A large deformation theory for rate-dependent elastic–plastic ...

International Journal of Plasticity 25 (2009) 1833–1878

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International Journal of Plasticity

journal homepage: www.elsevier .com/locate/ i jplas

A large deformation theory for rate-dependent elastic–plasticmaterials with combined isotropic and kinematic hardening

David L. Henann, Lallit Anand *

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

a r t i c l e i n f o

Article history:Received 24 August 2008Received in final revised form 25 November2008Available online 7 December 2008

Keywords:Large deformation viscoplasticityIsotropic hardeningKinematic hardeningTime-integration procedureFinite elements

0749-6419/$ - see front matter � 2008 Elsevier Ltdoi:10.1016/j.ijplas.2008.11.008

* Corresponding author. Tel.: +1 617 253 1635.E-mail address: [email protected] (L. Anand).

a b s t r a c t

We have developed a large deformation viscoplasticity theory withcombined isotropic and kinematic hardening based on the dualdecompositions F ¼ FeFp [Kröner, E., 1960. Allgemeine kontinu-umstheorie der versetzungen und eigenspannungen. Archive forRational Mechanics and Analysis 4, 273–334] and Fp ¼ Fp

enFpdis

[Lion, A., 2000. Constitutive modelling in finite thermoviscoplastic-ity: a physical approach based on nonlinear rheological models.International Journal of Plasticity 16, 469–494]. The elastic distor-tion Fe contributes to a standard elastic free-energy wðeÞ , while Fp

en,the energetic part of Fp , contributes to a defect energy wðpÞ – thesetwo additive contributions to the total free energy in turn lead tothe standard Cauchy stress and a back-stress. Since Fe ¼ FFp�1

and Fpen ¼ FpFp�1

dis , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolutionequations for Fp and Fp

dis – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-

integration procedure for the constitutive theory for implementa-tion in displacement-based finite element programs. The proce-dure that we develop is ‘‘simple” in the sense that it onlyinvolves the solution of one non-linear equation, rather than a sys-tem of non-linear equations. We show that our time-integrationprocedure is stable for relatively large time steps, is first-orderaccurate, and is objective.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In classical small deformation theories of metal plasticity, when attempting to model the Bausch-inger phenomenon (Bauschinger, 1886) and other phenomena associated with cyclic loading, kine-

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1834 D.L. Henann, L. Anand / International Journal of Plasticity 25 (2009) 1833–1878

matic strain hardening is often invoked by introducing a symmetric and deviatoric stress-like tensorialinternal variable Tback called the back-stress, which acts to oppose the Cauchy stress T in the formu-lation of an appropriate yield criterion and a corresponding flow rule for the material. The simplestmodel for the evolution of such a back-stress is Prager’s linear kinematic-hardening rule (Prager,1949):

1 Also2 Lion

responsFp ¼ ~Fi~F

_Tback ¼ B _Ep so that Tback ¼ BEp; ð1:1Þ

where B > 0 is a back-stress modulus and Ep is the plastic stain tensor in the standard decompositionE ¼ Ee þ Ep of the infinitesimal strain tensor E into elastic and plastic parts. A better description ofkinematic hardening is given by an evolution equation for the back-stress Tback proposed initially byArmstrong and Frederick (1966):1

_Tback ¼ B _Ep � cTback _ep; with _ep ¼def j _Epj; ð1:2Þ

where the term ðcTback _epÞ with c > 0 represents a dynamic recovery term, which is collinear with theback-stress Tback and is proportional to the equivalent plastic strain rate _ep. For monotonic uniaxialloading, the evolution of back-stress, instead of being linear as prescribed by (1.1), is then exponentialwith a saturation value. The Armstrong–Frederick model for kinematic hardening is widely used as asimple model to describe cyclic loading phenomena in metals. For a discussion of the Armstrong–Fred-erick model and its various extensions to model cyclic plasticity in the small deformation regime seeLemaitre and Chaboche (1990) and the recent review by Chaboche (2008).

While there have been numerous attempts to extend the Armstrong–Frederick rule to large deforma-tions, a suitable extension is still not widely agreed upon. Dettmer and Reese (2004) have recently re-viewed and numerically analyzed several previous models for large deformation kinematic hardeningof the Armstrong–Frederick type. One class of theories (cf., Dettmer and Reese, 2004, for a list of refer-ences to the literature) follow Prager and Armstrong and Frederick, and introduce a stress-like internalvariable to model kinematic hardening; these theories require suitable frame-indifferent evolutionequations to generalize an evolution equation resembling (1.2) to large deformations. An alternativeto such theories are theories based on a proposal by Lion (2000), who in order to account for energy storagemechanisms associated with plastic flow, introduced a kinematic constitutive assumption that the plasticdistortion Fp in the standard Kröner (1960) elastic–plastic decomposition F ¼ FeFp of the total deforma-tion gradient, may be further multiplicatively decomposed into energetic and dissipative parts Fp

en andFp

dis, respectively, as Fp ¼ FpenFp

dis (cf., Lion’s Fig. 5).2 The elastic distortion Fe contributes to a standard elas-tic free-energy wðeÞ, while Fp

en contributes a defect energy wðpÞ – these two additive contributions to the totalfree energy in turn lead to the standard Cauchy stress and a back-stress. Since Fe ¼ FFp�1 and Fp

en ¼ FpFp�1dis ,

the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evo-lution equations for Fp and Fp

dis – the two flow rules of the theory. An important characteristic of such a the-ory is that since Fp is invariant under a change in frame, then so also are Fp

en and Fpdis, and the resulting

evolution equations for the back-stress are automatically frame-indifferent, and do not require special con-siderations such as those required when generalizing equations of the type (1.2) to proper frame-indiffer-ent counterparts. Examples of kinematic-hardening constitutive theories based on Lion-type Fp ¼ Fp

enFpdis

decomposition may be found in Dettmer and Reese (2004), Menzel et al. (2005), Håkansson et al.(2005), Wallin and Ristinmaa (2005), and Vladimirov et al. (2008).

While Dettmer and Reese (2004) in Section 4.3 of their paper, and more recently Vladimirov et al.(2008) in Section 2.2 of their paper have presented a generalization of the Armstrong–Frederick typeof kinematic-hardening rate-independent plasticity theory using the Lion decomposition Fp ¼ Fp

enFpdis,

we find their presentation rather brief, and (in our view) not sufficiently expository. The primary focusof the papers by Dettmer and Reese and Vladimirov et al., was the development of suitable time-inte-gration algorithms, and they have developed several fully-implicit time-integration procedures for

recently published as Frederick and Armstrong (2007).uses the notation FM � F, Fe � Fe , Fin � Fp , Fine � Fp

en, Finp � Fpdis . Based on a statistical-volume-averaging argument for the

e of a polycrystalline aggregate, Clayton and McDowell (2003, Eq. (13)) have also introduced a Lion-type decompositionp; they call ~Fi � Fp

en a ‘‘meso-incompatibility tensor.”

D.L. Henann, L. Anand / International Journal of Plasticity 25 (2009) 1833–1878 1835

their model; one such integration procedure involves the solution of a system of 13 highly non-linearequations – cf., Eq. (53) of Vladimirov et al. (2008). Motivated by the work of these authors, it is thepurpose of this paper to:

(1) Develop a thermodynamically consistent rate-dependent plasticity theory with combined isotro-pic and kinematic hardening of the Armstrong–Frederick type, based on the dual decomposi-tions F ¼ FeFp and Fp ¼ Fp

enFpdis. The limit m! 0, where m is a material rate-sensitivity

parameter in our theory, corresponds to a rate-independent model. Importantly, we developthe theory based on the principle of virtual power and carefully lay down all assumptionsand specializations that we have adopted so that it may in the future be possible to generalizethe theory with other, more flexible isotropic and kinematic-hardening models.

(2) Develop a simple, stable, semi-implicit time-integration procedure for our rate-dependent con-stitutive theory for implementation in displacement-based finite element programs. The proce-dure that we develop is ‘‘simple” in the sense that it involves the solution of only one stiff non-linear equation,3 rather than a system of non-linear equations. We show that our time-integrationprocedure is stable for relatively large time steps, is first-order accurate, and is objective.

2. Kinematics

We consider a homogeneous body B identified with the region of space it occupies in a fixed refer-ence configuration, and denote by X an arbitrary material point of B. A motion of B is then a smooth one-to-one mapping x ¼ vðX; tÞ with deformation gradient, velocity and velocity gradient given by4

3 Witsensitivspecial

4 Notdivergethe poiFe�1 ¼and symby jAj ¼

5 Als

F ¼ rv; v ¼ _v; L ¼ grad v ¼ _FF�1: ð2:1Þ

To model the inelastic response of materials, we assume that the deformation gradient F may bemultiplicatively decomposed as (Kröner, 1960)

F ¼ FeFp: ð2:2Þ

As is standard, we assume that

J ¼ det F > 0;

and consistent with this we assume that

Je ¼def det Fe > 0; Jp ¼def det Fp > 0; ð2:3Þ

so that Fe and Fp are invertible. Here, suppressing the argument t:

� FpðXÞ represents the local inelastic distortion of the material at X due to a ‘‘plastic mechanism.” Thislocal deformation carries the material into – and ultimately ‘‘pins” the material to – a coherent struc-ture that resides in the structural space5 at X (as represented by the range of FpðXÞ);

� FeðXÞ represents the subsequent stretching and rotation of this coherent structure, and thereby rep-resents the corresponding ‘‘elastic distortion.”

h reference to the power-law model (9.35) in Section 9.3, the stiffness of the equations depends on the strain-rate-ity parameter m, and the stiffness increases to infinity as m tends to zero, the rate-independent limit. For small values of mcare is required to develop stable constitutive time-integration procedures; cf., e.g., Lush et al. (1989).ation: We use standard notation of modern continuum mechanics. Specifically: r and Div denote the gradient andnce with respect to the material point X in the reference configuration; grad and div denote these operators with respect tont x ¼ vðX; tÞ in the deformed body; a superposed dot denotes the material time-derivative. Throughout, we writeðFeÞ�1, Fp�> ¼ ðFpÞ�> , etc. We write trA, symA, skwA, A0, and sym0A, respectively, for the trace, symmetric, skew, deviatoric,

metric–deviatoric parts of a tensor A. Also, the inner product of tensors A and B is denoted by A : B, and the magnitude of AffiffiffiffiffiffiffiffiffiffiffiA : Ap

.o sometimes referred to as the ‘‘intermediate” or ‘‘relaxed” local space at X.


We refer to Fp and Fe as the inelastic and elastic distortions.

By (2.1)3 and (2.2),

L ¼ Le þ FeLpFe�1; ð2:4Þ

with

Le ¼ _FeFe�1; Lp ¼ _FpFp�1: ð2:5Þ

As is standard, we define the total, elastic, and plastic stretching and spin tensors through

D ¼ symL; W ¼ skwL;De ¼ symLe; We ¼ skwLe;

Dp ¼ symLp; Wp ¼ skwLp;

9>=>; ð2:6Þ

so that L ¼ DþW, Le ¼ De þWe, and Lp ¼ Dp þWp.The right and left polar decompositions of F are given by

F ¼ RU ¼ VR; ð2:7Þ

where R is a rotation (proper-orthogonal tensor), while U and V are symmetric, positive-definite ten-sors with

U ¼ffiffiffiffiffiffiffiffiF>F

p; V ¼

ffiffiffiffiffiffiffiffiFF>

p: ð2:8Þ

Also, the right and left Cauchy–Green tensors are given by

C ¼ U2 ¼ F>F; B ¼ V2 ¼ FF>: ð2:9Þ

Similarly, the right and left polar decompositions of Fe and Fp are given by

Fe ¼ ReUe ¼ VeRe; Fp ¼ RpUp ¼ VpRp; ð2:10Þ

where Re and Rp are rotations, while Ue, Ve, Up, Vp are symmetric, positive-definite tensors with

Ue ¼ffiffiffiffiffiffiffiffiffiffiffiffiFe>Fe

p; Ve ¼

ffiffiffiffiffiffiffiffiffiffiffiffiFeFe>

p; Up ¼

ffiffiffiffiffiffiffiffiffiffiffiffiFp>Fp

p; Vp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiFpFp>

p: ð2:11Þ

Also, the right and left elastic Cauchy–Green tensors are given by

Ce ¼ Ue2 ¼ Fe>Fe; Be ¼ Ve2 ¼ FeFe>; ð2:12Þ

and the right and left plastic Cauchy–Green tensors are given by

Cp ¼ Up2 ¼ Fp>Fp; Bp ¼ Vp2 ¼ FpFp>: ð2:13Þ

2.1. Incompressible, irrotational plastic flow

We make two basic kinematical assumptions concerning plastic flow:

(i) First, we make the standard assumption that plastic flow is incompressible, so that

Jp ¼ det Fp ¼ 1 and trLp ¼ 0: ð2:14Þ

Hence, using (2.2) and (2.14)1,

e
J ¼ J: ð2:15Þ


(ii) Second, from the outset we constrain the theory by limiting our discussion to circumstancesunder which the material may be idealized as isotropic. For isotropic elastic–viscoplastic theo-ries utilizing the Kröner decomposition it is widely assumed that the plastic flow is irrotationalin the sense that6

6 A nbased oGreen aspace. T

in whictural frindifferwith thfacet ofmaterialution oUsing twork foality, thHere, wdeform

7 Howaggrega

Wp ¼ 0: ð2:16Þ

Then, trivially, Lp � Dp and

_Fp ¼ DpFp; with trDp ¼ 0: ð2:17Þ

Thus, using (2.1), (2.4), (2.5), and (2.17), we may write (2.4) for future use as

grad v ¼ Le þ FeDpFe�1; with trDp ¼ 0: ð2:18Þ

2.2. Decomposition of Fp into energetic and dissipative parts

Next following Lion (2000), in order to account for energy storage mechanisms associated with plasticflow, we introduce an additional kinematic constitutive assumption that Fp may be multiplicativelydecomposed into an energetic Fp

en and a dissipative part Fpdis as follows:

Fp ¼ FpenFp

dis; det Fpen ¼ det Fp

dis ¼ 1: ð2:19Þ

We call the range of FpdisðXÞ the local substructural space. Thus Fp

en maps material elements from thesubstructural space to the structural space (which is the range of FpðXÞ). Lion (2000) calls the substruc-tural space as the ‘‘intermediate configuration of kinematic hardening.” A ‘‘physical interpretation” ofsuch an ‘‘intermediate configuration” is not completely clear7; here we simply treat it as a mathematicalconstruct resulting from Lion’s presumed decomposition (2.19).

Then, from (2.5)

Lp ¼ Lpen þ Fp

enLpdisF

p�1en ð2:20Þ

with

Lpen ¼ _Fp

enFp�1en with trLp

en ¼ 0; Lpdis ¼ _Fp

disFp�1dis with trLp

dis ¼ 0; ð2:21Þ

and corresponding stretching and spin tensors through

ote on role of the plastic spin in isotropic solids: There are numerous publications discussing the role of plastic spin in theoriesn the Kröner decomposition (cf., e.g., Dafalias, 1998). Based on the classical principle of frame-indifference (cf., Section 3),nd Naghdi (1971) (cf., also Casey and Naghdi, 1980), introduced the notion of a change in frame of the intermediate structuralhis notion leads to transformation laws for Fe and Fp of the form

�Fp ¼ QFp; �F

e ¼ FeQ>;

h Q ðX; tÞ is an arbitrary time-dependent rotation of the structural space. Unfortunately, Green and Naghdi viewed struc-ame-indifference as a general principle; that is, a principle that stands at a level equivalent to that of conventional frame-ence. This view has been refuted by many workers (cf., e.g., Dashner, 1986, and the references therein). While we agreee view that structural frame-indifference is not a general principle, this hypothesis may be used to represent an importantthe behavior of a large class of polycrystalline materials based on the Kröner decomposition. Indeed, for polycrystallinels without texture the structural space is associated with a collection of randomly oriented lattices, and hence the evo-f dislocations through that space should be independent of the frame with respect to which this evolution is measured.

he notion of a change in frame of the structural space, Gurtin and Anand (2005, p. 1714) have shown that within a frame-r isotropic plasticity based on the Kröner decomposition (which we follow here), one may assume, without loss in gener-at the plastic spin vanishes Wp ¼ 0.e adopt the assumption Wp ¼ 0 from the outset; we do so based solely on pragmatic grounds: when discussing finite

ations for isotropic materials a theory without plastic spin is far simpler than one with plastic spin.ever, see Clayton and McDowell (2003) for a ‘‘statistical-volume-averaging” argument for the response of a polycrystalline

te.


Dpen ¼ symLp

en; Wpen ¼ skwLp

en;

Dpdis ¼ symLp

dis; Wpdis ¼ skwLp

dis;

)ð2:22Þ

so that Lpen ¼ Dp

en þWpen and Lp

dis ¼ Dpdis þWp

dis.Paralleling the assumption of plastic irrotationality for isotropic materials, Wp ¼ 0, we assume that

Wpdis ¼ 0: ð2:23Þ

Then, trivially Lpdis ¼ Dp

dis, and

_Fpdis ¼ Dp

disFpdis with trDp

dis ¼ 0: ð2:24Þ

Since

Wp ¼Wpen þ skw Fp

enDpdisF

p�1en

� �; ð2:25Þ

in order to enforce the constitutive constraint Wp ¼ 0, we require that both

Wpen ¼ 0; ð2:26Þ

and

skw FpenDp

disFp�1en

� �¼ 0: ð2:27Þ

Also, (2.20) reduces to

Dp ¼ Dpen þ sym Fp

enDpdisF

p�1en

� �; with trDp

en ¼ trDpdis ¼ 0: ð2:28Þ

The right and left polar decompositions of Fpen and Fp

dis are given by

Fpen ¼ Rp

enUpen ¼ Vp

enRpen; Fp

dis ¼ RpdisU

pdis ¼ Vp

disRpdis; ð2:29Þ

where Rpen and Rp

dis are rotations, while Upen, Vp

en, Updis, Vp

dis are symmetric, positive-definite tensors with

Upen ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp>

en Fpen

q; Vp

en ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp

enFp>en

q; Up

dis ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp>

disFpdis

q; Vp

dis ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp

disFp>dis

q: ð2:30Þ

Also, the corresponding right and left Cauchy–Green tensors are given by

Cpen ¼ Up2

en ¼ Fp>en Fp

en; Bpen ¼ Vp2

en ¼ FpenFp>

en ; ð2:31ÞCp

dis ¼ Up2dis ¼ Fp>

disFpdis; Bp

dis ¼ Vp2dis ¼ Fp

disFp>dis: ð2:32Þ

3. Frame-indifference

Changes in frame (observer) are smooth time-dependent rigid transformations of the Euclideanspace through which the body moves. We require that the theory be invariant under such transforma-tions, and hence under transformations of the form

v�ðX; tÞ ¼ Q ðtÞðvðX; tÞ � oÞ þ yðtÞ; ð3:1Þ

with Q ðtÞ a rotation (proper-orthogonal tensor), yðtÞ a point at each t, and o a fixed origin. Then, undera change in frame, the deformation gradient transforms according to

F� ¼ QF; ð3:2Þ

and hence

C� ¼ C that is; C is invariant; ð3:3Þ

also _F� ¼ Q _Fþ _QF, and by (2.1)3,

L� ¼ QLQ> þ _QQ>: ð3:4Þ


Thus,

D� ¼ QDQ>; W� ¼ QWQ> þ _QQ>: ð3:5Þ

Moreover, ðFeFpÞ� ¼ Q ðFeFpÞ, and therefore, since observers view only the deformed configuration,

Fe� ¼ QFe; Fp� ¼ Fp; and thus Fp is invariant: ð3:6Þ

Hence, by (2.5),

Le� ¼ QLeQ> þ _QQ>; ð3:7Þ

and

Lp; Dp; and Wp are invariant: ð3:8Þ

Further, by (2.10),

Fe ¼ ReUe ! QFe ¼ QReUe;

Fe ¼ VeRe ! QFe ¼ QVeQ>QRe;

and we may conclude from the uniqueness of the polar decomposition that

Re� ¼ QRe; Ve� ¼ QVeQ>; Ue� ¼ Ue: ð3:9Þ

Hence, from (2.12), Be and Ce transform as

Be� ¼ QBeQ> and Ce� ¼ Ce; ð3:10Þ

further, since Fp is invariant,

Bp and Cp are also invariant: ð3:11Þ

Finally, since Fp is invariant under a change in frame,

Fpen and Fp

dis are invariant;

as are all of the following quantities based on Fpen and Fp

dis:

Upen; Cp

en; Vpen; Bp

en; Rpen; Up

dis; Cpdis; Vp

dis; Bpdis; Rp

dis:

4. Development of the theory based on the principle of virtual power

Following Gurtin and Anand (2005), the theory presented here is based on the belief that

� the power expended by each independent ‘‘rate-like” kinematical descriptor be expressible in terms of anassociated force system consistent with its own balance.

However, the basic ‘‘rate-like” descriptors, namely, v, Le, Dp, Dpen, and Dp

dis are not independent, sinceby (2.18) and (2.28) they are constrained by

grad v ¼ Le þ FeDpFe�1; trDp ¼ 0; ð4:1Þ

and


enDpdisF

p�1en

� �; trDp

en ¼ trDpdis ¼ 0; ð4:2Þ

and it is not apparent what forms the associated force balances should take. It is in such situations thatthe strength of the principle of virtual power becomes apparent, since the principle of virtual powerautomatically determines the underlying force balances.


We denote by Pt an arbitrary part (subregion) of the deformed body with n the outward unit nor-mal on the boundary @Pt of Pt . The power expended on Pt by material or bodies exterior to Pt resultsfrom a macroscopic surface traction tðnÞ, measured per unit area in the deformed body, and a macro-scopic body force b, measured per unit volume in the deformed body, each of whose working accom-panies the macroscopic motion of the body; the body force b presumed to account for inertia; that is,granted the underlying frame is inertial,

b ¼ b0 � q _v; ð4:3Þ

with b0 the noninertial body force and q the mass density in the deformed body. We therefore writethe external power as

WextðPtÞ ¼Z@Pt

tðnÞ � vdaþZPt

b � vdv: ð4:4Þ

We assume that power is expended internally by an elastic stress Se power-conjugate to Le, a plasticstress Tp power-conjugate to Dp, an energetic stress Sen conjugate to Dp

en, and a dissipative stress Tdis

conjugate to symðFpenDp

disFp�1en Þ. Since Dp, Dp

en, and symðFpenDp

disFp�1en Þ are symmetric and deviatoric, we as-

sume that Tp, Sen, and Tdis are symmetric and deviatoric. We write the internal power asZ
WintðPtÞ ¼
Pt

Se : Le þ J�1 Tp : Dp þ Sen : Dpen þ Tdis : symðFp

enDpdisF

p�1en Þ

� �� dv;

¼ZPt

Se : Le þ J�1 Tp : Dp þ Sen : Dpen þ sym0ðF

p>en TdisF

p�>en Þ : Dp

dis

� �� dv:

The term J�1 arises because Tp : Dp þ Sen : Dpen þ Tdis : sym Fp

enDpdisFp�1

en

� �is measured per unit volume in

the structural space, but the integration is carried out within the deformed body. Defining a new stressmeasure

Sdis ¼def sym0ðF

p>en TdisF

p�>en Þ; ð4:5Þ

the internal power may more compactly be written as

WintðPtÞ ¼ZPt

Se : Le þ J�1 Tp : Dp þ Sen : Dpen þ Sdis : Dp

dis

� �� dv: ð4:6Þ

4.1. Principle of virtual power

Assume that, at some arbitrarily chosen but fixed time, the fields v, Fe, Fpen (and hence F, Fp, Fp

dis) areknown, and consider the fields v, Le, Dp, Dp

en, and Dpdis as virtual velocities to be specified independently

in a manner consistent with the constraints (4.1) and (4.2). That is, denoting the virtual fields by ~v, ~Le,~Dp, ~Dp

en, and ~Dpdis to differentiate them from fields associated with the actual evolution of the body, we

require that

grad ~v ¼ ~Le þ Fe ~DpFe�1; tr~Dp ¼ 0; ð4:7Þ

and

~Dp ¼ ~Dpen þ sym Fp

en~Dp

disFp�1en

� �; tr~Dp

en ¼ tr~Dpdis ¼ 0: ð4:8Þ

More specifically, we define a generalized virtual velocity to be a list

V ¼ ð~v; ~Le; ~Dp; ~Dpen;

~DpdisÞ;

consistent with (4.7) and (4.8). Then, writing

WextðPt;VÞ ¼Z@Pt

tðnÞ � ~v daþZPt

b � ~vdv;

WintðPt ;VÞ ¼ZPt

Se : ~Le þ J�1 Tp : ~Dp þ Sen : ~Dpen þ Sdis : ~Dp

dis

� �� dv;

)ð4:9Þ


respectively, for the external and internal expenditures of virtual power, the principle of virtual power isthe requirement that the external and internal powers be balanced. That is, given any part Pt ,

WextðPt ;VÞ ¼WintðPt ;VÞ for all generalized virtual velocities V: ð4:10Þ

4.1.1. Consequences of frame-indifferenceWe assume that the internal power WintðPt ;VÞ is invariant under a change in frame and that the

virtual fields transform in a manner identical to their nonvirtual counterparts. Then given a change inframe, invariance of the internal power requires that

W�intðP�t ;V�Þ ¼WintðPt ;VÞ; ð4:11Þ

where P�t and W�intðP�t ;V�Þ represent the region and the internal power in the new frame. In the new

frame Pt transforms rigidly to a region P�t , the stresses Se, Tp, Sen, and Sdis transform to Se�, Tp�, S�en, andS�dis, while ~Le transforms to

~Le� ¼ Q ~LeQ> þ _QQ>;

and ~Dp, ~Dpen, and ~Dp

dis are invariant. Thus, under a change in frame WintðPt;VÞ transforms to

W�intðP�t ;V�Þ ¼W�

intðPt ;V�Þ

¼ZPt

Se� : Q ~LeQ> þ _QQ>� �

þ J�1 Tp� : ~Dp þ S�en : ~Dpen þ S�dis : ~Dp

dis

� �� dv

¼ZPt

ðQ>Se�Q Þ : ~Le þ Se� : ð _Q Q>Þ þ J�1 Tp� : ~Dp þ S�en : ~Dpen þ S�dis : ~Dp

dis

� �� dv:

Then (4.11) implies that
ZPt
ðQ>Se�Q Þ : ~Le þ Se� : ð _QQ>Þ þ J�1 Tp� : ~Dp þ S�en : ~Dpen þ S�dis : ~Dp

dis

� �� dv

¼ZPt

Se : ~Le þ J�1 Tp : ~Dp þ Sen : ~Dpen þ Sdis : ~Dp

dis

� �� dv; ð4:12Þ

or equivalently, since the part Pt is arbitrary,

ðQ>Se�Q Þ : ~Le þ Se� : ð _Q Q>Þ þ J�1 Tp� : ~Dp þ S�en : ~Dpen þ S�dis : ~Dp

dis

� �¼ Se : ~Le þ J�1 Tp : ~Dp þ Sen : ~Dp

en þ Sdis : ~Dpdis

� �: ð4:13Þ

Also, since the change in frame is arbitrary, if we choose it such that Q is an arbitrary time-independentrotation, and that _Q ¼ 0, we find from (4.13) that

ðQ>Se�Q Þ � Se� �: ~Le þ J�1 ðTp� � TpÞ : ~Dp þ S�en � Sen

� �: ~Dp

en þ S�dis � Sdis� �

: ~Dpdis

h i¼ 0:

Since this must hold for all ~Le, ~Dp, ~Dpen, and ~Dp

dis, we find that the stress Se transforms according to

Se� ¼ QSeQ>; ð4:14Þ

while Tp, Sen, and Sdis are invariant

Tp� ¼ Tp; S�en ¼ Sen; S�dis ¼ Sdis: ð4:15Þ

Next, if we assume that Q ¼ 1 at the time in question, and that _Q is an arbitrary skew tensor, we findfrom (4.13), using (4.14), (4.15) that

Se : _Q ¼ 0;

or that the tensor Se is symmetric,

Se ¼ Se>: ð4:16Þ

Thus, the elastic stress Se is frame-indifferent and symmetric.


4.1.2. Macroscopic force balanceIn applying the virtual balance (4.10) we are at liberty to choose any V consistent with the con-

straints (4.7) and (4.8). Consider a generalized virtual velocity with V for which ~v is arbitrary, and~Dp ¼ ~Dp

en ¼ ~Dpdis � 0, so that

~Le ¼ grad ~v: ð4:17Þ

For this choice of V, (4.10) yields
Z@Pt
tðnÞ � ~vdaþZPt

b � ~vdv ¼ZPt

Se : grad ~vdv: ð4:18Þ

Further, using divergence theorem, (4.18) yields
Z@Pt
ðtðnÞ � SenÞ � ~vdaþZPt

ðdivSe þ bÞ � ~vdv ¼ 0: ð4:19Þ

Since (4.19) must hold for all Pt and all ~v, standard variational arguments yield the traction condition

tðnÞ ¼ Sen; ð4:20Þ

and the local force balance

divSe þ b ¼ 0: ð4:21Þ

This traction condition and force balance and the symmetry and frame-indifference of Se are classicalconditions satisfied by the Cauchy stress T, an observation that allows us to write

T ¼def Se ð4:22Þ

and to view

T ¼ T> ð4:23Þ

as the standard macroscopic Cauchy stress and (4.21) as the local macroscopic force balance. Since we areworking in an inertial frame, so that (4.3) is satisfied, (4.21) reduces to the local balance law for linearmomentum,

divTþ b0 ¼ q _v; ð4:24Þ

with b0 the noninertial body force.

4.1.3. Microscopic force balancesTo discuss the microscopic counterparts of the macroscopic force balance, first consider a general-

ized virtual velocity with

~v � 0 and ~Dpdis � 0; ð4:25Þ

also, choose the virtual field ~Dp arbitrarily, and let

~Le ¼ �Fe ~DpFe�1 and ~Dpen ¼ ~Dp; ð4:26Þ

consistent with (4.7) and (4.8). Thus

T : ~Le ¼ �ðFe>TFe�>Þ : ~Dp: ð4:27Þ

Next, define an elastic Mandel stress by

Me ¼def JFe>TFe�>; ð4:28Þ

which in general is not symmetric. Then, on account of our choice (4.25)1 the external power vanishes,so that, by (4.10), the internal power must also vanish, and satisfy


WintðPt ;VÞ ¼ZPt

J�1 Tp þ Sen �Með Þ : ~Dpdv ¼ 0:

Since this must be satisfied for all Pt and all symmetric and deviatoric tensors ~Dp, a standard argumentyields the first microforce balance

ð4:29Þ

Next consider a generalized virtual velocity with

~v � 0 and ~Dp � 0; ð4:30Þ

also, choose the virtual field ~Dpdis arbitrarily, and let

~Dpen ¼ �sym Fp

en~Dp

disFp�1en

� �ð4:31Þ

consistent with (4.8). Thus

Sen : ~Dpen ¼ �ðF

p>en SenFp�>

en Þ : ~Dpdis: ð4:32Þ

Next, define a new stress measure by

Mpen ¼

def Fp>en SenFp�>

en ; ð4:33Þ

which in general is also not symmetric. For lack of better terminology, we call Mpen a plastic Mandel

stress. Then, on account of our choice (4.30)1 the external power vanishes, so that, by (4.10), the inter-nal power must also vanish, and satisfy

WintðPt ;VÞ ¼ZPt

Sdis �Mpen

� �: ~Dp

disdv ¼ 0:

Since this must be satisfied for all Pt and all symmetric and deviatoric tensors ~Dpdis, we obtain the sec-

ond microforce balance

ð4:34Þ

5. Free-energy imbalance

We consider a purely mechanical theory based on a second law requiring that the temporal increasein free energy of any part Pt be less than or equal to the power expended on Pt . The second law thereforetakes the form of a dissipation inequality

_ZPt

wJ�1dv 6WextðPtÞ ¼WintðPtÞ; ð5:1Þ

with w the free energy, measured per unit volume in the structural space.Since J�1dv with J ¼ det F represents the volume measure in the reference configuration, and since

Pt deforms with the body,

_ZPt

wJ�1dv ¼ZPt

_wJ�1dv:

Thus, since Pt is arbitrary, we may use (4.6), (4.22), and (4.23) to localize (5.1); the result is the localfree-energy imbalance


_w� JT : De � Tp : Dp � Sen : Dpen � Sdis : Dp

dis 6 0: ð5:2Þ

We introduce a new stress measure

Te ¼def JFe�1TFe�>; ð5:3Þ

which is symmetric, since T is symmetric; Te represents a second Piola stress with respect to the inter-mediate structural space. Note that by using the definition (5.3), the Mandel stress defined in (4.28) isrelated to the stress measure Te by

Me ¼ CeTe: ð5:4Þ

Next, differentiating (2.12)1 results in the following expression for the rate of change of Ce,

_Ce ¼ Fe> _Fe þ _Fe>Fe:

Hence, since Te is symmetric,

Te : _Ce ¼ 2Te : ðFe> _FeÞ ¼ 2ðFeTeFe>Þ : ð _FeFe�1Þ ¼ 2ðFeTeFe>Þ : De;

or, using (5.3) we obtain

JT : De ¼ 12

Te : _Ce: ð5:5Þ

Use of (5.5) in (5.2) allows us to express the free-energy imbalance in an alternate convenient form as

_w� 12

Te : _Ce � Tp : Dp � Sen : Dpen � Sdis : Dp

dis 6 0: ð5:6Þ

Finally, we note that w is invariant under a change in frame since it is a scalar field, and on accountof the transformation rules discussed in Section 3, the transformation rules (4.14) and (4.15), and thedefinitions (4.28), (5.3), (4.33), and (5.4), the fields

Ce; Cpen; Bp

en; Dp; Dpen; Dp

dis; Te; Tp; Sen; Sdis; Me; and Mpen ð5:7Þ

are also invariant.

6. Constitutive theory

To account for the classical notion of isotropic strain hardening we introduce a positive-valued sca-lar internal state-variable S, which has dimensions of stress. We refer to S as the isotropic deformationresistance. Since S is a scalar field, it is invariant under a change in frame.

Next, guided by the free-energy imbalance (5.6), and by experience with previous constitutive the-ories, we assume the following special set of constitutive equations:

w ¼ �wðeÞðCeÞ þ �wðpÞðBpenÞ;

Te ¼ �TeðCeÞ;Sen ¼ �SenðBp

enÞ;Tp ¼ �TpðDp; SÞ;Sdis ¼ �SdisðDp

dis;dp; SÞ;

_S ¼ hðdp; SÞ;

9>>>>>>>>>=>>>>>>>>>;

ð6:1Þ

where

dp ¼def jDpj ð6:2Þ

is the scalar flow rate corresponding to Dp. Note that since w is the free energy per unit volume of thestructural space, �wðeÞ is chosen to depend on Ce and �wðpÞ on Bp

en, because Ce and Bpen are squared stretch-

like measures associated with the intermediate structural space. Also note that we have introduced a


possible dependence of Sdis on dp because for some materials we anticipate that Dpdis (and thereby Sdis)

may be constrained by the magnitude of dp.The energy �wðeÞ represents the standard energy associated with intermolecular interactions, and the

energy �wðpÞ is a ‘‘defect-energy” associated with plastic deformation. For metallic materials �wðpÞ may beattributed to an energy stored in the complex local microstructural defect state, which typically in-cludes dislocations, sub-grain boundaries, and local incompatibilities at second-phase particles andgrain boundaries. At the macroscopic continuum level, the ‘‘defect-energy” �wðpÞ leads to the develop-ment and evolution of the energetic stress Sen, and this allows one to phenomenologically account forstrain-hardening phenomena commonly called kinematic hardening. Isotropic hardening is modeled bythe evolution (6.1)6 of the isotropic deformation resistance S.

Finally, note that on account of the transformation rules listed in the paragraph containing (5.7)and since S is also invariant, the constitutive equations (6.1) are frame-indifferent.

6.1. Thermodynamic restrictions

From (6.1)1

8 Rec9 We

thermo

_w ¼ @�wðeÞðCeÞ@Ce : _Ce þ @

�wðpÞðBpenÞ

@Bpen

: _Bpen;

and, using8

_Bpen ¼ Dp

enBpen þ Bp

enDpen; ð6:3Þ

and the symmetry of Bpen and @�w=@Bp

en,

@�wðpÞ Bpen

� �@Bp

en: _Bp

en ¼@�wðpÞðBp

enÞ@Bp

en: Dp

enBpen þ Bp

enDpen

� �¼ 2

@�wðpÞðBpenÞ

@Bpen

Bpen

� �0

: Dpen:

Hence, satisfaction of the free-energy imbalance (5.6) requires that the constitutive equations (6.1)satisfy

12

�TeðCeÞ � @�wðeÞðCeÞ@Ce

� �: _Ce þ �SenðBp

enÞ � 2@�wðpÞðBp

enÞ@Bp

enBp

en

� �0

� �: Dp

en

þ �TpðDp; SÞ : Dp þ �SdisðDpdis; d

p; SÞ : Dp

dis P 0; ð6:4Þ

and hold for all arguments in the domains of the constitutive functions, and in all motions of the body.Thus, sufficient conditions that the constitutive equations satisfy the free-energy imbalance are that9

(i) the free-energy determine the stresses Te and Sen via the stress relations

�TeðCeÞ ¼ 2@�wðeÞðCeÞ@Ce ; ð6:5Þ

�SenðBpenÞ ¼ 2

@�wðpÞðBpenÞ

@Bpen

Bpen

� �0

: ð6:6Þ

(ii) the plastic distortion-rates Dp and Dpdis satisfy the dissipation inequality

Tp : Dp þ Sdis : Dpdis P 0: ð6:7Þ

We assume henceforth that (6.5) and (6.6) hold in all motions of the body. We assume further that thematerial is strictly dissipative in the sense that

all that Wpen ¼ 0, cf., (2.26).

content ourselves with constitutive equations that are only sufficient, but generally not necessary for compatibility withdynamics.


Tp : Dp > 0 whenever Dp–0; ð6:8Þ

and

Sdis : Dpdis > 0 whenever Dp

dis–0: ð6:9Þ

Some remarks: It is possible to enhance the theory by introducing another positive-valued dimen-sionless strain-like scalar internal variable j which contributes an additional defect energy �wðpÞisoðjÞ,which evolves as j evolves according to an evolution equation of the type _j ¼ hðdp

;jÞ. This will leadto a scalar internal stress KðjÞ ¼def

@�wðpÞisoðjÞ=@j, and the dissipation inequality (6.7) will be modified as

Tp : Dp þ Sdis : Dpdis � K _j P 0: ð6:10Þ

Although a term such as �wðpÞisoðjÞ might be needed for a more complete characterization of the ‘‘storedenergy of cold work,” and a more precise determination of dissipation and thereby heat generation (cf.,e.g., Rosakis et al., 2000; Ristinmaa et al., 2007), here, since our interest is in an isothermal theory, werefrain from introducing such additional complexity. However, we do wish to emphasize that ourinternal variable S, which represents an isotropic deformation resistance to plastic flow, is in no senseintroduced as being derived from a defect energy �wðpÞisoðjÞ – it is introduced on a purely pragmatic phe-nomenological basis. Note that Ristinmaa et al. (2007) consider only ‘‘isotropic” hardening theories,while the theory under consideration here is for combined isotropic and kinematic hardening, andthe term �wðpÞðBp

enÞ already accounts for some aspects of the ‘‘stored energy of cold work.”Vladimirov et al. (2008) have considered a combined isotropic and kinematic-hardening theory

with an additional defect energy wisoðjÞ, and write their corresponding stress as @wisoðjÞ=@j ¼ð�RðjÞÞ � KðjÞ. They associate the stress ð�RðjÞÞ as the strain-hardening contribution to the isotropicdeformation resistance, and take _j � ð

ffiffiffiffiffiffiffiffi2=3

pÞdp, so that j represents an accumulated plastic strain. It

is our opinion that for materials which show strong isotropic strain hardening, the theory of Reese andco-workers will grossly underestimate the amount of dissipation. This is the major reason why we do notfollow the approach of Reese and co-workers, but directly introduce our stress-like internal variable Sto represent isotropic strain hardening.

6.2. Isotropy

The following definitions help to make precise our notion of an isotropic material (cf., Anand andGurtin, 2003):

(i) Orthþ = the group of all rotations (the proper-orthogonal group);(ii) the symmetry group GR, is the group of all rotations of the reference configuration that leaves the

response of the material unaltered;(iii) the symmetry group GS at each time t, is the group of all rotations of the structural space that

leaves the response of the material unaltered;(iv) the symmetry group GSS at each time t, is the group of all rotations of the substructural space that

leaves the response of the material unaltered.

Note that there is a slight difficulty in attaching a ‘‘physical meaning” to a rotation in GSS. Since thesubstructural space itself is a mathematical construct defined as the range of the linear map Fp

dis, a rota-tion Q 2 GSS should be considered as a rotation in a ‘‘thought experiment” involving the substructuralspace; such a rotation may never be ‘‘physically attainable.” We now discuss the manner in which thebasic fields transform under such transformations, granted the physically natural requirement ofinvariance of the internal power or equivalently, the requirement that

Te : _Ce; Tp : Dp; Sen : Dpen; and Sdis : Dp

dis be invariant: ð6:11Þ

6.2.1. Isotropy of the reference spaceLet Q be a time-independent rotation of the reference configuration. Then F! FQ , and hence


Fp ! FpQ ; Fpdis ! Fp

disQ and Fe and Fpen are invariant; and hence Ce and Cp

en are invariant;

ð6:12Þ

so that, by (2.5), (2.12), and (2.21),

_Ce; Dp; Dpen; and Dp

dis are invariant:

We may therefore use (6.11) to conclude that

Te; Tp; Sen; and Sdis are invariant: ð6:13Þ

Thus, with reference to the special constitutive functions (6.1), we note the these functions are not af-fected by rotations Q 2 GR.

6.2.2. Isotropy of the structural spaceNext, let Q , a time-independent rotation of the intermediate structural space, be a symmetry transfor-

mation. Then F is unaltered by such a rotation, and hence

Fe ! FeQ ; Fp ! Q>Fp; Fpen ! Q>Fp

en; and Fpdis is invariant; ð6:14Þ

and also

Ce ! Q>CeQ ; _Ce ! Q> _CeQ ; Dp ! Q>DpQ ; Bpen ! Q>Bp

enQ ;

Dpen ! Q>Dp

enQ ; and Dpdis is invariant: ð6:15Þ

Then (6.15) and (6.11) yield the transformation laws

Te ! Q>TeQ ; Tp ! Q>TpQ ; Sen ! Q>SenQ ; and Sdis is invariant: ð6:16Þ

Thus, with reference to the constitutive equations (6.1) we conclude that

�wðeÞðCeÞ ¼ �wðeÞðQ>CeQ Þ;�wðpÞðBp

enÞ ¼ �wðpÞðQ>BpenQ Þ;

Q>�TeðCeÞQ ¼ �TeðQ>CeQ Þ;Q>�SenðBp

enÞQ ¼ �SenðQ>BpenQ Þ;

Q>�TpðDp; SÞQ ¼ �TpðQ>DpQ ; SÞ;

9>>>>>>=>>>>>>;

ð6:17Þ

must hold for all rotations Q in the symmetry group GS at each time t, while the constitutive functions�SdisðDp

dis; dp; SÞ, and hðdp

; SÞ are invariant to such rotations.We refer to the material as one which is structurally isotropic if

GS ¼ Orthþ; ð6:18Þ

so that the response of the material is invariant under arbitrary rotations of the intermediate struc-tural space at each time t. Henceforth, we assume that (6.18) holds. In this case, the response functions�wðeÞ, �wðpÞ, �Te, �Sen, and �Tp must each be isotropic.

6.2.3. Isotropy of the substructural spaceFinally, let Q , a time-independent rotation of the substructural space, be a symmetry transformation.

Then F, Fe, and Fp are unaltered by such a rotation, and hence

Fpen ! Fp

enQ ; Fpdis ! Q>Fp

dis; ð6:19Þ

and also

Dpdis ! Q>Dp

disQ ; and Dpen is invariant: ð6:20Þ

Then (6.20) and (6.11) yield the transformation laws

Sdis ! Q>SdisQ ; and Sen is invariant: ð6:21Þ


Thus, with reference to the constitutive equations (6.1) we conclude that

Q>�SdisðDpdis; d

p; SÞQ ¼ �SdisðQ>Dp

disQ ; dp; SÞ ð6:22Þ

must hold for all rotations Q in the symmetry group GSS at each time t, while the other constitutivefunctions are invariant to such rotations.

We refer to the material as one which is substructurally isotropic if

GSS ¼ Orthþ; ð6:23Þ

so that the response of the material is also invariant under arbitrary rotations of the substructuralspace at each time. Henceforth, we assume that (6.23) also holds at each time, so that �Sdis is an isotro-pic function.

Thus, by assumption, all the tensorial response functions in (6.1) are isotropic. We confine ourattention to materials which may be adequately defined by such isotropic functions, and refer to suchmaterials as isotropic.

6.3. Consequences of isotropy of the free energy

Since �wðeÞðCeÞ is an isotropic function of Ce, it has the representation

�wðeÞðCeÞ ¼ ~wðeÞðICeÞ; ð6:24Þ

where

ICe ¼ I1ðCeÞ; I2ðCeÞ; I3ðCeÞ� �

is the list of principal invariants of Ce. Thus, from (6.5),

Te ¼ �TeðCeÞ ¼ 2@~wðeÞðICe Þ

@Ce ; ð6:25Þ

and �TeðCeÞ is an isotropic function of Ce. Then since the Mandel stress is defined by (cf., (5.4))

Me ¼ CeTe;

and �TeðCeÞ is isotropic, we find that Te and Ce commute,

CeTe ¼ TeCe; ð6:26Þ

and hence that the elastic Mandel stress Me is symmetric.Further the defect free energy has a representation

�wðpÞ ¼ ~wðpÞðIBpenÞ where IBp

en¼ I1ðBp

enÞ; I2ðBpenÞ; I3ðBp

enÞ� �

; ð6:27Þ

and this yields that

@~wðpÞðIBpenÞ

@Bpen

Bpen ð6:28Þ

is a symmetric tensor, and (6.6) reduces to

Sen ¼ �SenðBpenÞ ¼ 2

@~wðpÞðIBpenÞ

@Bpen

Bpen

!0

; ð6:29Þ

a symmetric and deviatoric tensor.Also the plastic Mandel stress defined in (4.33) is

Mpen ¼ 2Fp>

en

@~wðpÞðIBpenÞ

@Bpen

Bpen

!0

Fp�>en ;

10 The


¼ 2 Fp>en

@~wðpÞðIBpenÞ

@Bpen

Bpen

!Fp�>

en

" #0

;

¼ 2 Fp>en

@~wðpÞðIBpenÞ

@Bpen

!Fp

en

" #0

; ð6:30Þ

and since @~wðpÞðIBpenÞ=@Bp

en is symmetric, this stress is symmetric and deviatoric.

7. Flow rules

Henceforth, in accord with common terminology we call the symmetric and deviatoric tensor Sen aback-stress, and we denote an effective deviatoric Mandel stress by

Meeff

� �0 ¼

def Me0 � Sen: ð7:1Þ

Then, upon using the constitutive relation (6.1)4 and the first microforce balance (4.29), together withsymmetry of the Mandel stress discussed above, a central result of the theory is the first flow rule

Meeff

� �0 ¼ �TpðDp; SÞ: ð7:2Þ

Next, on account of the symmetry and deviatoric nature of the plastic stress Mpen (cf., (6.30)), the second

microforce balance (4.34) and the constitutive equation (6.4)5 yield the second flow rule of the theory:

Mpen ¼ �SdisðDp

dis;dp; SÞ: ð7:3Þ

We now make two major assumptions concerning the plastic flow for isotropic materials:

(1) Codirectionality hypotheses: Recall

dp ¼ jDpj; and let dpdis ¼ jD

pdisj;

also let

Np ¼def Dp

dp and Npdis ¼

def Dpdis

dpdis

ð7:4Þ

denote the directions of plastic flow whenever Dp–0 and Dpdis–0. Then the dissipation inequality

(6.7) may be written as

�Tpðdp;Np; SÞ : Np� �

dp þ �Sdisðdpdis;N

pdis; d

p; SÞ : Np

dis

� �dp

dis P 0: ð7:5Þ

Guided by (7.5), we assume henceforth that the dissipative flow stress Tp is parallel to andpoints in the same direction as Np so that
�Tpðdp
;Np; SÞ ¼ Yðdp;Np; SÞNp; ð7:6Þ

where

Yðdp;Np; SÞ ¼ �Tpðdp

;Np; SÞ : Np ð7:7Þ

represents a scalar flow strength of the material.Similarly we assume that dissipative flow stress Sdis is parallel to and points in the same direc-tion as Np

dis so that

�Sdisðdpdis;N

pdis;d

p; SÞ ¼ Ydisðdp

dis;Npdis;d

p; SÞNp

dis; ð7:8Þ

where

Ydisðdpdis;N

pdis; d

p; SÞ ¼ �Sdisðdp

dis;Npdis;d

p; SÞ : Np

dis ð7:9Þ

represents another scalar flow strength of the material.We refer to the assumptions (7.6) and (7.8) as the codirectionality hypotheses.10

se assumptions corresponds to the classical notion of maximal dissipation in Mises-type theories of metal plasticity.


(2) Strong isotropy hypotheses:We also assume that the scalar flow strength Yðdp

;Np; SÞ is independent of the flow direction Np,so that

Yðdp; SÞ: ð7:10Þ

Similarly, we also assume that the scalar flow strength Ydisðdpdis;N

pdis; d

p; SÞ is independent of the

flow direction Npdis, so that

Ydisðdpdis;d

p; SÞ: ð7:11Þ

We refer to the assumptions (7.10) and (7.11) as the strong isotropy hypotheses.

Thus, using (7.10) and (7.6), the flow rule (7.2) reduces to,

ðMeeffÞ0 ¼ Yðdp

; SÞNp; ð7:12Þ
which immediately gives
Np ¼ ðMeeffÞ0

jðMeeffÞ0j

; ð7:13Þ

and

jðMeeff Þ0j ¼ Yðdp

; SÞ: ð7:14Þ

When jðMeeff Þ0j and S are known, (7.14) serves as an implicit equation for the scalar flow rate dp.

Next, using (7.11) and (7.8), the flow rule (7.3) reduces to,

Mpen ¼ Ydisðdp

dis;dp; SÞNp

dis; ð7:15Þ

which gives

Npdis ¼

Mpen

jMpenj; ð7:16Þ

and

jMpenj ¼ Ydisðdp

dis; dp; SÞ: ð7:17Þ

When jðMeeff Þ0j and S are known, (7.17) serves as an implicit equation for the scalar flow rate dp

dis.Finally, using (7.6), (7.10), and (7.14), as well as (7.8), (7.11), and (7.17) the dissipation inequality

(6.7) reduces to

jðMeeff Þ0jd

p þ jMpenjd

pdis P 0: ð7:18Þ

8. Summary of the constitutive theory

In this section we summarize our constitutive theory.

(1) Free energy.

w ¼ ~wðeÞðICe Þ þ ~wðpÞðIBpenÞ; ð8:1Þ

where ICe and IBpen

are the lists of the principal invariants of Ce and Bpen, respectively.

(2) Cauchy stress.

T ¼def J�1 FeTeFe>� �where Te ¼ 2

@~wðeÞðICeÞ@Ce : ð8:2Þ

(3) Mandel stresses. Back-stress. The elastic Mandel is given by

Me ¼ CeTe; ð8:3Þ


and is symmetric.The back-stress is given by

Sen ¼ 2@~wðpÞðIBp

enÞ

@Bpen

Bpen

!0

; ð8:4Þ

and is symmetric deviatoric. The stress difference

Meeff ¼Me � Sen ð8:5Þ

is called an effective elastic Mandel stress.The plastic Mandel stress is given by

Mpen ¼ 2 Fp>

en

@~wðpÞðIBpenÞ

@Bpen

!Fp

en

" #0

; ð8:6Þ

and is symmetric and deviatoric.(4) Flow rules. The evolution equation for Fp is

_Fp ¼ DpFp; ð8:7Þ

with Dp given by

Dp ¼ dpNp; Np ¼ ðMeeff Þ0

jðMeeff Þ0j

; ð8:8Þ

where the scalar flow rate dp is obtained by solving the scalar strength relation

jðMeeffÞ0j ¼ Yðdp

; SÞ; ð8:9Þ

for given ðMeeffÞ0 and S, where Yðdp

; SÞ is a rate-dependent flow strength. The evolution equationfor Fp

dis is

_Fpdis ¼ Dp

disFpdis; ð8:10Þ

with Dpdis given by

Dpdis ¼ dp

disNpdis; Np

dis ¼Mp

en

jMpenj; ð8:11Þ

where the scalar flow rate dpdis is obtained by solving the scalar strength relation

jMpenj ¼ Ydisðdp

dis;dp; SÞ; ð8:12Þ

for given Mpen, dp, and S, where Ydisðdp

dis; dp; SÞ is another rate-dependent flow strength.

(5) Evolution equation for S.

_S ¼ hðdp; SÞ: ð8:13Þ

The evolution equations for Fp, Fpdis, and S need to be accompanied by initial conditions. Typical ini-

tial conditions presume that the body is initially (at time t ¼ 0, say) in a virgin state in the sense that

FðX;0Þ ¼ FpðX; 0Þ ¼ FpdisðX;0Þ ¼ 1; SðX;0Þ ¼ S0 ð¼ constantÞ; ð8:14Þ

so that by F ¼ FeFp and Fp ¼ FpenFp

dis we also have FeðX;0Þ ¼ 1 and FpenðX;0Þ ¼ 1.

9. Specialization of the constitutive equations

In this section, based on experience with existing recent theories of viscoplasticity with isotropicand kinematic hardening, we specialize our constitutive theory by imposing additional constitutiveassumptions.


9.1. Free-energy wðeÞ

The spectral representation of Ce is

Ce ¼X3

i¼1

xei re

i � rei ; with xe

i ¼ ke2i ; ð9:1Þ

where ðre1; r

e2; r

e3Þ are the orthonormal eigenvectors of Ce and Ue, and ðke

1; ke2; k

e3Þ are the positive eigen-

values of Ue. Instead of using the invariants ICe , the free-energy wðeÞ for isotropic materials may bealternatively expressed in terms of the principal stretches as

wðeÞ ¼ �wðeÞðke1; k

e2; k

e3Þ: ð9:2Þ

Then, by the chain-rule and (8.2)2, the stress Te is given by

Te ¼ 2@�wðeÞðke

1; ke2; k

e3Þ

@Ce ¼ 2X3

i¼1

@�wðeÞðke1; k

e2; k

e3Þ

@kei

@kei

@Ce ¼X3

i¼1

1ke

i

@�wðeÞðke1; k

e2; k

e3Þ

@kei

@xi

@Ce : ð9:3Þ

Assume that the squared principal stretches xei are distinct, so that the xe

i and the principal directionsre

i may be considered as functions of Ce; then

@xei

@Ce ¼ rei � re

i ; ð9:4Þ

and, granted this, (9.4) and (9.3) imply that

Te ¼X3

i¼1

1ke

i

@�wðeÞðke1; k

e2; k

e3Þ

@kei

rei � re

i : ð9:5Þ

Further, from (8.2)1,

T ¼ J�1FeTeFe> ¼ J�1ReUeTeUeRe> ¼ J�1ReX3

i¼1

kei@�wðeÞðke

1; ke2; k

e3Þ

@kei

rei � re

i

!Re>: ð9:6Þ

Next, since Me ¼ CeTe (cf., (8.3)), use of (9.1) and (9.5) gives the Mandel stress as

Me ¼X3

i¼1

kei@�wðeÞðke

1; ke2; k

e3Þ

@kei

rei � re

i : ð9:7Þ

Let

Ee ¼def ln Ue ¼X3

i¼1

Eei re

i � rei ; ð9:8Þ

denote the logarithmic elastic strain with principal values

Eei ¼

def ln kei ; ð9:9Þ

and consider an elastic free-energy function of the form

�wðeÞðke1; k

e2; k

e3Þ ¼ �wðeÞðEe

1; Ee2; E

e3Þ; ð9:10Þ

so that, using (9.7),

Me ¼X3

i¼1

@�wðeÞðEe1; E

e2; E

e3Þ

@Eei

rei � re

i : ð9:11Þ


We consider the following simple generalization of the classical strain-energy function of infinitesimalisotropic elasticity which uses a logarithmic measure of finite strain11

11 VlaHere, aHenckymoderaprocedu

�wðeÞðEeÞ ¼ GjEe0j

2 þ 12

KðtrEeÞ2; ð9:12Þ

where

G > 0 and K > 0; ð9:13Þ

are the shear modulus and bulk modulus, respectively. Then, (9.11) gives

Me ¼ 2GEe0 þ KðtrEeÞ1 ð9:14Þ

and on account of (9.6), (9.7), and (9.14),

T ¼ J�1ReMeRe>: ð9:15Þ

9.2. Free-energy wðpÞ

The spectral representation of Bpen is

Bpen ¼

X3

i¼1

bili � li; with bi ¼ a2i ; ð9:16Þ

where ðl1; l2; l3Þ are the orthonormal eigenvectors of Bpen and Vp

en, and ða1; a2; a3Þ are the positive eigen-values Vp

en. Instead of using the invariants IBpen

, the free-energy wðpÞ may alternatively be expressed as

wðpÞ ¼ �wðpÞða1; a2; a3Þ: ð9:17Þ

Then, by the chain-rule

@�wðpÞða1; a2; a3Þ@Bp

en¼X3

i¼1

@�wðpÞða1; a2; a3Þ@ai

@ai

@Bpen¼ 1

2

X3

i¼1

1ai


@bi

@Bpen: ð9:18Þ

Assume that bi are distinct, so that the bi and the principal directions li may be considered as functionsof Bp

en. Then,

@bi

@Bpen¼ li � li; ð9:19Þ

and, granted this, (9.18) implies that

@�wðpÞða1; a2; a3Þ@Bp

en

¼ 12

X3

i¼1

1ai


li � li: ð9:20Þ

Also, use of (9.16) and (9.20) in (8.4) gives the deviatoric back-stress as

Sen ¼X3

i¼1

ai@�wðpÞða1; a2; a3Þ

@aili � li

!0

: ð9:21Þ

Let

Epen ¼

def ln Vpen ¼

X3

i¼1

ln aili � li: ð9:22Þ

dimirov et al. (2008) employ a Neo-Hookean free-energy function modified for elastic compressibility (cf., their Eq. (28)).s an alternate, we use the classical strain-energy function for infinitesimal elasticity, but employ the large deformation-logarithmic elastic strain Ee . Anand (1979, 1986) has shown that the simple free-energy function (9.12) is useful fortely large elastic stretches. Additionally, use of the logarithmic strain helps in developing implicit time-integrationres based on the exponential map (cf., Appendix A).


denote a logarithmic energetic strain with principal values ðln aiÞ. Note that since trVpen ¼ a1a2a3 ¼ 1,

12 We

trðEpenÞ ¼ ln a1 þ ln a2 þ ln a3 ¼ lnða1a2a3Þ ¼ 0; ð9:23Þ

and hence the strain tensor Epen is traceless, Next, consider the following simple defect energy:

�wðpÞðEpenÞ ¼ BjEp

enj2 ¼ B ðln a1Þ2 þ ðln a2Þ2 þ ðln a3Þ2

h i; ð9:24Þ

with B > 0, which, since Epen is deviatoric, is analogous to (9.12). Then using (9.21), (9.22), and (9.23)

Sen ¼ 2BEpen; ð9:25Þ

we call the positive-valued constitutive parameter B the back-stress modulus.Then the plastic Mandel stress (4.33) becomes

Mpen ¼ Fp>

en SenFp�>en ¼ Rp>

en Vpenð2BEp

enÞVp�1en

� �Rp

en;

or

Mpen ¼ Rp>

en SenRpen: ð9:26Þ

9.3. Strength functions

First consider the strength relation

jðMeeff Þ0j ¼ Yðdp

; SÞ; ð9:27Þ

appearing in (8.9). Define an equivalent shear stress by

�s ¼def 1ffiffiffi2p jðMe

effÞ0j; ð9:28Þ

and an equivalent shear strain rate by

mp ¼def ffiffiffi2p

dp ¼ffiffiffi2pjDpj; ð9:29Þ

respectively. Using the definitions (9.28) and (9.29), we rewrite the strength relation (9.27) as12

�s ¼ Yðmp; SÞ; ð9:30Þ

and assume that the shear flow strength Yðmp; SÞ has the simple form

Yðmp; SÞ ¼ gðmpÞS; ð9:31Þ

where S now represents an isotropic flow resistance in shear, and

gð0Þ ¼ 0; gðmpÞ is a strictly increasing function of mp: ð9:32Þ

We refer to the dimensionless function gðmpÞ as the rate-sensitivity function. Next, by (9.32) the func-tion gðmpÞ is invertible, and the inverse function f ¼ g�1 is strictly increasing and hence strictly positivefor all nonzero arguments; further f ð0Þ ¼ 0. Hence the relation (9.30) may be inverted to give anexpression

mp ¼ g�1 �sS

� �� f

�sS

� �ð9:33Þ

for the equivalent plastic shear strain rate.An example of a commonly used rate-sensitivity function is the power-law function

gðmpÞ ¼ mp

m0

� �m

; ð9:34Þ

absorb inconsequential factors offfiffiffi2p

in writing (9.30).


where m > 0, a constant, is a rate-sensitivity parameter and m0 > 0, also a constant, is a reference flow-rate. The power-law function allows one to characterize nearly rate-independent behavior, for which mis very small. Further, granted the power-law function (9.34), the expression (9.33) has the specificform

13 Aga

mp ¼ m0�sS

� �1m

: ð9:35Þ

Next consider the strength relation

jMpenj ¼ Ydisðdp

dis;dp; SÞ; ð9:36Þ

appearing in (8.12). Define another equivalent shear stress by

�sen ¼def 1ffiffiffi2p jMp

enj; ð9:37Þ

and another equivalent shear strain rate by

mpdis ¼

def ffiffiffi2p

dpdis ¼

ffiffiffi2pjDp

disj; ð9:38Þ

respectively. Using the definitions (9.37) and (9.38), we rewrite the strength relation (9.36) as13

�sen ¼ Ydisðmpdis; m

p; SÞ: ð9:39Þ

A specialization of Ydisðmpdis; m

p; SÞ which leads to Armstrong and Frederick (1966) type kinematic hard-ening is

Ydisðmpdis; m

p; SÞ ¼ Bcmp

� �mp

dis; ð9:40Þ

where B is the back-stress modulus (cf., (9.25)), and c P 0 is a dimensionless constant. Using (9.40),(9.39) becomes

�sen ¼B

cmp

� �mp

dis; ð9:41Þ

so that the term ðB=cmpÞ represents a ‘‘pseudo”-viscosity (Dettmer and Reese, 2004). The relation(9.41) may be inverted to give

mpdis ¼

�sen

B

� �cmp; ð9:42Þ

which shows that mpdis ¼ 0 when either c ¼ 0 or when mp ¼ 0. Thus note that the constitutive assump-

tion (9.42) constrains the theory such that Dpdis ¼ 0 whenever Dp ¼ 0, and this in turn will lead to no

change in the back-stress Sen when Dp ¼ 0, that is when there is no macroscopic plastic flow.

9.4. Evolution equation for S

The evolution equation for the isotropic deformation resistance is specialized in a rate-independentform as

_S ¼ hðSÞmp ð9:43Þ

with hðSÞ a hardening function, which we take to be given by (Brown et al., 1989):

hðSÞ ¼ h0 1� SS�

� �a for S0 6 S 6 S�;

0 for S P S�;

(ð9:44Þ

in absorbing inconsequential factors offfiffiffi2p

in writing (9.39).


where S�, a, and h0 are constant moduli with S� > S0, a P 1, and h0 > 0. The hardening function (9.44)is strictly decreasing for S0 6 S 6 S� and vanishes for S P S�.

9.5. Dissipation inequality

Finally, using (9.28), (9.29), (9.37), (9.38), and (9.42), the dissipation inequality (7.18) reduces to

x ¼ vðF ¼ rF ¼ Fe

Fe; Je

Fp; Jp

Fe ¼ RDp ¼ sFp ¼ FFp

en; dFp

dis;

Fpen ¼

Dpdis ¼

T ¼ T>

w,

�sþ cBð�senÞ2

� �mp P 0: ð9:45Þ

9.6. Summary of the specialized constitutive model

In this section, we summarize the specialized form of our theory, which should be useful in appli-cations. The theory relates the following basic fields:

X; tÞ, motion;v; J ¼ det F > 0, deformation gradient;Fp, elastic–plastic decomposition of F;¼ det Fe ¼ J > 0, elastic distortion;¼ det Fp ¼ 1, inelastic distortion;

eUe, polar decomposition of Fe;ymð _FpFp�1Þ, plastic stretching corresponding to Fp;penFp

dis, elastic–plastic decomposition of Fp;et Fp

en ¼ 1, energetic part of Fp;det Fp

dis ¼ 1, dissipative part of Fp;Vp

enRpen, polar decomposition of Fp

en;symð _Fp

disFp�1dis Þ, plastic stretching corresponding to Fp

dis;, Cauchy stress;

free-energy density per unit volumeof intermediate structural space;

(1) Free energy.We consider a separable free energy

S > 0; scalar internal variable.

w ¼ wðeÞ þ wðpÞ: ð9:46Þ

With

Ue ¼X3

i¼1

kei re

i � rei ; ð9:47Þ

denoting the spectral representation of Ue, and with3

Ee ¼Xi¼1

ln kei re

i � rei ; ð9:48Þ

denoting an elastic logarithmic strain measure, we adopt the following special form for the free-energy wðeÞ:

wðeÞ ¼ GjEe0j

2 þ 12

KðtrEeÞ2; ð9:49Þ
where
G > 0; and K > 0; ð9:50Þ

are the shear modulus and bulk modulus, respectively.Further, with

Vpen ¼

X3

i¼1

aili � li; ð9:51Þ


denoting the spectral representation of Vpen, and with

Epen ¼

X3

i¼1

ln aili � li; trEpen ¼ 0; ð9:52Þ

we adopt a free-energy wðpÞ of the form

wðpÞ ¼ BjEpenj

2; ð9:53Þ

where the positive-valued parameter

B P 0; ð9:54Þ

is a back-stress modulus.(2) Cauchy stress. Mandel stresses. Back-stress. Effective stress.

Corresponding to the special free-energy functions considered above, the Cauchy stress is givenby

T ¼ J�1ReMeRe>; ð9:55Þ

where

Me ¼ 2GEe0 þ KðtrEeÞ1; ð9:56Þ

is an elastic Mandel stress.The symmetric and deviatoric back-stress is defined by

Sen ¼ 2BEpen; ð9:57Þ

and the driving stress for Dp is the effective stress given by

ðMeeffÞ0 ¼ Me

0 � Sen: ð9:58Þ

The corresponding equivalent shear stress is given by

�s ¼ 1ffiffiffi2p jðMe

effÞ0j: ð9:59Þ

The driving stress for Dpdis is the stress measure

Mpen ¼ Rp>

en SenRpen: ð9:60Þ

The corresponding equivalent shear stress is given by

�sen ¼1ffiffiffi2p jMp

enj: ð9:61Þ

(3) Flow rules.The evolution equation for Fp is

_Fp ¼ DpFp; FpðX; 0Þ ¼ 1;

Dp ¼ mp ðMeeff Þ02�s

� �;

mp ¼ m0�sS

� �1=m;

9>>>=>>>; ð9:62Þ

where m0 is a reference plastic strain rate with units of 1=time, and m is a strain-rate-sensitivityparameter.The evolution equation for Fp

dis is

_Fpdis ¼ Dp

disFpdis; Fp

disðX;0Þ ¼ 1;

Dpdis ¼ mp

disMp

en2�sen

� �;

mpdis ¼ c �sen

B

� �mp;

9>>=>>; ð9:63Þ

where c P 0 is a dimensionless constant.


(4) Evolution equation for the isotropic-hardening variable S.The internal variable S is taken to obey the evolution equation

_S ¼ hðSÞmp; SðX;0Þ ¼ S0;

hðSÞ ¼ h0 1� SS�

� �a for S0 6 S 6 S�;

0 for S P S�;

(9>>=>>; ð9:64Þ

where S�, a, and h0 are constant moduli with S� > S0, a P 1, and h0 > 0.

10. Some numerical solutions

In Appendix A, we develop a semi-implicit time-integration procedure, and obtain an (approxi-mate) algorithmic tangent for the special constitutive theory summarized in the previous section.The time-integration procedure and the associated algorithmic tangent have been implemented inthe commercial finite element program Abaqus/Standard (2008) by writing a user material subroutine(UMAT). In this section, we investigate some salient properties of the model and the accompanyingtime-integration procedure.

10.1. Simple tension/compression

First, in order to demonstrate that the constitutive theory captures the various isotropic and kine-matic-hardening effects, we carried out a numerical simulation of a symmetric strain-cycle in simpletension and compression. The material parameters used in our numerical study are:

G ¼ 80 GPa; K ¼ 175 GPa;m0 ¼ 0:001 s�1; m ¼ 0:02;S0 ¼ 100 MPa; h0 ¼ 1250 MPa; a ¼ 1 S� ¼ 250 MPa;B ¼ 40 GPa; c ¼ 400;

ð10:1Þ

the value of m ¼ 0:02 is intended to represent a nearly rate-independent material.The axial strain-cycle was imposed between true strain limits of � ¼ �0:02 at an absolute axial true

strain rate of j _�j ¼ 0:01 s�1; a fixed time step Dt ¼ 0:01 s was used in the numerical simulations. The cal-culations were performed for: (i) no hardening, h0 ¼ 0 and B ¼ 0; (ii) isotropic hardening only, h0–0 andB ¼ 0; (iii) kinematic hardening only, h0 ¼ 0 and B–0; and finally (iv) combined kinematic and isotropichardening, h0–0 and B–0. Plots of the axial stressr versus the axial strain � are shown in Fig. 1(a). With nohardening, we obtain an elastic-perfectly-plastic response, as expected. When isotropic hardening only isallowed, we observe an increase in the flow stress, and the stress level at which plastic flow recommenceson strain reversal is equal in magnitude to the stress level from which the reversal in strain was initiated.In the case of kinematic hardening only, we again observe an increase in the flow stress, but this time themagnitude of the stress level at which plastic flow recommences on strain reversal is substantially smallerthan the stress level from which the reversal in strain direction was initiated; a clear manifestation of theBauschinger effect. When isotropic hardening is combined with kinematic hardening, we see the contin-ual evolution in the flow stress associated with combined isotropic and kinematic hardening during for-ward straining, and a clear Bauschinger effect due to the kinematic hardening upon strain reversal.

10.2. Simple shear

Next, we concentrate on numerical solutions related to homogeneous simple shear. With respect toa rectangular Cartesian coordinate system with origin o and orthonormal base vectors feiji ¼ 1;3g, asimple shearing motion is described by

x ¼ Xþ ðmtÞX2e1; ð10:2Þ

with m a shear strain rate, and C ¼ mt is the amount of shear. Throughout this section we use the mate-rial parameters summarized in (10.1).

−0.02 −0.01 0 0.01 0.02−600

−400

−200

0

200

400

600

Axial strain, ε

Axi

al s

tres

s, σ

(M

Pa)

No hardeningIsotropic onlyKinematic onlyCombined

0 0.01 0.02 0.03 0.04 0.05−300

−200

−100

0

100

200

300

Amount of shear, Γ

Str

ess,

T12

(M

Pa)

No hardeningIsotropic onlyKinematic onlyCombined

a

b

Fig. 1. (a) Comparison of axial stress r versus axial strain � for various types hardening in a symmetric strain-cycle simulationin simple tension and compression. (b) Comparison of shear stress T12 versus shear strain C for various types of hardening inreversed simple shear.


The specific goals of our numerical solutions related to simple shear are as follows:

� To further demonstrate that the constitutive theory captures the major features of isotropic andkinematic hardening in the case of simple reversed shear.

� To investigate the convergence and stability properties of the time-integration scheme in simplereversed shear.

� To demonstrate that the constitutive equations do not suffer from oscillations in the stress responseat large shear strains.

� To demonstrate that the constitutive equations and the time-integration procedure are indeedobjective.


First, we wish to further demonstrate that the constitutive theory captures the various isotropicand kinematic-hardening effects. To this end, we consider simple shear, as described above, for a timeof 5 s at a shear strain rate of m ¼ 0:01 s�1 for a total shear strain C ¼ mt ¼ 0:05, and then at a shearstrain rate of m ¼ �0:01 s�1 for another 5 s to complete one reversal of strain, while using a fixed timestep Dt ¼ 0:01 s. As for simple tension/compression, the calculations were performed for: (i) no hard-ening, h0 ¼ 0 and B ¼ 0; (ii) isotropic hardening only, h0–0 and B ¼ 0; (iii) kinematic hardening only,h0 ¼ 0 and B–0; and finally (iv) combined kinematic and isotropic hardening, h0–0 and B–0, and theresults are shown in Fig. 1(b). Again, we observe the important qualitative features of each case in sim-ple reversed shear, i.e. elastic-perfectly-plastic response in the case of no hardening, a continual in-crease in the flow stress in cases involving isotropic hardening, and a clear Bauschinger effect incases involving kinematic hardening.

Next, we investigated the stability and convergence properties of our time-integration procedure.Since our integration procedure is semi-implicit rather than fully-implicit, one might be concernedthat there may be a restrictive time step constraint for stability. To ascertain any time step restrictions,we performed a convergence test. Reversed simple shear to a maximum shear strain of C ¼ 0:05 at ashear strain rate of jmj ¼ 0:01 s�1 was considered while increasingly coarsening the time step. The spe-cific time steps Dt of

1 10�3; 1 10�2; 2:5 10�2; 1 10�1; 2:5 10�1; and 1 s

were considered; these time steps translate to shear strain increments of

1 10�5; 1 10�4; 2:5 10�4; 1 10�3; 2:5 10�3; and 1 10�2;

respectively. Fig. 2 shows the stress–strain curves corresponding to these cases. For increasing timesteps, no evidence of instability is observed, but of course the accuracy of the solution degrades, espe-cially in the regions of elastic–plastic transitions. The order of convergence is demonstrated in Fig. 3;since no exact solution is available, the solution obtained using a time step of Dt ¼ 1 10�3 s was trea-ted as the ‘‘exact” solution and used to calculate the pointwise error for the cases corresponding to thelarger time steps. The error E was then obtained by taking the L-infinity norm of the pointwise error.As seen in Fig. 3, the time-integration procedure is first-order accurate. Thus, while no stability prob-lems are encountered, the accuracy of the integration procedures deteriorates, especially when largetime increments are taken during periods of rapidly changing elastic–plastic transitions.

0 0.01 0.02 0.03 0.04 0.05−300

−200

−100

0

100

200

300

Amount of shear, Γ

Str

ess,

T12

(M

Pa)

Δ t = 1×10−3 s

Δ t = 1×10−2 s

Δ t = 1×10−1 sΔ t = 1 s

Fig. 2. Comparison of solutions for reversed simple shear for various time steps.

−2.5 −2 −1.5 −1 −0.5 0 0.56

6.5

7

7.5

8

8.5

log(Δ t)

log(

E)

11

Fig. 3. Convergence diagram for the time-integration procedures demonstrating first-order accuracy.


We wish next to demonstrate that the constitutive equations do not produce shear-stress oscilla-tions at large strains – a feature which has plagued numerous previous large deformation constitutivetheories for kinematic hardening. To this end, we performed a monotonic simple shear simulation to avery large final shear strain of C ¼ 10, at a shear strain rate of m ¼ 0:01 s�1 using a fixed time stepDt ¼ 0:1 s. The shear T12 and normal components, T11 and T22, of the Cauchy stress T are plotted inFig. 4(a). Clearly there are no oscillations in the T12 versus C response; the shear stress rises and satu-rates (because of the chosen value of the material parameters) at a shear strain of C 1:5. We have alsocarried out a similar calculation with all other material parameters the same as before, but by setting thedynamic recovery parameter c ¼ 0; this corresponds to the case of ‘‘linear” kinematic hardening. Thevariation of the shear and normal components of the Cauchy stress T for this case are plotted inFig. 4(b). Here, since there is no dynamic recovery for the kinematic hardening, the stress levels for thisphysically unrealistic case get quite large, but the T12 versus C curve still does not show any oscillations;the peak in this curve is due to the use of a logarithmic energetic strain lnðVp

enÞ and not due to any oscil-latory behavior in the evolution of the back-stress.

Finally, in order to demonstrate the frame-indifference of the constitutive equations and numericalalgorithm, we have carried out a numerical calculation for simple shear with superposed rigid rota-tion, which is described by Weber et al. (1990):

x ¼ Q ðtÞ Xþ ðmtÞX2e1½ �; ð10:3Þ

with

Q ðtÞ ¼ ðe1 � e1 þ e2 � e2Þ cosðxtÞ þ ðe1 � e2 � e2 � e1Þ sinðxtÞ þ e3 � e3; ð10:4Þ

a rotation about the e3-axis. In performing our numerical calculation, we used the material parameterslisted in (10.1), and the calculation was performed with x ¼ 0:1p radians per second and m ¼ 0:01 persecond, for t 2 ½0;20�, so that h � xt 2 ½0;2p� and C � mt 2 ½0;0:2�. The calculation was carried by usinga fixed time step of Dt ¼ 0:1 s, which corresponds to a shear strain increment of DC ¼ 0:001 and a rota-tion increment of Dh ¼ 3:6�. Snapshots of the initial and deformed geometry at a few representativestages of the simple shear plus superposed rotation are shown in Fig. 5. Let the solution for the stress cor-responding to the motion (10.3) be denoted by T�ðtÞ. The result of the shear component of the ‘‘unrotat-ed” Cauchy stress ðQ ðtÞ>T�ðtÞQ ðtÞÞ versus the amount of shear C, obtained from the numericalcalculation is shown in Fig. 6. Also shown in this figure is T12 versus C for the motion (10.3) withQ ðtÞ ¼ 1. The two shear stress versus shear strain curves are indistinguishable from each other; the

0 2 4 6 8 10−50

0

50

100

150

200

250

300

350

400

Amount of shear, Γ

Str

ess

(MP

a)T

12

T11

T22

0 2 4 6 8 10−1.5

−1

−0.5

0

0.5

1

1.5 x 105

Amount of shear, Γ

Str

ess

(MP

a)

T12

T11

T22

a

b

Fig. 4. Simple shear to large shear strain (a) with dynamic recovery c–0; and (b) without dynamic recovery, c = 0. Note thatthere are no shear-stress oscillations in either case.


excellent agreement between the two calculations verifies the objectivity of the constitutive equationsand time-integration procedure.

10.3. Cyclic loading of a curved bar

Next, in order to exercise the algorithmic tangent in a case of inhomogeneous deformation, we considerthe cyclic deformation of a curved bar of circular cross-section. A finite element mesh for the curved bar isshown in Fig. 7(a). The bar, which has a diameter of 50 mm, curves through a radius of 100 mm for 90�, andthen extends straight for another 100 mm. The mesh consists of 2184 Abaqus-C3D8R elements. The nodeson the face denoted as ‘‘fixed”-face in Fig. 7(a) have prescribed null displacements, u1 ¼ u2 ¼ u3 ¼ 0, whilethe boundary conditions on the face denoted as the ‘‘moved”-face were specified as

Fig. 5. Simple shear with superposed rigid rotation. The deformed geometry at each stage is shown with a solid line, while thedashed line represents the initial geometry.


u1 ¼ u3 ¼ 0; and u2ðtÞ ¼ a sinðxtÞ; with a ¼ 20 mm and x ¼ 0:2p s�1: ð10:5Þ

The calculation was run to a final time of tf ¼ 30 s, so that the u2-displacement on the ‘‘moved”-facegoes through three complete cycles. The material properties used in the calculation were the same asthose in the previous calculations (10.1). The deformed mesh at the point of maximum deflection isalso shown in Fig. 7(a).

In this (and subsequent) multi-element calculations, a variable time-stepping integration proce-dure was employed. As in the paper by Lush et al. (1989), the calculation was performed with aslightly modified version of the Abaqus/Standard static analysis procedure. The modification wasintroduced to enhance the automatic time-stepping procedure in Abaqus to control the accuracy ofthe constitutive time-integration. This was done by using as a control measure the maximum equiva-lent plastic shear strain increment

Dcpmax ¼ ðDtÞðmp

nþ1Þmax

occurring at any integration point in the model during an increment. The value of Dcpmax was kept close

to a specified nominal value Dcps . From Fig. 2, we note that the accuracy of the solution begins to

0 0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

Amount of shear, Γ

Str

ess,

T12

(M

Pa)

Shear and rotationShear only

Fig. 6. Comparison of results for T12 versus C curves from shear with superposed rotation against shear with no superposedrotation.


degrade for shear strain increments of around 1 10�3; accordingly, we chose Dcps to be half this va-

lue, 0:5 10�3. The automatic time-stepping algorithm operated to keep the ratio

14 Aftewhetheincremeused to

Note threcalcu

R � Dcpmax

Dcps

ð10:6Þ

close to 1.0 by adjusting the size of the time increments.14

As the curved bar is cyclically displaced in the ðe2; e3Þ-plane at the ‘‘moved-face”, the section ofthe bar near the ‘‘fixed”-face is subjected to reversed-torsion, while the part of the bar towards the‘‘moved”-face is subjected to reversed-bending, resulting in states of shear, tension, and compres-sion in various parts of the body. With increasing displacement magnitude at the ‘‘moved”-face,inhomogeneous plastic deformation initiates and spreads in various disparate regions of the curvedbar. The integrated reaction force in the 2-direction on the ‘‘moved”-face is plotted against the pre-scribed u2-displacement in Fig. 7(b). This resulting overall cyclic load–displacement curve is the re-sult of the complex evolution of the internal variables controlling the combined isotropic andkinematic hardening at the various sections of the bar which are undergoing plastic deformation.The overall load versus displacement response is smooth, and one can clearly see the effects ofboth kinematic hardening in the prominent Bauschinger effect and isotropic hardening in the con-tinual increase in the magnitude of the load with cyclic deformation.

This example shows that our semi-implicit integration procedure, when coupled with our heuris-tic automatic time-stepping algorithm and our approximate algorithmic tangent, is effective in

r an equilibrium solution for a time increment Dtn ¼ tnþ1 � tn was found, the value of the ratio R was checked to determiner this solution would be accepted or not. If R was greater than 1.25, then the solution was rejected, and a new timent was used that was smaller by the factor ð0:85=RÞ. If R 6 1:25, then the solution was accepted, and the value of R wasdetermine the first trial size of the next time increment. The following algorithm was used:

if 0:8 < R 6 1:25 then Dtnþ1 ¼ Dtn=R;

if 0:5 < R 6 0:8 then Dtnþ1 ¼ 1:25Dtn;

if R 6 0:5 then Dtnþ1 ¼ 1:50Dtn:

ð10:7Þ

at the measure Dcpmax was allowed to exceed the user specified value Dcp

s by up to 25%. This was done to avoid having tolate increments that came out just slightly above the specified nominal value, but were otherwise essentially acceptable.

−20 −10 0 10 20−150

−100

−50

0

50

100

150

Displacement, u2, (mm)

Load

(kN

)

a

b

Fig. 7. (a) Undeformed and deformed meshes for the cyclic loading of a curved bar. (b) Resulting tip load versus tipdisplacement.


obtaining an accurate, stable, and efficient solution in an inhomogeneous large deformation implicitfinite element calculation.

10.4. Bending and spring-back of an aluminum sheet

As our final numerical example, we consider a simple plane-strain sheet forming operation inwhich an aluminum sheet is plastically bent about a mandrel of fixed radius and then unloaded.We use the recent experimental data of Cao et al. (2008) for AA6111-T4 sheets to estimate the mate-rial parameters in our model. The fit of the model to the experimental data is shown in Fig. 8, and theestimated material parameters are listed in (10.8):

G ¼ 25:5 GPa; K ¼ 68 GPa;m0 ¼ 0:001 s�1; m ¼ 0:02;S0 ¼ 67:55 MPa; h0 ¼ 1200 MPa; a ¼ 1; S� ¼ 156:7 MPa;B ¼ 1:09 GPa; c ¼ 61:08:

ð10:8Þ

0 0.02 0.04 0.06 0.08 0.1 0.12−400

−300

−200

−100

0

100

200

300

400

True axial strain

True

axi

al s

tress

(MPa

)

ExperimentModel

Fig. 8. Fit of the model to experimental data for AA6111-T4 sheet material. The data is from Cao et al. (2008); note that the kinkin the reversed loading data for the smallest strain amplitude is actually reported by these authors, and is probably an artifact ofthe experimental difficulties associated with performing reversed loading experiments on sheet materials.


The initial configuration for our plane-strain sheet forming simulation is shown in Fig. 9(a). Weconsider a 1 mm thick sheet with a length of 75 mm, which is to be to bent to a radius of 50 mmbetween a pair of matched-rigid dies; accordingly, the radius of the top and bottom dies is pre-scribed to be 50.5 mm and 49.5 mm, respectively. Due to the symmetry of the problem, we con-sider only half of the geometry with suitable boundary conditions at the symmetry-plane. Thesheet is modeled using a mesh consisting of 1880 Abaqus-CPE4H plane-strain elements with sevenelements through the sheet thickness, and the dies are modeled using rigid surfaces. Contact be-tween the sheet and the dies was modeled as frictionless. In the first step of the sheet formingsimulation, the top die is moved downward to bend the sheet completely around the bottomdie. Fig. 9(b) shows the geometry of the initial and fully bent geometries of the sheet. In the sec-ond step, the top die is moved back upwards, and the sheet is allowed to spring-back. The un-loaded sprung-back geometry is also shown in Fig. 9(b).

It is widely believed that plasticity models which do not account for kinematic hardening tend toincorrectly predict the amount of spring-back in simulations of sheet-metal forming operations (cf.,e.g., Zhao and Lee, 2001). Thus, our large deformation theory which not only includes both kinematicand isotropic hardening but also accounts for large rotations, should be useful in improved modelingof spring-back phenomena in sheet forming operations.

11. Concluding remarks

In this paper, we have formulated a large deformation constitutive theory for combined isotropicand kinematic hardening based on the dual decomposition F ¼ FeFp and Fp ¼ Fp

enFpdis. We now show

that a similar kinematic-hardening theory may also be constructed without using the decompositionFp ¼ Fp

enFpdis. Recall the relation (6.3) for the evolution of Bp

en,

_Bpen ¼ Dp

enBpen þ Bp

enDpen; ð11:1Þ

and the kinematical equation (2.28)

0 10 20 30 40−20

−15

−10

−5

0

5

x (mm)

x (

mm

)

UndeformedFully bentSprung−back

2

a

b

Fig. 9. (a) Initial configuration for plane-strain matched-die sheet bending simulations. (b) Undeformed and fully bent sheetgeometries, together with the sprung-back geometry.



enDpdisF

p�1en

� �: ð11:2Þ

The relation (11.2) gives

Dpen ¼ Dp � sym Fp

enDpdisF

p�1en

� �; ð11:3Þ

¼ Dp � mpdis

2�sen

� �sym Fp

enMpenFp�1

en

� �ðusing ð9:63ÞÞ;

¼ Dp � mpdis

2�sen

� �sym Fp

enRp>en SenRp

enFp�1en

� �ðusing ð9:60ÞÞ;

¼ Dp � mpdis

2�sen

� �sym Vp

enSenVp�1en

� �;

¼ Dp � mpdis

2�sen

� �sym Vp

enðB ln BpenÞV

p�1en

� �ðusing ð9:57Þ and Ep

en � ð1=2Þ ln BpenÞ;


¼ Dp � Bmpdis

2�sen

� �ðln Bp

enÞ;

¼ Dp � 12cmp

� �ðln Bp

enÞ ðusing ð9:42ÞÞ: ð11:4Þ

Using (11.4) in (11.1) gives

_Bpen ¼ Dp � 1

2cmp

� �ðln Bp

enÞ� �

Bpen þ Bp

en Dp � 12cmp

� �ðln Bp

enÞ� �

;

or

_Bpen ¼ DpBp

en þ BpenDp � cmpBp

enðln BpenÞ: ð11:5Þ

Thus, if we set

A � Bpen;

we obtain the evolution equation

_A ¼ DpAþ ADp � cmpAðln AÞ: ð11:6Þ

Therefore, instead of using the decomposition Fp ¼ FpenFp

dis and developing a theory which gives rise toa back-stress Sen because of a defect free-energy �wðpÞðBp

enÞ, one may develop an alternate theory basedon an internal variable A, with a corresponding defect free-energy �wðpÞðAÞ, and an evolution equationfor A obeying (11.6); indeed, this is the track taken recently by Anand et al. (2008). However, we donote that the development of an implicit time-integration procedure for a constitutive theory based onA is not as straightforward as the one developed here for a theory utilizing the Fp ¼ Fp

enFpdis

decomposition.Finally, as is abundantly clear from the extensive literature on hardening models for metal plastic-

ity (cf., e.g., Chaboche, 2008), the simple theory with combined isotropic and kinematic hardeningdeveloped in this paper is only foundational in nature, and there are numerous specialized enhance-ments/modifications to the theory that need to be incorporated in order to match actual experimentaldata for different metals; we leave such work to the future.

Acknowledgements

This work was supported by the National Science Foundation under Grant Nos. CMS-0555614 andCMMI-062524.

Appendix A

In this Appendix we develop a semi-implicit time-integration procedure, and obtain an (approxi-mate) algorithmic tangent for the theory formulated in this paper. The time-integration procedure andthe associated tangent are intended for use in the context of ‘‘implicit” finite element procedures. Wehave implemented the procedure described below in the commercial finite element program Abaqus/Standard (2008) by writing a user material subroutine (UMAT), and it is using this finite element pro-gram that we have carried out the calculations whose results are reported in the main body of thepaper.

A.1. Time-integration procedure

The constitutive time-integration problem is that given

fTn; Fpn; Sn; ðSenÞn; ðF

pdisÞng; as well as Fn and Fnþ1;

at time tn, we need to calculate


fTnþ1; Fpnþ1; Snþ1; ðSenÞnþ1; ðF

pdisÞnþ1g

at time tnþ1 ¼ tn þ Dt.The evolution equation for _Fp ¼ DpFp is integrated by means of an exponential map (Weber and

Anand, 1990)

Fpnþ1 ¼ exp DtDp

nþ1

� �Fp

n; with Dpnþ1 ¼ D̂p

nþ1ðððMeeffÞnþ1Þ0; Snþ1Þ; ð12:1Þ

the inverse of Fpnþ1 is then

Fp�1nþ1 ¼ Fp�1

n exp �DtDpnþ1

� �: ð12:2Þ

Next, using Fe ¼ FFp�1 and (12.1)1, the elastic deformation gradient at the the end of the step isgiven by

Fenþ1 ¼ Fe

trial exp �DtDpnþ1

� �; ð12:3Þ

where

Fetrial ¼

def Fnþ1Fp�1n ðtrial values are evaluated with plastic flow frozenÞ ð12:4Þ

is a trial value of Fe at the end of the step. The elastic right Cauchy–Green tensor and its trial value atthe end of the step are

Cenþ1 ¼ Fe>

nþ1Fenþ1; Ce

trial ¼ Fe>trialF

etrial: ð12:5Þ

Thus, using (12.3),

Cenþ1 ¼ exp �DtDp

nþ1

� �Ce

trial exp �DtDpnþ1

� �: ð12:6Þ

To proceed further we make our first approximation:

(A1) To first order in Dt, we approximate

exp �DtDpnþ1

� �¼: 1� DtDp

nþ1

� �: ð12:7Þ

Hence, (12.6) becomes

Cenþ1¼

: 1� DtDpnþ1

� �Ce

trial 1� DtDpnþ1

� �;

¼: Cetrial � DtDp

nþ1Cetrial � DtCe

trialDpnþ1;

¼: Cetrial 1� DtCe�1

trialDpnþ1Ce

trial � DtDpnþ1

� �: ð12:8Þ

Next, consider the term Ce�1trialD

pnþ1Ce

trial

� �; with EeG

trial ¼ 12 ðC

etrial � 1Þ denoting a Green strain correspond-

ing to Cetrial, we have Ce

trial ¼ 1þ 2EeGtrial, and hence

Ce�1trialD

pnþ1Ce

trial¼: 1� 2EeG

trial

� �Dp

nþ1 1þ 2EeGtrial

� �;

¼: Dpnþ1 þ 2Dp

nþ1EeGtrial � 2EeG

trialDpnþ1 � 4EeG

trialDpnþ1EeG

trial: ð12:9Þ

We now make our second approximation:

(A2) We assume that jEeGtrialj 1, so that (12.9) may be approximated as

Ce�1trialD

pnþ1Ce

trial ¼: Dp

nþ1: ð12:10Þ

Using (12.10), we find that (12.8) reduces to

Cenþ1¼

: Cetrial 1� 2DtDp

nþ1

� �: ð12:11Þ

Further, using the symmetry of Cenþ1 we note that the symmetric tensors Ce

trial and 1� 2DtDpnþ1

� �com-

mute, and hence share the same principal directions. Thus taking the logarithm of (12.11) we obtain

15 Subintegrat


Eenþ1¼

: Eetrial þ

12

ln 1� 2DtDpnþ1

� �; ð12:12Þ

where

Eenþ1 ¼

12

ln Cenþ1 and Ee

trial ¼12

ln Cetrial: ð12:13Þ

Next, we make our third approximation:

(A3) Recalling the series representation of the logarithm of a tensor, we assume that

ln 1� 2DtDpnþ1

� �¼: � 2DtDp

nþ1

for small DtDpnþ1. Thus, under the assumptions (A1)–(A3) of small Dt and small jEeG

trialj, (12.6) yields theimportant relation

Eenþ1¼

: Eetrial � DtDp

nþ1: ð12:14Þ

Next, let

C ¼def 2G I� 13

1� 1� �

þ K1� 1 ð12:15Þ

denote the elasticity tensor, and let

Menþ1 ¼ C½Ee

nþ1� and Metrial ¼ C½Ee

trial�; ð12:16Þ

respectively, denote the elastic Mandel stress at the end of the time step, as well as its trial value.Then, operating on (12.14) by C gives

Menþ1 ¼Me

trial � DtC½Dpnþ1�: ð12:17Þ

Subtracting ðSenÞnþ1 from the left-hand side and15

ðSenÞtrial � ðSenÞn; ðtrial values are evaluated with plastic flow frozenÞ

from the right-hand side of (12.17), and writing

ðMeeffÞtrial ¼

def Metrial � ðSenÞtrial; ð12:18Þ

gives

ðMeeffÞnþ1 ¼ ðM

eeffÞtrial � DtC½Dp

nþ1�: ð12:19Þ

Next, from (9.62) we have

Dpnþ1 ¼

1ffiffiffi2p mp

nþ1Npnþ1; Np

nþ1 ¼ððMe

effÞnþ1Þ0ffiffiffi2p

�snþ1; �snþ1 ¼

1ffiffiffi2p jððMe

effÞnþ1Þ0j; ð12:20Þ

where

mpnþ1 ¼

ffiffiffi2pjDp

nþ1j: ð12:21Þ

Using (12.15) and (12.20) in (12.19) we obtain

ðMeeffÞnþ1 ¼ ðM

eeffÞtrial �

ffiffiffi2p

GðDtmpnþ1ÞN

pnþ1: ð12:22Þ

Since Npnþ1 is deviatoric, the deviatoric and spherical parts of (12.22) give

tracting ðSenÞn rather than ðSenÞnþ1 from the right-hand side of (12.17) to arrive at (12.19), is a critical step in our time-ion procedure. It allows us to decouple the update of Fp and Fp

dis into to two separate, yet otherwise implicit updates.


ðMeeffÞnþ1

� �0 ¼ ððM

eeffÞtrialÞ0 �

ffiffiffi2p

GðDtmpnþ1ÞN

pnþ1;

trðMeeffÞnþ1 ¼ trðMe

effÞtrial: ð12:23Þ

Using (12.20)2, (12.23)1 may be arranged as
ffiffiffi2p
�snþ1 þ GðDtmpnþ1Þ

� �Np

nþ1 ¼ ððMeeffÞtrialÞ0: ð12:24Þ

Next, defining

Nptrial ¼

def ððMeeffÞtrialÞ0ffiffiffi2p

�strial

; �strial ¼def 1ffiffiffi

2p jððMe

effÞtrialÞ0j; ð12:25Þ

(12.24) may be written as

�snþ1 þ GðDtmpnþ1Þ

� �Np

nþ1 ¼ �strialNptrial; ð12:26Þ

which immediately gives

Npnþ1 ¼ Np

trial;

�snþ1 þ GðDtmpnþ1Þ ¼ �strial: ð12:27Þ

Thus the direction of plastic flow at the end of the step Npnþ1 is determined by the trial direction of plastic

flow Nptrial.

Next, the implicit form of the flow strength relation (9.30) is

�snþ1 ¼ Y mpnþ1; Snþ1

� �: ð12:28Þ

We integrate the evolution equation for S in a semi-implicit fashion as

Snþ1 ¼ Sn þ hðSnÞðDtmpnþ1Þ: ð12:29Þ

Since the hardening function h typically does not change rapidly, we do not expect the semi-implicitintegration of the evolution equation for S to have a significant effect on the stability of our algorithm.

Using (12.29) in (12.28), we have

�snþ1 ¼ Y mpnþ1; Sn þ hðSnÞðDtmp

nþ1Þ� ��

: ð12:30Þ

Finally, using (12.30) in (12.27)2 gives the following implicit equation for mpnþ1:

Gðmpnþ1Þ ¼ �strial � GðDtmp

nþ1Þ � Y mpnþ1; Sn þ hðSnÞðDtmp

nþ1Þ� ��

¼ 0: ð12:31Þ

Once a solution for mpnþ1 is obtained by solving (12.31), Snþ1 is easily evaluated by using (12.29). The

elastic Mandel stress Menþ1 is obtained from (12.17) using

Dpnþ1 ¼

1ffiffiffi2p mp

nþ1Nptrial; ð12:32Þ

as

Menþ1 ¼Me

trial �ffiffiffi2p

GðDtmpnþ1ÞN

ptrial: ð12:33Þ

The update for Fp is obtained from (12.1)1 using the result (12.32):

Fpnþ1 ¼ exp

1ffiffiffi2p ðDtmp

nþ1ÞNptrial

� �Fp

n: ð12:34Þ

Next, calculating

Fenþ1 ¼ Fnþ1Fp�1

nþ1; ð12:35Þ

and performing a polar decomposition of Fenþ1 gives Re

nþ1. Finally, using Renþ1, and the relations (12.33)

and (9.55) gives the update for the Cauchy stress as

Tnþ1 ¼ J�1nþ1ðR

enþ1ÞðM

enþ1ÞðR

enþ1Þ

> with Jnþ1 ¼ det Fnþ1: ð12:36Þ


We turn next to updating Sen and Fpdis. The evolution equation for _Fp

dis ¼ DpdisF

pdis is also integrated by

using an exponential map and (9.63) as

ðFpdisÞnþ1 ¼ exp DtðDp

disÞnþ1

� �ðFp

disÞn with ðDpdisÞnþ1 ¼

c2B

� �mp

nþ1ðMpenÞnþ1; ð12:37Þ

the inverse of ðFpdisÞnþ1 is then

ðFpdisÞ

�1nþ1 ¼ ðF

pdisÞ

�1n exp �DtðDp

disÞnþ1

� �: ð12:38Þ

Hence, using Fpen ¼ FpFp�1

dis , the energetic part of Fp at the end of the step is given by

ðFpenÞnþ1 ¼ ðF

penÞtrial exp �DtðDp

disÞnþ1

� �; ð12:39Þ

where

ðFpenÞtrial ¼

def Fpnþ1ðF

pdisÞ

�1n ð12:40Þ

is a trial value of Fpen at the end of the step.

The tensors ðFpenÞnþ1 and ðFp

enÞtrial admit the polar decompositions

ðFpenÞnþ1 ¼ ðR

penÞnþ1ðU

penÞnþ1; ðFp

enÞtrial ¼ ðRpenÞtrialðU

penÞtrial: ð12:41Þ

Using (12.41) in (12.39) and rearranging, we obtain

ðRpenÞnþ1ðU

penÞnþ1 exp DtðDp

disÞnþ1

� �¼ ðRp

enÞtrialðUpenÞtrial: ð12:42Þ

Next, from (9.57) and (9.60)

Mpen ¼ Rp>

en SenRpen ¼ Rp>

en ð2B ln VpenÞR

pen ¼ 2B ln Up

en: ð12:43Þ

Thus the principal directions of ðMpenÞnþ1, and hence from (12.37)2 those of ðDp

disÞnþ1, are the same asthose ðUp

enÞnþ1. Hence

ðUpenÞnþ1 exp DtðDp

disÞnþ1

� �is symmetric: ð12:44Þ

Then, because of the uniqueness of the polar decomposition theorem

ðRpenÞnþ1 ¼ ðR

penÞtrial; ð12:45Þ

ðUpenÞnþ1 exp DtðDp

disÞnþ1

� �¼ ðUp

enÞtrial: ð12:46Þ

Eq. (12.46) implies that ðUpenÞnþ1 and ðUp

enÞtrial have the same principal directions. Thus taking thelogarithm on both sides and rearranging we have

lnðUpenÞnþ1 ¼ lnðUp

enÞtrial � DtðDpdisÞnþ1: ð12:47Þ

Multiplying (12.47) through by 2B and using (12.43) gives

ðMpenÞnþ1 ¼ ðM

penÞtrial � 2BDtðDp

disÞnþ1; where ðMpenÞtrial ¼

def 2B lnðUpenÞtrial: ð12:48Þ

Finally, using (9.60), (12.37)2, (12.45), and pre-multiplying (12.48) by ðRpenÞtrial and post-multiplying by

ðRpenÞ>trial gives

ðSenÞnþ1 ¼ ðSenÞtrial � cðDtmpnþ1ÞðSenÞnþ1; where ðSenÞtrial ¼

def 2B lnðVpenÞtrial: ð12:49Þ

Thus, the back-stress Sen is updated as

ðSenÞnþ1 ¼1

1þ cðDtmpnþ1Þ

!ðSenÞtrial: ð12:50Þ

Correspondingly,

ðMpenÞnþ1 ¼ ðR

penÞ>trialðSenÞnþ1ðR

penÞtrial; ð12:51Þ

use of which in (12.37) provides the update ðFpdisÞnþ1 for Fp

dis.


Remark. Due to the use of the exponential map in integrating the evolution equations for Fp and Fpdis,

the constraint of plastic incompressibility is exactly maintained by our time-integration procedure.This is easily verified by recalling the identity detðexp AÞ ¼ expðtrAÞ and recognizing the deviatoricnature of Dp and Dp

dis in (12.1) and (12.37), respectively.

A.1.1. Summary of the time-integration procedure

� Given: fTn; Fpn; Sn; ðSenÞn; ðF

pdisÞng, as well as Fn and Fnþ1, at time tn.

� Calculate: fTnþ1; Fpnþ1; Snþ1; ðSenÞnþ1; ðF

pdisÞnþ1g at time tnþ1 ¼ tn þ Dt.

Step 1. Calculate the trial elastic deformation gradient

e p�1
Ftrial ¼ Fnþ1Fn : ð12:52Þ
Step 2. Perform the polar decomposition

Fetrial ¼ Re

trialUetrial: ð12:53Þ

Step 3. Calculate the trial elastic strain

Eetrial ¼ ln Ue

trial: ð12:54Þ

Step 4. Calculate the trial Mandel stress and the trial effective Mandel stress

Metrial ¼ C½Ee

trial�; ð12:55ÞðMe

effÞtrial ¼Metrial � ðSenÞn: ð12:56Þ

Step 5. Calculate the trial mean normal pressure, the deviatoric part of the trial effective Mandelstress, the trial equivalent shear stress, and the trial direction of plastic flow

�ptrial ¼ �13

trðMeeffÞtrial; ð12:57Þ

ððMeeffÞtrialÞ0 ¼ ðM

eeffÞtrial þ �ptrial1; ð12:58Þ

�strial ¼ffiffiffi12

rððMe

effÞtrialÞ0 ; ð12:59Þ

Nptrial ¼

ððMeeff ÞtrialÞ0ffiffiffi2p

�strial

: ð12:60Þ

Step 6. Calculate mpnþ1 by solving

Gðmpnþ1Þ ¼ �strial � GðDtmp

nþ1Þ � Y mpnþ1; Sn þ hðSnÞðDtmp

nþ1Þ� ��

¼ 0: ð12:61Þ

Step 7. Update Dp

Dpnþ1 ¼

ffiffiffiffiffiffiffiffiffiffið1=2

pÞmp

nþ1Nptrial: ð12:62Þ

Step 8. Update Fp

Fpnþ1 ¼ exp DtDp

nþ1

� �Fp

n: ð12:63Þ

Step 9. Update the Mandel stress Me

Menþ1 ¼Me


GðDtmpnþ1ÞN

ptrial: ð12:64Þ

Step 10. Update Fe

Fenþ1 ¼ Fnþ1Fp�1

nþ1: ð12:65Þ


Fenþ1 ¼ Re

nþ1Uenþ1: ð12:66Þ


Step 12. Update the Cauchy stress

Jnþ1 ¼ det Fnþ1ð Þ; ð12:67ÞTnþ1 ¼ J�1

nþ1ðRenþ1ÞðM

enþ1ÞðR

enþ1Þ

>: ð12:68Þ

Step 13. Update S

Snþ1 ¼ Sn þ hðSnÞðDtmpnþ1Þ: ð12:69Þ

Step 14. Calculate ðFpenÞtrial

ðFpenÞtrial ¼ Fp

nþ1ðFpdisÞ

�1n : ð12:70Þ


ðFpenÞtrial ¼ ðV

penÞtrialðR


Step 16. Calculate the trial back-stress

ðSenÞtrial ¼ 2B lnðVpenÞtrial: ð12:72Þ

Step 17. Update the back-stress Sen

ðSenÞnþ1 ¼1

1þ cðDtmpnþ1Þ

!ðSenÞtrial: ð12:73Þ

Step 18. Update the plastic Mandel stress Mpen

ðMpenÞnþ1 ¼ ðR

penÞ>trialðSenÞnþ1ðR


Step 19. Update Dpdis

ðDpdisÞnþ1 ¼

c2B

� �mp

nþ1ðMpenÞnþ1: ð12:75Þ

Step 20. Update Fpdis

ðFpdisÞnþ1 ¼ exp DtðDp

disÞnþ1

� �ðFp

disÞn: ð12:76Þ

A.2. Jacobian matrix

In typical ‘‘implicit” finite element procedures utilizing a Newton-type iterative solution method,one needs to compute an algorithmically consistent tangent, often called the Jacobian matrix. We ob-tain an estimate for our Jacobian matrix below. From the outset we note that Jacobian matrices areused only in the search for the global finite element solution, and while an approximate Jacobianmight affect the rate of convergence of the global iteration scheme, it will not impair the accuracyof our constitutive time-integration algorithm.

Consider the Cauchy stress at the end of the increment:

Tnþ1 ¼ ðdet Fenþ1Þ

�1Renþ1Me

nþ1ðRenþ1Þ

>: ð12:77Þ

Then with D denoting a variation, (12.77) gives

DTnþ1 � ðDRenþ1ÞðR

enþ1Þ

>� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}incremental elastic rotation

Tnþ1 þ Tnþ1 ðDRenþ1ÞðR

enþ1Þ

>� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}incremental elastic rotation

þ tr DFenþ1

� �Fe�1

nþ1

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

incremental elastic vol: change

Tnþ1

¼ ðdet Fenþ1Þ

�1Renþ1 DMe

nþ1

� �ðRe

nþ1Þ>: ð12:78Þ

We assume that the variations ðDRenþ1ÞðR

enþ1Þ

>� �and tr DFe

nþ1

� �Fe�1

nþ1

� �of the incremental elastic rota-

tion and volume change are reasonably well-estimated by a commercial finite element program, such


as Abaqus/Standard (2008), in which we have implemented our constitutive model by writing a usermaterial subroutine (UMAT).16. Thus, here we concentrate on evaluating the variation DMe

nþ1 in (12.78),which is computed from

16 Aba

DMenþ1 ¼ C DEe

trial

� �; ð12:79Þ

where the fourth-order tensor

C ¼def @Menþ1

@Eetrial

ð12:80Þ

is the important constitutive contribution to the global Jacobian matrix.Recall that the time-integration procedure gives

Menþ1 ¼Me


GðDtmpnþ1ÞN

ptrial: ð12:81Þ

Using (12.27)2, (12.81) may be written as

Menþ1 ¼Me

trial þffiffiffi2pð�snþ1 � �strialÞNp

trial: ð12:82Þ

By the product rule

C ¼ @Metrial

@Eetrialþ

ffiffiffi2pð�snþ1 � �strialÞ

@Nptrial

@Eetrialþ Np

trial �ffiffiffi2p @�snþ1

@Eetrial� @

�strial

@Eetrial

� �� : ð12:83Þ

Since

Metrial ¼ C Ee

trial

� �; ð12:84Þ

we have

@Metrial

@Eetrial¼ C: ð12:85Þ

Next, since

Nptrial ¼

ððMeeff ÞtrialÞ0ffiffiffi2p

�strial

; ð12:86Þ

we have

@Nptrial

@Eetrial¼

ffiffiffi12

r1

�strial

@ððMeeffÞtrialÞ0

@Eetrial

� ððMeeffÞtrialÞ0�s2

trial

� @�strial

@Eetrial

� �: ð12:87Þ

Now,

ðMeeffÞtrial ¼Me

trial � ðSenÞn¼ C Ee

trial

� �� ðSenÞn ð12:88Þ

¼ 2G I� 13

1� 1� �

þ K1� 1� �

Eetrial

� �� ðSenÞn:

Hence,

ððMeeffÞtrialÞ0 ¼ 2G I� 1

31� 1

� �� Ee

trial

� �� ðSenÞn; ð12:89Þ

qus uses a hypoelastic constitutive equation for the stress of the type

T�¼ _T�WeTþTWe¼ _T�WTþTW|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}

for Wp¼0

¼C½D�Dp �:


and therefore

@ððMeeff ÞtrialÞ0

@Eetrial

¼ 2G I� 13

1� 1� �

: ð12:90Þ

Further,

�strial ¼ffiffiffi12

rððMe

effÞtrialÞ0 : ð12:91Þ

Therefore, by the chain-rule

@�strial

@Eetrial¼

ffiffiffi12

r@ððMe

effÞtrialÞ0@Ee

trial

� �>@jððMe

effÞtrialÞ0j@ððMe

effÞtrialÞ0

�ð12:92Þ

¼ffiffiffi12

r2G I� 1

31� 1

� �> ððMeeffÞtrialÞ0

jððMeeffÞtrialÞ0j

�; ð12:93Þ

¼ffiffiffi2p

G I� 13

1� 1� �

ððMeeffÞtrialÞ0

jððMeeffÞtrialÞ0j

�ð12:94Þ

¼ffiffiffi2p

GððMe

eff ÞtrialÞ0jððMe

eff ÞtrialÞ0jð12:95Þ

¼ffiffiffi2p

GNptrial: ð12:96Þ

Thus, using (12.90) and (12.96) in (12.87) we obtain

@Nptrial

@Eetrial¼ 2Gffiffiffi

2p

�strial

I� 13

1� 1� �

� Nptrial � Np

trial

� �; ð12:97Þ

and substituting (12.85) and (12.97) in (12.83), we have

C ¼ Cþ�snþ1

�strial� 1

� �2G I� 1

31� 1

� �� Np

trial � Nptrial

� �

þ Nptrial �

ffiffiffi2p @�snþ1

@Eetrial

�ffiffiffi2p

GNptrial

� �� : ð12:98Þ

Noting that Eetrial enters the equations through �strial, we have, using (12.96)

@�snþ1

@Eetrial¼ @

�snþ1

@�strial

@�strial

@Eetrial¼ @

�snþ1

@�strial

ffiffiffi2p

GNptrial: ð12:99Þ

Finally, substituting (12.99) into (12.98), using (12.15), and rearranging, we have

C ¼ ~C� 2G�snþ1

�strial� @

�snþ1

@�strial

� �Np

trial � Nptrial ð12:100Þ

where

~C ¼ 2~G I� 13

1� 1� �

þ K1� 1 with ~G ¼def �snþ1

�strialG: ð12:101Þ

Thus, it remains to determine the derivative

@�snþ1

@�strial

in (12.100). First, we rewrite the updates for �snþ1; (12.27)2, and Snþ1, (12.69), as

�snþ1 ¼ �strial � GDtf�snþ1

Snþ1

� �;

Snþ1 ¼ Sn þ DthðSnÞf�snþ1

Snþ1

� �;

ð12:102Þ


respectively, where f is the flow function of (9.33). Using the definitions

17 Her

X ¼def �snþ1

Snþ1

� �; and Y ¼def �strialf g; ð12:103Þ

the system of equations (12.102) may be written as

X ¼ GðX; YÞ; ð12:104Þ

where

G ¼strial � GDtf �snþ1

Snþ1

� �Sn þ DthðSnÞf �snþ1

Snþ1

� �8><>:

9>=>;: ð12:105Þ

Differentiating (12.104) with respect to Y (at the solution point) we obtain

@X@Y¼ @GðX; YÞ

@Yþ @GðX; YÞ

@X@X@Y

; ð12:106Þ

from which

@X@Y¼ A� @GðX; YÞ

@Y

� �; ð12:107Þ

where17

A� ¼ I � @GðX; YÞ@X

� ��1

: ð12:108Þ

Straightforward calculations show

@GðX; YÞ@X

¼�GDt@f

@�s

tnþ1

�GDt@f@S

tnþ1

DthðSnÞ@f@�s

tnþ1

DthðSnÞ@f@S

tnþ1

264

375; and

@GðX; YÞ@Y

¼10

�: ð12:109Þ

Thus, we obtain that

@�snþ1

@�strial¼ A�11; ð12:110Þ

substitution of which in (12.100) completes our estimate for the Jacobian matrix C.

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