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

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A cyclic two-surface thermoplastic damage model with applicationto metallic plate dampers
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 Korea
b
Department of Mechanical and Aerospace Engineering, University at Buffalo, The State University of New York, Buffalo, NY 14260, USA
c
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
Theobjectiveofthisstudyistodevelopanewconstitutivemodelforcyclicresponseofmetalswithmuchbroader 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 of elasto-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 plasticstraintothestressstate. Thenormalityruleensuresthat theincre-mental plastic strain approaches the normal level of the yield sur-faceat the currentloadpoint. Thehardeningruleis usedtopredictchanges 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 rateindependent 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 yieldsurfaces, were developed to solve some deﬁciencies of the singleyield surface model. The single yield surface model assumed thatelasticdomainassumedtobelarge comparedwiththeexperimentand is also difﬁcult to express the sudden change from elastic toplastic and fromplastic to elastic. As regards the multisurface typemodel, Mroz [5] assumed that the multiple encircled surfaces con-tact consecutivelyand push eachother, and extendedsuch 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,Banerjeeet al. [9] and Changand Lee [10] developed a two-surface
plasticitymodel to represent bothkinematic 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] reviewed
Mroz’s multisurface model.Finally, another type of plasticity model was srcinated fromthe Armstrong–Frederick model [15]. The primary issue associatedwiththeArmstrong–Fredericktypeplasticitymodelisthequestionof how to control changes in back stress. As a consequence of this
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).Engineering Structures 52 (2013) 608–620
Contents lists available at SciVerse ScienceDirect
Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
issue, manyresearchers have modiﬁedthedynamicrecoveryterm.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 of materials 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 continuum
of 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],andTanietal.[44]appliedtheirdamagemodelfor
fatigue. 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 of Lemaitre and Chaboche [32], an existing two-surface model,and experimental data.
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 plasticbehavior or topredict theresponseof material un-der large, orcyclic, loadingbeyondtheelasticlimitcausingperma-nent plastic strain. Manyconstitutivemodelshave beendevelopedto understand the behavior of engineering materials and to solveengineering problems. A number of different cyclic plasticity mod-elsmaybealsoapplicable.Atwo-surfaceplasticitymodelthatrep-resents both kinematic and isotropic hardening behavior wasdeveloped by Banerjee et al. and, Chang and Lee. This two-surfacemodelischaracterizedbyaninnersurfacethatfollowsakinematichardening 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].Totalstrainatagivenstresscanbesplitintotwoparts,namely,elastic strain and plastic strain. For the multiaxial case, this can begeneralized as an incremental form
_
e
ij
¼
_
e
eij
þ
_
e
pij
ð
1
Þ
The elastic part of the equation is related to stress tensor usingthe linear elastic equations, such that
_
r
ij
¼
C
eijkl
_
e
ekl
ð
2
Þ
Substituting Eq. (1) into Eq. (2), one ﬁnds
_
r
ij
¼
C
eijkl
ð
_
e
kl
_
e
pkl
Þ ð
3
Þ
where
C
e
is the elastic constitutive matrix,
_
e
e
is the elastic strainincrement, and
_
e
p
is the plastic strain increment. A two-surfacemodelrequires(a)yieldcriteria,whichpredictwhetherthesolidre-spondselasticallyorplastically, (b) thestrainhardeningrule, whichdeterminesevolutionof yieldfunctionduetoinelasticdeformation,and(c) theﬂowrule, whichdetermines relationship betweenstressand plastic strain. Finally, the constitutive relation of a two-surfacemodel can be represented using the following equations:
For elastic loading or unloading,
_
r
ij
¼
kd
ij
_
e
kk
þ
2
l
_
e
ij
ð
4
Þ
For inelastic loading inside the outer surface,
_
r
ij
¼
kd
ij
_
e
kk
þ
2
l
_
e
ij
3
l
S
ij
S
kl
_
e
kl
r
L y
2
1
þ
H
p
3
l
h i
ð
5
Þ
For inelastic loading on the outer surface,
_
r
ij
¼
kd
ij
_
e
kk
þ
2
l
_
e
ij
3
l
S
ij
S
kl
_
e
kl
r
B y
2
1
þ
H
p
3
l
h i
ð
6
Þ
where
k
is the Lamé coefﬁcients,
d
ij
is the Kronecker delta,
_
e
ij
is thevolumetricstrain,
r
L y
istheinneryieldstrength,
r
B y
istheouteryieldstrength,
S
ij
is the deviatoric stress,
S
ij
is the deviatoric overstress(
r
ij
X
ij
, stress minus backstress) and
H
p
ð¼
h
B
2
r
B y
f
2
r
B y
n
Þ
is a harden-ing modulus, dependent on the hardening parameters
H
B
0
;
h
B
1
, and
n
.
Fig. 1.
Two-surface model deﬁnition.
D. Kim et al./Engineering Structures 52 (2013) 608–620
609
A two-surface model is given as the state of stress in three separateregions(a)elasticregion,(b) transition(ormeta-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, butwithintheboundingsurface. Oncetheinnersurfaceapproaches theboundingsurface,behaviorisgovernedbyanisotropicexpansionof theoutersurface,andtheinnersurfaceistranslatedsimultaneouslyto retain contact with the outer surface. Finally, hardening in theouter plastic region is associated with the isotropic hardening of the 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 (
r
L y
). Mean-while, theoutersurface, whichalwayscontains theinner surface, islocated on the center of stress space with a radius represented bythe outer yield strength (
r
B 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),andthe secondlawof thermodynamics (theentropylaw). Theﬁrstlaw 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
q
edV
¼
Z
V
@
q
e
@
t dV
¼
Z
S
J
e
d
X
þ
Z
V
q
rdV
ð
7
Þ
where
e
is the energy per unit mass,
J
e
is the energy ﬂux per unitsurface, and unit time,
r
, is the distributed internal heat source of strength 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 of heat from external sources and the entropy produced inside thesystem [51].
dS
¼
dS
e
þ
dS
i
ð
8
Þ
For any reversible transformation, the entropy source must bezero. For irreversible transformations, the entropy source mustbe positive. Thus,
dS
i
can be shown to satisfy
dS
i
P
0
ð
9
Þ
Thus,
dS
i
can be zero, positive depending on the interactions of the 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 srci-nated with Kachanov [52], and was further developed by Krajci-novic, among others. At each material point, the scalar quantity
D
is interpreted as a dimensionless number between zero andone,where
D
=0correspondstoanundamagedstateand
D
=1rep-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 ð
1
D
Þ
r
ð
10
Þ
Basaran and Yan introduced an entropy-based damage evolu-tion function founded on the principles of thermodynamics.Accordingtothesecondlaw, entropyisamonotonicallyincreasingfunction, 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
k
0
ln
W
ð
11
Þ
or
s
¼
Rm
s
ln
W
ð
12
Þ
where
k
0
is the Boltzmann constant and
W
is a disorder parameter,whichcanbedescribedastheprobabilitythatthesystemexistsinagiven state compared with all possible states. In Eq. (12), the entro-py per unit mass and its relationto the disorder parameter is given,where
R
is thegas constant and
m
s
is thespeciﬁcmass. BasaranandYan indicate that the damage parameter
D
is deﬁned as the ratio of change in the disorder parameter from the initial reference statedisorder as follows:
D
¼
D
cr
D
W W
0
¼
D
cr
1
e
ð
m
s
=
R
Þ
D
s
ð
13
Þ
where
D
cr
is introduced as the critical damage parameter, which iscalculated by deﬁning the fully damaged state from experiments.Although theoretically
D
cr
=1, in practice engineers often select
D
cr
<1 to more effectively represent materials that are near fail-ure.Theentropyproduction(
D
s
),whichappearsinEq.(13),iscalcu-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,
D
s
¼
Z
t t
0
r
: _
e
p
T
q
dt
þ
Z
t t
0
kT
2
q
j
gradT
j
2
!
dt
þ
Z
t t
0
r T dt
ð
14
Þ
where
q
is density and
T
is absolute temperature. As shown in Eq.(14),
D
=0 when
D
s
=0, and
D
=
D
cr
0
when
D
s
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
610
D. Kim et al./Engineering Structures 52 (2013) 608–620
elastic constitutive relationship considering thermoplastic relationis written using Hooke’s law in a rate form as
_
r
¼
C
e
ð
_
e
_
e
p
_
e
th
Þ ð
15
Þ
With
e
thij
¼
d
ij
a
D
T
ð
16
Þ
where
C
e
is the elastic constitutive tensor,
_
e
is the total strainincre-ment,
_
e
p
is the plastic strain increment,
_
e
th
is the thermal strainincrement, and
a
is the coefﬁcient of thermal expansion. Second,the yield surfaces for the two-surface model are deﬁned as
f
L
¼
12
ð
S
ij
X
ij
Þð
S
ij
X
ij
Þ
13
r
L y
2
ð
17
Þ
f
B
¼
12
S
ij
S
ij
13
r
B y
2
¼
J
2
13
r
B y
2
ð
18
Þ
where
S
ij
isadeviatoricstresstensor,
X
ij
is abackstresstensor,
r
L y
isthe inner yield strength, and
r
B y
is the outer yield strength. As theloadingsurfacecorrespondstokinematichardeningandthebound-ing surface produces isotropic hardening, two yield functions aredeﬁned, as shown in Eqs. (17) and (18). Following this, the plastic
ﬂow rule is deﬁned to calculate the evolution of the plastic strain.Thus,
_
e
p
¼
_
k
@
f
@
r
ð
19
Þ
where
_
k
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
23
_
e
pij
_
e
pij
q
is 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.
Basedonthe constitutive relationship, ﬂowrule, hardeningruleand 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
C
e
_
e
_
k
@
f
@
r
_
e
th
þ
@
f
@
e
p
_
k
@
f
@
r
¼
0
ð
20
Þ
Thus,
_
k
and
_
e
p
is simpliﬁed as
_
k
¼
@
f
@
r
C
e
_
e
@
f
@
r
C
e
_
e
th
@
f
@
r
C
e
@
f
@
r
@
f
@
e
@
f
@
r
ð
21
Þ
_
e
p
¼
_
k
@
f
@
r
¼
@
f
@
r
C
e
@
f
@
r
_
e
@
f
@
r
C
e
@
f
@
r
_
e
th
@
f
@
r
C
e
@
f
@
r
@
f
@
e
@
f
@
r
ð
22
Þ
Using Eq. (22), Eq. (15) is expressed by
_
r
¼
C
e
_
e
C
e
@
f
@
r
C
e
@
f
@
r
_
e
@
f
@
r
C
e
@
f
@
r
_
e
th
@
f
@
r
C
e
@
f
@
r
@
f
@
e
p
@
f
@
r
C
e
_
e
th
ð
23
Þ
In tensor form, Eq. (23) becomes
_
r
ij
¼
C
eijkl
_
e
kl
C
eijmn
@
f
@
r
mn
@
f
@
r
pq
C
e pqkl
_
e
kl
@
f
@
r
mn
C
emnpq
@
f
@
r
pq
C
e pqkl
_
e
thkl
@
f
@
r
mn
C
emnpq
@
f
@
r
pq
@
f
@
e
p pq
@
f
@
r
pq
0@1A
C
eijkl
_
e
thkl
ð
24
Þ
_
r
ij
¼
C
eijkl
_
e
kl
2
ll
ij
2
ll
pq
_
e
kl
2
ll
ij
S
kl
_
e
thkl
2
ll
ij
S
kl
23
H
p
S
ij
S
ij
!
C
eijkl
_
e
thkl
ð
25
Þ
_
r
ij
¼
C
eijkl
_
e
kl
4
l
2
S
ij
S
kl
_
e
kl
2
r
y
3
2
3
l
þ
2
r
y
3
2
H
p
þ
3
l
2
r
y
3
2
_
e
thkl
2
r
y
3
2
3
l
þ
2
r
y
3
2
H
p
C
eijkl
_
e
thkl
ð
26
Þ
_
r
ij
¼
kd
ij
_
e
kk
þ
2
l
_
e
ij
3
ll
ij
S
kl
_
e
kl
ð
r
y
Þ
2
1
þ
H
p
3
l
2
H
p
3
l
þ
H
p
ð
3
k
þ
2
ll
Þ
ij
a
D
T
ð
27
Þ
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 ð
1
f
y
D
Þ
_
r
¼ ð
1
f
y
D
Þ
C
e
_
e
e
¼ ð
1
f
y
D
Þ
C
e
ð
_
e
_
e
p
_
e
th
Þ ð
28
Þ
where
C
e
is the elastic constitutive matrix,
_
e
e
is the elastic strainincrement,
_
e
p
is the plastic strain increment,
_
e
th
is the incrementalthermal strain, and
f
y
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 showninFig. 2. Theproposedthermoplastictwo-surfacedamagemodelisaccordingly 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 applicabilityfor avarietyofmetalsthataresubjecttoprogressivedamageundercyc-lic loading. Thermal strain is also included in this two-surfacemodel to consider thermal effects, as indicated in Fig. 2.
Fig. 2.
Proposed thermoplastic two-surface model.
D. Kim et al./Engineering Structures 52 (2013) 608–620
611
5. Finite element implementation of thermoplastic damagemodel
5.1. Introduction and preparation for implementation
The two-surface damage model described above was imple-mentedasusersubroutines(UMATandUMATHT)intheABAQUSﬁ-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-orderadaptivestepsizeRunge–Kuttamethodisappliedtounder-take analysis and integrate the constitutive equations until a highlevelofaccuracyismaintained.Priortoinitiatingincrementalanal-ysis to obtain the solution, all equations must be expressed in anincrementalformtoensurethatthemodelisimplementedwithoutdifﬁcultyasaFortrancode, asshowninthefollowingequations:
S
n
þ
1
ij
¼
S
nij
þ
2
l
ð
1
D
Þ
D
e
n
þ
1
;
eij
ð
29
Þ
r
n
þ
1
ij
¼
S
n
þ
!
ij
þ
13
d
ij
r
n
þ
1
kk
ð
30
Þ
r
nij
¼
S
nij
þ
13
d
ij
r
nkk
ð
31
Þ
e
nij
¼
e
nij
13
d
ij
e
nkk
ð
32
Þ
where
D
e
ij
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
r
ij
¼
2
l
ð
1
D
Þ
D
e
ij
13
d
ij
D
e
kk
D
e
pij
þ
13
d
ij
D
e
pkk
D
e
thij
þ
13
d
ij
D
e
thkk
þ
13
d
ij
D
r
kk
ð
33
Þ
Using a ﬂow rule and Eq. (16), Eq. (33) yields
D
r
ij
¼
2
l
ð
1
D
Þ
D
e
ij
13
d
ij
D
e
kk
_
k
@
f
@
r
þ
13
d
ij
_
k
@
f
@
r
a
d
ij
D
T
þ
13
d
ij
d
kk
a
D
T
þ
13
d
ij
D
r
kk
ð
34
Þ
and then
D
r
ij
¼
2
l
ð
1
D
Þ
D
e
ij
_
k
@
f
@
r
a
d
ij
D
T
13
d
ij
ð
1
D
Þ
2
l
D
e
kk
_
k
@
f
@
r
d
kk
a
D
T
D
r
kk
ð
35
Þ
or
D
r
ij
¼ ð
1
D
Þ
2
l
D
e
ij
3
l
S
ij
S
kl
D
_
e
kl
ð
r
y
Þ
2
1
þ
H
p
3
l
2
a
d
ij
D
T
0B@1CA
13
ð
1
D
Þ
d
ij
2
l
D
e
kk
3
l
S
ij
S
kk
D
_
e
kk
ð
r
y
Þ
2
1
þ
H
p
3
l
2
d
kk
a
D
T
2
l
D
e
kk
0B@
3
k
D
e
kk
þ
3
l
S
ij
S
kk
D
_
e
kk
ð
r
y
Þ
2
1
þ
H
p
3
l
2
H
p
3
l
þ
H
p
ð
3
k
þ
2
l
Þ
d
kk
a
D
T
1CA
ð
36
Þ
and ﬁnally
D
r
ij
¼ ð
1
D
Þ
2
l
D
e
ij
þ
kd
ij
D
e
kk
3
l
S
ij
S
kl
D
e
kl
ð
r
y
Þ
2
1
þ
H
p
3
l
2
0B@
H
p
3
l
þ
H
p
ð
3
k
þ
2
l
Þ
d
ij
a
D
T
ð
37
Þ
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
+
D
t
) 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:
t
þ
D
t
~
r
i
1
¼
t
~
r
þ
Z
t
þ
D
t t
d
~
r
ð
38
Þ
The tangent constitutive matrix (
t
þ
D
t
C
i
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
t
þ
D
t
K
i
1
¼
X
e
Z
V
e
B
t t
þ
D
t
C
i
1
BdV
ð
39
Þ
t
þ
D
t
~
F
i
1
¼
X
e
Z
V
e
B
t t
þ
D
t
~
r
i
1
dV
ð
40
Þ
Finally, the solution is obtained in terms of the incremental no-dal displacements (
D
~
U
i
) by solving the set of equations
t
þ
D
t
K
i
1
D
~
U
i
¼
t
þ
D
t
R
t
þ
D
t
~
F
i
1
ð
41
Þ
Then, incremental nodal displacements are added to nodaldisplacements
t
þ
D
t
~
U
i
¼
t
þ
D
t
~
U
i
1
þ
D
~
U
i
ð
42
Þ
and strains are calculated from these updated displacements.
Additionally, aconvergencecheckisperformedbytheNewton–Raphson method in ABAQUS to determine whether the solution isdivergingorhasconvergedforsolvingnonlinearequilibriumequa-tions. If
U
n
is the current estimate, then the next estimate is givenby
U
n
þ
1
¼
U
n
f
ð
U
n
Þ
f
’
ð
U
n
Þ ð
43
Þ
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. If the 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 determinetheaccuracyandconvergenceofsolutionfornonlinearsolutionarespeciﬁed. Analysis is deemed complete when the convergencecheck satisﬁes the convergence criteria.
612
D. Kim et al./Engineering Structures 52 (2013) 608–620

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