Engineering Structures 52 (2013) 608–620

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Engineering Structures

journal homepage: www.elsevier .com/locate /engstruct

A cyclic two-surface thermoplastic damage model with applicationto metallic plate dampers

0141-0296/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2013.02.030

⇑ Corresponding author. Tel.: +82 42 870 5741; fax: +82 42 870 5999.E-mail addresses: dkkzone@gmail.com (D. Kim), gdargush@buffalo.edu

(G.F. Dargush), cjb@buffalo.edu (C. Basaran).

Dongkeon Kim a,⇑, Gary F. Dargush b, Cemal Basaran c

a Central Research Institute, Korea Hydro & Nuclear Power, 70, 1312 Beon-gil, Yuseong-daero, Yuseong-gu, Daejeon 305-343, Republic of Koreab Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USAc Department of Civil, Structural and Environmental Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA

a r t i c l e i n f o

Article history:Received 7 February 2012Revised 20 February 2013Accepted 23 February 2013Available online 20 April 2013

Keywords:Two-surface modelThermoplasticityDamageEntropy productionConstitutive model

a b s t r a c t

The objective of this study is to develop a new constitutive model for cyclic response of metals with muchbroader applicability. Accordingly, a two-surface damage thermoplasticity model is proposed to under-stand inelastic behavior and to evaluate a potential damaged state of the metals. This model, whichderived from small strain theory, is formulated through a thermodynamic approach to damage mechan-ics based on entropy production. A simple shear problem was utilized to examine several effects of thismodel, such as fatigue by cyclic loading and temperature, and to allow for the thermal effects on metals.Following this, the proposed cyclic damage model is implemented as a user subroutine in the finite ele-ment software ABAQUS. Finally, numerical results of energy dissipation devices are compared withexperimental data for validity of this model.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

A large number of researchers have developed a wide range ofelasto-plastic models under monotonic, cyclic, and complex load-ings [1,2]. The theory of rate independent plasticity has basic fun-damental features, such as the existence of yield, a plastic flowrule, normality rule and hardening rule. If the small strain theoryis considered, total strain is divided to elastic strain and plasticstrain. The yield surface divides the elastic and plastic region onthe basis of yield function and the flow rule relates the plasticstrain to the stress state. The normality rule ensures that the incre-mental plastic strain approaches the normal level of the yield sur-face at the current load point. The hardening rule is used to predictchanges in the yield criterion and flow equation.

There are three kinds of plasticity models depending on theircharacteristics about the yield surface. Firstly, plasticity modelswith internal variables were developed by Valanis’ endochronictheories [3,4] that rendered the response rate independent by con-sidering intrinsic time and deals with the plastic response bymeans of memory integrals.

The multi-surface and the Armstrong–Frederick kinematichardening type models are the primary types of plasticity models.Secondly, multi-surface model, basically has more than two yield

surfaces, were developed to solve some deficiencies of the singleyield surface model. The single yield surface model assumed thatelastic domain assumed to be large compared with the experimentand is also difficult to express the sudden change from elastic toplastic and from plastic to elastic. As regards the multisurface typemodel, Mroz [5] assumed that the multiple encircled surfaces con-tact consecutively and push each other, and extended such modelsto multidimensional cases to describe cyclic effects and smoothtransition between elastic and plastic region. Dafalias and Popov[6] developed a two-surface model, applied for metals, based onthe concept that the plastic modulus varies. Further, Krieg [7] pro-posed a two-surface plasticity model using a loading surface and alimit surface. Hashiguchi [8] developed the subsurface model todescribe the plastic strain rate inside the yield surface. Further,Banerjee et al. [9] and Chang and Lee [10] developed a two-surfaceplasticity model to represent both kinematic and isotropic harden-ing behavior characterized by an inner surface that follows a kine-matic hardening rule and an outer surface that provides forisotropic hardening. Dargush and Soong [11] applied this modelfacilitate their understanding of inelastic behavior of steel platedampers. Similar two-surface models were developed by Megahed[12] and McDowell [13], while Jiang and Sehitoglu [14] reviewedMroz’s multisurface model.

Finally, another type of plasticity model was originated fromthe Armstrong–Frederick model [15]. The primary issue associatedwith the Armstrong–Frederick type plasticity model is the questionof how to control changes in back stress. As a consequence of this

D. Kim et al. / Engineering Structures 52 (2013) 608–620 609

issue, many researchers have modified the dynamic recovery term.Models have been developed by many researchers such as Chab-oche et al. [16], Chaboche and Rousselier [17], Chaboche [18],McDowell [19], Jiang and Sehitoglu [20], Ohno and Abdel-Karim[21], Kang et al. [22], Voyiadjis and Abu Al-rub [23], Chen and Jiao[24], and Abdel-Karim [25].

Damage in metals results primarily from the initiation andgrowth processes of micro-cracks and cavities [26]. Generally,damage is regarded as the progressive or sudden deterioration ofmaterials prior to the failure of material due to loadings, thermalor chemical effects [27]. There are three main types of damages(a) ductile damage [28,29,30], (b) fatigue damage [31], and (c)creep damage [32,33]. The concept of macroscopic damage wasdeveloped by Kachanov [34] and Rabotnov [35]. This continuumof damage mechanics was further developed by Chaboche [36],Krajcinovic [37], Lemaitre and Chaboche [38]. Basaran and Yan[39], Basaran and Nie [40], and Basaran and Lin [41] used thermo-dynamic approach to formulate the damage model. Wang and Lou[42], Wang [43], and Tani et al. [44] applied their damage model forfatigue. Nechnech et al. [45], and Laborde and Hatzigeorgiou [46]developed damage model for high temperature.

In many applications, plasticity models are analytically veri-fied with experimental results to understand uncertain inelasticbehavior of the material by examining response under cyclicloading [47,48]. As a necessary tool to predict inelastic behavior,an appropriate cyclic plasticity model is required and should beverified under cyclic loading with complex loading historiessuch as repeated loading and unloading, with changing magni-tude. As the aforementioned existing plasticity models cannotpredict any failure or damage, such as low cycle fatigue failure,this research extends the two-surface model to allow thermaleffects and material degradation processes to be considered byincorporating concepts of damage mechanics. Recently, Basaranand Nie developed a damage model that utilized a damageparameter based on the second law of thermodynamics todetermine the fatigue life of material. In this study, we proposea two-surface thermoplasticity model that incorporates a dam-age parameter to understand inelastic behavior of metals andto consider thermal effects and material damage. This damagemodel is implemented as a user subroutine in the finite elementsoftware ABAQUS [49]. Finally, numerical results are comparedwith the well-known nonlinear kinematic hardening model ofLemaitre and Chaboche [32], an existing two-surface model,and experimental data.

Fig. 1. Two-surface model definition.

2. Constitutive model

2.1. Two-surface plasticity model

Linear elastic model such as Hooke’s law, is commonly used forsimplicity; however, it cannot be used to design structures thatconsider plastic behavior or to predict the response of material un-der large, or cyclic, loading beyond the elastic limit causing perma-nent plastic strain. Many constitutive models have been developedto understand the behavior of engineering materials and to solveengineering problems. A number of different cyclic plasticity mod-els may be also applicable. A two-surface plasticity model that rep-resents both kinematic and isotropic hardening behavior wasdeveloped by Banerjee et al. and, Chang and Lee. This two-surfacemodel is characterized by an inner surface that follows a kinematichardening rule and an outer surface that provides for isotropichardening. Dargush and Soong [11] applied this model to facilitatetheir understanding of inelastic behavior of steel plate dampers. Itcorrelated well with experimental force–displacement data for theinitial cycles [50].

Total strain at a given stress can be split into two parts , namely,elastic strain and plastic strain. For the multiaxial case, this can begeneralized as an incremental form

_eij ¼ _eeij þ _ep

ij ð1Þ

The elastic part of the equation is related to stress tensor usingthe linear elastic equations, such that

_rij ¼ Ceijkl

_eekl ð2Þ

Substituting Eq. (1) into Eq. (2), one finds

_rij ¼ Ceijklð _ekl � _ep

klÞ ð3Þ

where Ce is the elastic constitutive matrix, _ee is the elastic strainincrement, and _ep is the plastic strain increment. A two-surfacemodel requires (a) yield criteria, which predict whether the solid re-sponds elastically or plastically, (b) the strain hardening rule, whichdetermines evolution of yield function due to inelastic deformation,and (c) the flow rule, which determines relationship between stressand plastic strain. Finally, the constitutive relation of a two-surfacemodel can be represented using the following equations:

For elastic loading or unloading,

_rij ¼ kdij _ekk þ 2l _eij ð4Þ

For inelastic loading inside the outer surface,

_rij ¼ kdij _ekk þ 2l _eij �3l�Sij

�Skl _ekl

rLy

� �21þ Hp

3l

h i ð5Þ

For inelastic loading on the outer surface,

_rij ¼ kdij _ekk þ 2l _eij �3lSijSkl _ekl

rBy

� �21þ Hp

3l

h i ð6Þ

where k is the Lamé coefficients, dij is the Kronecker delta, _eij is thevolumetric strain, rL

y is the inner yield strength, rBy is the outer yield

strength, Sij is the deviatoric stress, Sij is the deviatoric overstress

(rij � Xij, stress minus backstress) and Hpð¼ hB 2rBy�f

2rBy

� �nÞ is a harden-

ing modulus, dependent on the hardening parameters HB0;h

B1, and n.

610 D. Kim et al. / Engineering Structures 52 (2013) 608–620

A two-surface model is given as the state of stress in three separateregions (a) elastic region, (b) transition (or meta-elastic) region, and(c) plastic region. In the elastic region, a stress point moves until itreaches the inner yield surface and strains are fully recoverable. Inthe meta-elastic region, the stress point is on the inner surface, butwithin the bounding surface. Once the inner surface approaches thebounding surface, behavior is governed by an isotropic expansion ofthe outer surface, and the inner surface is translated simultaneouslyto retain contact with the outer surface. Finally, hardening in theouter plastic region is associated with the isotropic hardening ofthe outer surface. Fig. 1 shows two distinct yield surfaces in devia-toric-stress space. The inner surface, which separates the elasticrange and inelastic range, is composed of its center and radius ex-pressed by the back stress (a) and inner yield strength (rL

y). Mean-while, the outer surface, which always contains the inner surface, islocated on the center of stress space with a radius represented bythe outer yield strength (rB

y). Translation of the inner surface corre-sponds to kinematic hardening, while expansion of outer surfaceproduces isotropic hardening.

3. Thermodynamics and damage mechanics

3.1. Introduction of thermodynamics

Thermodynamics, the study of energy conversions betweenheat and other types of energy, has developed into a general areaof science encompassing the mechanical, chemical, and electricalfields. Thermodynamics is based on two fundamental laws: thefirst law of thermodynamics (the law of conservation of energy),and the second law of thermodynamics (the entropy law). The firstlaw of thermodynamics relates the work done on the system andthe heat transfer into the system to the change in the internal en-ergy of the system. As shown below, the total energy in an arbi-trary volume V in the system can only change if energy flowsinto or out of the volume considered through the boundary X.Thus, one may write

ddt

Z V

qedV ¼Z V @qe

@tdV ¼ �

Z S

JedXþZ V

qrdV ð7Þ

where e is the energy per unit mass, Je is the energy flux per unitsurface, and unit time, r, is the distributed internal heat source ofstrength per unit mass.

According to the second law of thermodynamics, total entropyof the system always increases over time. Entropy variation canbe written as the sum of the entropy derived from the transfer ofheat from external sources and the entropy produced inside thesystem [51].

dS ¼ dSe þ dSi ð8Þ

For any reversible transformation, the entropy source must bezero. For irreversible transformations, the entropy source mustbe positive. Thus, dSi can be shown to satisfy

dSi P 0 ð9Þ

Thus, dSi can be zero, positive depending on the interactions ofthe surrounding systems. Accordingly, entropy of the universe in-creases or remains constant in all natural systems, despite de-creases of entropy as a result of a net increase in a relatedsystem. A net decrease in entropy of all related systems was notfound. Further, entropy production in a system is an irreversibleprocess. Finally, Basaran and Yan [41] developed a damage evolu-tion model based on the concept of irreversible entropy produc-tion, which is introduced in the next section.

3.2. Damage evolution function

To consider the deterioration of structural steel members ingeneral, and in particular during cyclic loading, a scalar field vari-able D is introduced as a damage index at each point. Within thepresent model and under constant amplitude cyclic loading, thecomponent tends to deteriorate gradually but at an increasing rateuntil failure occurs. This cumulative damage concept is suitable topredict damage to a component or structure as it encompasses arange of failure mechanisms, such as the growth of microcracksand microcavities. The concept of basic damage mechanics origi-nated with Kachanov [52], and was further developed by Krajci-novic, among others. At each material point, the scalar quantityD is interpreted as a dimensionless number between zero andone, where D = 0 corresponds to an undamaged state and D = 1 rep-resents a fully damaged state or fracture. Thus, the relation be-tween an effective damaged stress (�r) and undamaged stress (r)can be expressed by

�r ¼ ð1� DÞr ð10Þ

Basaran and Yan introduced an entropy-based damage evolu-tion function founded on the principles of thermodynamics.According to the second law, entropy is a monotonically increasingfunction, which is always positive for irreversible transformationsof the system. Consequently, entropy production can be used fordevelopment of accumulative damage. Boltzmann [53] expresseddisorder and entropy of a system via the relations

s ¼ k0 ln W ð11Þ

or

s ¼ Rms

ln W ð12Þ

where k0 is the Boltzmann constant and W is a disorder parameter,which can be described as the probability that the system exists in agiven state compared with all possible states. In Eq. (12), the entro-py per unit mass and its relation to the disorder parameter is given,where R is the gas constant and ms is the specific mass. Basaran andYan indicate that the damage parameter D is defined as the ratio ofchange in the disorder parameter from the initial reference statedisorder as follows:

D ¼ DcrDWW0

� �¼ Dcr 1� e�ðms=RÞDs

� �ð13Þ

where Dcr is introduced as the critical damage parameter, which iscalculated by defining the fully damaged state from experiments.Although theoretically Dcr = 1, in practice engineers often selectDcr < 1 to more effectively represent materials that are near fail-ure.The entropy production (Ds), which appears in Eq. (13), is calcu-lated by the summation of mechanical dissipation, thermaldissipation due to conduction of heat, and thermal dissipation dueto internal heat source per unit mass (r). Thus,

Ds ¼Z t

t0

�r : _�ep

Tqdt þ

Z t

t0

k

T2qjgradTj2

!dt þ

Z t

t0

rT

dt ð14Þ

where q is density and T is absolute temperature. As shown in Eq.(14), D = 0 when Ds = 0, and D = Dcr0 when Ds tends to infinity.Thus, D is always larger than zero with dissipation as the changein entropy has a nonnegative value.

4. Thermoplastic damage model formulation

A thermoplastic damage model formulated on the basis of atwo-surface plasticity model and damage evolution function, asdiscussed in the last two sections, is presented here. First, the

D. Kim et al. / Engineering Structures 52 (2013) 608–620 611

elastic constitutive relationship considering thermoplastic relationis written using Hooke’s law in a rate form as

_r ¼ Ceð _e� _ep � _ethÞ ð15Þ

With

ethij ¼ dijaDT ð16Þ

where Ce is the elastic constitutive tensor, _e is the total strain incre-ment, _ep is the plastic strain increment, _eth is the thermal strainincrement, and a is the coefficient of thermal expansion. Second,the yield surfaces for the two-surface model are defined as

fL ¼12ðSij � XijÞðSij � XijÞ �

13

rLy

� �2ð17Þ

fB ¼12

SijSij �13

rBy

� �2¼ J2 �

13

rBy

� �2ð18Þ

where Sij is a deviatoric stress tensor, Xij is a back stress tensor, rLy is

the inner yield strength, and rBy is the outer yield strength. As the

loading surface corresponds to kinematic hardening and the bound-ing surface produces isotropic hardening, two yield functions aredefined, as shown in Eqs. (17) and (18). Following this, the plasticflow rule is defined to calculate the evolution of the plastic strain.Thus,

_ep ¼ _k@f@r

ð19Þ

where _k ¼ffiffiffiffiffiffiffiffiffiffiffiffi23

_epij_ep

ij

qis the magnitude of the plastic strain increment.

Finally, this model should be enforced to satisfy consistency condi-tions as _f L ¼ 0 and _f B ¼ 0.

Based on the constitutive relationship, flow rule, hardening ruleand damage evolution function, a two-surface damage model isformulated, see below. By substituting Eq. (15) with the consis-tency condition, the following equation is obtained:

@f@r

Ce _e� _k@f@r� _eth

� � þ @f@ep

_k@f@r¼ 0 ð20Þ

Thus, _k and _ep is simplified as

_k ¼@f@r Ce _e� @f

@r Ce _eth

@f@r Ce @f

@r�@f@e

@f@r

ð21Þ

_ep ¼ _k@f@r¼

@f@r Ce @f

@r_e� @f

@r Ce @f@r

_eth

@f@r Ce @f

@r�@f@e

@f@r

ð22Þ

Using Eq. (22), Eq. (15) is expressed by

_r ¼ Ce _e� Ce@f@r Ce @f

@r_e� @f

@r Ce @f@r

_eth

@f@r Ce @f

@r�@f@ep

@f@r

� Ce _eth ð23Þ

In tensor form, Eq. (23) becomes

Fig. 2. Proposed thermoplas

_rij ¼ Ceijkl

_ekl �Ce

ijmn@f

@rmn

@f@rpq

Cepqkl

_ekl � @f@rmn

Cemnpq

@f@rpq

Cepqkl

_ethkl

@f@rmn

Cemnpq

@f@rpq� @f

@eppq

@f@rpq

0@

1A

� Ceijkl

_ethkl ð24Þ

_rij ¼ Ceijkl

_ekl �2llij2llpq

_ekl � 2llijSkl _ethkl

2llijSkl � 23 HpSijSij

!� Ce

ijkl_eth

kl ð25Þ

_rij ¼ Ceijkl

_ekl �4l2SijSkl _ekl

2ry

3

� �23lþ 2ry

3

� �2Hpþ

3l 2ry

3

� �2_eth

kl

2ry

3

� �23lþ 2ry

3

� �2Hp

� Ceijkl

_ethkl ð26Þ

_rij ¼ kdij _ekk þ 2l _eij �3llijSkl _ekl

ðryÞ2 1þ Hp

3l

� �2 �Hp

3lþ Hp ð3k

þ 2llÞijaDT ð27Þ

Finally, the equation above is formulated assuming a two-sur-face model and temperature. The damage parameter is added toEq. (28) which constitutes the next step toward developing atwo-surface damage plasticity model. Following this, the constitu-tive relation is written as follows in terms of the undamaged anddamaged stress:

_�r ¼ ð1� fyDÞ _r ¼ ð1� fyDÞCe _�ee ¼ ð1� fyDÞCeð _�e� _�ep � _�ethÞ ð28Þ

where Ce is the elastic constitutive matrix, _�ee is the elastic strainincrement, _�ep is the plastic strain increment, _�eth is the incrementalthermal strain, and fy is the reduction factor, which correlates theelastic modulus degradation to the damage parameter.

By substituting Eqs. (4)–(6) into Eq. (28) instead of the undam-aged incremental stress, the coupled damaged stress–strain rela-tionship of a two-surface plasticity model is formulated as shownin Fig. 2. The proposed thermoplastic two-surface damage model isaccordingly characterized as a two-surface plasticity model with ayield surface, flow rule and hardening rule on both loading andbounding surfaces, and a damage evolution function based on en-tropy production. The resulting model has broad applicability for avariety of metals that are subject to progressive damage under cyc-lic loading. Thermal strain is also included in this two-surfacemodel to consider thermal effects, as indicated in Fig. 2.

tic two-surface model.

612 D. Kim et al. / Engineering Structures 52 (2013) 608–620

5. Finite element implementation of thermoplastic damagemodel

5.1. Introduction and preparation for implementation

The two-surface damage model described above was imple-mented as user subroutines (UMAT and UMATHT) in the ABAQUS fi-nite element software. Once the small increment of strain is given,new updated state variables such as stress, back stress and plasticstrain, are obtained by integrating constitutive equations. A high-er-order adaptive step size Runge–Kutta method is applied to under-take analysis and integrate the constitutive equations until a highlevel of accuracy is maintained. Prior to initiating incremental anal-ysis to obtain the solution, all equations must be expressed in anincremental form to ensure that the model is implemented withoutdifficulty as a Fortran code, as shown in the following equations:

Snþ1ij ¼ Sn

ij þ 2lð1� DÞDenþ1;eij ð29Þ

rnþ1ij ¼ Snþ!

ij þ13

dijrnþ1kk ð30Þ

rnij ¼ Sn

ij þ13

dijrnkk ð31Þ

enij ¼ en

ij �13

dijenkk ð32Þ

where Deij is the deviatoric strain increment, and l is the shearmodulus. Subtracting Eq. (30) from Eq. (31) with the strain decom-position, and the relationship between strain and deviatoric strainin Eq. (32), one finds

D�rij ¼ 2lð1

� DÞ Deij �13

dijDekk � Depij þ

13

dijDepkk � Deth

ij þ13

dijDethkk

� �

þ 13

dijD�rkk

ð33Þ

Using a flow rule and Eq. (16), Eq. (33) yields

D�rij ¼ 2lð1� DÞ Deij �13

dijDekk � _k@f@rþ 1

3dij

_k@f@r� adijDT

�

þ13

dijdkkaDT�þ 1

3dijD�rkk ð34Þ

and then

D�rij ¼ 2lð1� DÞ Deij � _k@f@r� adijDT

� �� 1

3dijð1

� DÞ 2lDekk � _k@f@r� dkkaDT � D�rkk

� �ð35Þ

or

D�rij ¼ ð1� DÞ 2lDeij �3lSijSklD _ekl

ðryÞ2 1þ Hp

3l

� �2 � adijDT

0B@

1CA

� 13ð1� DÞdij 2lDekk �

3lSijSkkD _ekk

ðryÞ2 1þ Hp

3l

� �2 � dkkaDT � 2lDekk

0B@

�3kDekk þ3lSijSkkD _ekk

ðryÞ2 1þ Hp

3l

� �2 �Hp

3lþ Hp ð3kþ 2lÞdkkaDT

1CA ð36Þ

and finally

D�rij ¼ ð1� DÞ 2lDeij þ kdijDekk �3lSijSklDekl

ðryÞ2 1þ Hp

3l

� �2

0B@

� Hp

3lþ Hp ð3kþ 2lÞdijaDT�

ð37Þ

Eq. (37) is an incremental form of Fig. 2 , and this is going to beapplied to UMAT in ABAQUS for thermoplastic analysis.

5.2. Procedure for implementation

As the incremental form is established, a procedure for incre-mental analysis is initiated, as introduced below. Firstly, timeshould be initialized. Secondly, the solution (t~U, nodal displace-ments) is assumed and other internal variables such as strain (t~e)and stress (t~r) are stored at time (t). Finally, deformation variablesfor time (t + Dt) must be initialized and the iteration counter mustbe set to 1. Following this, the stresses and internal variables arecalculated through the integration. For example, the stress is up-dated from a known converged solution, as follows:

tþDt~ri�1 ¼ t~rþZ tþDt

td~r ð38Þ

The tangent constitutive matrix (tþDtCi�1), consistent with theintegration process, is also updated. Following this, the tangentstiffness matrix and nodal force vector for each element are calcu-lated using a Gaussian quadrature, in the following manner

tþDtKi�1 ¼X

e

ZVe

Bt tþDtCi�1BdV ð39Þ

tþDt~Fi�1 ¼X

e

ZVe

Bt tþDt~ri�1dV ð40Þ

Finally, the solution is obtained in terms of the incremental no-dal displacements (D~Ui) by solving the set of equations

tþDtKi�1D~Ui ¼ tþDtR� tþDt~Fi�1 ð41Þ

Then, incremental nodal displacements are added to nodaldisplacements

tþDt~Ui ¼ tþDt~Ui�1 þ D~Ui ð42Þ

and strains are calculated from these updated displacements.Additionally, a convergence check is performed by the Newton–

Raphson method in ABAQUS to determine whether the solution isdiverging or has converged for solving nonlinear equilibrium equa-tions. If Un is the current estimate, then the next estimate is givenby

Unþ1 ¼ Un �f ðUnÞf ’ðUnÞ

ð43Þ

If the solution is diverging, the algorithm reduces the time andstarts the solution process again. Conversely, if the solution is pro-gressing well, but has not yet converged, the iteration counter(i = i + 1) must be increased and the process must be repeated. Ifthe solution has converged for the current time step, the timeincrement must be modified and the solution process must be ini-tiated once again. For iteration control, parameters that determinethe accuracy and convergence of solution for nonlinear solution arespecified. Analysis is deemed complete when the convergencecheck satisfies the convergence criteria.

Table 1Cash–Karp parameters for embedded Runge–Kutta.

i ci c�i

1 37/378 2825/27,6482 0 03 250/621 18,575/48,3844 125/594 13,525/55,2965 0 277/15,3366 512/1771 1/4

D. Kim et al. / Engineering Structures 52 (2013) 608–620 613

5.3. Numerical method for integration

Each iteration process should produce a valid solution if theconstitutive model is integrated accurately. Generally, intermedi-ate results do not constitute accurate solutions due to improperstress distribution. Accordingly, many numerical methods such asthe radial return method, forward Euler integration method, andbackward Euler method are proposed to ensure accurate integra-

X1

X2

d=2m

X1

X2

T1=300KT1=300K

d=2m

T2=1300KT2=1300K

0.1m

(a) Case 1 (d = 2.0m, T1 = 300K, T2=1300K

X1

X2

X1

X2

T1=300KT1=300K

d=1m

T2=1300KT2=1300K

0.05md=1m

(b) Case 2 (d = 1.0m, T1 = 300K, T2=1300K

X1

X2

X1

X2

T1=300KT1=300K

d=1m

T2=1300KT2=1300K

0.1md=1m

(c) Case 3 (d = 1.0m, T1 = 300K, T2=1300

X1

X2

X1

X2

T1=300KT1=300K

d=0.2m

T2=500KT2=500K

0.01md=0.2m

(d) Case 4 (d = 0.2m, T1 = 300K, T2=500K

Fig. 3. The geometry of the

tion. The Cash–Karp method is one of several embedded Runge–Kutta methods and is used as a numerical method to integrate con-stitutive equations. This embedded Runge–Kutta method permitsaccurate determination of the state of stress for tþDtFi�1 and forthe constitutive tensor (tþDtCi�1). This method can also producenumerical errors by round-off and truncation errors in numericalcalculation, and it finally can cause stability problem which leadsto meaningful solution when errors accumulated. Thus, this meth-od used adaptive step size integration algorithm, and employs sixevaluations to calculate a combination of the fourth order methodand fifth order method that are associated with the Cash–Karpmethod. This embedded fourth-order formula is

y�nþ1 ¼ yn þ c�1k1 þ c�2k2 þ c�3k3 þ c�4k4 þ c�5k5 þ c�6k6 þ Oðh5Þ ð44Þ

and accordingly the error estimate is

D � ynþ1 � y�nþ1 ¼X6

i¼1

ðci � c�1Þki ð45Þ

0.1m 0.1m 0.1m

, Max. displacement = 0.05d = 0.1m)

0.05m 0.05m 0.05m

, Max. displacement = 0.05d = 0.05m)

0.1m 0.1m 0.1m

K, Max. displacement = 0.1d = 0.1m)

0.01m 0.01m 0.01m

, Max. displacement = 0.05d = 0.01m)

simple shear problem.

-2.5E+07-2.0E+07-1.5E+07-1.0E+07-5.0E+060.0E+005.0E+061.0E+071.5E+072.0E+072.5E+07

0 1 2 3 4 5

Time (sec)

Forc

e(N

)

Two surface modelTwo surface damage model MTwo surface damage mode MT

Fig. 6. Coupled temperature-displacement analysis with mechanical and thermaldissipation (Case 2, d = 1.0 m, T1 = 300 K, T2 = 1300 K, Max.displacement = 0.05d = 0.05 m).

614 D. Kim et al. / Engineering Structures 52 (2013) 608–620

Step h1 produces D0, and the step h0 is estimated as

h0 ¼ h1D0

D1

��������

0:2

ð46Þ

where D0 is the desired accuracy. All parameters for the Cash–Karpmethod are shown in Table 1.

6. Numerical results

6.1. Numerical results of a simple shear problem

6.1.1. Displacement analysis and coupled displacement temperatureanalysis

A simple shear problem was initially used to consider severaleffects of this model, such as (a) fatigue by cyclic loading, (b) fati-gue by temperature, and (c) thermal effects. At room temperature(around 300 K), the mechanical properties of structural steel are

Fig. 4. Loading function (ratio is one at maximum displacement).

Table 2Material properties and model parameters for A36 structural steel.

Young’s modulus (E): 200,000 MPa Poisson’s ratio (t): 0.3Inner yield strength ðrL

yÞ: 198 MPa Outer yield strength (rBy ): 427 MPa

Hardening parameter ðHB0Þ: 6450 MPa hB

1: �8.47, n: �10.4; Hardeningparameter

Coefficient of thermal expansion (a):10.8 � 10�6/�C

Density (q): 7800 kg/m3

Specific heat: 460 J/kg �C Thermal conductivity: 50 W/m kGas constant (R):

8.314 m2 s�2 kg mole�1 K�1Boltzmans’ constant (K) :1.38E�23 kg m2 s�2 k�1

Avogadro’s number (N0): 6.02E23/mole Specific mass ( �ms): 180 g/mole

-1.0E+08-8.0E+07-6.0E+07-4.0E+07-2.0E+070.0E+002.0E+074.0E+076.0E+078.0E+071.0E+08

0 1 2 3 4 5

Time (sec)

Forc

e(N

)

Two surface modelTwo surface damage model MTwo surface damage mode MT

Fig. 5. Coupled temperature-displacement analysis with mechanical and thermaldissipation (Case 1, d = 2.0 m, T1 = 300 K, T2 = 1300 K, Max.displacement = 0.05d = 0.1 m).

-3.0E+07

-2.0E+07

-1.0E+07

0.0E+00

1.0E+07

2.0E+07

3.0E+07

0 1 2 3 4 5

Time (sec)

Forc

e(N

)

Two surface modelTwo surface damage model MTwo surface damage mode MT

Fig. 7. Coupled temperature-displacement analysis with mechanical and thermaldissipation (Case 3, d = 1.0 m, T1 = 300 K, T2 = 1300 K, Max.displacement = 0.1d = 0.1 m).

-1.0E+06-8.0E+05-6.0E+05-4.0E+05-2.0E+050.0E+002.0E+054.0E+056.0E+058.0E+051.0E+06

0 1 2 3 4 5

Time (sec)

Forc

e(N

)

Two surface modelTwo surface damage model MTwo surface damage mode MT

Fig. 8. Coupled temperature-displacement analysis with mechanical and thermaldissipation (Case 4, d = 0.2 m, T1 = 300 K, T2 = 500 K, Max.displacement = 0.05d = 0.01 m).

stable, and it exhibits its full capacity; however, higher tempera-tures can affect and deteriorate the mechanical properties of struc-tural steel. For example, during strong cyclic loading, thedissipated energy can be converted to another heat source and ele-vate the temperature of the surrounding metal by redistributingthe heat energy. Structural steel is generally used at temperaturesaround room temperature and the melting point of carbon steel isin the range of 1700 K. Naturally, full capacity of the mechanicalproperties of structural steel is not shown as the temperature ap-proaches melting point. Thermal effects are dependent on material

Fig. 9. Entropy production for Case 2.

Fig. 10. Damage parameter for Case 2.

D. Kim et al. / Engineering Structures 52 (2013) 608–620 615

type, size and geometry of the specimen, and strain rate. For thepurpose of this analysis, the element size (0.2 m, 1 m, and 2 m)and temperature (300 K for room temperature, 500 K for moderatetemperature, 1300 K for high temperature) are considered as twovariables to investigate and develop an understanding of the ther-mal effects of structural steel.

The geometry and dimensions of the undeformed and deformedelements of this shear problem are shown in Fig. 3, which includesfour 8-node linear brick elements (C3D8) for static displacementanalysis and 8-node trilinear displacement temperature elements(C3D8T) for coupled displacement temperature analysis, if temper-ature is considered. As regards to the boundary condition, elements

Fig. 11. Deformed and undeformed shape (mesh 1, 104 elements).

Fig. 12. Deformed and undeformed shape (mesh 2, 176 elements).

616 D. Kim et al. / Engineering Structures 52 (2013) 608–620

cannot experience deflection at the bottom, because nodes at thebottom of elements are pinned to its support. Also, elements can-

not produce any moments, because nodes at the bottom of ele-ments are free to rotate. As shown in Fig. 4, the loading and

Fig. 13. Deformed and undeformed shape (mesh 3, 1568 elements).

Fig. 14. Deformed and undeformed shape (mesh 4, 11,264 elements).

D. Kim et al. / Engineering Structures 52 (2013) 608–620 617

unloading at the top is a displacement controlled sine functionwith a maximum displacement of 0.05d for Case 1, Case 2 and Case4, and a maximum displacement of 0.1d for Case 3. Material prop-erties and model parameters for this simple shear problem usingthe A36 structural steel are shown in Table 2.

Fig. 5 shows the cyclic response of undamaged and damagedmaterial using a two-surface model and a two-surface damagemodel that considers temperature. After several cycles, entropyproduction is accumulated by mechanical dissipation. Conse-quently, the damage parameter increases gradually at first and

Fig. 15. Damage parameter contour.

-350

-175

0

175

350

-0.42 -0.21 0 0.21 0.42

Rad

Forc

e(k

N)

mesh1mesh2mesh3mesh4

Fig. 16. Chaboche model.

-350

-175

0

175

350

-0.42 -0.21 0 0.21 0.42

Rad

Forc

e(k

N)

mesh1mesh2mesh3mesh4

Fig. 17. Two-surface plasticity model.

-350

-175

0

175

350

-0.42 -0.21 0 0.21 0.42

Rad

Forc

e (k

N)

mesh1_dmesh2_dmesh3_dmesh4_d

Fig. 18. Two-surface damage model.

-350

-175

0

175

350

-0.42 -0.21 0 0.21 0.42

rad

Forc

e (k

N)

Chaboche modelExperimentExperiment (After fracture)

Fig. 19. Chaboche model comparison with experimental results [50].

618 D. Kim et al. / Engineering Structures 52 (2013) 608–620

then more rapidly, until failure of the material is identified as thedamage parameter approaches one. Thermal dissipation, in addi-

tion to mechanical dissipation, impacts entropy production. In-deed, the material can be damaged more severely and more

-350

-175

0

175

350

-0.42 -0.21 0 0.21 0.42

Rad

Forc

e (k

N)

Two surface model

Two surface damage modelExperiment

Experiment (After fracture)

Fig. 20. Two-surface damage model comparison with experimental results [50].

D. Kim et al. / Engineering Structures 52 (2013) 608–620 619

rapidly in a context where thermal dissipation is occurring, ascompared to a case where only mechanical dissipation is takingplace, as shown in Fig. 5. Although damage by thermal dissipationis comparatively smaller than that caused by mechanical dissipa-tion in Fig. 5, it does have the capacity to be significant, especiallyif there are high temperatures or strong thermal gradients. As d(size of element) of Case 1 approaches 0.5d of Case 2, entropy pro-duction by the thermal gradient gets larger, as shown in Fig. 6.Accordingly, if the specimen is small-scaled, the result is governedby thermal dissipation as opposed to mechanical dissipation. Case3 exhibits twice as much displacement as compared with Case 2.Entropy production by mechanical dissipation is also larger in Case3 as compared to Case 2, as shown in Fig. 7. Even though temper-ature differences between the top and bottom are much smallerthan Case 1, Case 2 and Case 3, entropy production by thermal dis-sipation increases because the temperature gradient increases, asshown in Fig. 8. Thus, based on these results, smaller sized speci-mens are more easily deteriorated by thermal gradients. For theapplication of electrical chips in electrical engineering, the effectof element dimension by thermal dissipation is more importantthan metallic dampers in civil engineering and more study shouldbe needed for the small sized application.

6.1.2. Entropy production and damage parameterFig. 9 shows the distribution of entropy production for Case 2 by

defining entropy production at each integration point. Fig. 10 alsoshows the distribution of the damage parameter for Case 2 bydefining the damage parameter. The left side of the model, whichindicates the bottom of the element in Fig. 3b, is pinned support,while the right side of the model, which indicates the top of theelement also in Fig. 3b, is a free edge. Results are plotted at 4.5 sas Case 2 has maximum displacement at that time. Total entropyproduction as well as damage parameter are accumulated bymechanical dissipation and thermal dissipation, beginning fromt = 0 s. As shown in Fig. 10, the damage parameter is as large as0.96 at the middle, 0.92 at the left edge and, 0.52 at the free edge.Consequently, failure is likely to occur at between the left edge and

Table 3Comparison with experimental results.

Chaboche model

Max. force at 1st cycle (kN) 179.48Max. force after fracture occurred (kN)Positive force at zero radian of 1st cycle (kN) 146.17Positive force at zero radian after fracture occurred (kN)Negative force at zero radian of 1st cycle (kN) �145.66Negative force at zero radian after fracture (kN) occurred

the middle of the element. The damage parameter can be usefullyemployed as an index for a damaged state of a material.

6.2. Numerical results of energy dissipation devices

6.2.1. The modeling and mesh sensitivity analysis of energy dissipationdevices

The energy dissipation devices, which were loaded by applyingcyclical enforced displacements with constant amplitude as shownin Fig. 11, were modeled by 8-node solid brick elements (C3D8) inABAQUS. For the finite element analysis, a user subroutine (UMAT)was developed to incorporate the two-surface damage model.Material properties and model parameters for the A36 structuralsteel are shown in Table 2. The loading and unloading at the freeedge is specified as a displacement-controlled sine function withthe maximum displacement at the top and bottom set at 0.1L,where L represents the length of the plate. Four types of metallicdamper models, which are labeled mesh 1 (104 elements), mesh2 (176 elements), mesh 3 (1568 elements), and mesh 4 (11,264 ele-ments) are modeled for mesh sensitivity analysis to check robust-ness of the model. The deformed shape plots for the four meshlevels are shown in Figs. 11–14.

The damage parameter, as shown in Fig. 15, is also at its largestat the fixed edge and is quite small at the free edge as the strainenergy by mechanical dissipation at the fixed edge is larger thanthat at the free edge. Accordingly, a damage parameter can beeffectively employed as an index to express the damaged configu-ration. From Figs. 16–18, force–displacement responses of the fourtypes of meshes are shown dependent on the type of plastic model(Chaboche model, two-surface model, and two-surface damagemodel). As the meshes become finer, the numerical results con-verge with the results of mesh 3 and mesh 4, which mesh 3 andmesh 4 does not have many differences.

6.2.2. Comparison with experimental resultsForce–displacement plots of the numerical models (Chaboche

model, two-surface model and two-surface damage model) andexperimental results are plotted in Figs. 19 and 20. The results plot-ted in the figures below are based on mesh 4 (finest mesh). Analy-sis of the energy dissipation devices by a Chaboche model showthat the strength is increasing as the cycles progress. This modelcannot predict any damage and degradation of strength, as shownin Fig. 19.

Although a standard two-surface plasticity model cannot pre-dict any damage, it correlates well with the undamaged cyclic re-sponse. Two surface model has the stiffening effect whichnonlinear geometry theory is considered but Chaboche model can-not consider nonlinear geometry effect. Thus, two surface modelhas larger peak force than that by Chaboche model. A two-surfacedamage model can predict damage of material and correlates wellwith the damaged cyclic response. The comparison with analyticaland experimental results of maximum force at 1st cycle, force afterfracture occurred, positive or negative force at zero radian, andpositive or negative force at zero radian after fracture occurredare shown in Table 3. Thus, the two-surface model and two-surface

Two-surface model Two-surface damage model Experimental results

209.87 197.85 198161.86 156

161.07 153.15 148121.36 113

�157.15 �152.58 �162�118.04 �118

620 D. Kim et al. / Engineering Structures 52 (2013) 608–620

damage model produced mostly positive correlations with experi-ments both before and after fracture occurred, though some differ-ences exist, as shown in Fig. 20 and in Table 3.

7. Conclusions

This paper presents a two-surface thermoplastic damage modelthat is formulated and implemented as a UMAT subroutine withinABAQUS. The Chaboche model, two-surface plasticity model, andtwo-surface damage model are used for the simple shear problem,and the inelastic behavior of energy dissipation devices, subject totwo different types of loadings. Additionally, the finite elementsolutions correlate reasonably well with experimental results bothbefore, and after, damage had occurred. Although the Chabochemodel cannot predict any damage or strength degradation, thismodel offers a fast analysis time and simple numerical algorithmand can be used effectively for approximate analysis. The two-sur-face plasticity model correlates well with experimental results.Nevertheless it cannot predict any damage or strength degrada-tion. Finally, the proposed two-surface damage model effectivelypredicts cyclic response and correlates well with experimentalresults. This model is also found to predict failure of material andstructural components as it uses a damage evolution functionbased on entropy production. Therefore, this approach offers a ma-jor contribution toward understanding the cyclic behavior of struc-tural steel. Additionally, depending on the size of the device andthe level of temperature gradients generated by the dissipated en-ergy, thermal effects may represent another consideration.

Acknowledgments

This work was supported by the National Research Foundationof Korea (NRF) grant funded by the Korean government (MEST)(No. 20110001395) and by the Korean government (MEST) (No.20110028794).

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