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Existence and Uniqueness for Rate-Independent Infinitesimal Gradient Plasticity with Isotropic Hardening and Plastic Spin François Ebobisse and Patrizio Neff Mathematics and Mechanics of Solids 2010 15: 691 originally published online 21 August 2009 DOI: 10.1177/1081286509342269

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Downloaded from mms.sagepub.com at Univ. Duisburg-Essen on March 4, 2013 Existence and Uniqueness for Rate-Independent Infinitesimal Gradient Plasticity with Isotropic Hardening and Plastic Spin

FRANÇOIS EBOBISSE Department of Mathematics and Applied Mathematics, University of Cape Town, 7701 Rondebosch, South Africa

PATRIZIO NEFF Chair of Nonlinear Analysis and Modelling, Facultät für Mathematik, Universität Duisburg-Essen, Campus Essen, Universitätsstr. 2, 45141 Essen, Germany

(Received 24 September 2007 accepted 14 November 2007)

Abstract: Existence and uniqueness for infinitesimal dislocation based rate-independent gradient plasticity with linear isotropic hardening and plastic spin are shown using convex analysis and variational inequality methods. The dissipation potential is extended non-uniquely from symmetric plastic rates to non-symmetric plastic rates and three qualitatively different formats for the dissipation potential are distinguished.

Key words: Plasticity, gradient plasticity, dislocations, plastic spin, quasistatic evolution, rate-independent process, variational inequality

1. INTRODUCTION

We study the existence of solutions of quasistatic initial boundary value problems arising in gradient plasticity with isotropic hardening. The models we study use constitutive equations with internal variables to describe the deformation behavior of metals at small strain [1–4]. While gradient plasticity is of high current interest especially for describing novel effects at very small length scales [5–11], mathematical studies of the time continuous higher gradient plasticity problem are still rather scarce. Recently Reddy [12] (see also Ebobisse [13]) treated a geometrically linear model introduced by Gurtin [6], essentially different from the model we consider here. We mention also the very successful energetic approach of Mielke [14, 15] which led to existence results in both small strain and finite strain models. Our model has been introduced in Neff et al. [16]. The model features a non-symmetric plastic distortion p 3 and second spatial gradients of the plastic distortion D2 p acting

Mathematics and Mechanics of Solids 15: 691–703, 2010 DOI: 10.1177/1081286509342269 The Author(s), 2010. Reprints and permissions: http://www.sagepub.co.uk/journalsPermissions.nav

Downloaded from mms.sagepub.com at Univ. Duisburg-Essen on March 4, 2013 692 F. EBOBISSE and P. NEFF as dislocation based kinematical backstresses which are directly related to the geometrically necessary dislocations (GNDs) which induce a long-range interaction. Furthermore, local kinematical hardening is incorporated as well. Uniqueness of classical solutions for rate- independent and rate-dependent formulations of this model is shown in [17]. The more difficult existence question for the rate-independent model in terms of a weak reformulation is addressed in [16]. First numerical results for a simplified rate-independent irrotational formulation (no plastic spin, i.e. symmetric plastic distortion p) are presented in [18]. A distinguishing feature of this model with kinematical hardening is that, similar to classical approaches, only the symmetric part p : sym p of the plastic distortion appears in the local part of the Eshelby stress tensor hsym p, while the higher order stresses are the only source of non-symmetry. Here, is the symmetric Cauchy stress tensor and h is the local kinematical hardening modulus. For more on the basic invariance questions related to this issue, see [19, 20]. Here, we modify the previous model. On the one hand we skip the local kinematical hardening, i.e. we put h 0. On the other hand we augment the model with classical isotropic hardening based on the equivalent plastic strain concept which we understand to be related to statistically stored dislocations (SSDs).1 Thus both different physical mechanisms related to GNDs and SSDs contribute in their specific way to the hardening regime. The related viscoplastic formulation of dislocation based gradient plasticity with kinematical hardening is treated in [25]. Note that equivalent (or accumulated) plastic strain means classically that a measure of t 2 the total plastic strain, i.e. t 0 3 p dt, is driving the increase of the yield limit 0 y y k and thus leads to isotropic hardening. This is tantamount to prescribing the 2 plastic dissipation to be related to p p . In our case, where the plastic distortion p is not necessarily symmetric, the generalization of the former is not unique. Therefore, we introduce a parameter 0 and weight the contribution to the dissipation according to sym p2 skew p2. If we had chosen, instead, to use the concept of equivalent plastic work, so that t t 2 0 3 p dt as a measure of plastic work is driving the increase of the yield limit y, then the generalization to non-symmetric plastic distortions would be unique since t t 2 t 2 t 2 0 3 p dt 0 3 sym p skew p dt 0 3 sym p dt because the Cauchy stress tensor is symmetric. Therefore, from a modelling point of view, in order to have both concepts of equivalent plastic strain and equivalent plastic work being based on the same primitive variable, the parameter value 0 would suggest itself. However, we are presently unable to include a full analysis for 0. The outline of this contribution is as follows: first, we briefly provide a motivation for the model and derive the strong form of equations based on a Cahn–Allen type energetic approach. Then we reformulate the model in a weak sense as a variational inequality. Existence and uniqueness are shown in a suitable Hilbert space.

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2. STRONG FORMULATION

2.1. The Balance Equation

The conventional macroscopic force balance leads to the equation of equilibrium

div f 0 (2.1)

in which is the infinitesimal symmetric Cauchy stress tensor and f is the body force.

2.2. Constitutive Relations

The constitutive equations are obtained from a free energy imbalance together with a flow law that characterizes plastic behaviour. Since the model under study involves plastic spin, we consider an additive decomposition of the displacement gradient u into elastic and plastic distortions e and p,sothat

ui u e puij (2.2) x j

with the non-symmetric plastic distortion p incapable of sustaining volumetric changes that is, tr p 0 p 3 (2.3)

We define e : sym e sym u p as the infinitesimal elastic strain and p : sym p as the plastic strain, while : sym u u uT 2 is the total strain. Equation (2.2) induces the well-known additive decomposition of strains as follows:

e p (2.4)

We consider here a quadratic free energy given in the additively decoupled format

u p p lin lin p Curl : e e curl Curl iso (2.5) elastic energy defect energy GND hardening energy SSD

where

1 L2 1 lin : lin Curl p c Curl p2 k 2 e e 2 e e curl 2 iso 2

Lc is an energetic intrinsic length scale and k is the positive isotropic hardening constant. The defect energy is related to GNDs present through the dislocation density tensor Curl p [26–28], where the operator Curl is acting on the rows of any second order tensor X such that Curl u 0. The isotropic hardening energy is related to SSDs phenomenologically through a measure of equivalent plastic strain .

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For isotropic media the fourth order elasticity tensor is given by

X 2 X tr XI (2.6) for any symmetric second order tensor X,where and are the Lamé moduli supposed to satisfy

0and3 20

These conditions suffice for pointwise ellipticity of the elasticity tensor in the sense that there exists a constant m0 0 such that for any symmetric second order tensor we have

2 X Xm0X (2.7) the magnitude of any second order tensor X being given by the Frobenius matrix norm XX X12. The anisotropic case is included by allowing a general to satisfy (2.7).2 The local free-energy imbalance states that

ut 0 e p0 e p0 (2.8)

Now we expand , substitute (2.5) and get

2 e e p Lc Curl p Curl p k 0 (2.9) which, using arguments from thermodynamics,3 implies on the one hand the elasticity relation

e 2 sym u p tr u p I (2.10) and the reduced dissipation inequality

2 p Lc Curl p Curl p k 0 (2.11)

Now we integrate (2.11) over and get 2 0 p Lc Curl p Curl p k dV 2 p Lc Curl Curl p p 3 d div L2 pi Curl pi k dV c dt i1

Using the divergence theorem we obtain

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2 LcCurl Curl p p k dV 3 d 2 i i Lc p Curl p n dS 0 (2.12) dt i1

where n is the unit outward normal on . In order to obtain a dissipation inequality in the spirit of classical plasticity, we assume that the infinitesimal plastic distortion p satisfies the so-called linearized insulation condition 3 d 2 i i Lc p Curl p n dS 0 (2.13) dt i1

In order for the insulation condition to be satisfied we will impose the following boundary conditions on the plastic distortion:

px t px 0 on D and Curl p 0on D (2.14)

where D is that part of the boundary where zero Dirichlet conditions on the displace- ments u are prescribed. Under (2.13), we then obtain the dissipation inequality lin [ curl p g ]dV 0 (2.15)

where

lin 2 curl : Lc Curl Curl p and g k (2.16)

For further use we define the non-symmetric stress tensor

lin E : curl (2.17)

lin the non-symmetry relating only to the non-local term curl.

2.3. The Plastic Flow Law: Dual Formulation

The classical yield function for isotropic hardening von Mises plasticity is given by

p :dev g y

p for the generalized stress g.

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Here, given a parameter [0 ], we consider a yield function 1 defined for p every generalized stress E g by 2 1 2 0 dev sym E dev skew E g if 0 y " 0 33 p dev E g y if E Msym : (2.18) 1 if 0 otherwise ! 0 dev sym E g y if

0 Here, y is the initial yield stress of the material. So the set of admissible (elastic) generalized stresses is # $ p p : E g : 1 0 (2.19)

If we let p pthen the maximum dissipation principle gives the normality law

p p N (2.20)

p p where N denotes the normal cone to at , which is the set of generalised strain rates p that satisfy p p0forall (2.21)

2.4. The Plastic Flow Law: Primal Formulation

Using convex analysis (Legendre transformation) the relation (2.20) can be inverted and we find that p p p p N (2.22) where is the one-homogeneous dissipation function for rate-independent processes which in this case is defined by # $ q sup E q g 1 E g 0 g 0 (2.23)

Straightforward calculations lead to the expressions for the dissipation function given in Table 1. Such a case distinction for the dissipation function has been introduced in Mielke [29] for finite plasticity. p p p In (2.22), denotes the subdifferential of evaluated at .Thatis, p means that

p p p M M for any M (2.24)

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Table 1. The dissipation function q for [0 ].

[0 ] q " 2 2 2 2 y sym q skew q if sym q skew q 0 otherwise " ysym q if sym q 0 otherwise " 33 yq if q M and q sym otherwise

That is,

lin q p curl q p g for any q (2.25)

We refer to as the irrotational (p is symmetric), 1 as the equal spin and 0 as the free spin case.

2.5. Strong Formulation of the Model

To summarize, we have obtained the following strong formulation for the model of infinitesimal gradient plasticity with isotropic hardening and plastic spin: Find 1 ! 1 3 (i) the displacement u H 0 T H0 D satisfying zero Dirichlet conditions on D, (ii) the infinitesimal plastic distortion p H 10 T ! L2 3 for the case 0 (while only sym p H 10 T ! L2 3 for the case 0andp symmetric and p H 10 T ! L2 3 for )withCurlpt L2 M33 and Curl Curl pt L2 M33, (iii) the isotropic hardening variable (the equivalent plastic strain) H 10 T ! L2 such that the relations and conditions of Table 2 hold.

3. WEAK FORMULATION OF THE MODEL

Assume that the problem in Section 2.5 has a solution u p.Let! H 1 3 with ! ! D 0 in the sense of traces. Multiply the equilibrium equation by u and integrate in space to get ! udV f ! udV (3.1)

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Table 2. Strong primal formulation of the model. Equilibrium equation: div f 0in [0 T ] Elasticity relation: sym u p lin Constitutive relation: curl g p where lin 2 curl Lc Curl Curl p and

is dened in (2.23) ux t 0on D Boundary conditions: p x t p x 0 on D ! Curl p 0on D ux 0 u x 0 Initial conditions: p x 0 p0 x ! x 0 0 x

Using the symmetry of and the elasticity relation we get sym u p sym ! udV f ! udV (3.2)

2 Now, for any q C 3 such that q 0onD and any L ,weintegrate (2.25) over , integrate by parts the term with Curl Curl using the boundary conditions

q p 0onD Curl p 0on D and get lin q dV p dV curl q p g dV

p dV sym u p sym q pdV 2 Lc Curl Curl p q p dV k dV

p dV sym u p sym q pdV 2 Lc Curl p Curl q p dV k dV (3.3)

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Now summing (3.2) and (3.3) we arrive at the following weak formulation of the problem in Section 2.5 in the form of a variational inequality: sym u p sym ! q sym u p 2 Lc Curl p Curl q p k dV

q dV p dV f ! udV (3.4)

for all ! q

4. EXISTENCE AND UNIQUENESS

4.1. The Case 0 (With Spin) To prove the existence and uniqueness of a solution of the variational inequality (3.4) for 0, we introduce the following function spaces:

1 3 "! 1 3 ! # V : H0 D H : D 0 (4.1) # $ $ 33 Q : q : 3 : q Hcurl M qD 0 (4.2) " : L2 (4.3)

Z : V Q " (4.4) % & W : z ! q Z: sym q2 skew q2 a e in (4.5)

equipped with the norms

%!% %!% % %2 % %2 % %2 V : L2 q Q : q L2 Curl q L2 (4.6) % %2 %!%2 % %2 % %2 ! z Z : V q Q L2 for z q Z (4.7)

and the functionals # ! 2 a z : sym u p sym q Lc Curl p Curl q k dV (4.8) q dV if sym q2 skew q2 a e in jz : (4.9) ! otherwise

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$ z : f ! dV (4.10) for # u pand z ! q . Thus assuming, for instance, that the model is initially unstressed and undeformed and so corresponds to f x 0 0andu0 0, p0 0and 0 0, the weak formulation of 1 the model in the case 0 reads as follows: find # u p H 0 T ! Z such that #0 0, # t W for a.e. t [0 T ] and the variational inequality

a# z# jz j# $ z# for every z Wa and for a e t [0 T ] (4.11)

The existence and uniqueness of the solution of (4.11) are obtained from the following abstract result by Han and Reddy [2].

Theorem 4.1. Let H be a Hilbert space K H a non-empty, closed, convex cone a: H H $ a bilinear form that is symmetric, bounded on H and coercive on K $ H 10 T ! H & with $0 0 and j : H $ non-negative, convex, positively homogeneous, and Lipschitz continuous on K . Then there exists a unique solution # H 10 T ! H of the problem

a#t z #t jz j# t $t z #t'z K (4.12) satisfying #0 0 and # t K .

While it is easy to see that the set W, the functionals j and $ satisfy the assumptions of Theorem 4.1 and that the bilinear form a is continuous on Z, let us show that a is coercive on W. Let, therefore, z ! q W.Then

% ! %2 2 % %2 % %2 a z z m0 sym sym q 2 Lc Curl q 2 k 2 m0 0 is from 2 7 % !%2 % %2 ! m0 sym 2 sym q 2 2 sym sym p

2 % %2 % %2 Lc Curl q 2 k 2 ' ( 1 m %sym !%2 %sym q%2 % %sym !%2 %sym q%2 0 2 2 2 % 2

1 1 L2 %Curl q%2 k %sym q%2 k %skew q%2 k % %2 c 2 2 2 2 2 2 2

2 2 2 using Young’s inequality and sym q skew q from W ' ( 1 1 m 1 %%sym !%2 m 1 k %sym q%2 k %skew q%2 0 2 0 % 2 2 2 2 1 L2 %Curl q%2 k % %2 c 2 2 2

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So, choosing % such that

1 %&1 k 1 2m0

and using Korn’s first inequality, there exist two positive constants C1m0k Lc 0 C2m0k Lc0suchthat %!%2 % %2 % %2 % %2 % %2 a z z C1 V sym q 2 skew q 2 Curl q 2 2 %!%2 % %2 % %2 % %2 % %2 C2 V q 2 Curl q 2 2 C2 z Z

'z ! q W (4.13)

Remark that the pointwise control of the skewsymmetric part of q comes from 0 here. In addition, for 0 we could as well treat the completely ‘microfree case’, i.e., the boundary condition for the plastic distortion Curl p 0on.

4.2. The Case (Irrotational, No Spin)

To prove the existence and uniqueness of the solution of the variational inequality (3.4) for is, in fact, quite easy. We only have to remark that necessarily the plastic distortions p remain symmetric throughout and have to adapt the Hilbert space framework accordingly. An irrotational model with linear kinematic hardening in the spirit of the present approach has been dealt with in [16, 18]. For another irrotational model see [12] and [13], where the full gradient of the plastic strain is involved in the primal formulation of the flow 1 33 law, leading to the space H Msym as the appropriate space for infinitesimal plastic strains. Another irrotational gradient plasticity model has been considered in [23, 24], where gradient effects appear only through a dependence of the yield limit on spatial gradients 0 ' of the equivalent plastic strain ,i.e. y y k k3 . This amounts to using 1 2 1 2 iso 2 k 2 k3 and setting Lc 0.

5. OPEN PROBLEM: THE CASE 0 (FREE SPIN)

We have seen that for 0 the precise form of the defect energy did not really matter. We 2 lin Lc % %2 could also replace the defect energy by a full gradient term p 2 p without affecting the mathematical development at all. This changes completely for the free spin case 0, in which the form of the defect energy matters. The interplay between, on the one hand, a pointwise control of the symmetric part of the plastic distortion sym p and, on the other hand, only a non-local control of the skewsymmetric part skew p by the Curl p contribution provides for challenging questions.

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In order to prove the existence and uniqueness of the solution of the variational inequality (3.4) for 0, the first task is to provide a suitable Hilbert space environment where one is able to apply Theorem 4.1. For 0 we infer from (3.4) the estimate %!%2 % %2 % %2 % %2 a z z C1 V sym q 2 Curl q 2 2 (5.1)

Thus, taking into account the structure of the bilinear form a and the set W0, it is natural to sym ! 33 consider, for the space of infinitesimal plastic distortions, the completion Hcurl D M of the linear space

33 "q C M : tr q 0 q 0onD# with respect to the norm

% %2 % %2 % %2 q sym curl : sym q 2 Curl q 2 (5.2)

Despite appearances, this quadratic form indeed defines a norm as shown in [16]. Thus sym ! 33 Theorem 4.1 carries over by replacing Q with Hcurl D M and using the norm (5.2). However, in this space it is not immediately obvious how to define a linear bounded tangential trace operator. We will investigate the detailed properties of this Hilbert space and the regularity of the obtained solution in a future contribution.

NOTES

1. The isotropic increase of the yield surface will depend only on local measures of equivalent plastic strain . Non-local extensions have been discussed in [9, 21–24]. lin 2 2. The defect energy may also be more general, as long as an estimate curl Curl p c Curl p is satisfied.

3. The rate e can be chosen arbitrarily.

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