variabledefined in these models and there is no distinction inthe aspect of choosing a reduction factor between uniaxialand biaxial loadings The plastic hardening rate is generallydefined by the equivalent plastic strain so the sum of theprincipal damage values (in three directions 119909 119910 and 119911)seems reasonable to use as the reduction factor of the plastichardening rate undermultiaxial stresses By introducing sucha reduction factor the widely used plastic yield functionssuch as those applied by Lee and Fenves [7] Wu et al [9]and Lubliner et al [18] can be used in the true stress space

In the present work a coupled plastic damage model isformulated in the framework of thermodynamics The con-stitutive formulations are developed by considering an incre-ment in the concrete compliance due to microcrack propa-gation The plasticity part is based on the true stress using ayield function with tensile and compressive hardening func-tions The plasticity yield function widely used in effective

stress space is modified to be applied in this study byconsidering a reduction in the plastic hardening rate Finallythe proposedmodel is applied to plain concrete Comparisonbetween the experimental data and the numerical simulationsshows that the proposed model is able to represent the mainfeatures of mechanical performances observed in concretematerial under uniaxial biaxial and cyclic loadings

2 General Framework of CoupledPlastic Damage Model

Themodel presented in this work is thermodynamically con-sistent and is developed using internal variables to representthe material damage state The assumption of small strainswill be adopted in this work In the isothermal conditionsthe state variables are composed of the total strain tensor120576 scalar damage variables 119889+ and 119889

minus plastic strain 120576119901 andinternal variables for plastic hardening 120581+ and 120581minus The totalstrain tensor is decomposed into an elastic part 120576119890 and plasticpart 120576119901

120576 = 120576119890

+ 120576119901

(1)

To establish the constitutive law a thermodynamicpotential for the damaged elastoplastic material should beintroduced as a function of the internal state variablesConsidering that the damage process is coupled with plas-tic deformation and plastic hardening the thermodynamicpotential can be expressed by

120595 = 120595119890

(119889plusmn

120576119890

) + 120595119875

(119889plusmn

120581plusmn

)

120595119890

=

1

2

120576119890

S 120576119890(2)

where 120595119890 and 120595119875 are the elastic and plastic energy for plastichardening of damaged material respectively S is the fourth-order stiffness tensor of the damaged material

According to the second principle of thermodynam-ics any arbitrary irreversible process satisfies the Clausius-Duhem inequality as

120590 minus ge 0 (3)

Substituting the time derivative of the thermodynamicpotential into the inequality yields

(120590 minus

120597120595119890

120597120576119890) 119890

+ 120590 119901

minus

120597120595

120597119889+

119889+

minus

120597120595

120597119889minus

119889minus

minus

120597120595119901

120597120581++

minus

120597120595119901

120597120581minusminus

ge 0

(4)

Because the inequality must hold for any value of 119890 119901

119889+ 119889minus + and minus the constitutive equality can be obtained

as follows

120590 =

120597120595119890

120597120576119890= S 120576119890 = S (120576 minus 120576119901) (5)



120576 minus 120576119901

= C 120590 S = Cminus1 (6)




C = C0 + C119889 (7)


120590 = 120590+

+ 120590minus

(8)


120590+

= P+ 120590 120590minus

= 120590 minus 120590+

= Pminus 120590 (9)


P+ = sum

119894







119889 Pminus (11)





119889and Cminus

119889 Based on the


C+119889=

119889+M+ Cminus

119889=



119889and



119884+

119889= minus

120597120595

120597119889+= minus

1

2

(120576 minus 120576119901

)

120597S120597119889+ (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889= minus

120597120595


1

2

(120576 minus 120576119901

)

120597S120597119889minus (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(13)


120597S120597119889+= minusS 120597C

120597119889+ S

120597S120597119889minus= minusS 120597C

120597119889minus S

(14)


119884+

119889=

1

2

120590

120597C120597119889+ 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889=

1

2

120590

120597C120597119889minus 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(15)


= C +

119889+ 120597C120597119889+ 120590 +

119889minus 120597C120597119889minus 120590 +

119901

(16)



119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

le 0

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


le 0

(17)



119889+

=120582+

119889

120597119892+

119889

120597119884+

119889

=120582+

119889

119889minus

=120582minus

119889

120597119892minus

119889

120597119884minus

119889

=120582minus

119889(18)

in which 120582+

119889ge 0 and

120582minus



120582+

119889and

120582minus



119892+

119889(119884+

119889 119889+

) le 0120582+

119889ge 0

120582+

119889119892+

119889= 0

119892minus

119889(119884minus

119889 119889minus

) le 0120582minus

119889ge 0

120582minus

119889119892minus

119889= 0

(19)


120582+

119889and

120582minus

119889can be


120597119884+

119889

120597120590

Cminus1

+120582+

119889(

120597Cminus1

120597119889+ 120576119890

minus

120597119903+

120597119889+) +

120582minus

119889(

120597Cminus1

120597119889minus 120576119890

) = 0

120597119884minus

119889

120597120590

Cminus1

+120582minus

119889(

120597Cminus1

120597119889minus 120576119890

minus

120597119903minus

120597119889minus) +

120582+

119889(

120597Cminus1

120597119889+ 120576119890

) = 0

(20)






119891119901(120590 119888plusmn

) le 0 (21)


119888+

(119889+

120581+

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581+

119888minus

(119889minus

120581minus

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581minus

(22)



119901

=120582119901

120597119865119901(120590 119889plusmn 120581plusmn)

120597120590

(23)

119891119901(120590 119888plusmn

) = 0120582119901ge 0 119891

119901(120590 119889plusmn

120581plusmn

)120582119901= 0 (24)


119889= 0 and

120582minus


120582119901can


120597120590

Cminus1

+120582119901

[

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119891119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119891119901

120597120590

) minus

120597119891119901

120597120590

Cminus1 120597119891119901

120597120590

] = 0

(25)





minus119909119899


120590trial119899+1

= 120590119899+ S119899 Δ120576119899+1

(26)


Δ120576 = C119899 Δ120590 + Δ119889

+ 120597C120597119889+ 120590 + Δ119889

minus 120597C120597119889minus 120590 + Δ120576

119901

= 0

(27a)

(119891119901)119899+1

= 119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119891119901

120597120581+)

trial

119899+1

Δ120581+

+ (

120597119891119901

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27b)


(119892+

119889)119899+1

= (119892+

119889)

trial119899+1

+ (

120597119892+

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892+

119889

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119892+

119889

120597120581+)

trial

119899+1

Δ120581+

= 0

(27c)

(119892minus

119889)119899+1

= (119892minus

119889)

trial119899+1

+ (

120597119892minus

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892minus

119889

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119892minus

119889

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27d)


Δ120576119901

= minus ((119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+(

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times (

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

(28)


Δ120590 = ((119891trial119899+1

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

minus Δ119889+ 120597C120597119889+ 120590


]

minus1

(29)



120597119891119901

120597120590

)

trial

119899+1

otimes

120597119865119901

120597120590

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

(30)





M+ = 1

1198640

[(1 + V0)

120590+otimes120590+

120590+ 120590+minus ]0

120590+otimes 120590+

120590+ 120590+]

Mminus = 1

1198640

[(1 + V0)





(31)


0is the initial


119894119895119896119897= (12)(120590

plusmn

119894119897120590plusmn

119895119896+ 120590plusmn

119894119896120590plusmn

119895119897)


119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

0(1 + 119861

+

119889+

)

minus119860+

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


0[1 + 119861



)

minus119860minus

(32)








119891119901= 1205721198681+ radic3119869


minus

(33)

where 1198681= 12059011+ 12059022+ 12059033





120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus



minus

1198870119891minus



120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)





120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+


119867] 120595119875minus

0(120581minus

) (36)



119875



120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)





120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)



+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901


119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)





119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)





120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)


119867= 1198631+1198632+1198633 and

11986311198632 and119863



H =[

[

[


0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)


120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)




119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120576 minus 120576119901

= C 120590 S = Cminus1 (6)




C = C0 + C119889 (7)


120590 = 120590+

+ 120590minus

(8)


120590+

= P+ 120590 120590minus

= 120590 minus 120590+

= Pminus 120590 (9)


P+ = sum

119894







119889 Pminus (11)





119889and Cminus

119889 Based on the


C+119889=

119889+M+ Cminus

119889=



119889and



119884+

119889= minus

120597120595

120597119889+= minus

1

2

(120576 minus 120576119901

)

120597S120597119889+ (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889= minus

120597120595


1

2

(120576 minus 120576119901

)

120597S120597119889minus (120576 minus 120576

119901

) minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(13)


120597S120597119889+= minusS 120597C

120597119889+ S

120597S120597119889minus= minusS 120597C

120597119889minus S

(14)


119884+

119889=

1

2

120590

120597C120597119889+ 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889+

119884minus

119889=

1

2

120590

120597C120597119889minus 120590 minus

120597120595119875(119889plusmn 120581plusmn)

120597119889minus

(15)


= C +

119889+ 120597C120597119889+ 120590 +

119889minus 120597C120597119889minus 120590 +

119901

(16)



119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

le 0

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


le 0

(17)



119889+

=120582+

119889

120597119892+

119889

120597119884+

119889

=120582+

119889

119889minus

=120582minus

119889

120597119892minus

119889

120597119884minus

119889

=120582minus

119889(18)

in which 120582+

119889ge 0 and

120582minus



120582+

119889and

120582minus



119892+

119889(119884+

119889 119889+

) le 0120582+

119889ge 0

120582+

119889119892+

119889= 0

119892minus

119889(119884minus

119889 119889minus

) le 0120582minus

119889ge 0

120582minus

119889119892minus

119889= 0

(19)


120582+

119889and

120582minus

119889can be


120597119884+

119889

120597120590

Cminus1

+120582+

119889(

120597Cminus1

120597119889+ 120576119890

minus

120597119903+

120597119889+) +

120582minus

119889(

120597Cminus1

120597119889minus 120576119890

) = 0

120597119884minus

119889

120597120590

Cminus1

+120582minus

119889(

120597Cminus1

120597119889minus 120576119890

minus

120597119903minus

120597119889minus) +

120582+

119889(

120597Cminus1

120597119889+ 120576119890

) = 0

(20)






119891119901(120590 119888plusmn

) le 0 (21)


119888+

(119889+

120581+

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581+

119888minus

(119889minus

120581minus

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581minus

(22)



119901

=120582119901

120597119865119901(120590 119889plusmn 120581plusmn)

120597120590

(23)

119891119901(120590 119888plusmn

) = 0120582119901ge 0 119891

119901(120590 119889plusmn

120581plusmn

)120582119901= 0 (24)


119889= 0 and

120582minus


120582119901can


120597120590

Cminus1

+120582119901

[

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119891119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119891119901

120597120590

) minus

120597119891119901

120597120590

Cminus1 120597119891119901

120597120590

] = 0

(25)





minus119909119899


120590trial119899+1

= 120590119899+ S119899 Δ120576119899+1

(26)


Δ120576 = C119899 Δ120590 + Δ119889

+ 120597C120597119889+ 120590 + Δ119889

minus 120597C120597119889minus 120590 + Δ120576

119901

= 0

(27a)

(119891119901)119899+1

= 119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119891119901

120597120581+)

trial

119899+1

Δ120581+

+ (

120597119891119901

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27b)


(119892+

119889)119899+1

= (119892+

119889)

trial119899+1

+ (

120597119892+

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892+

119889

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119892+

119889

120597120581+)

trial

119899+1

Δ120581+

= 0

(27c)

(119892minus

119889)119899+1

= (119892minus

119889)

trial119899+1

+ (

120597119892minus

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892minus

119889

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119892minus

119889

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27d)


Δ120576119901

= minus ((119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+(

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times (

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

(28)


Δ120590 = ((119891trial119899+1

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

minus Δ119889+ 120597C120597119889+ 120590


]

minus1

(29)



120597119891119901

120597120590

)

trial

119899+1

otimes

120597119865119901

120597120590

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

(30)





M+ = 1

1198640

[(1 + V0)

120590+otimes120590+

120590+ 120590+minus ]0

120590+otimes 120590+

120590+ 120590+]

Mminus = 1

1198640

[(1 + V0)





(31)


0is the initial


119894119895119896119897= (12)(120590

plusmn

119894119897120590plusmn

119895119896+ 120590plusmn

119894119896120590plusmn

119895119897)


119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

0(1 + 119861

+

119889+

)

minus119860+

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


0[1 + 119861



)

minus119860minus

(32)








119891119901= 1205721198681+ radic3119869


minus

(33)

where 1198681= 12059011+ 12059022+ 12059033





120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus



minus

1198870119891minus



120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)





120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+


119867] 120595119875minus

0(120581minus

) (36)



119875



120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)





120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)



+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901


119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)





119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)





120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)


119867= 1198631+1198632+1198633 and

11986311198632 and119863



H =[

[

[


0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)


120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)




119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119889+

=120582+

119889

120597119892+

119889

120597119884+

119889

=120582+

119889

119889minus

=120582minus

119889

120597119892minus

119889

120597119884minus

119889

=120582minus

119889(18)

in which 120582+

119889ge 0 and

120582minus



120582+

119889and

120582minus



119892+

119889(119884+

119889 119889+

) le 0120582+

119889ge 0

120582+

119889119892+

119889= 0

119892minus

119889(119884minus

119889 119889minus

) le 0120582minus

119889ge 0

120582minus

119889119892minus

119889= 0

(19)


120582+

119889and

120582minus

119889can be


120597119884+

119889

120597120590

Cminus1

+120582+

119889(

120597Cminus1

120597119889+ 120576119890

minus

120597119903+

120597119889+) +

120582minus

119889(

120597Cminus1

120597119889minus 120576119890

) = 0

120597119884minus

119889

120597120590

Cminus1

+120582minus

119889(

120597Cminus1

120597119889minus 120576119890

minus

120597119903minus

120597119889minus) +

120582+

119889(

120597Cminus1

120597119889+ 120576119890

) = 0

(20)






119891119901(120590 119888plusmn

) le 0 (21)


119888+

(119889+

120581+

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581+

119888minus

(119889minus

120581minus

) =

120597120595119901(119889plusmn 120581plusmn)

120597120581minus

(22)



119901

=120582119901

120597119865119901(120590 119889plusmn 120581plusmn)

120597120590

(23)

119891119901(120590 119888plusmn

) = 0120582119901ge 0 119891

119901(120590 119889plusmn

120581plusmn

)120582119901= 0 (24)


119889= 0 and

120582minus


120582119901can


120597120590

Cminus1

+120582119901

[

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119891119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119891119901

120597120590

) minus

120597119891119901

120597120590

Cminus1 120597119891119901

120597120590

] = 0

(25)





minus119909119899


120590trial119899+1

= 120590119899+ S119899 Δ120576119899+1

(26)


Δ120576 = C119899 Δ120590 + Δ119889

+ 120597C120597119889+ 120590 + Δ119889

minus 120597C120597119889minus 120590 + Δ120576

119901

= 0

(27a)

(119891119901)119899+1

= 119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119891119901

120597120581+)

trial

119899+1

Δ120581+

+ (

120597119891119901

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27b)


(119892+

119889)119899+1

= (119892+

119889)

trial119899+1

+ (

120597119892+

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892+

119889

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119892+

119889

120597120581+)

trial

119899+1

Δ120581+

= 0

(27c)

(119892minus

119889)119899+1

= (119892minus

119889)

trial119899+1

+ (

120597119892minus

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892minus

119889

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119892minus

119889

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27d)


Δ120576119901

= minus ((119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+(

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times (

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

(28)


Δ120590 = ((119891trial119899+1

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

minus Δ119889+ 120597C120597119889+ 120590


]

minus1

(29)



120597119891119901

120597120590

)

trial

119899+1

otimes

120597119865119901

120597120590

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

(30)





M+ = 1

1198640

[(1 + V0)

120590+otimes120590+

120590+ 120590+minus ]0

120590+otimes 120590+

120590+ 120590+]

Mminus = 1

1198640

[(1 + V0)





(31)


0is the initial


119894119895119896119897= (12)(120590

plusmn

119894119897120590plusmn

119895119896+ 120590plusmn

119894119896120590plusmn

119895119897)


119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

0(1 + 119861

+

119889+

)

minus119860+

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


0[1 + 119861



)

minus119860minus

(32)








119891119901= 1205721198681+ radic3119869


minus

(33)

where 1198681= 12059011+ 12059022+ 12059033





120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus



minus

1198870119891minus



120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)





120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+


119867] 120595119875minus

0(120581minus

) (36)



119875



120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)





120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)



+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901


119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)





119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)





120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)


119867= 1198631+1198632+1198633 and

11986311198632 and119863



H =[

[

[


0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)


120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)




119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(119892+

119889)119899+1

= (119892+

119889)

trial119899+1

+ (

120597119892+

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892+

119889

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119892+

119889

120597120581+)

trial

119899+1

Δ120581+

= 0

(27c)

(119892minus

119889)119899+1

= (119892minus

119889)

trial119899+1

+ (

120597119892minus

119889

120597120590

)

trial

119899+1

Δ120590

+ (

120597119892minus

119889

120597119889minus)

trial

119899+1

Δ119889minus

+ (

120597119892minus

119889

120597120581minus)

trial

119899+1

Δ120581minus

= 0

(27d)


Δ120576119901

= minus ((119891trial119899+1

+ (

120597119891119901

120597120590

)

trial

119899+1

Δ120590

+(

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times (

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

(28)


Δ120590 = ((119891trial119899+1

+ (

120597119891119901

120597119889+)

trial

119899+1

Δ119889+

+ (

120597119891119901

120597119889minus)

trial

119899+1

Δ119889minus

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

)

120597119865119901

120597120590

minus Δ119889+ 120597C120597119889+ 120590


]

minus1

(29)



120597119891119901

120597120590

)

trial

119899+1

otimes

120597119865119901

120597120590

)

times (

120597119891119901

120597119888+

120597119888+

120597120581+(

120597120581+

120597120576119901

120597119865119901

120597120590

) +

120597119891119901

120597119888minus

120597119888minus

120597120581minus

times(

120597120581minus

120597120576119901

120597119865119901

120597120590

))

minus1

(30)





M+ = 1

1198640

[(1 + V0)

120590+otimes120590+

120590+ 120590+minus ]0

120590+otimes 120590+

120590+ 120590+]

Mminus = 1

1198640

[(1 + V0)





(31)


0is the initial


119894119895119896119897= (12)(120590

plusmn

119894119897120590plusmn

119895119896+ 120590plusmn

119894119896120590plusmn

119895119897)


119892+

119889(119884+

119889 119889+

) = 119884+

119889minus 119903+

0(1 + 119861

+

119889+

)

minus119860+

119892minus

119889(119884minus

119889 119889minus

) = 119884minus


0[1 + 119861



)

minus119860minus

(32)








119891119901= 1205721198681+ radic3119869


minus

(33)

where 1198681= 12059011+ 12059022+ 12059033





120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus



minus

1198870119891minus



120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)





120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+


119867] 120595119875minus

0(120581minus

) (36)



119875



120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)





120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)



+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901


119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)





119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)





120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)


119867= 1198631+1198632+1198633 and

11986311198632 and119863



H =[

[

[


0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)


120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)




119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119891119901= 1205721198681+ radic3119869


minus

(33)

where 1198681= 12059011+ 12059022+ 12059033





120572 =

(119891minus

1198870119891minus

0) minus 1

2 (119891minus

1198870119891minus

0) minus 1

(34)

where 119891minus

1198870and 119891

minus



minus

1198870119891minus



120573 = (1 minus 120572)

119888minus

119888+minus (1 + 120572) (35)





120595119875

(119889plusmn

120581plusmn

) = [1 minus 120601+

119867] 120595119875+

0(120581+


119867] 120595119875minus

0(120581minus

) (36)



119875



120595119875+

0= 119891+

0120581+

+

1

2

ℎ+

(120581+

)

2

120595119875minus

0= 119891minus

0120581minus

+

1

2

ℎminus

(120581minus

)

2

(37)





120581+

= int +

119889119905 120581minus

= int minus

119889119905 (38)



+

= 119877120576

119901

max = 119877120582119901

120597119865119901

120597max

minus

= minus (1 minus 119877)120576

119901

min = (119877 minus 1)120582119901

120597119865119901

120597min

(39)

where 120576119901max and 120576119901


119877 =

sum119894⟨119894⟩

sum119894

1003816100381610038161003816119894

1003816100381610038161003816

(40)





119888+

=

120597120595119901

120597120581+= [1 minus 120601

+

119867] (119891+

0+ ℎ+

120581+

)

119888minus

=

120597120595119901

120597120581minus= [1 minus 120601

minus

119867] (119891minus

0+ ℎminus

120581minus

)

(41)





120595119890

=

1 + V0

21198640

119867119894119895119904119895119896119867119896119897119904119897119894+

3 (1 minus 2V0)

21198640

1205902

119867

1 minus 119863119867

(42)


119867= 1198631+1198632+1198633 and

11986311198632 and119863



H =[

[

[


0 0

0 (1 minus 1198632)minus12

0

0 0 (1 minus 1198633)minus12

]

]

]

(43)


120576119890

=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (44)




119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119862119894119895119896119897120590119896119897=

1 + V0

1198640

(119867119894119896119904119896119897119867119897119895)

119863

+

1 minus 2V0

1198640

120590119867

1 minus 119863119867

120575119894119895 (45)


21205901= 120574 and 120590


120601+


120601+

119867= 1 minus

(1 minus 2V0)

119889+(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(46)


be calculated

120601minus

119867= 1 minus

(1 minus 2V0)

119889minus(1 minus ((120574 + V

0120574) (1 + 120574

2))) + (1 minus 2V

0)

(47)


120601+

119867= 1 minus

(1 minus 2V0)

119889++ (1 minus 2V

0)

120601minus

119867= 1 minus

(1 minus 2V0)


0)

(48)




0 V0 119891+0 and 119891

minus






minus

0) are





119903+

0=

1198912

119905

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119905=

(1 + 120572) 119891+

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

119903minus

0=

1198912

119888

21198640

(

1205744minus 21205742V0+ 1

1205742+ 1

)

119891119888=

(1 + 120572) 119891minus

0

120572 (1 + 120574) + radic1205742minus 120574 + 1

(49)











minus)


826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










826

mm

826mm

(a) (b)

(c) (d)


1205901

1205901

f+0

120576+0 1205761

A

E0 0

(a)

1205902

1205902

fminus0

120576minus01205762

E0 0A

(b)



00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

Experimental

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

A+ = 06

A+ = 08

A+ = 10

ExperimentalA+ = 06

A+ = 08

A+ = 10 A+ = 06

A+ = 08

A+ = 10

Redu

ctio

n fa

ctor

120601+ H

(a)

Experimental

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain00000 00001 00002 00003 00004 00005

00

02

04

06

08

10

Strain

B+ = 12

B+ = 16

B+ = 20

ExperimentalB+ = 12

B+ = 16

B+ = 20 B+ = 12

B+ = 16

B+ = 20Re

duct

ion

fact

or120601

+ H

(b)

00000 00001 00002 00003 00004 000050

1

2

3

Stre

ss (M

Pa)

Strain

00000 00001 00002 00003 0000400

05

10

15

20

25

30

35

Stre

ss (M

Pa)

Plastic strain

00000 00001 00002 00003 00004 0000500

02

04

06

08

10

Strain

Redu

ctio

n fa

ctor

120601+ H


h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa h+ = 3170MPah+ = 10000MPa

h+ = 25000MPa

(c)





119891+

0= 347MPaV0= 02

ℎ+

= 3170MPa119903+

0= 190 times 10

minus4MPa119860+

= 08

119861+

= 16


0= 150MPaV0= 02

ℎminus

= 50000MPa119903minus

0= 375 times 10


= 198

119861minus

= 32


0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain00000 00005 00010 00015 00020 00025

0

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Aminus = 198

Aminus = 206


Aminus = 198


Aminus = 198

Aminus = 206

Redu

ctio

n fa

ctor

120601minus H

(a)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Bminus = 32

Bminus = 35


Bminus = 32


Bminus = 32

Bminus = 35Re

duct

ion

fact

or120601

minus H

(b)

0000 0001 0002 0003 0004 00050

10

20

30

Stre

ss (M

Pa)

Strain

00000 00005 00010 00015 00020 000250

10

20

30

Stre

ss (M

Pa)

Plastic strain

0000 0001 0002 0003 0004 000500

02

04

06

08

10

Strain


Redu

ctio

n fa

ctor

120601minus H

hminus = 40000MPa




(c)










21205901= minus1minus1 and 120590

21205901= 0052minus1)


0


plusmn

0can



0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

10

20

30

40St

ress

(MPa

)

0000 0001 0002 0003 0004 0005Strain

(a)

0

10

20

30

40

50

Stre

ss (M

Pa)

0000 0001 0002 0003 0004 0005Strain

(b)

0

10

20

30

40


0000 0001 0002 0003 0004 0005Strain

Stre

ss (M

Pa)

(c)


0000 0001 0002 0003 00040

10

20

30

Stre

ss (M

Pa)

Strain

(d)


21205901=


21205901= 0052 minus 1







1198640= 31000MPa

119891+

0= 30MPa

119891minus

0= 150MPaV0= 02

ℎ+

= 3100MPaℎminus

= 80000MPa

119860+

= 06

119861+

= 14

119860minus

= 138

119861minus

= 20


5 Conclusions



00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00000 00001 00002 00003 00004 000050

1

2

3

Strain

Stre

ss (M

Pa)


(a)


0000 0001 0002 0003 0004 00050

10

20

30

Strain

Stre

ss (M

Pa)

(b)








Acknowledgment


References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a coupled plastic damage model for ...several plastic damage models of concrete...

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