Stress computation in finite thermoviscoplasticity

International Journal of Plasticity 22 (2006) 1699–1727

www.elsevier.com/locate/ijplas

Stress computation in finite thermoviscoplasticity

Dirk Helm *

Institute of Mechanics, Department of Mechanical Engineering, University of Kassel,

Monchebergstr. 7, D-34109 Kassel, Germany

Received 3 April 2005; received in final revised form 7 November 2005Available online 29 March 2006

Abstract

In this article, a constitutive theory in the framework of continuum thermomechanics is intro-duced to represent the viscoplastic behavior of metals at finite deformations. In particular, the exper-imentally observed thermomechanical coupling phenomena are described by the theory. The modelis based on the assumption that the occurring viscoplastic deformations are isochoric.

For the numerical integration of the constitutive theory, a backward Euler method is applied. Asa matter of fact, the application of the original backward Euler scheme does not preserve the prop-erty of the viscoplastic deformations to be isochoric. A major topic of the present article is the devel-opment of an improved numerical integrator on the basis of the original backward Euler method,which preserves exactly the incompressibility of the occurring viscoplastic deformations.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Viscoplasticity; Stress computation; Backward Euler scheme; Geometric integration; Stored energy ofcold work

1. Introduction

The inelastic material behavior of metals is often involved in technical processes likemetal forming. The material properties of metals are well investigated (see e.g. Benallalet al., 1989; Bruhns et al., 1992; Chaboche and Lemaitre, 1990; Haupt and Lion, 1995;Khan and Jackson, 1999; Krempl, 1979; Krempl and Khan, 2003; Lion, 1994) and the

0749-6419/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijplas.2006.02.007

* Tel.: +49 561 804 2824; fax: +49 561 804 2720.E-mail address: [email protected]: http://www.ifm.maschinenbau.uni-kassel.de/~helm.

mailto:[email protected]

http://www.ifm.maschinenbau.uni-kassel.de/~helm

1700 D. Helm / International Journal of Plasticity 22 (2006) 1699–1727

origin of the viscoplastic behavior is well understood. Basically, the distortion of themetallic lattice as well as the production and motion of dislocations leads to a static hys-teresis and non-linear rate dependence. It is worthy to be mentioned that the rate-depen-dent behavior of metals is already observable at moderate strain rates (cf. Haupt and Lion,1995; Krempl, 1979; Lion, 1994).

In order to represent the inelastic behavior of metals a lot of constitutive theories weredeveloped: see e.g. Bertram (2003), Bucher et al. (2004), Bruhns et al. (1992), Green andNaghdi (1965), Haupt and Kamlah (1995), Miehe and Stein (1992), Lehmann (1983),Rajagopal and Srinivasa (1998a), Scheidler and Wright (2001), Simo and Miehe (1992),Tsakmakis (1996), and Tsakmakis (2004) for a common overview see the textbooks Haupt(2002) and Lubliner (1990). Furthermore, the viscoplastic behavior of metals is discussedin Chaboche (1977), Haupt and Lion (1995), Krempl (1987), Lion (2000), and Perzyna(1963).

Many different concepts are proposed in the literature for representing the thermome-chanical behavior of material bodies: e.g. Jou et al. (1996), Maugin and Muschik (1994),Meixner and Reik (1959), Muller and Ruggeri (1998), Truesdell and Noll (1965), and Zie-gler (1977). In the present article, the development of the constitutive equations follows athermomechanical framework, which was established by Coleman and Gurtin (1967),Coleman and Noll (1963), and Truesdell and Noll (1965). In this theory, the second lawof thermodynamics is represented by the local form of the Clausius–Duhem inequalityand implies a constraint, which must be satisfied by the constitutive relations for everyadmissible thermodynamic process (cf. the discussions in Jou et al. (1996) and Mauginand Muschik (1994)). In contrast to this concept, Germain et al. (1983) introduces apseudopotential of dissipation and Green and Naghdi (1978) as well as Ziegler (1977) sug-gest a constitutive relation for the rate of dissipation (cf. the remarks in Maugin andMuschik (1994)). In these frameworks, the evolution equations of internal variables areoften determined by using the postulate of maximum dissipation (cf. Deseri and Mares,2000; Miehe, 1998; Rajagopal and Srinivasa, 1998b), which can be traced back in its ther-momechanical formulation to Ziegler (1963) and Ziegler (1977). In plasticity theory, arelated concept was formulated by von Mises (1928) and Hill (1950).

In addition to the modeling of the mechanical behavior, the representation of the ther-momechanical coupling phenomena is of special interest (Bodner and Lindenfeld, 1995;Chaboche, 1993b; Hakansson et al., 2005; Haupt et al., 1997; Helm, 1998; Jansohn,1997; Kamlah and Haupt, 1998; Mollica et al., 2001; Tsakmakis, 1998): it is well knownthat metals show thermoelastic coupling phenomena. Moreover, a part of the workrequired to produce plastic deformations is stored as internal energy in the microstructureof the material. Since the basic results of Taylor and Quinney (1934), this phenomenon hasbeen investigated in detail: cf. Bever et al. (1973), Chrysochoos et al. (1989), Helm (1998),Oliferuk et al. (1985), and Rosakis et al. (2000). In the first studies (cf. Taylor and Quin-ney, 1934), a small amount of approximately 11% of the introduced cold work is stored inthe microstructure. In contrast to this, subsequent studies (e.g. Bever et al., 1973; Chryso-choos et al., 1989; Helm, 1998; Oliferuk et al., 1985) led to the result that the stored energydepends on the deformation history and that more than 11%, up to approximately 70%, ofthe cold work is stored in the microstructure of the material.

In the present article, a simple constitutive theory for metal viscoplasticity is developed,which incorporates the description of finite deformations, non-linear isotropic hardening,as well as the thermomechanical coupling phenomena. For simplicity, the model does not

D. Helm / International Journal of Plasticity 22 (2006) 1699–1727 1701

include kinematic hardening in order to obtain a simple constitutive theory. Such a theoryis suitable for problems, which are dominated by monotonous loading processes. In thediscussed theory it is assumed that the viscoplastic deformations occur in an isochoricmanner.

For solving initial-boundary value problems, the stress state must be calculated by inte-grating a system of ordinary differential equations. In the framework of numerical meth-ods like the finite element method, the system of differential equations is numericallysolved by an appropriate integration scheme. In the framework of implicit finite elementmethods, the most popular strategy to integrate plastic or viscoplastic models is the back-ward Euler scheme (see e.g. Hartmann et al., 1997; Simo and Miehe, 1992; Simo andHughes, 1998) as well as the exponential algorithm (see e.g. Miehe and Stein, 1992; Simo,1993), which was originally proposed in finite viscoplasticity by Weber and Anand (1990).The exponential algorithm has the advantage that the incompressibility of the inelasticdeformations is always fulfilled (cf. Remark 2). In contrast to this, the backward Eulermethod does not preserve the plastic incompressibility at finite step sizes. Consequently,the backward Euler scheme leads to incommensurate errors if large step sizes are applied(see e.g. the numerical studies in Tsakmakis and Willuweit (2003)). In Simo et al. (1985),Simo and Miehe (1992) and also in Luhrs et al. (1997) the error is eliminated due to anadditional assumption, which leads to a cubic equation for a scalar-valued variable. Inaddition to the proposed constitutive theory, a major topic of the present article is the pre-sentation of a numerical integration scheme on the basis of the backward Euler methodthat preserves exactly the incompressibility of inelastic deformations.

2. Finite viscoplasticity

In this section, a simple constitutive theory describing the viscoplastic behavior of met-als is introduced. The proposed theory is formulated in the context of finite deformationsand implies the description of isotropic hardening.

2.1. Kinematics

In continuum mechanics, a material body B consists of material points P and itsmotion is represented by a continuous sequence of configurations. Due to the selectionof a reference configuration, the motion is depicted by the one-to-one mapping:x = vR(X,t). In addition to the current position of the material body, its local changesin space and time are of interest. The local changes in space are represented by the defor-mation gradient, F(X,t) = GradvR(X,t), which is the Frechet-derivative of the motionx = vR(X,t). The deformation gradient is a linear mapping transforming material line ele-ments dX of the reference configuration in material line elements of the current configu-ration: dx = F dX.

In order to represent the inelastic behavior of different types of materials, the multipli-cative decomposition of the deformation gradient (cf. Gurtin and Anand, 2005; Haupt,2002; Lee, 1969; Lee and Liu, 1967; Lubliner, 1985)

F ¼ FeFi ð1Þinto an elastic part Fe and an inelastic part Fi is often applied. Already in the comprehen-sive work of Kroner (1960) about a continuum theory of dislocations and internal stresses


the multiplicative decomposition (1) was proposed. Alternative concepts were introduced,for example, by Green and Naghdi (1965) and Bertram (1998). The applied multiplicativedecomposition (1) implies a stress-free intermediate configuration Ki. Already in the the-ory of Eckart (1948) (cf. the remark in Lee and Liu (1967)) and likewise in Kroner (1958)within the context of crystal plasticity, the concept of stress-free configurations was intro-duced to define plastic states.

It should be mentioned (cf. the discussion in Casey and Naghdi (1980), Green and Nag-hdi (1971), and Haupt (2002)) that both tensors cannot be interpreted as the gradient of adeformation and the multiplicative decomposition (1) is not unique, because the multipli-cative decomposition F ¼ F�eF�i with F�e ¼ Fe

~QT and F�i ¼ ~QFi is also admissible if ~Q is aproper orthogonal tensor. Therefore, it is evident that the material line elementsdx ¼ Fi dX in the intermediate configuration Ki are not Frechet-differentials and all con-stitutive equations must be additionally invariant with respect to rotations ~Q of the inter-mediate configuration Ki.

If the multiplicative decomposition is used, it is well known that the Green strain tensorE = [C � 1]/2 cannot be separated into an elastic and inelastic part. Therein, C = FTF rep-resents the right Cauchy–Green tensor. However, if the Green strain tensor is transformedto the intermediate configuration Ki, the resulting strain tensor C is decomposable into anelastic part Ce and an inelastic part Ci:

C ¼ F�Ti EF�1

i ¼ Ce þ Ci. ð2ÞTherein, the strain tensors Ce and Ci are given by

Ce ¼1

2ðFT

e Fe � 1Þ ¼ 1

2ðCe � 1Þ ð3Þ

and

Ci ¼1

2ð1� F�T

i F�1i Þ ¼

1

2ð1� B�1

i Þ. ð4Þ

In analogy to the right and left Cauchy–Green tensor, C = FTF and B = FFT, the tensorsCe ¼ FT

e Fe and Bi ¼ FiFTi are introduced. In the applied concept, which is based on the

multiplicative decomposition of the deformation gradient (1) and implies the additivedecomposition of the strain measure C on the intermediate configuration Ki (cf. Eq.(2)), the intermediate configuration Ki is used to formulate a thermomechanically consis-tent constitutive theory.

In addition to the multiplicative decomposition of the deformation gradient accordingto Eq. (1), the distinction between volumetric and isochoric deformations is of specialinterest. Based on the suggestion of Flory (1961), the deformation gradient

F ¼ Fvol �F; ð5Þand also the elastic and inelastic part of F, can be decomposed into a volumetric and anisochoric part if the operators

ð�Þvol: A 7!Avol ¼ ðdet AÞ

131 and �ð�Þ : A 7! �A ¼ ðdet AÞ�

13A ð6Þ

are applied:

Fvol ¼ ðdet FÞ131 and �F ¼ ðdet FÞ�

13F. ð7Þ

On account of the property det �A ¼ 1, the tensor �A is a unimodular tensor.


2.2. Thermomechanical restrictions

In order to develop a thermomechanical consistent theory, it is advisable to consider thesecond law of thermodynamics from the beginning. Here, the balance of entropy in formof the Clausius–Duhem inequality is applied. In certain cases, it is expedient to separatethe Clausius–Duhem inequality (cf. Scheidler and Wright, 2001; Truesdell and Noll,1965) into the internal dissipation inequality

di ¼ � _w� _hgþ 1

qR

~T � _E P 0; ð8Þ

and the heat conduction inequality

hch ¼ �1

qRhqR �Gradh P 0. ð9Þ

According to Truesdell and Noll (1965), the postulation that di P 0 and ch P 0 must befulfilled is a special case of the Clausius–Duhem inequality. In general, it is a strongerinterpretation of the Clausius–Duhem inequality. Therein, w is the free energy, h the abso-lute thermodynamic temperature, g the entropy, qR the mass density in the reference con-figuration, qR the heat flux vector in the reference configuration, and Grad the materialgradient. If the Fourier model for heat conduction is used, qR = k(det F)C�1Grad h, theheat conduction inequality is always fulfilled if the heat conduction coefficient is non-neg-ative: k P 0.

For the following considerations the stress power must be formulated in terms of theintermediate configuration Ki:

~T � _E ¼ Fi~TFT

i � F�Ti

_EF�1i ¼ S � C

M

. ð10ÞIn the concept of dual variables, which was originally proposed by Haupt and Tsakmakis(1989) (for an entire overview, see Haupt, 2002), the stress tensor S ¼ Fi

~TFTi and the strain

tensor C are so-called dual variables. Furthermore, the associated objective time derivativeof C is given by the Oldroyd derivative

CM

¼ _Cþ LTi Cþ CLi with ð�Þ

M

¼ _ð�Þ þ LTi ð�Þ þ ð�ÞLi; ð11Þ

which is formulated relative to the intermediate configuration Ki, and therefore calculatedon the basis of the inelastic deformation rate Li ¼ _FiF

�1i . Due to the fact that the Oldroyd

derivative is a linear operator, the relation

CM

¼ CM

e þ CM

i; with CM

e ¼ _Ce þ LTi Ce þ CeLi and C

M

i ¼ _Ci þ LTi Ci þ CiLi ð12Þ

is as well valid.To develop a thermomechanically consistent constitutive theory describing viscoplastic-

ity, the following basic structure of the free energy function is proposed:

w ¼ wðCe; h; y; yeÞ ¼ weðCe; hÞ þ wsðy; ye; hÞ. ð13Þ

This constitutive assumption implies the description of different energy storage and releasemechanisms: the small changes in the atomic lattice due to thermomechanical processes arerepresented by the elastic strain state Ce and the influence of the temperature is given by h.


Moreover, the isotropic hardening effect is modeled by means of the scalar-valued internalvariable ye of strain type. It should be mentioned that the conjugate thermomechanicalquantity, which is an internal variable of stress type (cf. Eq. (19)), can be interpreted asan internal variable for representing isotropic hardening (cf. Section 2.4.2 for details).Furthermore, the dependence of the free energy function on the internal variables ye andy is introduced to represent the experimental observation that energy is stored in the mate-rial body during plastic deformations as discussed in the foregoing section. In general, it isexpedient to separate the free energy function into an elastic part we and a part ws, whichrepresents the energy storage during inelastic deformations.

First, the material time derivative of the free energy function (13) is inserted into theinternal dissipation inequality,

di ¼1

qR

S� qR

owe

oCe

" #� CM

e � gþ owoh

" #_h� ows

oy_y � ows

oye

_ye þ1

qR

S � CM

i

þ owe

oCe

� ½LTi Ce þ CeLi�P 0; ð14Þ

using Eqs. (10) and (12). On account of the identity CM

i ¼ ½LTi þ Li�=2, the last scalar prod-

uct in Eq. (14) can be represented in form of

owe

oCe

� ½LTi Ce þ CeLi� ¼ 2Ce

owe

oCe

" #� CM

i; ð15Þ

if owe=oCe is introduced as an isotropic tensor function depending on Ce. Furthermore, itis assumed that the introduced strain-like internal variable y is given by

y ¼ ye þ yd. ð16ÞIn a moment it becomes clear that this additive decomposition is introduced for represent-ing non-linear isotropic hardening (cf. Eq. (19) in combination with Eq. (35) in Section2.4.2). Considering the identity (15) and the decomposition (16), the internal dissipationis expressed as

di ¼1

qR

S� qR

owe

oCe

" #� CM

e � gþ owoh

" #_h� ows

oyþ ows

oye

" #_y þ ows

oye

_yd

þ 1

qR

Sþ qRCe

owe

oCe

" #� CM

i P 0. ð17Þ

For arbitrary rates of Ce, h, y, yd, and Ci, this inequality implies potential relations for thestress tensor S in the intermediate configuration Ki and the entropy g,

S ¼ qR

owe

oCe

and g ¼ � owoh; ð18Þ

if _y, _yd, and CM

i does not depend on CM

e and _h (cf. Coleman and Noll (1963) and the discus-sions in Jou et al. (1996), Liu (2002), Scheidler and Wright (2001), as well as Truesdell(1984)). Furthermore, the thermomechanical force R of stress type is defined:


R ¼ qR

ows

oye

. ð19Þ

After the introduction of the yield function in Eq. (26), it becomes clear that the thermo-dynamic force R represents isotropic hardening. Considering these relations and also thedefinition of the Mandel stress tensor (see Mandel, 1972; cf. Lubliner, 1990)

P ¼ ½1þ 2Ce�S ¼ CeS; ð20Þthe final form of the remaining dissipation inequality

qRdi ¼ � qR

ows

oyþ R

" #_y þ R _yd þ P � C

M

i P 0 ð21Þ

is a suitable basis to define evolution equations for the internal variables. First, it is as-sumed that the inelastic strain state Ci develops according to (cf. Haupt, 2002; Lion,2000; Lubliner, 1990; Tsakmakis, 1996)

CM

i ¼ ki

PD

kPDk. ð22Þ

Therein, ki P 0 is the inelastic multiplier, which will be specified in Section 2.4.1 and thesuperscript D denotes the deviator of a tensor: AD = A � (trA)/31. If the inelastic defor-mations are isochoric (det Fi = 1), the Oldroyd-rate of Ci must be deviatoric.

For the internal variable y, the evolution equation

_y ¼ffiffiffi2

3

rkCM

ik ¼ffiffiffi2

3

rki ð23Þ

is assumed, which measures the accumulated inelastic strains. Finally, it is postulated thatthe internal variable yd develops according to the evolution equation

_yd ¼ fR. ð24ÞTherein, f is also a non-negative multiplier (details in Section 2.4.2), which is introducedfor representing the non-linear isotropic hardening behavior. Resulting from these defini-tions of the evolution equations, the internal dissipation inequality is given by

qRdi ¼ ki kPDk �ffiffiffi2

3

rR

" #þ fR2 � qR

owoy

_y P 0. ð25Þ

In oder to guarantee the non-negative entropy production for the first term, it is assumedthat viscoplastic deformations occur (i.e., the inelastic multiplier ki is positive) if the yieldfunction

f ¼ kPDk �ffiffiffi2

3

r½kðhÞ þ R�; ð26Þ

which is of von Mises-type (cf. von Mises (1913)), is positive: f > 0. In contrast to this,thermoelastic deformations (i.e., ki = 0) occur if the yield function is non-positive: f 6 0.The introduced temperature-dependent function k(h) > 0 represents the yield radius with-out isotropic hardening. Here, the yield function depends on the Mandel stress tensor P.This dependence on P was originally proposed by Mandel (1972) (cf. Lubliner, 1990). A


non-negative internal dissipation is likewise contributed by the second term. After theintroduction of the free energy function in the next section, the third term in Eq. (25) willbe discussed.

2.3. Free energy

In metal plasticity, different types of energy storage and release mechanisms occur: e.g.,small distortions of the metallic lattice lead to thermoelastic energy contributions. Addi-tionally, plastic deformations are accompanied by energy storage and release phenomenadue to the movement as well as the production and annihilation of dislocations. In thecontext of a finite strain theory, the basic phenomena of thermoelasticity were represented,for example, by Glaser (1991), Helm (2001), Kamlah and Tsakmakis (1999), Miehe (1988),Simo and Miehe (1992), as well as Wriggers et al. (1992). In the present theory, the ther-moelastic part we ¼ weðCe; hÞ ¼ �weðCe; hÞ of the free energy function is given by

qRwe ¼ qR�weðCe; hÞ

¼ jðhÞ2

ln ðdet CeÞ12

h i� �2

þ lðhÞ2ðtr �

Ce � 3Þ � 3jðhÞaðhÞðh� h0Þ ln ðdet CeÞ12

h iþZ h

h0

cd0ð�hÞ d�hþ u0 � h

Z h

h0

cd0ð�hÞ�h

d�hþ g0

� �. ð27Þ

Therein, the first term represents the free energy due to volumetric deformations (cf. Simoand Pister, 1984) and the second term describes the energy storage as a result of isochoricthermoelastic strains (cf. Simo et al., 1985; Simo, 1988). On account of the definition in Eq.(6), the term

�Ce represents the unimodular part of Ce. The third term models the thermo-

elastic coupling phenomena and the energy storage owing to thermal effects are describedby the last terms. In the thermoelastic part of the free energy, j(h) tallies with the bulkmodulus and l(h) with the shear modulus. Moreover, a(h) defines the coefficient of ther-mal expansion, h0 represents the reference temperature, cd0

is the main part of the specificheat capacity at constant deformation, and u0 as well as g0 are necessary to specify the freeenergy at the reference temperature. In a single-phase material, the constants u0 and g0 arearbitrary, because these parameters influence merely the absolute value of the free energyand the entropy, but not the stress–strain relation or the heat production. In contrast tothis, these parameters play an important rule if a multiphase material with phase changesis regarded.

In addition to the thermoelastic part of the free energy, the inelastic part ws of the freeenergy must be defined (cf. Chaboche, 1996; Haupt et al., 1997; Jansohn, 1997; Tsakmakis,1998; Wallin and Ristinmaa, 2005),

qRws ¼ qRwsðy; ye; hÞ ¼cðhÞ

2y2

e þ /ðhÞkðhÞy; ð28Þ

in order to describe the energy storage due to viscoplastic deformations. The first part isnecessary to model non-linear isotropic hardening. Therein, the temperature-dependentfunction c(h) P 0 represents the initial slope of the internal variable R (cf. Eq. (19)). Incontrast to this, the other term represents the thermomechanical coupling phenomenonthat only a certain part of the performed inelastic work is converted into heat and theremaining part is stored in the material body. This fact is well known since the basic inves-tigations of Taylor and Quinney (1934) (cf. Chrysochoos et al., 1989; Helm, 1998; Oliferuk


et al., 1985). The idea to add the term /(h)k(h)y in the free energy was originated byTsakmakis (1998). In Haupt et al. (1997), Helm (1998), and Jansohn (1997) this suggestionwas successfully applied to represent the thermomechanical coupling phenomenon due toplastic deformations. Already in Coleman and Owen (1975), the free energy function de-pends in an ideal plastic material on the plastic strain state. In order to guarantee a non-negative entropy production (cf. Eq. (25)),

qRdi ¼ ki f þffiffiffi2

3

rkðhÞð1� /ðhÞÞ

" #þ fR2 P 0; ð29Þ

the introduced function /(h) is limited: /(h) 2 [0, 1]. Consequently, the proposed model isthermomechanically consistent.

Fig. 1 illustrates the influence of the term /(h) k(h)y on the thermomechanical couplingphenomena. This term is a part of the free energy according to Eq. (28) and represent acertain part of the stored energy of cold work. In contrast to this, the term k(h)(1 �/(h))y results from the time integration of the dissipation qRdi if the case of ideal plasticity(f! 0 and R = 0) is regarded (cf. Eq. (29)). Therefore, the constitutive theory is able tomodel the thermomechanical coupling phenomenon, that energy is stored in the materialbody during plastic deformations.

2.3.1. Implications of the introduced free energy function

On account of the defined free energy function, a few important implications should bediscussed. The stress state in the intermediate configuration Ki results from the potentialrelation according to Eq. (18):

S ¼ qR

ow

oCe

¼ 2qR

o�we

oCe

¼ jðhÞ ln ðdet CeÞ12

h iC�1

e þ lðhÞðdet CeÞ�13 1� 1

3ðtr CeÞC�1

e

� �� 3aðhÞjðhÞðh� h0ÞC�1

e . ð30Þ

Furthermore, the Mandel stress tensor

P ¼ CeS ¼ jðhÞ ln ðdet CeÞ12

h i1þ lðhÞ �CD

e � 3aðhÞjðhÞðh� h0Þ1 ð31Þ

Fig. 1. Interpretation of the material parameter /(h) in the case of ideal plasticity.


is required in the flow rule (cf. Eq. (22)). In the present theory, only isochoric elastic defor-mations influence the evolution of the inelastic deformations, because of PD ¼ lðhÞ �CD

e . Incontrast to this, the spherical part of P depends merely on the volumetric thermoelasticdeformations.

In order to represent the isotropic hardening behavior, the internal variable R of stresstype is required. Using the definition in Eq. (19), the constitutive equation for R is given by

R ¼ qR

ows

oye

¼ cðhÞye ¼ cðhÞðy � ydÞ. ð32Þ

2.4. Evolution equations for internal variables

Resulting from the foregoing thermomechanical considerations in Section 2.2, the basicstructure of the evolution equations for the internal variables Ci and yd of strain-type areknown. However, the scalar-valued multipliers ki and f in these evolution equations arestill undefined.

2.4.1. Evolution equation for the inelastic strains

On account of the defined yield function in Eq. (26), the flow rule is representable as anormality rule (cf. the previous definition in Eq. (22)):

CM

i ¼ kiPN with PN ¼of

oP¼ PD

kPDk. ð33Þ

In the present constitutive theory, the viscoplastic material behavior of metals is repre-sented by means of an inelastic multiplier,

ki ¼1

gdðhÞfrd

� �mdðhÞ

; ð34Þ

which was originally proposed by Perzyna (1963). Therein, the operator Æxæ = (jxj + x)/2 isknown as MacCauley-bracket. In addition, gd(h) > 0 and md(h) are temperature-depen-dent material parameters to represent the viscous behavior. Consequently, the inelasticmultiplier ki is a non-negative function. Inelastic deformations take place if the inelasticmultiplier ki is positive, i.e. f > 0. For md = 1, the inelastic multiplier merges into the sug-gestion of Hohenemser and Prager (1932).

2.4.2. Evolution equation for the internal variable yd

The multiplier f was introduced in the evolution equation for the internal variable yd (seeEq. (24)), which is a part of the non-linear isotropic hardening concept. Due to isotropichardening, the model incorporates the experimentally observed evolution of the yieldradius (cf. Bruhns et al., 1992; Chaboche, 1993a; Khan and Jackson, 1999; Lubliner, 1990).

On the basis of the thermomechanical considerations in Section 2.2, the evolution equa-tion for yd is assumed to be

_yd ¼ fR ¼ bðhÞcðhÞ _yR. ð35Þ

Therein, f ¼ ðbðhÞ=cðhÞÞ _y is defined to represent non-linear isotropic hardening (cf. Chab-oche, 1993a; Jansohn, 1997; Wallin and Ristinmaa, 2005): if the evolution equation (35) is


inserted into the time derivative of R (cf. Eq. (32)), an equivalent evolution equation forthe internal variable R is obtained:

_R ¼ cðhÞ _y � bðhÞ _y � 1

cðhÞdcdh

_h

� �R. ð36Þ

For temperature-independent material parameters, this ordinary differential equationhas the solution R(y) = (c/b) (1 � exp(�by)) if the initial condition R(y = 0) = 0 isapplied. Consequently, c represents the initial slope of R and the material param-eter b is responsible for the saturation value: R(y =1) = c/b.

Remark 1 (Stored energy of cold work). After the introduction of the constitutive theory,an important property should be discussed. For simplicity, it is assumed that all materialparameters are temperature-independent. Moreover, if a slow process or a vanishingviscosity is considered, the value of the yield function can be neglected (f! 0). In thisspecial case, the inelastic stress power wi is given by

wi ¼ P � CM

i ¼ ½k þ RðyÞ� _y ð37Þand the inelastic work can be calculated by using Eq. (36):

W i ¼Z t

0

wi d�t ¼Z y

0

½k þ Rð�yÞ� d�y ¼ k þ cb

� �y � RðyÞ

b. ð38Þ

The accompanied stored energy of cold work is known from Eq. (28):

Es ¼ qR~wsðyÞ ¼

1

2cRðyÞ2 þ /ky. ð39Þ

In the considered case, the isotropic hardening function is only a function of the internalvariable y: R(y) = (c/b) (1 � exp(�by)). In Fig. 2, the ratio of the stored energy to coldwork is calculated by means of the material parameters according to Table 3. This exampledemonstrates the influence of the material parameter / (cf. Eq. (28)) on the amount of en-

Fig. 2. Ratio of stored energy to cold work for different values of /.


ergy storage. Moreover, the dependence of the energy storage on the plastic deformationsis nearby the experimental facts (cf. Bever et al., 1973; Chrysochoos et al., 1989; Helm,1998; Oliferuk et al., 1985).

3. Numerical treatment

In the foregoing section, a simple constitutive theory to describe the material behaviorof metals during viscoplastic deformations was defined. The proposed theory is thermome-chanically consistent and represents phenomena like isotropic hardening and thermome-chanical coupling processes. On the basis of this simple theory, the major topic of thearticle is discussed in this section: the preservation of viscoplastic incompressibility in abackward Euler integration scheme.

3.1. Introduction to the backward Euler method

In the current context and also in different fields of natural sciences, ordinary differen-tial equations of the type

_yð�tÞ ¼ Aðyð�tÞ;�tÞyð�tÞ ð40Þ

are of interest (y 2 Rn). Depending on the operator A, it is necessary to solve this first or-der ordinary differential equation numerically. If a forward or backward Euler method isapplied, the time derivative is replaced by the difference quotient:

_yð�tÞ ¼ yð�t þ hÞ � yð�tÞh

�OðhÞ. ð41Þ

Through this, the local error in the time discretization is in the order of magnitude O(h),wherein O is a vector representing the big-O Landau symbol. If Eq. (40) is regarded at thetime �t ¼ t þ Dt with h = �Dt, the numerical procedure is called backward Euler methodand can be expressed in the form

yðt þ DtÞ ¼ yðtÞ þ DtAðyðt þ DtÞ; t þ DtÞyðt þ DtÞ þOðDt2Þ; ð42Þ

or in the equivalent equation

yðt þ DtÞ ¼ ½1� DtAðyðt þ DtÞ; t þ DtÞ��1½yðtÞ þOðDt2Þ�. ð43Þ

Therein, [1 � DtA(y(t + D t),t + Dt)]�1 is the numerical integration operator of the appliedbackward Euler method. In contrast to this, the forward Euler method is given by

yðt þ DtÞ ¼ yðtÞ þ DtAðyðtÞ; tÞyðtÞ þOðDt2Þ; ð44Þ

if Eq. (40) is considered at the time �t ¼ t and h = Dt. Both schemes are one-step integrationmethods and have the convergence order one. In contrast to the forward Euler method,the backward Euler method leads to a non-linear system of algebraic equations, whichmust be solved with an appropriate strategy.


3.2. Ordinary differential equation of a unimodular tensor

A specific ordinary differential equation is given by

_YðtÞ ¼ AðYðtÞ; tÞYðtÞ; ð45Þwhere Y(t) must be a unimodular tensor for all t, i.e. detY(t) = 1. Using the Frechet-deriv-ative of g(Y(t)) = detY(t), the time derivative of g(Y(t)) ð _gðYðtÞÞ ¼ 0Þ leads totrA(Y(t),t) = 0 for preserving detY(t) = 1 in the differential equation (45) (cf. Haireret al., 2002). According to the definitions in Hairer et al. (2002), the differential equation(45) under the constraint that Y(t) is a unimodular tensor can be interpreted as a differen-tial equation on the manifold M ¼ fYðtÞ 2 LinðVÞj det YðtÞ ¼ 1g, because any initial con-dition Yðt0Þ 2M implies YðtÞ 2M for all t.

If the backward Euler method (cf. Eq. (43)) is applied for integrating the differentialequation (45), the integration operator is given by

Iðt þ Dt;DtÞ ¼ ½1� DtAðYðt þ DtÞ; t þ DtÞ��1 ð46Þand Y(t + Dt) results from

Yðt þ DtÞ ¼ Iðt þ Dt;DtÞ½YðtÞ þOðDt2Þ�. ð47ÞIn the application of the backward Euler method, the local error O(Dt2), resulting from thetime discretization, is neglected. Consequently, the recursion formula

iY ¼ iI nY ¼ ½1� Dt iA��1 nY ð48Þmust be considered. Therein, i = n + 1 and the abbreviations iY for the numerical result ofY(t + Dt), nY for Y(t) as well as iI for Iðt þ Dt;DtÞ and iA for A(Y(t + Dt), t + Dt) are ap-plied. On account of the fact that the integration operator does not fulfill the conditionðdet iIÞ ¼ ðdet ½1� Dt iA�Þ ¼ 1 for Dt > 0, an integration step which starts with(det nY) = 1 results in a tensor iY, which is not a unimodular tensor. Consequently, thebackward Euler scheme calculates a solution, which is not on the manifold M.

In order to avoid incorrect or unphysical results, an important requirement of a numer-ical integrator is that the numerical solution lies exactly on the manifold (cf. Hairer, 2001).For the given problem, such a numerical integrator can be constructed if Eq. (47) is writtenin the following form:

Yðt þ DtÞ ¼ Iðt þ Dt;DtÞYðtÞ ½1þ YðtÞ�1OðDt2Þ�|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

eðYðtÞ;Dt2Þ

. ð49Þ

Therein, a function e(Y(t),Dt2) is defined, which implies the local error. The inverse of Y(t)exists because Y(t) should be a unimodular tensor. Thus, Eq. (49) leads to

det eðYðtÞ;Dt2Þ ¼ ½det Iðt þ Dt;DtÞ��1 ð50Þif (det Y(t + Dt)) = 1 is postulated. Therefore, the spherical part of e is determinable

e�ðYðtÞ;Dt2Þ ¼ ½det Iðt þ Dt;DtÞ��131. ð51Þ

On the basis of this relation, Eq. (49) results in

Yðt þ DtÞ ¼ �Iðt þ Dt;DtÞYðtÞ�eðYðtÞ;Dt2Þ ð52Þwith the integration operator (cf. the notation �I according to Eq. (6))


�Iðt þ Dt;DtÞ ¼ ½1� DtAðYðt þ DtÞ; t þ DtÞ��1

½det ð1� DtAðYðt þ DtÞ; t þ DtÞÞ��13

. ð53Þ

In Eq. (52), the tensor �eðYðtÞ;Dt2Þ is the unimodular part of e(Y(t),Dt2), which implies theremaining part of the local error. This term has the property

limDt!0

�eðYðtÞ;Dt2Þ ¼ 1. ð54Þ

Consequently, the resulting recursion formula

iY ¼ i �I nY ¼ ðdet ½1� Dt iA�Þ13½1� Dt iA��1 nY ð55Þ

preserves exactly the unimodularity of iY independent of the magnitude of thestep size if nY is a unimodular tensor. In this case, the integration operatori �I ¼ ðdet ½1� Dt iA�Þ

13½1� Dt iA��1 is a unimodular tensor, too. Thus, in the context of

the given differential equation (45), a frequently criticized shortcoming of the backwardEuler method is avoided, if the developed numerical strategy is performed.

Remark 2 (Exponential map). It is well known that the exponential map of a tensorcan be applied to integrate a differential equation of the type _YðtÞ ¼ AðYðtÞ; tÞYðtÞ (cf.Miehe and Stein, 1992; Simo, 1993; Weber and Anand, 1990). Due to the identitydet (exp(A)) = exp(trA), the exponential map transforms a deviatoric tensor into aunimodular tensor. Therefore, if Eq. (45) with A = AD and (det nY) = 1 is integrated bymeans of

iY ¼ exp½Dt iA� nY; ð56Þthe result iY fulfills the constraint (detiY) = 1 exactly. Depending on the tensor A(Y(t),t),the calculation of the exponential map requires a spectral decomposition iA. Furthermore,if the tensors iY, iA, and nY are not coaxial, a symmetric tensor nY results in a tensor iY,which is not symmetric. In order to preserve the symmetry of iY, additional assumptionsare required (cf. Dettmer and Reese (2004)).

In addition to the proposed integrator i �I of the modified backward Euler method (cf.Eq. (55)), the exponential map exp [Dt iA] in Eq. (56) is also a geometric numericalintegrator for the discussed problem.

Remark 3 (Standard projection methods). In order to fulfill a given constraint, Hairer(2001) and Hairer et al. (2002) propose so-called projection methods: in the standard pro-jection method, it is assumed that ny is on the manifold M ¼ fyjgðyÞ ¼ 0g. In the first stepof the algorithm, a value nþ1~y is determined by an arbritrary one-step method. In the sec-ond step, the value nþ1y 2M results from the projection of nþ1~y onto the manifold M. Thisprojection method is a common strategy to fulfill a given constraint.

Hairer et al. (2002) remark that the distance between nþ1~y and the manifold M is notlarger than the local error O(Dtp+1), wherein p is the order of the applied integrationscheme to obtain nþ1~y. Thus, the whole integration method has at least the order p.

3.3. Stress computation in finite viscoplasticity

In the present section, the proposed integration strategy is applied to finite viscoplastic-ity. In the framework of the finite element method, the current deformation gradient iF


and the temperature ih at the time t + Dt as well as all quantities at the time t, which aredenoted with n (e.g. nF), are known. The change in the deformation gradient is given by

DF ¼ iF nF�1. ð57ÞResulting from the definition of a yield function, the model consists of elastic and elasto-viscoplastic parts. Consequently, the stress computation is divided into an elastic predictorstep and a viscoplastic corrector step:

� In the elastic predictor step, it is assumed that the stress state is in the elastic range.Therefore, the inelastic part Fi of the deformation gradient F is constant: iFi = nFi.On the basis of this assumption, the trial state of Be is calculable:

tBe ¼ DF nFeðDF nFeÞT ¼ DF nBeDFT. ð58ÞThe notation t(Æ) (t is the abbreviation for trial-state) indicates that these quantities arebased on the assumption of purely elastic deformations in the predictor step. Using theidentity S ¼ FeSFT

e , the transformation of the stress tensor S from the intermediate con-figuration Ki into the weighted Cauchy stress tensor S = (det F)T (T: Cauchy stress ten-sor) leads to

S ¼ FeSFTe ¼ lðhÞ�BD

e þ jðhÞ lnffiffiffiffiffiffiffiffiffiffiffiffiffidet Be

p� 3aðhÞðh� h0Þ

h i1. ð59Þ

Consequently, the trial stress state is given by

tS ¼ lðihÞ t �BDe þ jðihÞ ln

ffiffiffiffiffiffiffiffiffiffiffiffiffiffidet tBe

p� 3aðihÞðih� h0Þ

h i1. ð60Þ

In order to verify the assumption of the predictor step, the trial value of the yield func-tion must be determined. On account of the fact that in the present model the weightedCauchy stress tensor S is an isotropic tensor function of Be, i.e. S and Be are coaxial, theinteresting identity (cf. Helm, 2001)

kPDk ¼ kSDk ð61Þis valid. This can be understood if the transformation

PðDÞ ¼ FePDFT

e ¼ BeSD ð62Þ

is considered. With respect to the yield function, the introduced Mandel stress tensor P,which operates in the intermediate configuration, becomes a physical meaning: due tothe yield function according to Eq. (26), viscoplastic deformations occur if the weightedCauchy stress state is larger than the current yield radius

f ¼ kSDk �ffiffiffi2

3

r½kðhÞ þ R� > 0. ð63Þ

Consequently, the assumption of the predictor step is correct if the trial-state of theyield function

tf ¼ ktSDk �ffiffiffi2

3

r½kðihÞ þ tR� 6 0 ð64Þ

is non-positive. Therein, the trial-state of the isotropic hardening function R is given by

tR ¼ cðihÞðny � nydÞ. ð65ÞIn the other case, tf > 0, inelastic deformations occur and a corrector step is required.


� The corrector step is computationally more expensive, because a system of ordinary dif-ferential equations must be solved. Up to now, the evolution of the inelastic strains isonly expressed on the intermediate configuration (cf. Eq. (33)). To apply an appropriatenumerical integration scheme, it is advisable to transform the tensorial evolution equa-tions to the reference configuration in order to obtain material time derivatives (cf. Pin-sky et al., 1983). The transformation of the evolution equation for the inelastic strainstate Ci to the reference configuration results in

_Ci ¼ 2FTi CM

iFi ¼ 2kiCi~PNCi with ~PN ¼ F�1

i PNF�Ti . ð66Þ

In order to preserve the viscoplastic incompressibility (det Fi = 1), the tensor Ci is a uni-modular tensor: detCi = 1. Consequently, the strategy of Section 3.2 is applied for pre-serving the unimodularity of Ci and for integrating the constitutive equation:

iCi ¼ ðdet ½1� 2 in iCii ~PN�Þ

13½1� 2 in iCi

iPN��1 nCi ð67Þusing in = Dt iki. In the present article, the entire problem is solved on the current con-figuration. Using the well-known identities

iBe ¼ iF iC�1i

iFT; nBe ¼ nF nC�1i

nFT; ð68Þand

tBe ¼ DF nBeDFT ¼ DF nF nC�1i

nFTDFT ¼ iF nC�1i

iFT ð69Þas well as the transformation rule (cf. Eq. (62))

iPN ¼ iF i ~PNiFT ¼

iBeiSD

kiSDk; ð70Þ

the recursion formula (67) can be transformed to KM:

iBe ¼ ðdet ½1� 2 in iB�1e

iPN�Þ�13 tBe½1� 2 in iB�1

eiPN�. ð71Þ

The final form of recursion formula is given by

iBe ¼ ðdet ½1� 2 in iN�Þ�13 tBe½1� 2 in iN�; ð72Þ

because of the identity

iN ¼ iB�1e

iPN ¼iSD

kiSDk. ð73Þ

Independent of the size of the time step, Eq. (71) and also Eq. (72) fulfills the necessarycondition

det iBe ¼ det tBe ¼ ðdet iFÞ2 ð74Þto preserve detCi = 1. In addition to the differential equation (66), the model containsanother differential equations according to Eq. (23) and also Eq. (35), which are neces-sary to represent the non-linear isotropic hardening behavior. The application of thebackward Euler method (cf. Section 3.1) leads to the integration steps

iy ¼ ny þffiffiffi2

3

rin ð75Þ


and

iyd ¼ nyd þffiffiffi2

3

rbðihÞcðihÞ

in iR. ð76Þ

In the general case, a system of seven non-linear equations must be solved to determineiBe as well as iyd. However, a few algebraic modifications reduce the problem. Due tothis, the solution of three non-linear equations for three unknown scalar-valued vari-ables is required. Similar strategies are applied e.g. in Hartmann et al. (1997), Luhrset al. (1997), Simo and Hughes (1998), Simo and Miehe (1992). The development ofthe improved algorithm starts with the calculation of the deviatoric stress state. There-fore, the recursion formula according to Eq. (72) as well as the definition

ia1 ¼ det ½1� 2 in iN� ð77Þis inserted into Eq. (59):

iSD ¼ lðihÞa�13

1t �BD

e � 2 inðt �Be

iSDÞD

kiSDk

" #. ð78Þ

In the next step, Eq. (78) is partially solved to obtain iSD:

iSD ¼ A�11 ðin; ia1; kiSDkÞR1ðin; ia1;

ia2; kiSDkÞ; ð79Þapplying the abbreviations

A1ðin; ia1; kiSDkÞ ¼ 1þ 2lðihÞ ia�1

31

in

kiSDkt �Be ð80Þ

and

R1ðin; ia1;ia2; kiSDkÞ ¼ ia

�13

1tSD þ 2

3

lðihÞ in ia2

kiSDk1

� �; ð81Þ

as well as the definition

ia2 ¼ tr ðt �BeiSDÞ. ð82Þ

Consequently, the current deviatoric stress state can be expressed as a function of sca-lar-valued variables, but the value of iiSDi is still undetermined. However, the norm ofiSD can be depicted as a function of in, ia1, and ia2: first, the yield function (Eq. (63)) issolved for iiSDi

kiSDk ¼ if þffiffiffi2

3

r½kðihÞ þ iR�. ð83Þ

Using Eq. (34), the current value of the yield function if is representable as a function ofin:

if ¼ f ðinÞ ¼ rd

gdðihÞ inDt

� � 1mdðihÞ

. ð84Þ

Furthermore, the current value of the isotropic hardening stress can be specified as afunction of in: inserting Eqs. (75) and (76) into Eq. (32), the result,


iR ¼ cðihÞ ny þffiffiffi2

3

rin� nyd �

ffiffiffi2

3

rbðihÞcðihÞ

in iR

" #; ð85Þ

is solved to iR:

iR ¼ RðinÞ ¼tRþ

ffiffi23

qcðihÞ in

1þffiffi23

qbðihÞ in

ð86Þ

using the definition according to Eq. (65). Consequently, iR is representable as a func-tion of in.

Inserting Eqs. (84) and (86) into Eq. (83), the norm of the current stress deviator islikewise a function of in:

is ¼ sðinÞ ¼ kiSDk ð87Þ

¼ rdgdðihÞ in

Dt

� � 1mdðihÞ þ

ffiffiffi2

3

rkðihÞ þ

tRþffiffi23

qcðihÞ in

1þffiffi23

qbðihÞ in

264

375. ð88Þ

Altogether, the deviator of the current stress state is representable as a function of threescalar-valued variables

iSD ¼ A�11 ðin; ia1ÞR1ðin; ia1;

ia2Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Nðin; ia1;

ia2Þ

ð89Þ

with

A1ðin; ia1Þ ¼ 1þ 2lðihÞ ia�1

31

inis

t �Be ð90Þ

and

R1ðin; ia1;ia2Þ ¼ ia

�13

1tSD þ 2

3

lðihÞ in ia2

is1

� �. ð91Þ

These three variables – in, ia1, and ia2 – are determined by solving the following systemof non-linear equations:

/1ðin; ia1;ia2Þ ¼ kNðin; ia1;

i2Þk � sðinÞ ¼ 0; ð92Þ

/2ðin; ia1;ia2Þ ¼ ia1 � det 1� 2 in

Nðin; ia1;ia2Þ

kNðin; ia1; ia2Þk

� �¼ 0; ð93Þ

/3ðin; ia1;ia2Þ ¼ ia2 � trðt �BeNðin; ia1;

ia2ÞÞ ¼ 0. ð94Þ

To solve this system of non-linear equations, an appropriate numerical method like aNewton method must be applied. If the scalar-valued variables in, ia1, and ia2 areknown, all other quantities are calculable: the current deviatoric stress state iSD is givenby Eq. (89), the current elastic strain state follows from (cf. Eq. (71))

iBe ¼ ia�1

31

tBe � 2 in tBe

Nðin; ia1;ia2Þ

kNðin; ia1; ia2Þk

� �; ð95Þ

as well as iy and iyd result from Eqs. (75) and (76). In contrast to the unmodified back-ward Euler method, the proposed algorithm on the basis of a modified backward Euler


scheme preserves the viscoplastic incompressibility for arbitrary step sizes and requiresmerely the solution of three non-linear equations for three unknown scalar-valuedvariables.

For the subsequent numerical studies, the developed algorithm is denoted by stressalgorithm A.

Remark 4 (Unmodified backward Euler scheme). The stress algorithm A reduces to theunmodified backward Euler method if

iBe ¼ tBe½1� 2 in iN� ¼ tBe 1� 2 inNðin; ia1;

ia2ÞkNðin; ia1; ia2Þk

� �ð96Þ

is applied instead of Eq. (71). Therefore, the computational effort is identical: a system ofthree non-linear equations (Eq. (92)–(94)) must be solved but the unmodified backwardEuler scheme does not preserve the viscoplastic incompressibility of the inelastic deforma-tions in finite step sizes. The algorithm based on the unmodified backward Euler method isdenoted by stress algorithm B.

Remark 5 (Small elastic strains). Of course, the viscoplastic behavior of metals is charac-terized by small elastic strains but large inelastic strains and finite local rotations. If the Tay-lor series of (det iBe) for Be � 1 is considered, the approximation (det iBe) � (trBe � 2) isvalid. If also the deviatoric stress state according to Eq. (59) is regarded in its linearized form

SD ¼ lðhÞBDe ; ð97Þ

an efficient stress algorithm can be specified: first, the numerical error due to the time dis-cretization is regarded in the integration scheme of the backward Euler method,

iBe ¼ tBe½1� 2 in iN� þOðDt2Þ; ð98Þin order to consider the approximation of small elastic strains in the stress algorithm. Theerror can be subdivided into a spherical and deviatoric part. In stress algorithm C, thespherical part is approximated by (tr iBe) � [(det iF)2 + 2]. Consequently, the recursionformula

iBe ¼ ½tBe � 2 in tBeiN�D þ 1

3½ðdet iFÞ2 þ 2�1 ð99Þ

is obtained. In contrast to the foregoing algorithm, this strategy results in a system ofmerely two non-linear equations

u1ðin; ia2Þ ¼ kNðin; ia2Þk � sðinÞ ¼ 0; ð100Þu2ðin; ia2Þ ¼ ia2 � tr ðtBeNðin; ia2ÞÞ ¼ 0; ð101Þ

using the relations N(in, ia2) = A1�1(in)R1(in, ia2) and

A1ðin; ia1Þ ¼ 1þ 2lðihÞ inis

tBe ð102Þ

as well as

R1ðin; ia1;ia2Þ ¼ tSD þ 2

3

lðihÞ in ia2

is1. ð103Þ


Remark 6 (Volume-preserving of viscoplastic deformations). In Simo et al. (1985), Simoand Miehe (1992) and also in Luhrs et al. (1997) a further strategy is proposed, whichcan be considered in the framework of the present constitutive theory if the assumptionof small elastic strains is applied: here, the scalar-valued variable ie in the recursionformula

iBe ¼ ½tBe � 2 in tBeiN�D þ ie1 ð104Þ

of Be is determined from the condition

ðdet iFÞ2 ¼ det ðtBe½1� 2 in iN� þ ie1Þ. ð105ÞThis condition results in a cubic polynomial in ie, which has at least one real root. Thus,the algorithm preserves exactly the incompressibility of the viscoplastic deformations.If the constitutive equation for the stress tensor depends only on the deviator of Be – inthe present model, this is the special case of small elastic deformations – the constraintaccording to Eq. (105) can be solved after the determination of BD

e . According to the def-inition of Hairer et al. (2002), this algorithm can be understood as standard projectionmethod (cf. Remark 3).

3.4. Numerical examples

In the foregoing section, a modified backward Euler method (stress algorithm A) wasdeveloped, which is able to preserve exactly the incompressibility of viscoplastic deforma-tions. In order to compare this strategy, the unmodified backward Euler scheme (stressalgorithm B) and an additional modification of the backward Euler scheme (stress algo-rithm C) were introduced. First, these algorithms are compared in the case of simple shear.Thereafter, the applicability of stress algorithm A is shown in a more complex finite ele-ment simulation.

3.4.1. Simple shear

The deformation gradient F = 1 + c12(t)e1 � e2 represents an isochoric deformation,which is known as simple shear. Here, a monotonous loading process to a maximumdeformation of c12(tmax) = c0 = 1 during a time interval of tmax = 100 s is regarded:c12(t) = (c0t)/tmax. To compare the different algorithms, a volumetric error measure � isdefined: � = j1 � detCij. Three different time step sizes are investigated: Dt = 5 s,Dt = 0.5 s, and Dt = 0.05 s. All calculations are carried out with the material parametersof a steel-like material listed in Tables 1 and 2.

Table 2Viscoplastic material parameters (simple shear)

gd (MPa/s) md (–) k(h) = k0 (MPa) c (MPa) b (–)

2 · 106 3 200 2500 25

Table 1Thermoelastic material parameters

l (MPa) j (MPa) a (1/K) cd0(J/(kg K)) k (W/(m K)) q (kg/m3)

80,000 170,000 1.1 · 10�5 465 58 7870


From theoretical point of view, the proposed stress algorithm A preserves exactly theunimodularity of the integration variable, but it is impossible to avoid numerical errorsin magnitude of the machine precision: therefore, the errors � in algorithm A are practicallyzero for all step sizes. In contrast to this algorithm, the errors in the stress algorithms B andC are depicted in Fig. 3. In the case of the unmodified backward Euler scheme (stress algo-rithm B), the error increases nearly linear if the deformation is increased. Furthermore, adecreasing time step size reduces the error slowly (cf. Fig. 3). Consequently, the unmodifiedbackward Euler method is not suitable to integrate Be at large time steps if the constitutivetheory depends on the volumetric part of Be (e.g. the Cauchy stress tensor according toEq. (59)). In opposition to that the error in the stress algorithm C depends mostly on geo-metric effects, because of the applied approximation tr iBe = (det iF)2 + 2 + O(iBe � 1i2).However, compared to the unmodified backward Euler method (stress algorithm B), theerror measure � is notedly reduced, because of the assumption made in algorithm C.

Finally, the influence of the integration error on the computed stress state is discussed:due to the capability of the stress algorithm A, the hydrostatic pressure according to Eq.(59) is practically zero. Consequently, this algorithm calculate the exact solution for thehydrostatic pressure. In contrast to this, the hydrostatic stress states of the algorithms B

and C are depicted in Fig. 4. These algorithms lead to significant hydrostatic compressionstresses, because both algorithms compute detCi > 1 for the applied simple shear deforma-tion. In particular, the stress algorithm B causes improper results. A very interesting resultis obtained if the norm of the stress deviator is regarded: for a given step size, all algo-rithms calculate nearly identical stresses. Representative results obtained by stress algo-rithm A are depicted in Fig. 5: in all algorithms, a rough time discretization leads toacceptable deviatoric stresses.

3.4.2. Necking in a tension test

A popular example to demonstrate the capability of a thermomechanical theory forfinite plasticity as well as viscoplasticity is the necking of a circular bar under tension load(e.g. Glaser, 1991; Lehmann and Blix, 1985; Simo and Miehe, 1992; Wriggers et al., 1992).In order to verify the developed theory in complex thermomechanical processes, the stress

Fig. 3. Error � = j1 � detCij vs. shear deformation c12 during simple shear.

Fig. 4. Hydrostatic pressure in stress algorithms B and C.

Fig. 5. Norm of the stress deviator resulting from stress algorithm A.


algorithm A, which preserves exactly the incompressibility of the viscoplastic deforma-tions, is implemented into the commercial finite element program ABAQUS/Explicit(cf. Hibbitt, Karlsson & Sorensen, Inc., 2002a,b). In general, the explicit strategy requiresvery small time step sizes. However, in the regarded case of quasi-static deformations, thetime increment can be effectively increased by using the mass scaling strategy. For theimplementation into an implicit finite element code the consistent linearization of the ther-momechanical problem would required (cf. Glaser, 1991; Miehe, 1988; Simo and Miehe,1992; Wriggers et al., 1992). In the present example, a circular bar with a diameter of10 mm and a length of 40 mm is considered. On both ends of the bar, displacement bound-ary conditions are applied, which stretch the bar in the axial direction with a magnitude of0.5 mm/s for 10 s. Furthermore, the initial temperature of the material body is 293 K andthis temperature is held constant at the ends of the bar during the whole deformation pro-cess. Thus, the thermal boundary conditions of a clamping device are simulated. If theload is applied and the model does not include a softening effect, a nearly homogeneousdeformation occurs in a thermomechanical simulation, because small thermoelastic strains


resulting from the inhomogeneous temperature field destroys the homogenous deforma-tion. Necking would occur if the transversal displacement at the ends are suppressed,an imperfection is introduced, or a softening mechanism is introduced in the constitutivemodel. In the present numerical example, the initial yield radius depends slightly on tem-perature (Glaser, 1991; Simo and Miehe, 1992):

kðhÞ ¼ k0½1� akðh� h0Þ� ð106Þwith k0 and also ak according to Table 3. Such a small softening effect is able to trigger thenecking phenomenon in a thermomechanical process with inhomogeneous temperaturedistribution.

Considering the symmetry of the thermomechanical problem, the simulation is carriedout with 900 axisymmetric 4-node elements using reduced integration techniques (cf. Hib-bitt, Karlsson & Sorensen, Inc., 2002b). Fig. 6 represents the used mesh. If the special caseof an isothermal process is regarded, the specimen does not show a necking phenomenon,i.e. the transversal deformation is constant. In Fig. 9 the transversal displacements u1 on

Table 3Viscoplastic material parameters (tension test)

gd (MPa/s) md (–) k0 (MPa) ak (1/K) c (MPa) b (–)

10,000 2 350 1 · 10�3 10,000 25

Fig. 6. Initial and final shape (thermomechanical coupled solution with / = 0.1).

Fig. 7. Reaction force vs. applied displacement.


the outer surface are depicted at the end of the deformation process (t = 10 s) and Fig. 8shows the accompanied temperature along the �y-coordinate (cf. Fig. 6). The reaction forceF2 due to the applied displacement u2 is depicted in Fig. 7. As known from Simo andMiehe (1992), stress softening occurs in this case.

In general, the temperature field changes due to the self-heating effect if the materialbody is loaded. In order to represent the material property that energy is stored in thematerial body due to inelastic deformations, the function /(h) 2 [0,1] was introduced(cf. Section 2 and Fig. 1). For larger values of /, the model predicts a larger part of energy,which is stored in the material during cold work. In Figs. 7–9, the value of / is varied. Forsmaller values of /, the temperature of the specimen increases. Due to the temperaturedependent yield radius (Eq. (106)), larger transversal displacements are observed for smal-ler values of /, because higher temperatures are involved in such cases (cf. Fig. 9).

Fig. 8. Temperature along the �y-coordinate.

Fig. 9. Transversal displacements u1 vs. the �y-coordinate.


4. Conclusions

Due to the lack of exact solutions, the numerical integration of differential equations isrequired in different fields of natural sciences. In some cases, the integration variable hassome specific properties, which should be conserved by the numerical integration method.The consideration of specific properties during numerical integration is known as geomet-ric integration. In the present context and also in other constitutive theories describinginelasticity, the inelastic deformations are assumed to be isochoric. Consequently, thisproperty should be preserved in the numerical integration procedure. In the present frame-work, the integration of a ordinary differential equation for a unimodular tensor isrequired. In order to fulfill the unimodularity of the integration variable, a modified back-ward Euler scheme was developed. Independent of the applied step size, this modifiedbackward Euler method preserves exactly the unimodularity of the integration variable(stress algorithm A).

The developed integration scheme is applied to a simple model, which describes thefinite viscoplastic behavior of metals. The constitutive theory is developed in the frame-work of continuum thermomechanics. In the suggested model, the viscoplastic deforma-tions are isochoric. In order to focus the attention on the integration algorithm, atheory with isotropic hardening and without kinematic hardening is introduced and inves-tigated numerically. However, if non-monotonous deformation processes take place, theconsideration of kinematic hardening is required. Also in this case, the application ofthe present integration scheme can be applied without problems.

As pointed out in the article, metals show strong thermomechanical coupling effects:during the inelastic deformations, a certain part of the inelastic work is stored in themicrostructure of the material. Consequently, the thermomechanical model should be rep-resent this material property. On account of the introduced parameter / the amount ofstored energy can be justified in a wide range without influence on the isothermal behaviorof the model.

Numerical examples show that the developed integration algorithm (stress algorithm A)is able to preserve exactly the incompressibility of the viscoplastic deformations. In thesimple shear example, the developed stress algorithm is compared with an unmodified


backward Euler scheme (stress algorithm B) and an additional modification of the back-ward Euler scheme (stress algorithm C). The stress algorithm C is based on the assumptionof small elastic deformations and does not require the solution of an additional non-linearequation. The resulting integration error is sufficiently small for different types of applica-tions. In contrast to this, the unmodified backward Euler method (stress algorithm B)requires very small step sizes to obtain (det Ci) � 1. In order to demonstrate that the devel-oped stress algorithm A is able to apply in non-linear finite element analyses, the neckingprocesses of a circular bar under tension load is simulated.

Acknowledgments

The author gratefully acknowledges for the support of this work by the German Re-search Foundation (DFG). Furthermore, the author thanks very much Prof. Dr. -Ing. Pe-ter Haupt and Priv. -Doz. Dr. -Ing. Stefan Hartmann for their profound discussions.

References

Benallal, A., Le Gallo, P., Marquis, D., 1989. An experimental investigation of cyclic hardening of 316 stainlesssteel and of 2024 aluminium alloy under multiaxial loadings. Nuclear Engineering and Design 114, 345–353.

Bertram, A., 1998. An alternative approach to finite plasticity based on material isomorphisms. InternationalJournal of Plasticity 52, 353–374.

Bertram, A., 2003. Finite thermoplasticity based on isomorphisms. International Journal of Plasticity 19, 2027–2050.

Bever, M., Holt, D., Titchener, A., 1973. The Stored Energy of Cold Work. Progress in Materials Science, vol. 17.Pergamon, Oxford.

Bodner, S.R., Lindenfeld, A., 1995. Constitutive modelling of the stored energy of cold work under cyclic loading.European Journal of Mechanics A – Solids 14, 333–348.

Bruhns, O., Lehmann, T., Pape, A., 1992. On the description of transient cyclic hardening behaviour of mild steelCK15. International Journal of Plasticity 8, 331–359.

Bucher, A., Gorke, U.-J., Kreißig, R., 2004. A material model for finite elasto-plastic deformations considering asubstructure. International Journal of Plasticity 20, 619–642.

Casey, J., Naghdi, P., 1980. A remark on the use of the decomposition F = FeFp in plasticity. Journal of AppliedMechanics 47, 672–675.

Chaboche, J.L., 1977. Viscoplastic constitutive equations for the description of cyclic and anisotropic behaviourof metals. Bulletin de L’Academie Polonaise des Sciences 25, 33–41.

Chaboche, J.-L., 1993a. Cyclic viscoplastic constitutive equations, Part I: A thermodynamically consistentformulation. Journal of Applied Mechanics 60, 813–821.

Chaboche, J.-L., 1993b. Cyclic viscoplastic constitutive equations, Part I: A thermodynamically consistentformulation, Part II: Stored energy – comparison between models and experiments. Journal of AppliedMechanics 60, 813–828.

Chaboche, J., 1996. Unified cyclic viscoplastic constitutive equations: development, capabilities, and thermo-dynamic framework. In: Krausz, A., Krausz, K. (Eds.), Unified Constitutive Laws of Plastic Deformation.Academic Press, New York, pp. 1–68.

Chaboche, J., Lemaitre, J., 1990. Mechanics of Solid Materials. Cambridge Press, Cambridge, New York.Chrysochoos, A., Maisonneuve, O., Martin, G., Caumon, H., Chezeaux, J., 1989. Plastic and dissipated work and

stored energy. Nuclear Engineering and Design 114, 323–333.Coleman, B., Gurtin, M., 1967. Thermodynamics with internal state variables. Journal of Chemical Physics 47,

597–613.Coleman, B., Noll, W., 1963. The thermodynamics of elastic materials with heat conduction and viscosity.

Archive for Rational Mechanics and Analysis 13, 167–178.Coleman, B., Owen, D., 1975. On thermodynamics and elastic–plastic materials. Archive for Rational Mechanics

and Analysis 59, 25–51.


Deseri, L., Mares, R., 2000. A class of viscoelastoplastic constitutive models based on the maximum dissipationprinciple. Mechanics of Materials 32, 389–403.

Dettmer, W., Reese, S., 2004. On the theoretical and numerical modelling of Armstrong–Frederick kinematichardening in the finite strain regime. Computer Methods in Applied Mechanics and Engineering 193, 87–116.

Eckart, C., 1948. The thermodynamics of irreversible processes IV: the theory of elasticity and anelasticity.Physical Review 73, 373–382.

Flory, P., 1961. Thermodynamic relations for high elastic materials. Transactions of the Faraday Society 57, 829–838.

Germain, P., Nguyen, Q.S., Suquet, P., 1983. Continuum thermodynamics. Transaction of the ASME 50, 1010–1020.

Glaser, S., 1991. Berechnung gekoppelter thermomechanischer Prozesse. Ph.D. Thesis, Institut fur Statik undDynamik, Bericht Nr. 91/3, Universitat Stuttgart.

Green, A.E., Naghdi, P.M., 1965. A general theory of an elastic–plastic continuum. Archive for RationalMechanics and Analysis 18, 251–281.

Green, A.E., Naghdi, P.M., 1971. Some remarks on elastic–plastic deformation at finite strain. InternationalJournal of Engineering Science, 1219–1229.

Green, A.E., Naghdi, P.M., 1978. The second law of thermodynamics and cyclic processes. Journal of AppliedMechanics 45, 487–492.

Gurtin, M., Anand, L., 2005. The decomposition F = FeFp, material symmetry, and plastic irrotationality forsolids that are isotropic-viscoplastic or amorphous. International Journal of Plasticity 21, 1686–1719.

Hairer, E., 2001. Geometric integration of ordinary differential equations on manifolds. BIT NumericalMathematics 41 (5), 996–1007.

Hairer, E., Lubich, C., Wanner, G., 2002. Geometric Numerical Integration. Springer, Berlin.Hakansson, P., Wallin, M., Ristinmaa, M., 2005. Comparison of isotropic hardening and kinematic hardening in

thermoplasticity. International Journal of Plasticity 21, 1435–1460.Hartmann, S., Luhrs, G., Haupt, P., 1997. An efficient stress algorithm with application in viscoplasticity and

plasticity. International Journal for Numerical Methods in Engineering 40, 991–1013.Haupt, P., 2002. Continuum Mechanics and Theory of Materials. Springer, Berlin.Haupt, P., Kamlah, M., 1995. Representation of cyclic hardening and softening properties using continuous

variables. International Journal of Plasticity 11, 267–291.Haupt, P., Lion, A., 1995. Experimental identification and mathematical modelling of viscoplastic material

behavior. Continuum Mechanics and Thermodynamics 7, 73–96.Haupt, P., Tsakmakis, C., 1989. On the application of dual variables in continuum mechanics. Continuum

Mechanics and Thermodynamics 1, 165–196.Haupt, P., Helm, D., Tsakmakis, C., 1997. Stored energy and dissipation in thermoviscoplasticity. ZAMM 77,

S119–S120.Helm, D., 1998. Experimentelle Untersuchung und phanomenologische Modellierung thermomechanischer

Kopplungseffekte in der Metallplastizitat. In: Hartmann, S., Tsakmakis, C. (Eds.), Aspekte der Kontinu-umsmechanik. Festschrift zum 60. Geburtstag von Professor Peter Haupt. Berichte des Instituts furMechanik, Institut fur Mechanik der Universitat Kassel, pp. 81–105.

Helm, D., 2001. Formgedachtnislegierungen: Experimentelle Untersuchung, phanomenologische Modellierungund numerische Simulation der thermomechanischen Materialeigenschaften. Ph.D. Thesis, Institut furMechanik, Universitat Gesamthochschule Kassel, Kassel.

Hibbitt, Karlsson & Sorensen, Inc., 2002a. ABAQUS Theory Manual. Version 6.3, Hibbitt, Karlsson &Sorensen, Inc.

Hibbitt, Karlsson & Sorensen, Inc., 2002b. ABAQUS/Explicit User’s Manual. Version 6.3, Hibbitt, Karlsson &Sorensen, Inc.

Hill, R., 1950. The Mathematical Theory of Plasticity. Oxford University Press, Oxford.Hohenemser, K., Prager, W., 1932. Uber die Ansatze der Mechanik isotroper Kontinua. ZAMM 12, 216–226.Jansohn, W., 1997. Formulierung und Integration von Stoffgesetzen zur Beschreibung großer Deformationen in

der Thermoplastizitat und -viskoplastizitat. Ph.D. thesis, Institut fur Materialforschung, ForschungszentrumKarlsruhe.

Jou, D., Casas-Vazquez, J., Lebon, G., 1996. Extended Irreversible Thermodynamics. Springer, Berlin/Heidelberg/New York.

Kamlah, M., Haupt, P., 1998. On the macroscopic description of stored energy and self-heating during plasticdeformations. International Journal of Plasticity 13, 893–911.


Kamlah, M., Tsakmakis, C., 1999. Use of isotropic thermoelasticity laws in finite deformation viscoplasticitymodels. Continuum Mechanics and Thermodynamics 36, 73–88.

Khan, A.S., Jackson, K.M., 1999. On the evolution of isotropic hardening with finite plastic deformation. Part I:compression/tension loading of OFHC copper cylinders. International Journal of Plasticity 15, 1265–1275.

Krempl, E., 1979. An experimental study of room-temperature rate-sensitivity, creep and relaxation of AISI Type304 stainless steel. Journal of Mechanics and Physics of Solids 27, 363–375.

Krempl, E., 1987. Models of viscoplasticity. Some comments on equilibrium (back) stress and drag stress. ActaMechanica 69, 25–42.

Krempl, E., Khan, F., 2003. Rate (time)-dependent deformation behavior: an overview of some properties ofmetals and solid polymers. International Journal of Plasticity 19, 1069–1095.

Kroner, E., 1958. Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin.Kroner, E., 1960. Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Archive for Rational

Mechanics and Analysis 4, 273–334.Lee, E., 1969. Elastic–plastic deformation at finite strains. Journal of Applied Mechanics 36, 59–67.Lee, E., Liu, D., 1967. Finite-strain elastic–plastic theory with application to plane-wave analysis. Journal of

Applied Physics 38, 19–27.Lehmann, T., 1983. On a Generalized Constitutive Law in ThermoplasticityPlasticity Today. Elsevier,

Amsterdam.Lehmann, T., Blix, U., 1985. On the coupled thermo-mechanical process in the necking problem. International

Journal of Plasticity 1, 175–188.Lion, A., 1994. Materialeigenschaften der Viskoplastizitat, Experimente, Modellbildung und Parameteridenti-

fikation. Ph.D. Thesis, Institut fur Mechanik, Bericht Nr. 1/1994, Universitat Gesamthochschule Kassel.Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear

rheological models. International Journal of Plasticity 16, 469–494.Liu, I.-S., 2002. Continuum Mechanics. Springer, Berlin.Lubliner, J., 1985. A model of rubber viscoelasticity. Mechanics Research Communications 12, 93–99.Lubliner, J., 1990. Plasticity Theory. Macmillan Publishing Company, New York, London.Luhrs, G., Hartmann, S., Haupt, P., 1997. On the numerical treatment of finite deformations in elastovisco-

plasticity. Computer Methods in Applied Mechanics and Engineering 144, 1–21.Mandel, J., 1972. Plasticite Classique et Viscoplasticite. CISM Courses, vol. 97. Springer, Berlin.Maugin, G., Muschik, W., 1994. Thermodynamics with internal variables Part I. General concepts. Part II.

Applications. Journal of Non-Equilibrium Thermodynamics 19, 250–289.Meixner, J., Reik, H., 1959. Thermodynamik der irreversiblen Prozesse. In: Flugge, S. (Ed.), Handbuch der

Physik, Band III/2. Springer, Berlin.Miehe, C., 1988. Zur numerischen Behandlung thermomechanischer Prozesse. Ph.D. Thesis, Fachbereich

Bauingenieur- und Vermessungswesen der Universitat Hannover.Miehe, C., 1998. A constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric.

International Journal of Solids Structures 35, 3859–3897.Miehe, C., Stein, E., 1992. A canonical model of multiplicative elasto-plasticity: formulation and aspects of the

numerical implementation. European Journal of Mechanics A – Solids 11, 25–43.Mollica, F., Rajagopal, K.R., Srinivasa, A.R., 2001. The inelastic behavior of metals subject to loading reversal.

International Journal of Plasticity 17, 1119–1146.Muller, I., Ruggeri, 1998. Rational Extended Thermodynamics. Springer, Berlin/Heidelberg/New York.Oliferuk, W., Gadaj, S., Grabski, W., 1985. Energy storage during tensile deformation of Armco iron and

austenitic steel. Materials Science and Engineering 70, 131–141.Perzyna, P., 1963. The constitutive equations for rate sensitive plastic materials. Quarterly of Applied

Mathematics 20, 321–332.Pinsky, P., Ortiz, M., Taylor, R., 1983. Operator split methods in the numerical solution of the finite deformation

elastoplastic dynamic problem. Computer and Structures 17, 345–359.Rajagopal, K.R., Srinivasa, A.R., 1998a. Mechanics of the inelastic behavior of materials – Part 1: Theoretical

underpinnings. International Journal of Plasticity 14, 945–967.Rajagopal, K.R., Srinivasa, A.R., 1998b. Mechanics of the inelastic behavior of materials – Part 2: Inelastic

response. International Journal of Plasticity 14, 969–995.Rosakis, P., Rosakis, A.J., Ravichandran, G., Hodowany, J., 2000. A thermodynamic internal variable model for

the partition of plastic work into heat and stored energy in metals. Journal of the Mechanics and Physics ofSolids 48, 581–607.


Scheidler, M., Wright, T., 2001. A continuum framework for finite viscoplasticity. International Journal ofPlasticity 17, 1033–1085.

Simo, J., 1988. A framework for finite strain elastoplasticity based on maximum plastic dissipation and themultiplicative decomposition: Part I: Continuum formulation. Computer Methods in Applied Mechanics andEngineering 66, 199–219.

Simo, J., 1993. Recent developments in the numerical analysis of plasticity. In: Stein, E. (Ed.), Progress inComputational Analysis of Inelastic Structures, CISM Report No. 321, Udine, Springer Verlag, Wien, NewYork, pp. 115–173.

Simo, J., Hughes, T., 1998. Computational Inelasticity. Springer.Simo, J., Miehe, C., 1992. Associative coupled thermoplasticity at finite strains formulation, numerical analysis

and implementation. Computer Methods in Applied Mechanics and Engineering 98, 41–104.Simo, J., Pister, K., 1984. Remarks on rate constitutive equations for finite deformation problems. Computer

Methods in Applied Mechanics and Engineering 46, 201–215.Simo, J., Taylor, R., Pister, K., 1985. Variational and projection methods for the volume constraint in finite

deformation elasto-plasticity. Computer Methods in Applied Mechanics and Engineering 51, 177–208.Taylor, G., Quinney, M., 1934. The latent energy remaining in a metal after cold working. Proceedings of the

Royal Society of London A 143, 307.Truesdell, C., 1984. Rational Thermodynamics, second ed. Springer, New York, Berlin.Truesdell, C., Noll, W., 1965. The Non-Linear Field Theories of Mechanics. In: Flugge, S. (Ed.), Handbuch der

Physik, Band III/3. Springer, Berlin.Tsakmakis, C., 1996. Kinematic hardening rules in finite plasticity Part I: a constitutive approach. Continuum

Mechanics and Thermodynamics 8, 215–231.Tsakmakis, C., 1998. Energiehaushalt des elastisch-plastischen Korpers. Unpublished paper presented in the

symposium: Aspekte der Kontinuumsmechanik und Materialtheorie. Kassel, 14 April.Tsakmakis, C., 2004. Description of plastic anisotropy effects at large deformations. Part I: restrictions imposed

by the second law and the postulate of Il’iushin. International Journal of Plasticity 20, 167–198.Tsakmakis, C., Willuweit, A., 2003. Use of the elastic predictor–plastic corrector method for integrating finite

deformation plasticity laws. In: Hutter, K., Baaser, H. (Eds.), Deformation and Failure in Metallic Materials.Springer, Berlin, Heidelberg, New York, pp. 79–108.

von Mises, R., 1913. Mechanik der festen Korper im plastisch-deformablen Zustand. Gottinger Nachrichtenmathematisch-physikalischer Klasse, 582–592.

von Mises, R., 1928. Mechanik der plastischen Formanderungen von Kristallen. ZAMM 8, 161–185.Wallin, M., Ristinmaa, M., 2005. Deformation gradient based kinematic hardening model. International Journal

of Plasticity 21, 2025–2050.Weber, G., Anand, L., 1990. Finite deformation constitutive equations and a time integration procedure for

isotropic, hyperelastic-viscoplastic solids. Computer Methods in Applied Mechanics and Engineering 79, 173–202.

Wriggers, P., Miehe, C., Kleiber, M., Simo, J., 1992. On the coupled thermomechanical treatment of neckingproblems via finite element methods. International Journal for Numerical Methods in Engineering 33, 869–883.

Ziegler, H., 1963. Some extremum principles in irreversible thermodynamics with applications to continuummechanics. In: Sneddon, I.N., Koiter, W. (Eds.), Progress in Solid Mechanics, vol. IV. North-Holland,Amsterdam, pp. 93–193.

Ziegler, H., 1977. An Introduction to Thermomechanics. North-Holland, Amsterdam.

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Stress computation in finite thermoviscoplasticity