A Large Deformation Theory for Rate Dependent Elastic Plastic

8/8/2019 A Large Deformation Theory for Rate Dependent Elastic Plastic

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A large deformation theory for rate-dependent elasticplastic

materials 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

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 FeFp [Krner, E., 1960. Allgemeine kontinu-umstheorie der versetzungen und eigenspannungen. Archive forRational Mechanics and Analysis 4, 273334] and Fp FpenF

pdis

[Lion, A., 2000. Constitutive modelling in finite thermoviscoplastic-ity: a physical approach based on nonlinear rheological models.International Journal of Plasticity 16, 469494]. The elastic distor-tion Fe contributes to a standard elastic free-energy we , while Fpen ,the energetic part ofFp , contributes to a defect energy wp thesetwo additive contributions to the total free energy in turn lead tothe standard Cauchy stress and a back-stress. Since Fe FFp1

and Fpen FpFp1dis , the evolution of the Cauchy stress and the back-

stress in a deformation-driven problem is governed by evolutionequations for Fp and Fpdis 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 only

involves 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-

0749-6419/$ - see front matter 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijplas.2008.11.008

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

International Journal of Plasticity 25 (2009) 18331878

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matic strain hardeningis 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 Pragers linear kinematic-hardening rule (Prager,1949):

_Tback B _Ep so that Tback BEp; 1:1where B > 0 is a back-stress modulus and Ep is the plastic stain tensor in the standard decompositionE Ee 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 B _Ep cTback _ep; with _ep defj _Epj; 1:2where the term cTback _ep 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 uniaxial

loading, the evolution of back-stress, instead of being linear as prescribed by (1.1), is then exponentialwith a saturation value. The ArmstrongFrederick model for kinematic hardening is widely used as asimple model to describe cyclic loading phenomena in metals. For a discussion of the ArmstrongFred-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 ArmstrongFrederick 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 ArmstrongFrederick 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 alternativetosuchtheoriesaretheoriesbasedonaproposalby 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 Krner (1960) elasticplastic decomposition F FeFp of the total deforma-tion gradient, may be further multiplicatively decomposed into energetic and dissipative parts Fpen andFpdis, respectively, as F

p FpenFp

dis (cf., Lions Fig. 5).2 The elastic distortion Fe contributes to a standard elas-

tic free-energywe, whileFpen contributes a defect energywp these two additive contributions to the total

free energy in turn lead to the standard Cauchy stress and a back-stress. Since Fe FFp1 and Fpen FpFp1dis ,the evolution of the Cauchy stress andthe back-stressin a deformation-driven problem is governed by evo-

lution equations forFp and Fpdis 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 Fpen 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 FpenFpdisdecomposition may be found in Dettmer and Reese (2004), Menzel et al. (2005), Hkansson 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 ArmstrongFrederick typeof kinematic-hardening rate-independent plasticity theory using the Lion decomposition Fp FpenF

pdis,

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

1 Also recently published as Frederick and Armstrong (2007).2 Lion uses the notation FM F, Fe Fe , Fin Fp , Fine Fpen, Finp Fpdis. Based on a statistical-volume-averaging argument for the

response of a polycrystalline aggregate, Clayton and McDowell (2003, Eq. (13)) have also introduced a Lion-type decomposition

Fp ~Fi~Fp ; they call ~Fi Fpen a meso-incompatibility tensor.

1834 D.L. Henann, L. Anand / International Journal of Plasticity 25 (2009) 18331878


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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-dependentplasticity theory with combined isotro-

pic and kinematic hardening of the ArmstrongFrederick type, based on the dual decomposi-tions F FeFp and Fp FpenF

p

dis. The limit m ! 0, where m is a material rate-sensitivityparameter 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 Xan arbitrary material point ofB. A motion ofB is then a smooth one-to-one mapping x vX; t with deformation gradient, velocity and velocity gradient given by4

F rv; v _v; L grad v _FF1: 2:1To model the inelastic response of materials, we assume that the deformation gradient F may be

multiplicatively decomposed as (Krner, 1960)

F FeFp: 2:2As is standard, we assume that

J det F > 0;and consistent with this we assume that

Je def det Fe > 0; Jp def det Fp > 0; 2:3so that Fe and Fp are invertible. Here, suppressing the argument t:

FpX represents the local inelastic distortion of the material at Xdue 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 ofFpX);

FeX represents the subsequent stretching and rotation of this coherent structure, and thereby rep-resents the corresponding elastic distortion.

3 With reference to the power-law model (9.35) in Section 9.3, the stiffness of the equations depends on the strain-rate-

sensitivity parameter m, and the stiffness increases to infinity as m tends to zero, the rate-independent limit. For small values of m

special care is required to develop stable constitutive time-integration procedures; cf., e.g., Lush et al. (1989).4 Notation: We use standard notation of modern continuum mechanics. Specifically: r and Div denote the gradient and

divergence with respect to the material point Xin the reference configuration; grad and div denote these operators with respect to

the point x vX; t in the deformed body; a superposed dot denotes the material time-derivative. Throughout, we writeF

e1 Fe1 , Fp> Fp> , etc. We write trA, symA, skwA, A0 , and sym0A, respectively, for the trace, symmetric, skew, deviatoric,and symmetricdeviatoric parts of a tensor A. Also, the inner product of tensors A and B is denoted by A : B, and the magnitude ofA

by jAj ffiffiffiffiffiffiffiffiffiffiffi

A : Ap

.5 Also sometimes referred to as the intermediate or relaxed local space at X.

D.L. Henann, L. Anand / International Journal of Plasticity 25 (2009) 18331878 1835
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We refer to Fp and Fe as the inelastic and elastic distortions.

By (2.1)3 and (2.2),

L Le FeLpFe1; 2:4

with

Le _FeFe1; Lp _FpFp1: 2:5

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

D symL; W skwL;De symLe; We skwLe;Dp symLp; Wp skwLp;

9>=>; 2:6

so that L D W, Le De We, and Lp Dp Wp.

The right and left polar decompositions of F are given by

F RU VR; 2:7

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

U ffiffiffiffiffiffiffiffi

F>Fp

; Vffiffiffiffiffiffiffiffi

FF>p

: 2:8Also, the right and left CauchyGreen tensors are given by

C U2 F>F; B V2 FF>: 2:9Similarly, the right and left polar decompositions of Fe and Fp are given by

Fe ReUe VeRe; Fp RpUp VpRp; 2:10where Re and Rp are rotations, while Ue, Ve, Up , Vp are symmetric, positive-definite tensors with

Ue ffiffiffiffiffiffiffiffiffiffiffiffi

Fe>Fep

; Ve ffiffiffiffiffiffiffiffiffiffiffiffi

FeFe>p

; Up ffiffiffiffiffiffiffiffiffiffiffiffi

Fp>Fpp

; Vp ffiffiffiffiffiffiffiffiffiffiffiffi

FpFp>p

: 2:11Also, the right and left elastic CauchyGreen tensors are given by

Ce Ue2 Fe>Fe; Be Ve2 FeFe>; 2:12and the right and left plastic CauchyGreen tensors are given by

Cp Up2 Fp>Fp; Bp Vp2 FpFp>: 2:13

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

det Fp

1 and tr Lp

0:

2:14

Hence, using (2.2) and (2.14)1,Je J: 2:15



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(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 elasticviscoplastic theo-ries utilizing the Krner decomposition it is widely assumed that the plastic flow is irrotationalin the sense that6

Wp

0: 2:16Then, trivially, Lp Dp and

_Fp DpFp; with tr Dp 0: 2:17

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

grad v Le FeDpFe1; with trDp 0: 2:18

2.2. Decomposition ofFp 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 Fpen and a dissipative part F

pdis as follows:

Fp FpenFpdis; det Fpen det Fpdis 1: 2:19We call the range of FpdisX the local substructural space. Thus F

pen maps material elements from the

substructural space to the structural space (which is the range ofFpX). 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 Lions presumed decomposition (2.19).

Then, from (2.5)

Lp Lpen FpenLpdisFp1en 2:20with

Lpen _FpenFp1en with trLpen 0; Lpdis _FpdisFp1dis with trLpdis 0; 2:21and corresponding stretching and spin tensors through

6 A note on role of the plastic spin in isotropic solids: There are numerous publications discussing the role of plastic spin in theories

based on the Krner decomposition (cf., e.g., Dafalias, 1998). Based on the classical principle of frame-indifference (cf., Section 3),Green and Naghdi (1971) (cf., also Casey and Naghdi, 1980), introduced the notion of a change in frame of the intermediate structural

space. This notion leads to transformation laws for Fe and Fp of the form

Fp QFp; Fe FeQ>;

in which QX; t is an arbitrary time-dependentrotation of the structural space. Unfortunately, Green and Naghdi viewed struc-tural frame-indifference as a general principle; that is, a principle that stands at a level equivalent to that of conventional frame-indifference. This view has been refuted by many workers (cf., e.g., Dashner, 1986, and the references therein). While we agreewith the view that structural frame-indifference is not a general principle, this hypothesis may be used to represent an importantfacet of the behavior of a large class of polycrystalline materials based on the Krner decomposition. Indeed, for polycrystallinematerials without texture the structural space is associated with a collection of randomly oriented lattices, and hence the evo-lution of dislocations through that space should be independent of the frame with respect to which this evolution is measured.Using the notion of a change in frame of the structural space, Gurtin and Anand (2005, p. 1714) have shown that within a frame-work for isotropic plasticity based on the Krner decomposition (which we follow here), one may assume, without loss in gener-ality, that the plastic spin vanishes Wp 0.Here, we adopt the assumption Wp 0 from the outset; we do so based solely on pragmatic grounds: when discussing finitedeformations for isotropic materials a theory without plastic spin is far simpler than one with plastic spin.

7 However, see Clayton and McDowell (2003) for a statistical-volume-averaging argument for the response of a polycrystalline

aggregate.

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Dpen symLpen; Wpen skwLpen;Dpdis symLpdis; Wpdis skwLpdis;

)2:22

so that Lpen Dpen W

pen and L

p

dis Dp

dis Wp

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

Wpdis 0: 2:23Then, trivially Lpdis D

pdis, and

_Fpdis DpdisFpdis with trDpdis 0: 2:24

Since

Wp Wpen skw FpenDpdisFp1en

; 2:25

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

Wpen

0;2:26

and

skw FpenDpdisF

p1en

0: 2:27

Also, (2.20) reduces to

Dp Dpen sym FpenDpdisFp1en

; with trDpen trDpdis 0: 2:28

The right and left polar decompositions of Fpen and Fp

dis are given by

Fpen RpenUpen VpenRpen; Fpdis RpdisUpdis VpdisRpdis; 2:29

where Rpen and Rpdis are rotations, while Upen, Vpen, Updis, Vpdis are symmetric, positive-definite tensors with

Upen ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Fp>en Fpen

q; Vpen

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiFpenF

p>en

q; Updis

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp>

disFp

dis

q; Vpdis

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiFp

disFp>

dis

q: 2:30

Also, the corresponding right and left CauchyGreen tensors are given by

Cpen Up2en Fp>en Fpen; Bpen Vp2en FpenFp>en ; 2:31Cpdis Up2dis Fp>disFpdis; Bpdis Vp2dis FpdisFp>dis: 2:32

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

vX; t QtvX; t o yt; 3:1

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

F QF; 3:2and hence

C

C that is

;C is invariant;

3:3

also _F Q_F _Q F, and by (2.1)3,

L QLQ> _Q Q>: 3:4



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Thus,

D QDQ>; W QWQ> _Q Q>: 3:5Moreover, FeFp QFeFp, and therefore, since observers view only the deformed configuration,

Fe QFe; Fp Fp; and thus Fp is invariant: 3:6Hence, by (2.5),

Le QLeQ> _Q Q>; 3:7and

Lp; Dp; and Wp are invariant: 3:8Further, by (2.10),

Fe ReUe ! QFe QReUe;Fe VeRe ! QFe QVeQ>QRe;

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

Re QRe; Ve QVeQ>; Ue Ue: 3:9Hence, from (2.12), Be and Ce transform as

Be QBeQ> and Ce Ce; 3:10further, since Fp is invariant,

Bp and Cp are also invariant: 3:11Finally, since Fp is invariant under a change in frame,

Fpen and Fpdis are invariant;

as are all of the following quantities based on Fpen and Fpdis:

Upen; Cpen; V

pen; B

pen; R

pen; U

pdis; C

pdis; V

pdis; B

pdis; R

pdis:

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 Dpdis are not independent, since

by (2.18) and (2.28) they are constrained by

grad v Le FeDpFe1; trDp 0; 4:1and

Dp

Dpen

sym FpenD

pdisF

p1en ; trDpen trD

pdis

0;

4:2

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.



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We denote by Pt an arbitrary part(subregion) of the deformed body with n the outward unit nor-mal on the boundary @Pt ofPt. The power expended on Pt by material or bodies exterior to Pt resultsfrom a macroscopic surface traction tn, 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 b0 q _v; 4:3with b0 the noninertial body force and q the mass density in the deformed body. We therefore writethe external power as

WextPt Z@Pt

tn vdaZPt

b vdv: 4:4

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 D

pen, and a dissipative stress Tdis

conjugate to symFpenDpdisF

p1en . Since D

p, Dpen, and symFpenD

pdisF

p1en are symmetric and deviatoric, we as-

sume that Tp, Sen, and Tdis are symmetric and deviatoric. We write the internal poweras

WintPt ZPt

Se : Le J1 Tp : Dp Sen : Dpen Tdis : symFpenDpdisFp1en

dv;

ZPt

Se : Le J1 Tp : Dp Sen : Dpen sym0Fp>enTdisFp>en : Dpdis

dv:

The term J1 arises because Tp : Dp Sen : Dpen Tdis : sym F

penD

pdisF

p1en

is measured per unit volume in

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

Sdis def sym0Fp>enTdisFp>en ; 4:5the internal power may more compactly be written as

WintPt ZPt

Se : Le J1 Tp : Dp Sen : Dpen Sdis : Dpdis

dv: 4:6

4.1. Principle of virtual power

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

known, and consider the fields v, Le, Dp, Dpen, 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, ~Dpen, and ~D

pdis to differentiate them from fields associated with the actual evolution of the body, we

require that

grad ~v ~Le Fe ~DpFe1; tr~Dp 0; 4:7and

~Dp ~Dpen sym Fpen ~DpdisFp1en

; tr~Dpen tr~Dpdis 0: 4:8

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

V ~v; ~Le; ~Dp; ~Dpen; ~Dpdis;consistent with (4.7) and (4.8). Then, writing

WextPt;V Z@Pt

tn ~vda ZPt

b ~vdv;

WintPt;V ZPt

Se : ~Le J1 Tp : ~Dp Sen : ~Dpen Sdis : ~Dpdis

dv;

)4:9

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

WextPt;V WintPt;V for all generalized virtual velocities V: 4:10

4.1.1. Consequences of frame-indifference

We assume that the internal power WintPt;V is invariant under a change in frame and that thevirtual fields transform in a manner identical to their nonvirtual counterparts. Then given a change inframe, invariance of the internal power requires that

WintPt;V WintPt;V; 4:11

where Pt andWintP

t;V

represent the region and the internal power in the new frame. In the newframePt transforms rigidly to a region P

t, the stresses S

e, Tp, Sen, and Sdis transform to Se, Tp, Sen, and

Sdis, while~Le transforms to

~Le Q~LeQ> _Q Q>;and ~Dp, ~Dpen, and ~Dpdis are invariant. Thus, under a change in frame WintPt;V transforms to

WintPt;V WintPt;V

ZPt

Se : Q~LeQ> _Q Q>

J1 Tp : ~Dp Sen : ~Dpen Sdis : ~Dpdis

dv

ZPt

Q>SeQ : ~Le Se : _Q Q> J1 Tp : ~Dp Sen : ~Dpen Sdis : ~Dpdis

dv:

Then (4.11) implies that

ZPt Q>SeQ : ~Le Se : _Q Q> J1 Tp : ~Dp Sen : ~Dpen Sdis : ~Dpdis dv

ZPt


dv; 4:12

or equivalently, since the part Pt is arbitrary,

Q>SeQ : ~Le Se : _Q Q> J1 Tp : ~Dp Sen : ~Dpen Sdis : ~Dpdis


: 4:13

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

Q>SeQ S

e : ~Le J1 Tp Tp : ~Dp Sen Sen : ~Dpen Sdis Sdis : ~Dpdish i 0:Since this must hold for all ~Le, ~Dp, ~Dpen, and ~D

pdis, we find that the stress S

e transforms according to

Se QSeQ>; 4:14while Tp, Sen, and Sdis are invariant

Tp Tp; Sen Sen; Sdis Sdis: 4:15Next, if we assume that Q 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

0;

or that the tensor Se is symmetric,

Se Se>: 4:16Thus, the elastic stress Se is frame-indifferent and symmetric.



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4.1.2. Macroscopic force balance

In 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 ~Dpen ~D

p

dis 0, so that

~Le grad ~v: 4:17For this choice ofV, (4.10) yieldsZ

@Pt

tn ~vdaZPt

b ~vdv ZPt

Se : grad ~vdv: 4:18

Further, using divergence theorem, (4.18) yieldsZ@Pt

tn Sen ~vdaZPt

divSe b ~vdv 0: 4:19

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

tn Sen; 4:20and the local force balance

divSe b 0: 4:21

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

TdefSe 4:22and to view

T T> 4:23as 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,

divT b0 q _v; 4:24with b0 the noninertial body force.

4.1.3. Microscopic force balances

To discuss the microscopic counterparts of the macroscopic force balance, first consider a general-ized virtual velocity with

~v 0 and ~Dpdis 0; 4:25also, choose the virtual field ~Dp arbitrarily, and let

~Le Fe ~DpFe1 and ~Dpen ~Dp; 4:26

consistent with (4.7) and (4.8). Thus

T: ~Le Fe>TFe> : ~Dp: 4:27Next, define an elastic Mandel stress by

Me defJFe>TFe>; 4:28which 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



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WintPt;V ZPt

J1 Tp Sen Me : ~Dpdv 0:

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

4:29

Next consider a generalized virtual velocity with

~v 0 and ~Dp 0; 4:30also, choose the virtual field ~Dpdis arbitrarily, and let

~Dpen sym Fpen ~DpdisFp1en

4:31

consistent with (4.8). ThusSen : ~D

pen Fp>en SenFp>en : ~Dpdis: 4:32

Next, define a new stress measure by

Mpen defFp>en SenFp>en ; 4:33which in general is also not symmetric. For lack of better terminology, we call Mpen a plastic Mandelstress. 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

Wint

Pt;V

ZPt Sdis Mpen : ~Dpdisdv 0:

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

ond microforce balance

4:34

5. Free-energy imbalance

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

_ZPt

wJ1dv 6WextPt WintPt; 5:1

with w the free energy, measured per unit volume in the structural space.Since J1dv with J det F represents the volume measure in the reference configuration, and since

Pt deforms with the body,

_

ZPt wJ1dv ZPt_wJ1dv:

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



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_w JT: De Tp : Dp Sen : Dpen Sdis : Dpdis 6 0: 5:2We introduce a new stress measure

Te defJFe1TFe>; 5:3which is symmetric, since Tis 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 CeTe: 5:4Next, differentiating (2.12)1 results in the following expression for the rate of change of C

e,

_Ce Fe> _Fe _Fe>Fe:Hence, since Te is symmetric,

Te : _Ce 2Te : Fe> _Fe 2FeTeFe> : _FeFe1 2FeTeFe> : De;or, using (5.3) we obtain

JT: De 12

Te : _Ce: 5:5

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 : Dpdis 6 0: 5:6

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; Bpen; D

p; Dpen; Dpdis; T

e; Tp; Sen; Sdis; Me; and Mpen 5:7are also invariant.

6. Constitutive theory

To account for the classical notion of isotropicstrain 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 weCe wpBpen;Te TeCe;Sen SenBpen;Tp TpDp; S;Sdis SdisDpdis;dp; S;_S hdp; S;

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

6:1

where

dp def jDpj 6:2

is the scalar flow rate corresponding to Dp. Note that since w is the free energy per unit volume of thestructural space, we is chosen to depend on Ce and wp on Bpen, because C

e and Bpen are squared stretch-like measures associated with the intermediate structural space. Also note that we have introduced a

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

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

energy wp is a defect-energy associated with plastic deformation. For metallic materials wp 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 wp 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 hardeningis 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

_w @weCe@Ce

: _Ce @wpBpen@Bpen

: _Bpen;

and, using8

_Bpen DpenBpen BpenDpen; 6:3and the symmetry of Bpen and @w=@B

pen,

@wp Bpen

@Bpen: _Bpen

@wpBpen@Bpen

: DpenBpen BpenDpen

2@wp

Bpen

@Bpen Bp

en 0: D

p

en:

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

1

2TeCe @

weCe@C

e

: _Ce SenBpen 2

@wpBpen@Bpen

Bpen

0

: Dpen

TpDp; S : Dp SdisDpdis; dp; S : Dpdis P 0; 6:4and 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

TeCe 2@weCe@Ce

; 6:5

SenBpen 2@wpBpen

@BpenBpen

0

: 6:6

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

Tp : Dp Sdis : Dpdis P 0: 6:7We assume henceforth that (6.5) and (6.6) hold in all motions of the body. We assume further that the

material is strictly dissipative in the sense that

8 Recall that Wpen 0, cf., (2.26).9 We content ourselves with constitutive equations that are only sufficient, but generally not necessary for compatibility with

thermodynamics.

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Tp : Dp > 0 whenever Dp0; 6:8and

Sdis : Dpdis > 0 whenever D

pdis0: 6:9

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 wpisoj,which evolves as j evolves according to an evolution equation of the type _j hdp;j. This will leadto a scalar internal stress Kj

def@w

p

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

Tp : Dp Sdis : Dpdis K_jP 0: 6:10Although a term such as wpisoj 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 sense

introduced as being derived from a defect energy

wp

isoj 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 wpBpen already accounts for some aspects of the stored energy of cold work.

Vladimirov et al. (2008) have considered a combined isotropic and kinematic-hardening theorywith an additional defect energy wisoj, and write their corresponding stress as @wisoj=@j Rj Kj. They associate the stress Rj as the strain-hardening contribution to the isotropicdeformation resistance, and take _j

ffiffiffiffiffiffiffiffi2=3

pd

p, so that j represents an accumulated plastic strain. Itis 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 S

to 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) Orth = 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 constructdefined as the range of the linear map Fpdis, 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 : D

pdis be invariant: 6:11

6.2.1. Isotropy of the reference space

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



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Fp ! FpQ; Fpdis ! FpdisQ and Fe and Fpen are invariant; and hence Ce and Cpen are invariant;6:12

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

_Ce

; Dp

; Dp

en; and Dp

dis are invariant:We may therefore use (6.11) to conclude that

Te; Tp; Sen; and Sdis are invariant: 6:13Thus, 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 space

Next, 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

F

e

! Fe

Q; F

p

! Q>Fp

; F

p

en ! Q>Fp

en; and F

p

dis is invariant; 6:14and also

Ce ! Q>CeQ; _Ce ! Q> _CeQ; Dp ! Q>DpQ; Bpen ! Q>BpenQ;Dpen ! Q>DpenQ; and Dpdis is invariant: 6:15

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

Te ! Q>TeQ; Tp ! Q>TpQ; Sen ! Q>SenQ; and Sdis is invariant: 6:16Thus, with reference to the constitutive equations (6.1) we conclude that

we

Ce

we

Q>CeQ

;

wpBpen wpQ>BpenQ;Q>TeCeQ TeQ>CeQ;Q>SenBpenQ SenQ>BpenQ;Q>TpDp; SQ TpQ>DpQ; S;

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

6:17

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

pdis; d

p; S, and hdp; S are invariant to such rotations.

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

GS Orth; 6:18

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 functionswe, wp, Te, Sen, and Tp must each be isotropic.

6.2.3. Isotropy of the substructural space

Finally, 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 ! FpenQ; Fpdis ! Q>Fpdis; 6:19and also

Dpdis

!Q>Dp

disQ; and Dpen is invariant:

6:20

Then (6.20) and (6.11) yield the transformation lawsSdis ! Q>SdisQ; and Sen is invariant: 6:21



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

Q>SdisDpdis;dp; SQ SdisQ>DpdisQ; dp; S 6:22must 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 Orth; 6:23so 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 weCe is an isotropic function of Ce, it has the representation

weCe ~weICe ; 6:24where

ICe I1Ce; I2Ce; I3Ce

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

Te TeCe 2@~weICe @C

e ; 6:25

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

Me CeTe;and TeCe is isotropic, we find that Te and Ce commute,

CeTe TeCe; 6:26and hence that the elastic Mandel stress Me is symmetric.

Further the defect free energy has a representation

wp ~wpIBpen where IBpen I1Bpen; I2Bpen; I3Bpen

; 6:27

and this yields that@~wpIBpen

@BpenBpen 6:28

is a symmetric tensor, and (6.6) reduces to

Sen SenBpen 2@~wpIBpen

@BpenBpen

0

; 6:29

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

Mpen 2Fp>en @~wpIBpen @Bpen

Bpen

0

Fp>en ;

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2 Fp>en@~wpIBpen

@BpenBpen

Fp>en

" #0

;

2 Fp>en@~wpIBpen

@Bp

en Fpen" #0; 6:30

and since @~wpIBpen =@Bpen 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

0defMe0 Sen: 7:1

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 TpDp; S: 7:2

Next, onaccount ofthesymmetry anddeviatoricnature oftheplastic stress Mpen (cf., (6.30)),thesecondmicroforce balance (4.34) and the constitutive equation (6.4)5 yield the second flow rule of the theory:

Mpen SdisDpdis;dp; S: 7:3We now make two major assumptions concerning the plastic flow for isotropic materials:

(1) Codirectionality hypotheses: Recall

dp jDpj; and let dpdis jDpdisj;

also let

Np def Dp

dp and N

pdis

def Dpdis

dpdis

7:4

denote the directions of plastic flow whenever Dp0 and Dpdis0. Then the dissipation inequality(6.7) may be written as

Tpdp;Np; S : Np dp Sdisdpdis;Npdis; dp; S : Npdis dpdis P 0: 7:5Guided by (7.5), we assume henceforth that the dissipative flow stress Tp is parallel to andpoints in the same direction as Np so that

Tpdp;Np; S Ydp;Np; SNp; 7:6where

Ydp;Np; S Tpdp;Np; S : Np 7:7represents 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 Npdis so that

Sdisdpdis;Npdis;dp; S Ydisdpdis;Npdis;dp; SNpdis; 7:8where

Ydisdpdis; Npdis;dp; S Sdisdpdis;Npdis;dp; S : Npdis 7:9represents another scalar flow strength of the material.We refer to the assumptions (7.6) and (7.8) as the codirectionality hypotheses.10

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



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(2) Strong isotropy hypotheses:We also assume that the scalar flow strength Ydp; Np; S is independent of the flow direction Np ,so that

Ydp; S: 7:10Similarly, we also assume that the scalar flow strength Ydisdpdis; N

pdis; d

p; S is independent of theflow direction Npdis, so that

Ydisdpdis;dp; S: 7:11We 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,

Meeff0 Ydp; SNp; 7:12which immediately gives

Np

Meeff

0

jMeeff0j ; 7:13and

jMeeff0j Ydp; S: 7:14When jMeeff0j and S are known, (7.14) serves as an implicit equation for the scalar flow rate d

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

Mpen Ydisdpdis;dp; SNpdis; 7:15which gives

Npdis Mp

enjMpenj ; 7:16

and

jMpenj Ydisdpdis;dp; S: 7:17When jMeeff0j and S are known, (7.17) serves as an implicit equation for the scalar flow rate d

p

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

jMeeff0jdp jMpenjdpdis P 0: 7:18

8. Summary of the constitutive theory

In this section we summarize our constitutive theory.

(1) Free energy.

w ~weICe ~wpIBpen ; 8:1where ICe and IBpen are the lists of the principal invariants of C

e and Bpen, respectively.(2) Cauchy stress.

Tdef

J

1

F

e

T

e

F

e> where Te 2@~we

ICe

@Ce : 8:2(3) Mandel stresses. Back-stress. The elastic Mandel is given by

Me CeTe; 8:3



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and is symmetric.The back-stress is given by

Sen 2@~wpIBpen

@BpenBpen

0

; 8:4

and is symmetric deviatoric. The stress difference

Meeff Me Sen 8:5is called an effective elastic Mandel stress.The plastic Mandel stress is given by

Mpen 2 Fp>en@~wpIBpen

@Bpen

Fpen

" #0

; 8:6

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

_Fp DpFp; 8:7with Dp given by

Dp dpNp; Np Meeff0

jMeeff0j; 8:8

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

jMeeff0j Ydp; S; 8:9for given Meeff0 and S, where Yd

p; S is a rate-dependent flow strength. The evolution equation

for F

p

dis is_Fp

dis DpdisFpdis; 8:10with Dpdis given by

Dpdis dpdisNpdis; Npdis MpenjMpenj

; 8:11

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

jMpenj Ydisdpdis;dp; S; 8:12for given Mpen, d

p, and S, where Ydisd

p

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

(5) Evolution equation for S._S hdp; S: 8:13

The evolution equations for Fp, Fpdis, and Sneed to be accompanied by initial conditions. Typical ini-tial conditions presume that the body is initially (at time t 0, say) in a virgin state in the sense that

FX;0 FpX; 0 FpdisX;0 1; SX;0 S0 constant; 8:14so that by F FeFp and Fp FpenF

pdis we also have F

eX; 0 1 and FpenX; 0 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.



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9.1. Free-energy we

The spectral representation of Ce is

C

e

X3

i1 x

e

i r

e

i re

i ; with x

e

i ke2

i ; 9:1where re1; r

e2; r

e3 are the orthonormal eigenvectors ofC

e and Ue, and ke1; ke2; k

e3 are the positive eigen-

values of Ue. Instead of using the invariants ICe , the free-energy we for isotropic materials may be

alternatively expressed in terms of the principal stretches as

we weke1; ke2; ke3: 9:2

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

T

e

2@we

ke1; k

e2; k

e3

@Ce 2X3

i1

@we

ke1; k

e2; k

e3

@kei@kei@Ce X

3

i1

1

kei

@we

ke1; k

e2; k

e3

@kei@xi@Ce : 9:3

Assume that the squared principal stretches xei are distinct, so that the xei and the principal directions

rei may be considered as functions of Ce; then

@xei@C

e rei rei ; 9:4

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

Te

X3

i1

1

kei

@weke1; ke2; ke3@kei

rei rei : 9:5

Further, from (8.2)1,

T J1FeTeFe> J1ReUeTeUeRe> J1ReX3i1

kei@weke1; ke2; ke3

@keirei rei

Re>: 9:6

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

Me X3i1

kei@weke1; ke2; ke3

@keirei rei : 9:7

Let

Ee def ln Ue X3i1

Eei rei rei ; 9:8

denote the logarithmic elastic strain with principal values

Eei def ln kei ; 9:9and consider an elastic free-energy function of the form

weke1; ke2; ke3 weEe1; Ee2;Ee3; 9:10

so that, using (9.7),

Me X3i1

@weEe1;Ee2; Ee3@Eei

rei rei : 9:11



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

weEe GjEe0j2 1

2KtrEe2; 9:12

where

G > 0 and K> 0; 9:13are the shear modulus and bulk modulus, respectively. Then, (9.11) gives

Me 2GEe0 KtrEe1 9:14and on account of (9.6), (9.7), and (9.14),

T J1ReMeRe>: 9:15

9.2. Free-energy wp

The spectral representation of Bpen is

Bpen X3i1

bili li; with bi a2i ; 9:16

where l1; l2; l3 are the orthonormal eigenvectors ofBpen and V

pen, and a1; a2; a3 are the positive eigen-

values Vpen. Instead of using the invariants IBpen , the free-energy wp may alternatively be expressed as

wp wpa1; a2; a3: 9:17Then, by the chain-rule

@wpa1; a2; a3@Bpen

X3i1

@wpa1; a2; a3@ai

@ai@Bpen

12

X3i1

1ai

@wpa1; a2; a3@ai

@bi@Bpen

: 9:18

Assume that bi are distinct, so that the bi and the principal directions li may be considered as functionsofBpen. Then,

@bi@Bpen

li li; 9:19

and, granted this, (9.18) implies that

@wpa1; a2; a3@B

p

en 1

2 X3

i1

1

ai

@wpa1; a2; a3@ai

li

li:

9:20

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

Sen X3i1

ai@wpa1; a2; a3

@aili li

0

: 9:21

Let

Epen def lnVpen X3i1

lnaili li: 9:22

11 Vladimirov et al. (2008) employ a Neo-Hookean free-energy function modified for elastic compressibility (cf., their Eq. (28)).

Here, as an alternate, we use the classical strain-energy function for infinitesimal elasticity, but employ the large deformation

Hencky-logarithmic elastic strain Ee . Anand (1979, 1986) has shown that the simple free-energy function (9.12) is useful for

moderately large elastic stretches. Additionally, use of the logarithmic strain helps in developing implicit time-integration

procedures based on the exponential map (cf., Appendix A).



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denote a logarithmic energetic strain with principal values ln ai. Note that since trVpen a1a2a3 1,

trEpen lna1 lna2 lna3 lna1a2a3 0; 9:23and hence the strain tensor Epen is traceless, Next, consider the following simple defect energy:

wpEpen BjEpenj2 B lna12 lna22 lna32h i; 9:24with B > 0, which, since Epen is deviatoric, is analogous to (9.12). Then using (9.21), (9.22), and (9.23)

Sen 2BEpen; 9:25we call the positive-valued constitutive parameter B the back-stress modulus.

Then the plastic Mandel stress (4.33) becomes

Mpen Fp>en SenFp>en Rp>en Vpen2BEpenVp1en

Rpen;

or

Mpen R

p>en SenR

pen: 9:26

9.3. Strength functions

First consider the strength relation

jMeeff0j Ydp; S; 9:27appearing in (8.9). Define an equivalent shear stress by

s def 1

ffiffiffi2p jMeeff0j; 9:28

and an equivalent shear strain rate by

mp defffiffiffi2

pdp

ffiffiffi2

pjDpj; 9:29

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

s Ymp; S; 9:30and assume that the shear flow strength Ymp; S has the simple form

Ymp; S gmpS; 9:31where S now represents an isotropic flow resistance in shear, and

g0 0; gmp

is a strictly increasing function of mp

: 9:32We refer to the dimensionless function gmp as the rate-sensitivity function. Next, by (9.32) the func-tion gmp is invertible, and the inverse function f g1 is strictly increasing and hence strictly positivefor all nonzero arguments; further f0 0. Hence the relation (9.30) may be inverted to give anexpression

mp g1 sS

f s

S

9:33

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

gmp mp

m0 m; 9:34

12 We absorb inconsequential factors offfiffiffi

2p

in writing (9.30).



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where m > 0, a constant, is a rate-sensitivity parameterand 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

mp m0 sS

1m: 9:35

Next consider the strength relation

jMpenj Ydisdpdis;dp; S; 9:36appearing in (8.12). Define another equivalent shear stress by

sen def 1ffiffiffi2

p jMpenj; 9:37

and another equivalent shear strain rate by

mpdis def ffiffiffi2p dpdis ffiffiffi2p jDpdisj; 9:38respectively. Using the definitions (9.37) and (9.38), we rewrite the strength relation (9.36) as13

sen Ydismpdis; mp; S: 9:39A specialization ofYdism

pdis; m

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

Ydismpdis; mp; S B

cmp

mp

dis; 9:40

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

(9.39) becomes

sen Bcmp

mpdis; 9:41

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

mpdis senB

cmp; 9:42

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

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

change in the back-stress Sen when Dp

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 hSmp 9:43with hS a hardening function, which we take to be given by (Brown et al., 1989):

h

S

h0 1 SS a

for S0 6 S6 S;

0 for SP S;( 9:4413 Again absorbing inconsequential factors of

ffiffiffi

2p

in writing (9.39).



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where S, a, and h0 are constant moduli with S > S0, aP 1, and h0 > 0. The hardening function (9.44)

is strictly decreasing for S0 6 S6 S and vanishes for SP 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

s cB

sen2

mp P 0: 9:45

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:

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

w we wp: 9:46With

Ue X3

i1

kei rei rei ; 9:47

denoting the spectral representation of Ue, and with

Ee X3i1

ln kei rei rei ; 9:48

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

we GjEe0j2

1

2KtrEe2; 9:49

where

G > 0; and K> 0; 9:50

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

Vpen X3i1

aili li; 9:51

x vX; t, motion;

F rv; J det F > 0, deformation gradient;F FeFp, elasticplastic decomposition of F;Fe; Je det Fe J> 0, elastic distortion;Fp; Jp det Fp 1, inelastic distortion;Fe ReUe, polar decomposition of Fe;Dp sym _FpFp1, plastic stretching corresponding to Fp;Fp FpenF

pdis, elasticplastic decomposition of F

p;Fpen; det F

pen 1, energetic part of F

p;Fpdis; det F

pdis 1, dissipative part of F

p;Fpen V

penR

pen, polar decomposition of F

pen;

Dpdis sym_FpdisF

p1dis , plastic stretching corresponding to F

pdis;

T T

>

, Cauchy stress;w, free-energy density per unit volumeof intermediate structural space;

S> 0; scalar internal variable.



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denoting the spectral representation of Vpen, and with

Epen X3i1

lnaili li; trEpen 0; 9:52

we adopt a free-energy wp

of the formwp BjEpenj2; 9:53where the positive-valued parameter

BP 0; 9:54is 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

J1ReMeRe>;

9:55

whereMe 2GEe0 KtrEe1; 9:56is an elastic Mandel stress.The symmetric and deviatoric back-stress is defined by

Sen 2BEpen; 9:57and the driving stress for Dp is the effective stress given by

Meeff0 Me0 Sen: 9:58

The corresponding equivalent shear stress is given bys 1ffiffiffi

2p jMeeff0j: 9:59

The driving stress for Dpdis is the stress measure

Mpen Rp>en SenRpen: 9:60The corresponding equivalent shear stress is given by

sen 1ffiffiffi2

p jMpenj: 9:61

(3) Flow rules.

The evolution equation for Fp is_Fp DpFp; FpX;0 1;Dp mp Meeff0

2s

;

mp m0 sS 1=m

;

9>>>=>>>; 9:62

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

_Fpdis DpdisFpdis; FpdisX;0 1;

Dpdis mpdis Mpen2sen

;

mpdis c senB

mp;

9>>=>>; 9:63where cP 0 is a dimensionless constant.



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

_S hSmp; SX;0 S0;

hS h0 1

S

S a for S0 6 S6 S;0 for SP S;

(9>>=>>; 9:64

where S, a, and h0 are constant moduli with S > S0, aP 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 80 GPa; K 175 GPa;m0 0:001 s1; m 0:02;S0

100 MPa; h0

1250 MPa; a

1 S

250 MPa;

B 40 GPa; c 400;10:1

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

strain rate ofj _j 0:01 s1; a fixed time stepDt 0:01 s was used in the numerical simulations. The cal-culations were performed for: (i) no hardening, h0 0 and B 0; (ii) isotropic hardening only, h00 andB 0; (iii) kinematic hardening only, h0 0 and B0; and finally (iv) combined kinematic and isotropichardening, h00and B0. Plots of the axial stressr versus the axial strain areshownin Fig.1(a).Withnohardening, 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 flowrecommences 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 1; 3g, a

simple shearing motion is described byx X mtX2e1; 10:2

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



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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 simple

reversed 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.

0.02 0.01 0 0.01 0.02600

400

200

0

200

400

600

Axial strain,

Axialstress,(

MPa

)

No hardening

Isotropic only

Kinematic only

Combined

0 0.01 0.02 0.03 0.04 0.05300

200

100

0

100

200

300

Amount of shear,

Stress,T12

(MPa)

No hardening

Isotropic only

Kinematic 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.



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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 ofm 0:01 s1 for a total shear strain C mt 0:05, and then at a shearstrain rate ofm 0:01 s1 for another 5 s to complete one reversal of strain, while using a fixed timestep Dt 0:01 s. As for simple tension/compression, the calculations were performed for: (i) no hard-ening, h0 0 and B 0; (ii) isotropic hardening only, h00 and B 0; (iii) kinematic hardening only,h0 0 and B0; and finally (iv) combined kinematic and isotropic hardening, h00 and B0, 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 ofC 0:05 at ashear strain rate ofjmj 0:01 s1 was considered while increasingly coarsening the time step. The spe-cific time steps Dt of

1 103; 1 102; 2:5 102; 1 101; 2:5 101; and 1 swere considered; these time steps translate to shear strain increments of

1 105; 1 104; 2:5 104; 1 103; 2:5 103; and 1 102;respectively. Fig. 2 shows the stressstrain 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 elasticplastic transitions. The order of convergence is demonstrated in Fig. 3;since no exact solution is available, the solution obtained using a time step ofDt 1 103 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 Ewas 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 elasticplastic transitions.

0 0.01 0.02 0.03 0.04 0.05300

200

100

0

100

200

300

Amount of shear,

Stress,

T12

(MPa)

t = 1103

s

t = 1102

s

t = 1101

s

t = 1 s

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



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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 ofC 10, at a shear strain rate of m 0:01 s1 using a fixed time stepDt 0:1 s. The shear T12 and normal components, T11 and T22, of the Cauchy stress T are plotted in

Fig. 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 ofC % 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 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 versusC curve still does not show any oscillations;the peak in this curve is due to the use of a logarithmic energetic strain lnVpen 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 Qt X mtX2e1 ; 10:3

with

Qt e1 e1 e2 e2 cosxt e1 e2 e2 e1 sinxt e3 e3; 10:4a 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 0:1p radians per second and m 0:01 persecond, for t 2 0; 20, so that h xt 2 0; 2p and C mt 2 0; 0:2. The calculation was carried by usinga fixed time step ofDt 0:1 s, which corresponds to a shear strain increment ofDC 0:001 and a rota-tion increment ofDh 3:6. Snapshots of the initial and deformed geometry at a few representative

stagesof 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 byTt. The result of the shear component of the unrotat-ed Cauchy stress Qt>TtQt 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) withQt 1. The two shear stress versus shear strain curves are indistinguishable from each other; the

2.5 2 1.5 1 0.5 0 0.56

6.5

7

7.5

8

8.5

log( t)

log(E)

1

1

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



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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 thealgorithmic tangentin a case of inhomogeneous deformation, we consider

the cyclic deformation of a curved bar of circular cross-section. A finite element mesh for the curved bar isshown in Fig.7(a).Thebar,whichhasadiameterof50 mm,curvesthrougharadiusof100 mmfor90,andthen extendsstraight for another 100 mm. The mesh consists of 2184 Abaqus-C3D8R elements.The nodesonthe face denoted as fixed-face in Fig.7(a)have prescribed null displacements, u1 u2 u3 0, whilethe boundary conditions on the face denoted as the moved-face were specified as

0 2 4 6 8 1050

0

50

100

150

200

250

300

350

400

Amount of shear,

Stress(MPa)

T12

T11

T22

0 2 4 6 8 101.5

1

0.5

0

0.5

1

1.5x 10

5

Amount of shear,

Stress(M

Pa)

T12

T11

T22

a

b

Fig. 4. Simple shear to large shear strain (a) with dynamic recovery c0; and (b) without dynamic recovery, c = 0. Note thatthere are no shear-stress oscillations in either case.



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u1

u3

0; and u2t

a sinxt

; with a

20 mm and x

0:2p s1:

10:5

The calculation was run to a final time of tf 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 Dtmpn1max

occurring at any integration point in the model during an increment. The value ofDcpmax was kept closeto a specified nominal value Dcps . From Fig. 2, we note that the accuracy of the solution begins to

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.



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degrade for shear strain increments of around 1 103; accordingly, we chose Dcps to be half this va-lue, 0:5 103. The automatic time-stepping algorithm operated to keep the ratio

R Dcpmax

Dcps10:6

close to 1.0 by adjusting the size of the time increments.14As the curved bar is cyclically displaced in the e2; e3-plane at the moved-face, the section of

the bar near the fixed-face is subjected to reversed-torsion, while the part of the bar towards themoved-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 loaddisplacement 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 of

both 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

14 After an equilibrium solution for a time increment Dtn tn1 tn was found, the value of the ratio R was checked to determinewhether this solution would be accepted or not. If R was greater than 1.25, then the solution was rejected, and a new time

increment was used that was smaller by the factor 0:85=R. If R 6 1:25, then the solution was accepted, and the value of R wasused to determine the first trial size of the next time increment. The following algorithm was used:

if 0:8 < R 6 1:25 then Dtn1

Dtn=R;

if 0:5 < R 6 0:8 then Dtn1 1:25Dtn;if R 6 0:5 then Dtn1 1:50Dtn:

10:7

Note that the measure Dcpmax was allowed to exceed the user specified value Dcps by up to 25%. This was done to avoid having to

recalculate increments that came out just slightly above the specified nominal value, but were otherwise essentially acceptable.

0 0.05 0.1 0.15 0.20

50

100

150

200

250

300

350

Amount of shear,

Stress,

T12

(MPa)

Shear and rotationShear only

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



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Dp Dpen sym FpenDpdisFp1en

: 11:2

The relation (11.2) gives

Dpen Dp sym FpenDpdisFp1en

; 11:3

Dp mpdis

2sen

sym FpenM

penF

p1en

using 9:63;

Dp mpdis

2sen

sym FpenR

p>en SenR

penF

p1en

using 9:60;

Dp mp

dis

2sen sym VpenSenVp1en ;

Dp mpdis

2sen

sym VpenB ln BpenVp1en

using 9:57 and Epen 1=2 ln Bpen;

0 10 20 30 4020

15

10

5

0

5

x (mm)

x(

mm)

Undeformed

Fully bent

Sprungback

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.



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Dp Bmpdis

2sen

ln Bpen;

Dp 12cmp

ln Bpen using 9:42: 11:4

Using (11.4) in (11.1) gives

_Bpen Dp 1

2cmp

ln Bpen

Bpen Bpen Dp

1

2cmp

ln Bpen

;

or

_Bpen DpBpen BpenDp cmpBpenln Bpen: 11:5Thus, if we set

A Bpen;we obtain the evolution equation

_A DpA ADp cmpAlnA: 11:6Therefore, instead of using the decomposition Fp FpenF

p

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

pen, one may develop an alternate theory based

on an internal variable A, with a corresponding defect free-energy wpA, 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 on

A is not as straightforward as the one developed here for a theory utilizing the Fp FpenFpdis

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 hardening

developed 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; Senn; Fpdisng; as well as Fn and Fn1;at time tn, we need to calculate

http://-/?-http://-/?-


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fTn1;Fpn1; Sn1; Senn1; Fpdisn1gat time tn1 tn Dt.

The evolution equation for _Fp DpFp is integrated by means of an exponential map (Weber andAnand, 1990)

Fpn1 exp DtDpn1

Fpn; with Dpn1 Dpn1Meeffn10; Sn1; 12:1

the inverse ofFpn1 is then

Fp1n1 Fp1n exp DtDpn1

: 12:2Next, using Fe FFp1 and (12.1)1, the elastic deformation gradient at the the end of the step is

given by

Fen1 Fetrial exp DtDpn1

; 12:3where

Fetrial def

Fn1Fp1n trial values are evaluated with plastic flow frozen 12:4is a trial value ofFe at the end of the step. The elastic right CauchyGreen tensor and its trial value atthe end of the step are

Cen1 Fe>n1Fen1; Cetrial Fe>trialFetrial: 12:5Thus, using (12.3),

Cen1 exp DtDpn1

Cetrial exp DtDpn1

: 12:6To proceed further we make our first approximation:

(A1) To first order inD

t, we approximateexp DtDpn1

: 1 DtDpn1 : 12:7Hence, (12.6) becomes

Cen1: 1 DtDpn1

Cetrial 1 DtDpn1

;

: Cetrial DtDpn1Cetrial DtCetrialDpn1;: Cetrial 1 DtCe1trialDpn1Cetrial DtDpn1

: 12:8

Next, consider the term Ce1trialDpn1C

etrial

; with EeGtrial

12 C

etrial 1 denoting a Green strain correspond-

ing to Ce

trial

, we have Ce

trial

1 2EeG

trial

, and hence

Ce1trialDpn1C

etrial: 1 2EeGtrial

Dpn1 1 2EeGtrial

;

: Dpn1 2Dpn1EeGtrial 2EeGtrialDpn1 4EeGtrialDpn1EeGtrial: 12:9We now make our second approximation:

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

Ce1trialDpn1C

etrial : Dpn1: 12:10

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

Cen1: Cetrial 1 2DtDpn1

: 12:11Further, using the symmetry ofCen1 we note that the symmetric tensors C

etrial and 1 2DtD

pn1

com-

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



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Een1: Eetrial 1

2ln 1 2DtDpn1

; 12:12

where

E

e

n1 1

2 ln C

e

n1 and Ee

trial 1

2 ln C

e

trial: 12:13Next, we make our third approximation:

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

ln 1 2DtDpn1 : 2DtDpn1

for small DtDpn1. Thus, under the assumptions (A1)(A3) of small Dt and small jEeGtrialj, (12.6) yields the

important relation

Een1: Eetrial DtDpn1: 12:14Next, let

C def 2G I 13

1 1

K1 1 12:15

denote the elasticity tensor, and let

Men1 CEen1 and Metrial CEetrial; 12:16respectively, 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

Men1

Metrial

DtC

Dpn1

:

12:17

Subtracting Senn1 from the left-hand side and15

Sentrial Senn; trial values are evaluated with plastic flow frozenfrom the right-hand side of(12.17), and writing

Meefftrial def Metrial Sentrial; 12:18gives

Meeffn1 Meefftrial DtCDpn1: 12:19Next, from (9.62) we have

Dpn1 1ffiffiffi

2p mpn1Npn1; Npn1

Meeffn10ffiffiffi2

psn1

; sn1 1ffiffiffi2

p jMeeffn10j; 12:20

where

mpn1 ffiffiffi

2p

jDpn1j: 12:21Using (12.15) and (12.20) in (12.19) we obtain

Meeffn1 Meefftrial ffiffiffi

2p

GDtmpn1Npn1: 12:22Since Npn1 is deviatoric, t

A Large Deformation Theory for Rate Dependent Elastic Plastic

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