NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN...

February 12, 2009 14:54 WSPC/103-M3AS 00344

Mathematical Models and Methods in Applied SciencesVol. 19, No. 2 (2009) 307–346c© World Scientific Publishing Company

NOTES ON STRAIN GRADIENT PLASTICITY:FINITE STRAIN COVARIANT MODELLING ANDGLOBAL EXISTENCE IN THE INFINITESIMAL

RATE-INDEPENDENT CASE

PATRIZIO NEFF∗,†, KRZYSZTOF CHELMINSKI‡

and HANS-DIETER ALBER∗,§

∗Fachbereich Mathematik, Technische Universitat DarmstadtSchlossgartenstrasse 7, 64289 Darmstadt, Germany

‡Department of Mathematics, Technical University of Warsawpl. Politechniki 1, 00-661 Warsaw, Poland

†[email protected]‡[email protected]

§[email protected]

Received 8 June 2007Revised 28 May 2008

Communicated by A. Mielke

We propose a model of finite strain gradient plasticity including phenomenological Pragertype linear kinematical hardening and nonlocal kinematical hardening due to dislocationinteraction. Based on the multiplicative decomposition, a thermodynamically admissibleflow rule for Fp is described involving as plastic gradient Curl Fp. The formulation iscovariant w.r.t. superposed rigid rotations of the reference, intermediate and spatialconfiguration but the model is not spin-free due to the nonlocal dislocation interactionand cannot be reduced to a dependence on the plastic metric Cp = FT

p Fp.The linearization leads to a thermodynamically admissible model of infinitesimal

plasticity involving only the Curl of the nonsymmetric plastic distortion p. Linearizedspatial and material covariance under constant infinitesimal rotations is satisfied.

Uniqueness of strong solutions of the infinitesimal model is obtained if two non-classical boundary conditions on the plastic distortion p are introduced: p · τ = 0 onthe microscopically hard boundary ΓD ⊂ ∂Ω and [Curl p] · τ = 0 on the microscopi-cally free boundary ∂Ω\ΓD, where τ are the tangential vectors at the boundary ∂Ω. Aweak reformulation of the infinitesimal model allows for a global in-time solution of therate-independent initial boundary value problem. The method is based on a mixed varia-tional inequality with symmetric and coercive bilinear form. We use a new Hilbert-spacesuitable for dislocation density dependent plasticity.

Keywords: Gradient plasticity; thermodynamics with internal variables; material andspatial covariance.

AMS Subject Classification: 74A35, 74A30, 74C05, 74C10

†Corresponding author

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308 P. Neff, K. Chelminski & H.-D. Alber

1. Introduction

This paper addresses the modeling and mathematical analysis of a geometricallylinear gradient plasticity model. There is an abundant literature on gradient plas-ticity formulations, in most cases letting the yield-stress depend also on somehigher derivative of a scalar measure of accumulated plastic distortion.6 Experimen-tally, the dependence of the yield stress on plastic gradients is well-documented.5

Several experimental facts testify to the length scale dependence of materials:the Hall–Petch scaling relates the grain size in polycrystals to the yield stress:σy = σ0

y + k+√grainsize

. The Taylor scaling relates the flow stress σy to the dislocationdensity: σy = σ0

y + k+‖Curl Fp‖. It is also known that the thinner the grains, thestiffer the material gets. As a rule one may say: the smaller the sample the stifferit gets (while unbounded stiffness can be excluded from atomistic calculations).

From a numerical point of view the incorporation of plastic gradients servesthe purpose of removing the mesh-sensitivity, either in the softening case, or, moredifficult to observe numerically, already in classical Prandtl–Reuss plasticity (shearbands and slip lines with ill-defined band width). The incorporation of a lengthscale, which is a natural by-product of gradient plasticity, has the potential toremove the mesh sensitivity.

The finite strain model we propose is based on the multiplicative decomposi-tion of the deformation gradient and does not modify the yield stress directly butincorporates, motivated by mechanism-based single crystal plasticity, the disloca-tion density into the thermodynamic potential. The corresponding flow rule can beextracted in a nonstandard way, based on the development in Maugin.22 Models,similar in spirit to our formulation, may be found in Refs. 17 and 24. The necessaryre-interpretation of the second law of thermodynamics can be found, e.g. in Refs. 22and 21. This development is based on ideas initially proposed in Ref. 27. In contrastto Refs. 17 and 24 we evaluate the extended dissipation principle in a more relaxedsense.

While gradient plasticity is of high current interest9–11 we have not been ableto locate any mathematical study of the time continuous higher gradient plastic-ity problem, apart for Reddy36 treating a geometrically linear model of Gurtin,9

different from our proposal.Gurtin includes Fp in the free energy and takes free variations with respect to

Fp, leading to a model with additional balance equations for microforces, similar,e.g. to a Cosserat or Toupin–Mindlin type model. In contrast, in our model Fp andthence CurlFp is treated as an inner variable included only in the thermodynamicpotential, leading to evolution equations for Fp of degenerate parabolic type.

In Mielke and Muller26 the time-incremental finite-strain problem is investi-gated. It is shown that once the update potential for one time step is establishedand known to be properly coercive and polyconvex as a function of Fe in the mul-tiplicative decomposition of the deformation gradient, then, adding a regularizingterm depending on Curl Fp is indeed enough to show the existence of minimizers inthis update problem for the new deformation and the new plastic variable.

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Notes on Strain Gradient Plasticity 309

Our contribution is organized as follows: first, we present a basic modelingof multiplicative gradient plasticity and show that the formulation is thermody-namically consistent in an extended sense. We impose sufficient conditions for theinsulation condition to hold. No conditions inside the bulk between elastic andplastic parts are imposed. Then we motivate full spatial, intermediate and referen-tial covariance under constant rotations. Thus our finite strain model is objectiveand isotropic w.r.t. both the reference configuration and the intermediate configu-ration. Nevertheless, the model cannot be reduced to depend on the plastic metricCp = FT

p Fp only because of the presence of plastic gradients.Subsequently, we repeat the derivation of the model in the geometrically

linear context. Strong solutions of the obtained model with general monotone,non-associative flow-rule together with suitable boundary conditions on the non-symmetric infinitesimal plastic distortion p can be shown to be unique. In theclassical case with rate-independent flow rule coming from the subdifferential ofa convex domain in stress-space (Prandtl–Reuss) we are able to reformulate theproblem as a mixed variational inequality and to show existence and uniqueness ina suitably defined nonstandard Hilbert space. The relevant notation is found in theAppendix.

2. Thermodynamics

2.1. Extended statement of the second law of thermodynamics

The following treatment is based on ideas apparently first proposed by Mullerin 196727 in an investigation of the form of the energy inequality. Muller intro-duced an extra energy flux into the second law of thermodynamics (Clausius–Duheminequality) which only has to satisfy natural invariance conditions. Elaborating onthese ideas, Maugin22 [Eq. (2.14)] considered gradient dependent plasticity mod-els within a general treatment of thermodynamics with internal variables (T.I.V ).There he introduced the idea of this “extra energy flux” to account for the hard-ening/softening diffusion process. This causes the Clausius–Duhem inequality toencompass an extra term restricted (only) by the condition that the dissipationdensity takes a suitable bilinear format. A recent account can be found in Ref. 21which we adapt freely for our purpose.

Assume constant mass density > 0 and constant temperature (Θ = const.,isothermal process) throughout. Let W be the Helmholtz free energy, depending on∇ϕ and a set of internal variables ζ. Then the classical nonlocalized statement ofthe second law of thermodynamics reads: for all control volumes V ⊆ Ω∫

V〈S1(F (t), ζ(t)),∇ϕt〉 −

ddt

W (∇ϕ(t), ζ(t))︸︷︷︸D

dV ≥ 0

⇔∫V

ddt

W (∇ϕ(t0), ζ(t))︸︷︷︸≤0 reduced dissipation inequality

dV ≤ 0 at frozen t0 ∈ R, (2.1)

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with D ≥ 0 the non-negative dissipation due to the evolution of the inner variables.Here, S1(F, ζ) = DF W (F, ζ) is the first Piola–Kirchhoff stress tensor and F =∇ϕ is the deformation gradient. For sufficiently smooth integrands one obtains bylocalization the classical statement of the local reduced dissipation inequality

∀x ∈ Ω, ∀ t ∈ R :ddt

W (∇ϕ(x, t0), ζ(x, t)) ≤ 0. (2.2)

Maugin21 extends this classical definition to cover also models depending on gra-dients of internal variables, i.e. models where also ∇xζ is involved. The extendednonlocal version can be read as: ∀ V ⊆ Ω∫

V〈S1(F (t), ζ(t),∇ζ(t)),∇ϕt〉 −

ddt

W (∇ϕ(t), ζ(t),∇ζ(t))

+ Div q(∇ϕ, ζ,∇ζ) dV ≥ 0

⇔∫V

ddt

W (∇ϕ(t0), ζ(t),∇ζ(t)) − Div q(∇ϕ, ζ,∇ζ) dV ≤ 0 at frozen t0 ∈ R,

(2.3)

where the extra energy flux q may be chosen suitably.a Again, after localizationfor smooth enough integrands, one is left with the extended reduced dissipationinequality ∀ x ∈ Ω, ∀ t ∈ R:

ddt

W (∇ϕ(x, t0), ζ(x, t),∇ζ(x, t)) − Div q(∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) ≤ 0. (2.4)

A major advantage of the new condition (2.3) is its efficiency in deriving evolutionequations for the inner variables.

2.2. A priori energy inequality

The condition (2.4) does not make a statement as far as boundary conditions forthe internal variables ζ are concerned and indeed Maugin does not propose suchconditions. However, for materials which depend on gradients of internal variablesthis question needs to be addressed. We propose as an additional condition thesatisfaction of an energy inequality over the bulk.b We require that the rate of

aMaugin21 (p. 99) writes: “. . . to select k (the extra energy flux) in such a form as to eliminateany true divergence term in the dissipation inequality.” Introducing this extra energy flux allowsto consider a local version of the second law of thermodynamics also for materials with nonlocalterms.bThis is the same as postulating the classical Clausius–Duhem inequality over the bulk, i.e.Z

Ω〈S1, Ft〉 −

d

dtW (F (x, t), ζ(x, t),∇ζ(x, t)) dV ≥ 0, (2.5)

i.e. the global rate of change of internal energy is bounded by the power supplied by externalforces.

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change of the free energy is bounded by the power of classical external forces, i.e.

ddt

∫Ω

W (∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) dV

≤∫

Ω

〈f(x, t), ϕt(x, t)〉 dV

+∫

∂Ω

〈ϕt(x, t), S1(∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) · n〉dS, (2.6)

with f the given volume forces. The condition (2.6) allows us to obtain the a priorienergy inequality:∫

Ω

W (∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) dV

≤∫

Ω

W (∇ϕ(x, 0), ζ(x, 0),∇ζ(x, 0)) dV +∫ t

0

∫Ω

〈f(x, t), ϕt(x, t)〉 dV dt

+∫ t

0

∫∂Ω

〈ϕt(x, t), S1(∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) · n〉dS dt. (2.7)

(The difference to equality in (2.7) is the time-integrated positive dissipation inMielke.25) Sincec

ddt

∫Ω

W (∇ϕ(x, t), ζ(x, t),∇ζ(x, t)) dV

=∫

Ω

〈f, ϕt〉 dV +∫

∂Ω

〈ϕt, S1 · n〉dS +∫

Ω

⟨DζW − Div D∇ζW,

ddt

ζ

⟩dV

+∫

∂Ω

⟨ddt

ζ, D∇ζW · n⟩

dS, (2.8)

inequality (2.7) can be satisfied if,∫Ω

⟨DζW − Div D∇ζW,

ddt

ζ

⟩dV +

∫∂Ω

⟨ddt

ζ, D∇ζW · n⟩

dS ≤ 0. (2.9)

Sufficient for this is, e.g.

ddt

ζ = f

−[DζW − Div D∇ζW ]︸︷︷︸Σ

,⟨f(X)X

⟩ ≥ 0, x ∈ Ω

D∇ζW · n = 0, x ∈ ∂Ω\ΓD micro-free condition, (2.10)

ζ = 0, x ∈ ΓD micro-hard condition.

cUse Div S1 = −f .

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Note that in the local model (absence of gradients) no possibility for boundaryconditions on ζ arises. Requiring simply∫

∂Ω

⟨ddt

ζ, D∇ζW · n⟩

dS =∫

Ω

Div(

D∇ζWT · d

dtζ

)dV = 0, insulation condition

(2.11)

can be interpreted as energy insulation condition on the internal variable in thebulk to the outer world which means that no nonlocal energy exchange across theexternal boundary is possible. Note that (2.10) is stronger than the local insulationcondition contrary to (2.11) which itself is sufficient for (2.9) and (2.7). The choicebetween the different boundary conditions which kill the appearing divergence termmay be guided by taking the weakest condition which still allows for uniqueness ofclassical solutions. Such a motivation has been pursued in Sec. 8.1 of Ref. 11. Thedifferent boundary conditions leading to the insulation condition can be subsumed,in the language of Gurtin, as microscopically powerless boundary conditions.

3. The Gradient Plasticity Model at Finite Strain

3.1. The multiplicative decomposition

Consider the well-known multiplicative decomposition13, 15 of the deformation gra-dient F = ∇ϕ into elastic and plastic parts

F = Fe · Fp. (3.1)

Recall that while F = ∇ϕ is a gradient, neither Fe nor Fp need to be gradientsthemselves. In this decomposition, usually adopted in single crystal plasticity, Fe

represents elastic lattice stretching and elastic lattice rotation while Fp representslocal deformation of the crystal due to plastic rearrangement by slip on glide planes.In the single crystal case one additionally assumes this split to be of the form

Fp : TxΩref → TxΩref , Fe : TxΩref → Tϕ(x)Ωact, F : TxΩref → Tϕ(x)Ωact,

(3.2)

where Ωref and Ωact are the referential and actual configuration respectively. More-over, Fp is uniquely constitutively determined by a flow rule describing the slipkinematics on preferred glide planes.16

At the continuum level, however, the multiplicative decomposition (3.1) ismerely a nonlinear generalization of the classical additive decomposition of the smallstrain tensor into elastic and plastic parts. In this case a constitutive assumptionlike (3.2) is not mandatory and the question of the uniqueness of this decompositionhas been raised several times in the literature, since, formally it is always possibleto write

F = Fe Fp = Fe QT Q Fp = F ∗e F ∗

p , (3.3)

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with Q ∈ SO(3) an arbitrary rigid rotation.d In this contribution, we considerthe split (3.1) as void of any additional microstructural information regarding thenature of Fp, contrary to (3.2). Since we assume that we do not know how tochoose the rigid rotation in (3.3) a specific choice of rotation should be without con-sequence. This leads to an additional invariance requirement on the intermediateconfiguration: all possible selections of the intermediate configuration are treated asequal. It is clear that such a strong invariance requirement is subject to discussion.In the case that the (local) material is, in addition, isotropic in its reference config-uration, it is well-established that the expression governing the energy storage ofthe elasto-plastic material can be reduced to an isotropic function W = W (C, Cp),where C = FTF and Cp = FT

p Fp, together with an isotropic flow rule for Cp, seee.g. Ref. 19. Here, Fp itself does not appear anymore.

Remark 3.1. (Why invariance requirements?) In all of the following the readershould bear in mind that we restrict our modeling proposal to a fully isotropicsetting. From an application oriented view this might seem unrealistic. However,our point is that if anisotropies of whatever kind are to be included, they shouldappear explicitly through the use of a corresponding invariant framework or struc-tural tensors reflecting the given material symmetries.19 If this is not done but theformulation is not fully rotationally form-invariant, then one introduces inconsider-ate anisotropic behavior which lacks any real physical basis (see p. 220 of Ref. 20).

3.2. The plastic indifference of the elastic response

In this work we concentrate on the hyperelastic formulation of finite strain plasticity,i.e. we are concerned with finding the appropriate energy storage terms governingthe elastic and plastic behavior. Moreover, we restrict our attention to materialswhich are homogeneous in their purely elastic state.e In general, we assume thenthat the total stored energy can be expressed as a sum (this is already a constitutiveassumption)

W (F, Fp, Curl Fp) = We(F, Fp)︸︷︷︸elastic energy

+ Wph(Fp)︸︷︷︸linear kinematical hardening

+ Wcurl(Fp, Curl Fp)︸︷︷︸dislocation processes

.

(3.4)

Here, the local hardening potential Wph is a purely phenomenological energy storageterm formally consistent with a Prager type constant linear hardening behavior. Thedislocation potential Wcurl instead is a microscopically motivated energy storageterm due to dislocation processes which is the ultimate physical reason for anyhardening behavior. It acts on Curl Fp which is the curl applied to the rows of Fp.

dThe split is, formally, also invariant under any invertible B(x) ∈ GL+(3) in the sense that

F = Fe Fp = Fe B−1 B Fp = F e F

p , but F e leads to a different elastic response and cannot be

considered as “equivalent” to Fe.eNo further x-dependence in (3.4): the material behavior is the same in all material points.

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An additional constitutive assumption in finite strain plasticity is the plasticindifferencef condition.25 Plastic indifference implies that the elastic response ofthe material, governed by the elastic strain energy part, is invariant w.r.t. arbitraryprevious plastic deformation F 0

p see Ref. 39. This means that

∀ Fp ∈ GL+(3) : We(F, Fp) = We(FF 0p , FpF

0p ). (3.5)

Using (3.5) it is easily seen, by specifying F 0p = F−1

p , that

We(F, Fp) = We(FF−1p ,1) = We(Fe). (3.6)

Thus plastic indifference is a constitutive statement about the elastic response andnot a fundamental physical invariance law. Plastic indifference reduces the effectiveelastic dependence of We on Fe alone. It is based on the experimental evidence forsome materials that unloading and consecutive loading below the yield limit hasthe same elastic response as the initial virgin response.

Remaining in the context of finite strain plasticity, we propose, followingBertram2 a model based on evolution equations for Fp and not on a plastic met-ric Cp = FT

p Fp. In order to adapt the micromechanically motivated multiplica-tive decomposition to a description on the continuum level, one usually includesan assumption on the plastic spin, i.e. skew(Fp

ddtF

−1p ) = 0, a no-spin multiplica-

tive plasticity model based on Fp results but we will retain from imposing such acondition.

Many of the existing gradient plasticity theories do not involve plastic rotationseither, however, Gurtin and Anand10 note: “unless the plastic spin is (explicitly)constrained to be zero, constitutive dependencies on the Burgers tensor necessarilyinvolve dependencies on the (infinitesimal) plastic rotation.”

3.3. Illustration of the phenomenological multiplicative

decomposition based on the chain rule

For illustration purposes let us introduce symbolically a compatible reference config-uration Ωref , a fictitious compatible intermediate configuration Ωint and a compatibledeformed configuration Ωact, together with mappings

Ψp : x ∈ Ωref ⊂ E3 → Ψp(Ωref) = Ωint ⊂ E

3, Ψe : η ∈ Ωint ⊂ E3 → Ωact ⊂ E

3,

ϕ : x ∈ Ωref ⊂ E3 → Ωact ⊂ E

3,

ϕ(x) = Ψe(Ψp(x)), ∇xϕ(x) = ∇ηΨe(Ψp(x))∇xΨp(x) = Fe(x) Fp(x).

(3.7)

This means that we assume to realize the total deformation ϕ by two subsequentcompatible deformations Ψp and Ψe and interpret Fe and Fp as the respective

fConceptually similar is the assumption of isomorphic elastic ranges as discussed by Bertram.2

The restrictive assumption of isomorphic elastic ranges fits well for aluminum but not really for

copper.

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deformation gradients. A guiding question for this exposition can be stated as: whatform of constitutive restrictions are implied by respecting a possible gradient struc-ture inherent in (3.7). In this sense, this work extends the investigation in Ref. 4from homogeneous deformations (suitable for the local situation without gradientson plastic variables) to the inhomogeneous situation.

4. Referential Isotropy of Material Response

Many polycrystalline materials, in particular many metals, can be considered, evenafter plastically deforming (but before significant texture development occurs) tobehave (at least approximately) elastically isotropic. Restricting ourselves to suchmaterials in this work, we assume that the total stored energy W is isotropic withrespect to the intermediate configuration,g moreover, it will turn out that the Pragerlinear kinematic hardening potential Wph must be an isotropic function of Cp =FT

p Fp. This conclusion can already be found in Ref. 4, for a recent account seeRefs. 7, 18 and 19. However, we need to clarify in what sense a gradient plasticitymodel can be considered to be isotropic.

There is agreement in the literature as far as the meaning of elastic isotropy orisotropy w.r.t. the intermediate configuration is concerned. In this case the elasti-cally stored energy function should be isotropic w.r.t. rotations of the intermediateconfiguration, i.e.

We(Fe Q) = We(Fe) ∀ Q ∈ SO(3). (4.1)

Concerning the determination of the plastic distortion Fp, following Maugin23

(p. 110), it should hold for referentially isotropic materials (nothing to do withisotropy w.r.t. the intermediate configuration) without gradients on the plastic dis-tortion, that, comparing initial conditions

Fp(x, 0) versus Fp(x, 0) QT, (4.2)

differing only by one constant proper rotation QT ∈ SO(3), that the respectivesolutions of the model, at all later times t ∈ R are

Fp(x, t) versus Fp(x, t) QT. (4.3)

For full isotropy w.r.t. the reference configuration the material symmetry groupis SO(3). We note that the “plastic distortion rate” Fp

ddt [F−1

p ] is form-invariantunder Fp → Fp QT, thus the use of this rate is consistent with the require-ment (4.3). It remains then to show that the energy related parts of the modelwill satisfy a suitably extended version of (4.3), taking also plastic gradients intoaccount.

gThe subsequent development shows that assuming form-invariance of the energy with respect to

superposed rotations on the intermediate configuration already implies elastic isotropy.

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We refer to referential isotropyh if the model is form-invariant under the trans-formation

(F, Fe, Fp) → (FQ, Fe, FpQ), (4.4)

that means if (F, Fe, Fp) is a solution, then (FQ, Fe, FpQ) is a solution to rotateddata, where the rotation Q is understood to be one and the same rigid rotationof the whole body. Since Fe is left unaltered we see that the referential isotropycondition does not restrict the elastic response of the material but restricts ourPrager hardening potential to a functional dependence of the form

Wph(FpQ) = Wph(Fp) ∀ Q ∈ SO(3) ⇒ Wph(Fp) = Wph(FpFTp ), (4.5)

which can be seen by considering the left polar decomposition of Fp =√

FpFTp Rp

and specifying Q = RTp .

In order to motivate the restrictions of referential isotropy for the gradientplasticity model we present a basic observation valid for homogeneous, isotropichyperelasticity. In classical hyperelastic finite-strain elasticity and assuming homo-geneous material, isotropy (a priori now referential isotropy) can be viewed as aconsequence of the form-invariance of the free-energy under a rigid rotation of thereferential coordinate system. To see this consider the variational problem∫

Ω

W (∇ϕ(x)) dx → min . ϕ, (4.6)

for ϕ : Ωref ⊂ R3 → R

3. Consider also a transformed coordinate system, thetransformation being given by a diffeomorphism

ζ : Ωref → ζ(Ωref) = Ω∗, ζ(x) = ξ. (4.7)

By the transformation formula for integrals the problem (4.6) can be transformed tothis new configuration. We define the same function ϕ expressed in new coordinates(pull back of ϕ)

ϕ∗(ζ(x)) := ϕ(x) ⇒ ∇ξϕ∗(ζ(x))∇ζ(x) = ∇ϕ(x) (4.8)

and consider∫ξ∈Ω∗

W (∇ξϕ∗(ξ)∇xζ(ζ−1(ξ)))| det∇xζ−1|dξ → min . ϕ∗. (4.9)

The transformed free energy W ∗ for functions defined on Ω∗ is, therefore, given as

W ∗(ξ,∇ξϕ∗(ξ)) = W (∇ξϕ

∗(ξ)∇xζ(ζ−1(ξ)))| det∇xζ−1|. (4.10)

In the case that the transformation is only a rigid rotation, i.e. ζ(x) = Q · x, theformer turns into

W ∗(∇ξϕ∗(ξ)) = W (∇ξϕ

∗(ξ) Q). (4.11)

hPapadopoulos19 calls this “referential covariance” which is an extended statement of materialsymmetry relative to the given reference configuration.

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Now, in the classical hyperelastic context for homogeneous materials, isotropy isequivalent to form-invariance of W under a rigid rotation of the referential coordi-nates, i.e.

W ∗(X)form-invariance︷︸︸︷

= W (X) ⇔ ∀ Q ∈ SO(3) : W (XQ)isotropy︷︸︸︷

= W (X). (4.12)

In order to extent isotropy to the multiplicative decomposition, we make one pre-liminary assumption for illustration: following our guideline, Fp is viewed formally(for a moment) as being a plastic gradient ∇Ψp with Ψp : Ωref ⊂ R

3 → R3. Next,

the transformation of Ψp to a rigidly rotated configuration is obtained as

Ψ∗p(Qx) := Ψp(x) ⇒ ∇x∗Ψ∗

p(x∗) Q = ∇xΨp(x). (4.13)

This motivates to assume the transformation rule for Fp under a rigid transforma-tion of the reference configuration as being given as

F ∗p (x∗) = F ∗

p (Qx) = Fp(x) QT. (4.14)

Note that the argument of F ∗p must also be changed accordingly.

It is not immediately obvious what type of transformation law for CurlFp isinduced under the transformation Fp(x) → Fp(x)QT, even for constant rotations,since

Curlx[Fp(x) QT] = [Curlx Fp(x)] QT. (4.15)

On the other hand, it is simple to consider the transformation Fp(x) → Q Fp(x)and to deduce the corresponding transformation law Curl Fp → Q Curl Fp, since

Curlx[Q Fp(x)] = Q [Curlx Fp(x)]. (4.16)

Nevertheless, using the transformation law (4.14) for Fp we are in a position toextend the invariance condition (4.3) also to include plastic gradients: for referentialisotropy we postulate the form-invariance of the energy storage terms under a rigidrotation of the referential coordinates. First, for the local hardening contributionWph we have, from the transformation of integral formula, assuming the gradientstructure ∫

x∈Ω

Wph(Fp(x)) dx =∫

ξ∈Ω∗Wph(F ∗

p (ξ)Q) det QTdξ, (4.17)

such that

W ∗ph(F ∗

p (ξ)) := Wph(F ∗p (ξ)Q), (4.18)

and form-invariance demands that

W ∗ph(X) = Wph(X) ⇒ ∀ Q ∈ SO(3) : Wph(XQ) = Wph(X), (4.19)

which coincides with the result already obtained in (4.5).

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However, we can now apply the same consideration of form-invariance to thedislocation energy storage. In this case, then, the transformed energy reads, similarto (4.17) ∫

x∈Ω

Wcurl(Fp(x), Curlx[Fp(x)]) dx

=∫

ξ∈Ω∗Wcurl(F ∗

p (ξ)Q, Curlx[F ∗p (ξ)Q]) det QTdξ, (4.20)

such that

W ∗curl(F

∗p (ξ), Curlξ[F ∗

p (ξ)]) := Wcurl(F ∗p (ξ)Q, Curlx[F ∗

p (ξ)Q]). (4.21)

From a lengthy calculation in indicial notation, it holdsi

Curlx[F ∗p (Qx) Q]) = [Curlξ F ∗

p (ξ)] Q, (4.22)

which leads to

W ∗curl(F

∗p (ξ), Curlξ[F ∗

p (ξ)]) := Wcurl(F ∗p (ξ)Q, [Curlξ F ∗

p (ξ)] Q). (4.23)

Form-invariance for the dislocation energy storage demands therefore that

W ∗curl(X, Y ) = Wcurl(X, Y ) ⇒ Wcurl(XQ, Y Q]) = Wcurl(X, Y ) ∀ Q ∈ SO(3),

(4.24)

which is satisfied e.g. for Wcurl(X, Y ) = ‖X−1Y ‖2.j

5. Simultaneous Rigid Rotation of the Materialand Spatial Coordinates

Let the reference configuration Ωref be rigidly rotated through one constant rota-tion Q2 ∈ SO(3). We let Ω∗

ref = Q2 · Ωref together with the rotated coordinates

iThe (1, 1)-component in the matrix from the left-hand side of (4.22) is equal to

∂

∂x2(F ∗

p )1k(Qx)Qk3 − ∂

∂x3(F ∗

p )1k(Qx)Qk2 =

∂

∂ξl(F ∗

p )1k(ξ)[Ql2Qk

3 − Ql3Qk

2 ].

By orthogonality of the matrix Q we obtain that the (1, 1)-component is equal to»∂

∂ξ2(F ∗

p )13(ξ) − ∂

∂ξ3(F ∗

p )12(ξ)

–Q1

1 +

»∂

∂ξ3(F ∗

p )11(ξ) − ∂

∂ξ1(F ∗

p )13(ξ)

–Q1

2

+

»∂

∂ξ1(F ∗

p )12(ξ) − ∂

∂ξ2(F ∗

p )11(ξ)

–Q1

3,

which is the (1, 1)-component of the matrix from the right-hand side of (4.22). The proof for theother components is similar.jAlso satisfied for ‖Y XT‖2 leading to a dislocation energy storage of the form‖[Curl Fp(x)]FT

p (x)‖2, cf. (6.2).

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x∗ = Q2 · x. Assume also that the spatial coordinate system is rigidly rotated by Q1.The deformation w.r.t. the rotated coordinate systems is denoted by ϕ∗(x∗). It holds

ϕ∗(x∗) = Q1 ϕ(x) i.e. ⇔ ϕ∗(Q2 · x) = Q1 · ϕ(x), ∀x ∈ Ωref , (5.1)

whether or not the material response is isotropic.k From (5.1) we obtain from thechain rule

∇x[ϕ∗(x∗)] = ∇x[Q1ϕ(x)] ⇔ ∇x[ϕ∗(Q2.x)] = Q1∇xϕ(x)

⇔ ∇x∗ [ϕ∗(x∗)]Q2 = Q1∇xϕ(x) ⇔ QT1 ∇x∗ [ϕ∗(x∗)]Q2 = ∇xϕ(x).

(5.2)

The rotated free energy is denoted by W ∗(F ∗) with F ∗ = ∇x∗ [ϕ(x∗)]. It must bedefined such that the “rotated” minimization problem based on W ∗(F ∗) furnishesthe rotated solution and that the energy of the materially and spatially rotatedsolution is equal to the energy of the unrotated solution. Thus

W ∗(F ∗) = W ∗(∇x∗ϕ∗(x∗)) = W ∗(Q1 ∇ϕ(x)QT2 ) := W (∇ϕ(x))

⇒ W ∗(F ∗) = W (QT1 F ∗Q2). (5.3)

This identity is used to define the rotated energy. If W happens to be material frame-indifferentl and isotropic, i.e. form-invariant w.r.t. left and right multiplicationby (not necessarily equal) constant rotation matrices (material and spatial form-invariance under rigid rotations), then W (QT

1 XQ2) = W (X) and from (5.3) follows

W ∗(X) = W (X), (5.4)

which is the desired form-invariance in case of purely elastic behavior.We now repeat the transformation under rigid rotations for each function

appearing in (3.7) separately, i.e. it follows

Ψ∗p(Qp

2 · x) = Qp3Ψp(x) ⇒ Qp,T

3 ∇x∗ [Ψ∗p(x∗)] Qp

2 = ∇xΨp(x),

Ψ∗e(Qe

3 · η) = Qe1Ψe(η) ⇒ Qe,T

1 ∇η∗ [Ψ∗e(η∗)] Qe

3 = ∇ηΨe(η),

ϕ∗(Q2 · x) = Q1ϕ(x) ⇒ QT1 ∇x∗ [ϕ∗(x∗)] Q2 = ∇xϕ(x).

(5.5)

By choosing Q1 = Qe1, Q2 = Qp

2, Q3 = Qp3 we observe that the composition of

mappings carries over

ϕ∗(x∗) = Q1 ϕ(x) = Q1 Ψe(Ψp(x)) = Q1 Ψe(QT3 Ψ∗

p(x∗))

= Q1 [QT1 Ψ∗

e(Q3 [QT3 Ψ∗

p(x∗)])] = Ψ∗e(Ψ∗

p(x∗)), (5.6)

which implies

∇x∗ϕ∗(x∗) = ∇η∗Ψ∗e(Ψ∗

p(x∗))∇x∗Ψ∗p(x∗), F ∗(x∗) = F ∗

e (η∗) F ∗p (x∗). (5.7)

kFor homogeneous isotropy, the reference configuration (the referential coordinate system) isrotated but the spatial system is not; nevertheless, if the material is isotropic, the resulting responseis the same.lSee Refs. 3 and 38 for an in depth discussion of the different concepts subsumed in the notprecisely defined term “objectivity”.

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This suggests to identify

F ∗e (η∗) = ∇η∗Ψ∗

e(Ψ∗p(x∗)), F ∗

p (x∗) = ∇x∗Ψ∗p(x∗). (5.8)

However,

Q1 ∇ϕ(x) QT2 = Q1 ∇ηΨe(Ψp(x))∇xΨp(x) QT

2

= Q1 ∇ηΨe(Ψp(x)) QT3 Q3 ∇xΨp(x) QT

2

Q1 F (x) QT2 = Q1 Fe(x) QT

3 (Q3 Fp(x) QT2 )

= Q1 Fe(x) Fp(x) QT2 ,

(5.9)

but F (x) = Fe(x) Fp(x) implies

Q1 F (x) QT2︸︷︷︸

=F∗(x∗)

= Q1 Fe(x)QT3 Q3 Fp(x) QT

2 = Q1 Fe(x) QT3︸︷︷︸

=:F∗e (η∗)

(Q3 Fp(x) QT2 )︸︷︷︸

=F∗p (x∗)

,

(5.10)

which shows, how the rotated elastic deformation gradient F ∗e must be related to

the unrotated elastic deformation gradient Fe under rigid rotations of the reference,intermediate and spatial coordinates if the assumed gradient structure is to berespected. In fact, the former equation is used as a definition of F ∗

e in terms of Fe

under rigid rotation of the intermediate and spatial configuration.Hence we conclude that under a rigid rotation of the reference configuration

and simultaneous rotation of the intermediate and spatial coordinates the followingobtains

F ∗(x∗) = Q2 F (x) QT1 , F ∗

e (η∗) = Q1 Fe(η) QT3 ,

F ∗p (x∗) = Q3 Fp(x) QT

1 = Q3 Fp(QT2 x∗) QT

1 .(5.11)

Since such changes of coordinates leave the composition of mappings invariant werequire that the model be form-invariant under these transformations. We refer tothis as elasto-plastic transformation invariance requirement.m Note that the postu-lated transformation law (5.11) for the multiplicative decomposition is not meantto read:

F (x) = Fe(x)Fp(x) ⇒ Q1(x)F (x)Q2(x)T

= Q1(x)Fe(x)QT3 (x)Q3(x)Fp(x)Q2(x)T (5.12)

⇒ F (x) = Fe(x)QT3 (x)Q3(x)Fp(x) = F ∗

e (x) F ∗p (x),

for all non-constant rotations Q1,2,3(x) ∈ SO(3) since this would destroy com-patibility. Requiring (5.12) would reduce the theory (including higher gradients)necessarily to a model based only on the plastic metric Cp(x) = FT

p (x)Fp(x) and acompatibility measure for Cp as gradient contribution. The much more restrictive

mIt should be noted that in the case of the parametrization of shells with a planar referenceconfiguration, the compatible Fp introduces nothing else than the stress free reference configurationof the curved shell.

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condition (5.12) is sometimes motivated by the observation that the multiplicativesplit is locally unique only up to a local rotation Q3(x) ∈ SO(3), which, viewedwithout compatibility requirements, is self-evident. It has already been observedby Casey/Naghdi4 [Eq. (13)] that full local “objectivity”-requirements on the mul-tiplicative decomposition (i.e. to allow non-constant rotation matrices in (5.11))reduces the model to an isotropic formulation in C = FTF and Cp = FT

p Fp. Ourdevelopment, however, shows that this conclusion is strictly constraint to the localtheory without gradients on the plastic distortion: the local theory simply cannotdistinguish between local rotations and global rigid rotations.

For a fully rotationally form-invariant model the form-invariant transformationbehavior of the elastic energy (5.4) under (5.11) will be postulated for all contribu-tions separately, i.e. for all constant Q1,2,3 ∈ SO(3) it must be satisfied

W ∗e (X) := We(QT

1 XQ3) = We(X),

W ∗ph(X) := Wph(QT

3 XQ2) = Wph(X),

W ∗curl(X, Curlξ X) := Wcurl(QT

3 XQ2, Curlx[QT3 XQ2])

= Wcurl(QT3 XQ2, Q

T3 [Curlξ X ]Q2])

= Wcurl(X, Curlξ X),

Wcurl(QT3 XQ2, Q

T3 Y Q2]) = Wcurl(X, Y ).

(5.13)

Thus We and Wph must be isotropic and frame-indifferent functions of their argu-ments which is easily met whenever the functional dependence can be reduced toisotropic functions of Ce = FT

e Fe and Cp = FTp Fp. A sufficient condition for Wcurl

to satisfy (5.13) is given, e.g. by taking

Wcurl(X, Y ) = H(X−1Y ) or H(XTY ) with

H(QTZQ) = H(Z) ∀ Q ∈ SO(3). (5.14)

For the remainder we choose H(Z) = ‖Z‖2 and Z = X−1Y .

6. A Form-Invariant Finite Strain Thermodynamic Potential

We have arrived at postulating the invariance of the plasticity model according to(5.11) and (5.4). To be specific let us write down a modified Saint-Venant Kirchhoffisotropic quadratic energy We in the elastic stretches FT

e Fe, augmented with localself-hardening Wph and a contribution accounting for plastic gradients Wcurl

W (Fe, Fp, Curl Fp) = We(Fe) + Wph(Fp) + Wcurl(Fp, Curl Fp),

We(Fe) =µ

4

∥∥∥∥ FTe Fe

det Fe2/3

− 1

∥∥∥∥2

+λ

4

((det Fe − 1)2 +

(1

det Fe− 1)2)

,

Wph(Fp) =µ H0

4

∥∥∥∥∥ FTp Fp

det Fp2/3

− 1

∥∥∥∥∥2

,

Wcurl(Fp, Curl Fp) =µ L2

c

2‖F−1

p Curl Fp‖2.

(6.1)

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Note that in this setting Fp is not needed to be volume preserving. Here, µ, λ are theclassical isotropic Lame-parameters, H0 is the dimensionless hardening modulus, Lc

is the internal plastic length. This energy is materially and spatially form-invariantand satisfies the plastic indifference condition. The elastic energy We is additivelydecoupled into a volumetric and isochoric contribution, it is frame-indifferent andisotropic w.r.t. the intermediate configuration. The term Wph accounts for phe-nomenological local plastic hardening in the spirit of Prager constant linear harden-ing. It is fully form-invariant and indifferent to plastic volume changes. The termWcurl represents energy storage due to dislocations. Its argument GR = F−1

p Curl Fp

is the referential version of the tensor G = 1detFp

(Curl Fp)FTp , called the geomet-

ric dislocation density tensor in the intermediate configuration. G represents theincompatibility of the intermediate configuration Fp relative to the associated sur-face elements. The tensor G has the virtue to be form-invariant under compatiblechanges in the reference configuration.9, 37 It transforms as G(QFp) = QG(Fp)QT

for all rigid rotations Q. This tensor introduces the influence of geometrically (kine-matically) necessary d islocations (GND’s). In Gurtins notation this is GT, and herefers to this tensor as the local Burgers tensor in the lattice configuration measuredper unit surface area in this configuration. As such it corresponds “conceptually” toan objective tensor in the “actual” configuration, like the finite strain Cauchy stresstensor σ, which satisfies as well the invariances ∀ Q ∈ SO(3) : σ(QF ) = Qσ(F )QT.Note that our corresponding referential measure9 [Eq. (6.1)] GR is given by

GR = F−1p Curl Fp, G =

1det Fp

(Curl Fp)FTp =

1det Fp

Fp GR FTp . (6.2)

The referential measure GR is easily seen to be invariant under a compatible (homo-geneous) change of the intermediate configuration, i.e.

F (x) = Fe(x)Fp(x) = Fe(x)B−1BFp(x) = Fe(x) Fp(x),

GR(BFp(x)) = GR(Fp(x)), ∀ B ∈ GL+(3), (6.3)

while the local plastic self-hardening would be invariant under Fp → R+ SO(3) ·Fp

only.

6.1. Derivation of the finite strain evolution equations

Having postulated as thermodynamic potential (6.1) it remains to motivate theflow rule. Relevant in this respect are the thermodynamical driving forces actingon the internal variable Fp. Therefore, locally in a space point x ∈ Ω

ddt

We(F (t0), F−1p (t)) =

⟨DWe(FF−1

p ), Fddt

F−1p

⟩=⟨DWe(Fe), FF−1

p Fpddt

F−1p

⟩=⟨FT

e DWe(Fe), Fpddt

F−1p

⟩=⟨

Σe, Fpddt

F−1p

⟩,

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⟨DWph(Fp),

ddt

Fp

⟩=⟨DWph(Fp),

[ddt

Fp

]F−1

p Fp

⟩=⟨DWph(Fp)FT

p ,ddt

FpF−1p

⟩=⟨−DWph(Fp)FT

p , Fpddt

F−1p

⟩=⟨

Σsh, Fpddt

F−1p

⟩, (6.4)

where we have used FpddtF

−1p = − d

dtFpF−1p . For the nonlocal term, the prod-

uct 〈A, B〉 also includes integration over the domain Ω and we will use the self-adjointness of the Curl-operator provided the boundary conditions are suitablychosen, i.e. at least the appropriate insulation conditionn does hold. Then⟨

DWcurl(F−1p Curl Fp), F−1

p Curlddt

Fp +ddt

F−1p Curl Fp

⟩=⟨

Curl(F−Tp DWcurl(F−1

p Curl Fp),ddt

Fp

⟩+⟨

DWcurl(F−1p Curl Fp)(Curl Fp)T,

ddt

F−1p

⟩=⟨

Curl(F−Tp DWcurl(F−1

p Curl Fp)FTp ,−Fp

ddt

F−1p

⟩+⟨

F−Tp DWcurl(F−1

p Curl Fp)(Curl Fp)T, Fpddt

F−1p

⟩=⟨

Σcurl, Fpddt

F−1p

⟩. (6.6)

Let us also define the elastic domain in stress-space K := Σ ∈ M3×3|‖ dev Σ‖ ≤

σy, with yield stress σy, corresponding indicator function

χ(Σ) =

0 ‖ dev Σ‖ ≤ σy,

∞ otherwise,(6.7)

nIn the finite-strain case the insulation condition readsZΩ

*F−T

p DWcurl(F−1p Curl Fp)| z

Σ

, Curl

»d

dtFp

–+dV

=

ZΩ

fiCurl[F−T

p DWcurl(F−1p Curl Fp)],

d

dtFp

fldV +

Z∂Ω

3Xi=1

*Σi ×

„d

dtFp

«i

,n

+dS

| z insulation condition=0

,

(6.5)

and can be satisfied, if, e.g. Fp(x, t) · τ = Fp(x, 0) · τ on x ∈ ΓD

`implies d

dtFp · τ = 0 there

´and

Curl Fp · τ = 0 on ∂Ω\ΓD .

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and subdifferential in the sense of convex analysiso

∂χ(Σ) =

0 ‖ dev Σ‖ < σy ,

R+0

dev Σ‖ dev Σ‖ ‖ dev Σ‖ = σy ,

∅ ‖ dev Σ‖ > σy .

(6.10)

Note that choosing dev Σ instead of dev sym Σ in (6.7) allows for plastic spin.Putting together the associative flow rule the finite strain model in a classicalformulation reads: find

ϕ ∈ C1([0, T ]; C1(Ω, R3)), Fp ∈ C1([0, T ]; C(Ω, SL(3)),

Curl Fp(t) ∈ L2(Ω, M3×3), Σcurl(t) ∈ L2(Ω, M3×3), (6.11)

such that

Div S1(Fe) = −f, S1(Fe) = DF [We(FF−1p )] = DWe(Fe) · F−T

p ,

Fpddt

[F−1p ] ∈ −∂χ(Σ), Σ = Σe + Σsh + Σcurl,

Σe = FTe DWe(Fe), Σsh = −DWph(Fp)FT

p ,

Σcurl = F−Tp DWcurl(F−1

p Curl Fp) (Curl Fp)T

− Curl(F−Tp DWcurl(F−1

p Curl Fp))FTp ,

ϕ(x, t) = gd(x, t), Fp(x, t) · τ = Fp(x, 0) · τ, x ∈ ΓD,

0 = [Curl Fp(x, t)] · τ, x ∈ ∂Ω\ΓD, Fp(x, 0) = F 0p (x).

(6.12)

The chosen boundary conditions on the plastic distortion Fp imply the local insu-lation condition. Note that the local contributions are fully rotationally invariant(isotropic and objective) with respect to the transformation

∀Q1,2,3 ∈ SO(3) :

(F, Fe, Fp) → (Q1(x)FQ2(x)T, Q1(x)FeQ(x)T3 , Q3(x)FpQ2(x)T), (6.13)

and the nonlocal dislocation potential is still invariant with respect to the corre-sponding rigid transformation

∀ Q1,2,3 ∈ SO(3) : (F, Fe, Fp) → (Q1FQ2T, Q1FeQ3

T, Q3FpQ2T). (6.14)

oThe subdifferential for a convex function χ : M3×3 → R

+ is defined as ∂χ(Σ) = ∅ for Σ : χ(Σ) =∞ and otherwise through

ζ ∈ ∂χ(Σ) ⇔ ∀ H ∈ M3×3 : χ(Σ + H) ≥ χ(Σ) + 〈ζ, H〉. (6.8)

Note that whenever Σ+H ∈ K then χ(Σ+H) = ∞ and the last inequality generates no constrainton ζ. Therefore we have the equivalent characterization

ζ ∈ ∂χ(Σ) ⇔ ∀ H ∈ sl(3) : χ(Σ + H) ≥ χ(Σ) + 〈ζ, H〉. (6.9)

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Since We and Wph satisfy We(Q(x)FeQ(x)T) = We(Fe) and Wph(Q(x)FpQ(x)T) =Wph(Fp) the following statements are automatically true:

Σe(Q(x)FeQ(x)T) = Q(x)Σe(Fe)Q(x)T, Σe ∈ Sym(3),

Σsh(Q(x)FpQ(x)T) = Q(x)Σsh(Fp)Q(x)T, Σsh ∈ Sym(3),

Σcurl(QFp Curl[QFp]) = QΣcurl(Fp, Curl Fp)QT, Q = const.,

Σcurl(QTXQ, QTY Q]) = QTΣcurl(X, Y )Q, Q = const., (6.15)

In general, Σcurl is not symmetric, thus, the plastic inhomogeneity is responsible forthe plastic spin contribution in this covariant formulation. Since ∂χ is monotone,the formulation is thermodynamically admissible. This remains true if we replace∂χ with a general flow function f : M

3×3 → M3×3 which is only pre-monotone1

meaning that 〈f(Σ), Σ〉 ≥ 0 for all Σ ∈ M3×3.

6.2. Reformulation of the finite strain problem

Using the Legendre-transformation of the flow potential one can equivalently rewritethe flow rule

Fpddt

[F−1p ] ∈ −∂χ(Σ) ⇔ Σ ∈ ∂χ∗

(−Fp

ddt

[F−1p ])

⇔ χ∗(q) − χ∗(−Fp

ddt

[F−1p ])

≥⟨

Σ, q −[−Fp

ddt

[F−1p ]]⟩

∀ q ∈ sl(3). (6.16)

Inserting the constitutive relation for Σ, integrating over the domain Ω and partialintegration (using the conditions leading to the insulation condition (6.5)), showsthat the following must hold∫

Ω

χ∗(q(t, x)) − χ∗(−Fp

ddt

[F−1p ])

dV

≥∫

Ω

⟨Σe + Σsh + Σ1

dis, q + Fpddt

[F−1p ]⟩

−⟨

Σ2dis, Curl

[q + Fp

ddt

[F−1p ]Fp

]⟩dV ∀ q ∈ L2(0, T ; Hcurl(Ω, sl(3))),

Σcurl = Σ1dis − (Curl Σ2

dis)FTp , Σ1

dis = F−Tp DWcurl(F−1

p Curl Fp) (Curl Fp)T,

Σ2dis = F−T

p DWcurl(F−1p Curl Fp). (6.17)

Hence, solutions of (6.12) satisfy as well

0 =∫

Ω

〈S1(Fe),∇ξ〉 − 〈f, ξ〉dV ∀ ξ ∈ H1(Ω, R3), (6.18)

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S1(Fe) = DF [We(FF−1p )] = DWe(Fe) · F−T

p ,

0 ≤∫

Ω

χ∗(q(t, x)) − χ∗(−Fp

ddt

[F−1p ])−⟨

Σe + Σsh + Σ1dis, q + Fp

ddt

[F−1p ]⟩

+⟨

Σ2dis, Curl

[q + Fp

ddt

[F−1p ]Fp

]⟩dV ∀ q ∈ L2(0, T ; Hcurl(Ω, sl(3))),

(6.19)ϕ(x, t) = gd(x, t), Fp(x, t) · τ = Fp(x, 0) · τ, x ∈ ΓD, Fp(x, 0) = F 0

p (x).

Adding (6.18) and (6.19) yields a mixed variational inequality which is automati-cally satisfied by solutions of (6.12).

7. The Geometrically Linear Gradient Plasticity Model

The corresponding linearized model can either be obtained by linearizing all quan-tities, e.g. replacing Σe → Lin(Σe) = σ or, more elegantly by writing downthe corresponding quadratic potential in linearized quantities. Thus we expandF = 1 + ∇u, Fp = 1 + p + . . . , Fe = 1 + e + . . . and the multiplicative decomposi-tion (3.1) turns into

1 + ∇u = (1 + e + · · ·)(1 + p + · · ·) ∇u ≈ e + p + · · · ,FT

e Fe − 1 = 1 + 2 sym e + eTe − 1 2 sym e = 2 sym(∇u − p).(7.1)

Hence one obtains to highest order the additive decomposition13, 14 of the displace-ment gradient ∇u = e + p, with sym e = sym(∇u − p) the infinitesimal elasticlattice strain, skew e = skew(∇u − p) the infinitesimal elastic lattice rotation andκe = ∇ axl(skew e) the infinitesimal elastic lattice curvature24 and p the infinitesi-mal plastic distortion. The quadratic energy which corresponds to (6.1) is given by

W (∇u, p, Curl p) = W line (∇u − p) + Wph(p) + W lin

curl(Curl p),

W line (∇u − p) = µ‖ sym(∇u − p)‖2 +

λ

2tr [∇u − p]2, (7.2)

W linph (p) = µ H0‖ dev sym p‖2, W lin

curl(Curl p) =µ L2

c

2‖Curl p‖2.

The used free energy coincides with that in Ref. 24 (p. 1783) apart for our localkinematical hardening contribution. Note that the infinitesimal plastic distortionp : Ω ⊂ R

3 → M3×3 need not be symmetric, but that only its symmetric part, the

infinitesimal plastic strainp sym p, contributes to the local elastic energy expression.The infinitesimal plastic rotation skew p neither contributes locally to the elasticenergy nor contributes to the local plastic self-hardening but appears in the non-local hardening. The resulting elastic energy is invariant under infinitesimal rigidrotations ∇u → ∇u + A, A ∈ so(3) of the body. The invariance of the curvaturecontribution needs the homogeneity of the rotations.

pThe notation εp ∈ Sym(3) is reserved to the purely local theory.

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Provided that the infinitesimal plastic distortion p is known, (7.2) defines a linearelasticity problem with pre-stress for the displacement u. It remains to provide anevolution law for p which is consistent with thermodynamics. To this end we use anonlocal (integral) version of the second law of thermodynamics.

For any “nice” subdomain V ⊆ Ω consider for fixed t0 ∈ R the rate of changeof energy storage due to inelastic processes

ddt

∫V

W (∇u(x, t0), p(x, t), Curl p(x, t)) dV

=∫V

2µ

⟨sym(∇u − p),− d

dtp

⟩+ λtr [sym(∇u − p)] tr

[− d

dtp

]+ 2µ H0

⟨dev sym p, dev sym

ddt

p

⟩+ µ L2

c

⟨Curl p, Curl

ddt

p

⟩dV

=∫V

2µ

⟨sym(∇u − p),− d

dtp

⟩+ λtr [sym(∇u − p)]

⟨1,− d

dtp

⟩− 2µ H0

⟨dev sym p,− d

dtp

⟩+ µ L2

c

⟨Curl p, Curl

ddt

p

⟩dV

=∫V

⟨2µ sym(∇u − p) + λtr [∇u − p]1− 2µ H0 dev sym p,− d

dtp

⟩

+ µ L2c

⟨Curl[Curl p],

ddt

p

⟩+

3∑i=1

Div µ L2c

(ddt

pi × (curl p)i

)︸︷︷︸

“extra energy flux” q(pt,Curl p)

dV. (7.3)

Here and in the following, pi denotes the ith row of the matrix p.q Choosing con-stitutively as extra energy flux

qi = µ L2c

(ddt

pi × (curl p)i

), i = 1, 2, 3, (7.4)

shows that the extended (nonlocal) form of the reduced dissipation inequality atconstant temperature21 may be evaluated as follows:

0 ≥ZV

d

dtW (∇u(x, t0), p(x, t), Curl p(x, t)) − Div q(pt, Curl p) dV

=

ZV

fi2µ sym(∇u − p) + λtr [∇u − p]1− 2µH0 dev sym p,− d

dtp

fl

+ µ L2c

fiCurl[Curl p],

d

dtp

fldV

qThe extra energy flux term is needed to account for the possible nonlocal exchange of energyacross ∂V .

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=

ZV

*2µ sym(∇u − p) + λtr [∇u − p]1− 2µ H0 dev sym p − µ L2

c Curl[Curl p]| z =:Σ

,− d

dtp

+dV

=

ZV

fiσ − 2µ H0 dev sym p − µ L2

c Curl[Curl p],− d

dtp

fldV, (7.5)

where Σ is the linearized Eshelby stress tensor in disguise which is the driving forcefor the plastic evolution. Taking

ddt

p = f(Σ), (7.6)

where the function f : M3×3 → M

3×3 with f(0) = 0 satisfies the monotonicity inzero condition

∀Σ ∈ M3×3 : 〈f(Σ) − f(0), Σ − 0〉 = 〈f(Σ), Σ〉 ≥ 0, (7.7)

ensures the correct sign in (7.5) (positive dissipation) and thus the plastic evolutionlaw (7.6) is thermodynamically admissible. In the large scale limit Lc = 0, this isjust the class of pre-monotone type defined by Alber.1 The driving term Σ has thedimension of stress and Div(pt × Curl p) = 0 for purely elastic processes pt ≡ 0. Inthe case of associated plasticity, the function f may be obtained as subdifferential∂χ of a convex function χ. To this end, let us define the elastic domain in stress-space K := Σ ∈ M

3×3|‖dev Σ‖ ≤ σy with yield stress σy, corresponding indicatorfunction

χ(Σ) =

0 ‖dev Σ‖ ≤ σy

∞ otherwise,(7.8)

and subdifferential in the sense of convex analysis

∂χ(Σ) =

0 ‖dev Σ‖ < σy

R+0

dev Σ‖dev Σ‖ ‖dev Σ‖ = σy

∅ ‖dev Σ‖ > σy .

(7.9)

Choosing dev Σ instead of dev sym Σ in (7.8) allows for plastic spin.The remaining divergence term which has to be evaluated in order for an a priori

global energy inequality∫Ω

ddt

W (∇u(x, t0), p(x, t), Curl p(x, t)) dV ≤ 0 (7.10)

to hold over the entire body is given by the global insulation condition∫Ω

3∑i=1

Div(

ddt

pi × (curl p)i

)dV =

∫∂Ω

3∑i=1

⟨ddt

pi × (curl p)i),n⟩

dS = 0. (7.11)

The last condition is satisfied, e.g. if in each point of the boundary ∂Ω the localizedinsulation condition holds, i.e.

0 =⟨

ddt


, x ∈ ∂Ω, i = 1, 2, 3, (7.12)

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which may be satisfied by postulatingr

p(x, t) · τ = p(x, 0) · τ, x ∈ ΓD

(⇒ d

dtp(x, t) · τ = 0

),

Curl p(x, t) · τ = 0, x ∈ ∂Ω \ ΓD.

(7.13)

Following Gurtin and Anand10 on gradient plasticity: “Our goal is a theory thatallows for constitutive dependencies on (the dislocation density tensor) G, but thatotherwise does not depart drastically from the classical theory.” Therefore, ourguiding principles for the development of gradient plasticity may be stated as

• The large scale limit Lc → 0 with zero local hardening H0 = 0 should coincidewith the classical Prandtl–Reuss model with deviatoric von Mises flow rule.

• The large scale limit Lc → 0 should determine the plastic distortion to be irro-tational, i.e. only εp := sym p appears (zero plastic spin).

• The model for Lc > 0 should be well-posed. Existence and uniqueness should beobtained in suitable Hilbert-spaces.

• The model for Lc > 0 should contain maximally second-order derivatives in theevolution law.

• The model for Lc > 0 should be linearized materially and spatially covariant andthermodynamically consistent (in the extended sense).

• The model for Lc > 0 should be isotropic with respect to both, the referentialand intermediate configuration.

• The model for Lc > 0 should contain first-order boundary conditions on p onlyat the hard Dirichlet boundary ΓD ⊂ ∂Ω for the plastic distortion p only in termsof the plastic spin skew p there.

• The model for Lc > 0 may contain second-order boundary conditions on p likeCurl p · τ = 0 at the total external boundary ∂Ω or p · τ = 0, motivated, perhapsfrom thermodynamics and insulation conditions.

8. Strong Infinitesimal Gradient Plasticity with Plastic Spin

The infinitesimal strain gradient plasticity model reads now: find

u ∈ H1([0, T ]; H10 (Ω, ΓD, R3)), sym p ∈ H1([0, T ]; L2(Ω, sl(3)),

Curl p(t) ∈ L2(Ω, M3×3), dev Curl Curl p(t) ∈ L2(Ω, M3×3),(8.1)

such that

Div σ = −f, σ = 2µ sym(∇u − p) + λ tr [∇u − p]1,

p ∈ ∂χ(Σlin), Σlin = Σline + Σlin

sh + Σlincurl,

rIt is not immediately obvious how a boundary condition on p at ΓD can be posed keeping inmind that εp := sym p is not subject to any essential boundary condition in the classical theory. In

Gurtin11 Eq. (2.17) it is shown that the microscopically hard condition p · τ |ΓD= 0 has a precise

physical meaning: there is no flow of the Burgers vector across the boundary ΓD.

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Σline = 2µ sym(∇u − p) + λ tr [∇u − p]1 = σ,

Σlinsh = −2µ H0 dev sym p, Σlin

curl = −µ L2c Curl(Curl p),

u(x, t) = ud(x), p(x, t) · τ = p(x, 0) · τ, x ∈ ΓD,

0 = [Curl p(x, t)] · τ, x ∈ ∂Ω \ ΓD, p(x, 0) = p0(x). (8.2)

In general, Σlincurl is not symmetric even if p is symmetric. Thus, the plastic

inhomogeneity is responsible for the plastic spin contribution in this rotationallyinvariant formulation. Since ∂χ is monotone, the formulation is thermodynami-cally admissible. This remains true if we replace ∂χ with a general flow functionf : M

3×3 → M3×3 which is only pre-monotone. The mathematically suitable space

for symmetric p is the classical space Hcurl(Ω) := v ∈ L2(Ω), Curl v ∈ L2(Ω).The boundary conditions on the plastic distortion p serve only the purpose to fixideas.

In the large scale limit Lc → 0 we recover a classical elasto-plasticity modelwith local kinematic hardening of Prager-type. Observe that the term Σlin

curl =−µ L2

c Curl(Curl p) acts as nonlocal kinematical backstress and constitutes a crys-tallographically motivated alternative to merely phenomenologically motivatedbackstress tensors. The term −2µ H0 dev sym p is a symmetric local kinematicalbackstress. The model is therefore able to represent linear kinematic hardening andBauschinger-like phenomena. Moreover, the driving stress Σ is nonsymmetric dueto the presence of the second-order gradients, while the local contribution σ, dueto elastic lattice strains, remains symmetric.

Additionally, the infinitesimal local contributions are fully rotationally invari-ant (isotropic and objective) with respect to the transformation (∇u, p) →(∇u + A(x), p + A(x)) and the nonlocal dislocation potential is still invariant withrespect to the infinitesimal rigid transformation (∇u, p) → (∇u + A, p + A) whereA, A(x) ∈ so(3).

9. Mathematical Results

9.1. Uniqueness of strong solutions

Assume that strong solutions to the model (8.2) exist. We will show that thesesolutions are already unique. The aim of this paragraph is, moreover, to study theinfluence of the different boundary conditions for the plastic distortion p on thepossible uniqueness. In that way it is intended to identify the weakest boundarycondition which suffices for uniqueness. Possible boundary conditions (which aresufficient for the global insulation condition) are

pure micro-free : Curl p · τ = 0, x ∈ ∂Ω,

micro-hard/free :

Curl p · τ = 0, x ∈ ∂Ω\ΓD micro-free

p · τ = 0, x ∈ ΓD micro-hard

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spin micro-hard/free :

Curl p · τ = 0, x ∈ ∂Ω micro-free

[ skew p] · τ = 0, x ∈ ΓD spin micro-hard

pure micro-hard : p · τ = 0, x ∈ ∂Ω,

global insulation condition :∫

∂Ω

3∑i=1

⟨pi × (curl p)i,n

⟩dS = 0. (9.1)

We note that the global insulation condition is not additively stable, i.e. the differ-ence of two solutions p1−p2 which satisfy each individually the insulation conditionneed not satisfy the insulation condition. Thus the global insulation condition isnot a good candidate for establishing uniqueness.s

Here we follow closely the uniqueness proof given by Alber1 (p. 32), using thea priori energy estimate and the monotonicity for the difference of two solutions.We allow in this part the generality of a monotone flow function f instead of ∂χ.Assume that two strong solutions (u1, p2) and (u2, p2) of (8.2) exist (satisfying thesame boundary and initial conditions), notably

σ1 · n = σ2 · n = 0, x ∈ ∂Ω\ΓD,

u1 = u2 = ud, x ∈ ΓD.(9.2)

Insert the difference of the solutions into the total energy W , integrate over Ω andconsider the time derivative

ddt

∫Ω

W (∇(u1 − u2), p1 − p2, Curl(p1 − p2)) dV

=∫

Ω

⟨DW lin

e (∇(u1 − u2), p1 − p2),∇u1 −∇u2

⟩− ⟨DW lin

e (∇(u1 − u2), p1 − p2), p1 − p2

⟩+⟨DW lin

ph (p1 − p2), p1 − p2

⟩+⟨

DW lincurl(Curl(p1 − p2)), Curl

ddt

(p1 − p2)⟩

dV

=∫

Ω

〈σ(∇(u1 − u2), p1 − p2),∇u1 −∇u2〉

− 〈σ(∇(u1 − u2), p1 − p2), p1 − p2〉+ 〈2µ H0 dev sym(p1 − p2), p1 − p2〉

+⟨

µ L2c Curl(p1 − p2), Curl

ddt

(p1 − p2)⟩

dV

sIn the spirit of Gurtin10 the insulation condition is motivated by imposing boundary conditionsthat result in a “null expenditure of microscopic power”.

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= −∫

Ω

⟨Div σ(∇(u1 − u2), p1 − p2)︸︷︷︸

=0

, u1 − u2

⟩dV

+∫

∂Ω

〈σ(∇(u1 − u2), p1 − p2) · n, (u1 − u2)t〉︸︷︷︸=0 with (9.2)

dS

−〈σ(∇(u1 − u2), p1 − p2), p1 − p2〉 +∫

Ω

〈2µ H0 dev sym(p1 − p2), p1 − p2〉

+⟨

µ L2c Curl(p1 − p2), Curl

ddt

(p1 − p2)⟩

dV

= −0 + 0 +∫

Ω

〈2µ H0 dev sym(p1 − p2), p1 − p2〉

− 〈σ(∇(u1 − u2), p1 − p2), p1 − p2〉

+⟨

µ L2c Curl Curl(p1 − p2),

ddt

(p1 − p2)⟩

dV

=∫

Ω

⟨Σlin

2 − Σlin1 , p1 − p2

⟩dV

= −∫

Ω

⟨Σlin

2 − Σlin1 , p2 − p1

⟩dV

= −∫

Ω

⟨Σlin

2 − Σlin1 , f(Σlin

2 ) − f(Σlin1 )⟩

dV ≤ 0, (9.3)

due to the monotonicity of f. Hence, after integrating the last inequality in time wealso obtain the difference of two solutions∫

Ω

W (∇(u1 − u2)(t)), (p1 − p2)(t), Curl(p1 − p2)(t)) dV

≤∫

Ω

W (∇(u1 − u2)(0)), (p1 − p2)(0), Curl(p1 − p2)(0)) dV = 0. (9.4)

Thus we have ∫Ω

‖ sym(∇(u1 − u2)(t) − (p1 − p2)(t)‖2 dV = 0,∫Ω

‖ dev sym(p1 − p2)(t)‖2 dV = 0,∫Ω

‖Curl(p1 − p2)(t)‖2 dV = 0.

(9.5)

Since p1, p2 ∈ sl(3), it follows that sym(p1−p2) = 0 almost everywhere, i.e. p1−p2 ∈so(3). Moreover, from the micro-hard boundary condition p1 · τ = p2 · τ = 0 weobtain p1(x, t) ·τ = p2(x, t) ·τ = p(x, 0) ·τ which implies that (p1−p2) ·τ = 0 on ΓD

for two linear independent tangential directions τ . Since a skew-symmetric matrixA ∈ so(3) has either rank two or rank zero (in which case it is zero), we conclude

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that p1 − p2 = 0 on the Dirichlet-boundary ΓD due to the skew-symmetry of thedifference. However, Curl controls all first partial derivatives on skew-symmetricmatrices,34 i.e. it holds locally

∀A(x) ∈ so(3) : ‖Curl A(x)‖2 ≥ 12‖∇A(x)‖2, (9.6)

therefore p1 − p2 = 0 by Poincare’s inequality. Thus from Korn’s first inequality weobtain uniqueness also for the displacement u.

As a result, apart for the pure micro-free condition and the global insulationcondition all mentioned boundary conditions in (9.1) ensure uniqueness of classicalsolutions. In the case of the pure micro-free condition, the skew-symmetric partof the difference of two solutions remains indetermined up to a constant skew-symmetric matrix. The spin micro-hard condition has the advantage of not imposinga Dirichlet boundary condition on the symmetric plastic strain tensor sym p whichis also not present in the classical theory. Thus it is a candidate for the desiredweakest boundary condition.

9.2. Reformulation of the problem

The meaning of the flow rule is, recalling the definition of the subdifferential (6.8),

p ∈ ∂χ(Σlin) ⇔ χ(Σ) ≥ χ(Σlin) +⟨p, Σ − Σlin

⟩∀ Σ ∈ K. (9.7)

The Legendre transformation of χ is given, as is well known, by

R(ξ) = χ∗(ξ) =

σy ‖ξ‖ ξ ∈ sl(3)

∞ ξ ∈ sl(3),(9.8)

where R is also called the dissipation potential (notation: j in Ref. 12). Here, forrate-independent processes, R is convex and homogeneous of grade one. It holdsthat

sign(ξ) = ∂R(ξ)

= ∂χ∗(ξ) = [∂χ]−1(ξ) =

σy

ξ

‖ξ‖ ξ = 0, ξ ∈ sl(3)

ζ|‖ dev ζ‖ ≤ σy ξ = 0

∅ ξ ∈ sl(3),

(9.9)

and for a “gauge” g : X → R, g(x) ≥ 0, g(0) = 0 with g convex it holds12:ξ ∈ ∂g(x) ⇔ x ∈ ∂g∗(ξ). Hence, the flow rule can be formulated equivalently bywriting (note that Σlin is not necessarily deviatoric)

p ∈ ∂χ(Σlin) ⇔ Σlin ∈ [∂χ]−1(p) ⇔ Σlin ∈ ∂R(p)

⇔ R(q) ≥ R(p) +⟨Σlin, q − p

⟩ ∀ q ∈ L2([0, T ], sl(3)).(9.10)

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Integrating the former in space we note that∫Ω

R(q) dV ≥∫

Ω

R(p) +⟨Σlin, q − p

⟩dV (9.11)

is well defined for all q − p ∈ L2(0, T ; L1(Ω, sl(3)) (p as a solution is alreadysmoother), since we have that⟨

Σlin, q − p⟩

=⟨dev Σlin, q − p

⟩ ≤ σy‖q − p‖, (9.12)

from the flow rule. From (9.10)2 it is easy to still recover the pointwise constrainton the deviatoric stresses. Since q, p ∈ sl(3) it holds⟨

dev Σlin, q − p⟩

=⟨Σlin, q − p

⟩ ≤ R(q) − R(p) ≤ ‖R(q) − R(p)‖ ≤ σy‖q − p‖.(9.13)

Choosing q ∈ sl(3) such that q − p = dev Σlin shows ‖ dev Σlin‖ ≤ σy . We haveincluded this reverse calculation because this pointwise stress bound will be lostin our following weak reformulation due to the Curl-terms. This shows that therelation between the strong and weak formulation deserves detailed attention.

9.3. Strengthening of the boundary conditions for the plastic

distortion

In order for us to be able to show existence and uniqueness of a subsequent weakformulation, we need to strengthen the boundary and initial conditions. More pre-cisely, we assume p(x, 0) · τ = 0, x ∈ ΓD and, moreover, for all t ∈ (0, T )

sym p(x, t) · τ = 0, x ∈ ΓD,

skew p(x, t) · τ = 0, x ∈ ΓD ⇒ skew p(x, t) = 0, (9.14)

[ Curl p(x, t)] · τ = 0 [Curl skew p(x, t)] · τ = 0, x ∈ ∂Ω.

Remark 9.1. (Boundary conditions on the plastic distortion) The boundary con-ditions imposed by the insulation condition in the geometrically exact case are notnecessarily the same as those imposed by the need to show uniqueness/existence inthe geometrically linear setting. We could as well neglect the condition Fp(x, t) · τ =Fp(x, 0) · τ for x ∈ ΓD and assume then CurlFp · τ = 0 on ∂Ω. In this case, linearizeduniqueness will be lost.

9.4. The weak formulation of the problem

Let us re-formulate problem (8.2) with (9.11) in a weak sense. We set V =H1

0 (Ω, ΓD, R3), the space of displacements with incorporated Dirichlet boundaryconditions at ΓD ⊂ ∂Ω. Assume that solutions (u, p) to (8.2) exist with the regu-larity assumed in (8.1) so that all terms in (8.2) concerning the plastic variable p

have a meaning in the strong sense, notably Curl p · τ can be defined.

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Multiply the first equation in (8.2) with test functions (v − u) ∈L2([0, T ], H1

0 (Ω, ΓD, R3)), integrate in space and use the boundary condition (v −u)|ΓD

= 0 and the traction free condition on ∂Ω\ΓD in the sense of traces to obtain

0 =∫

Ω

〈σ(x, t),∇v −∇u〉 − 〈f(x, t), v − u〉 dV ∀ v − u ∈ L2([0, T ], V ),

σ = 2µ sym(∇u − p) + λ tr [∇u − p]1. (9.15)

Take (9.11) with

q = q ∈ H := H1([0, T ]; C1(Ω; sl(3), p|ΓD· τ = 0)), (9.16)

to obtain

0 ≤∫

Ω

R(q) − R(p) − ⟨Σlin, q − p⟩

dV ∀ q ∈ H, (9.17)

insert the constitutive relation (8.2)2 for Σlin and perform the (possible) partialintegration on the Curl Curl-term using the boundary conditions of (8.2) for p, i.e.Curl p · τ = 0 on ∂Ω\ΓD and p · τ = 0 on ΓD (for this step the satisfaction of theinsulation condition suffices) to get

0 ≤∫

Ω

R(q) − R(p) − 〈σ − 2µH0 dev sym p, q − p〉,

+ µ L2c〈Curl p, Curl[q − p]〉 dV ∀ q ∈ H. (9.18)

Adding (9.15) and (9.18) we obtain the variational inequality: for almost all t ∈(0, T )∫

Ω

〈f(x, t), v − u〉dV ≤∫

Ω

〈σ(x, t),∇v −∇u〉 dV

+∫

Ω

R(q) − R(p) − 〈σ − 2µ H0 dev sym p, q − p〉

+ µ L2c〈Curl p, Curl[q − p]〉dV ∀ (v, q) ∈ H1([0, T ]; V ) × H.

(9.19)

On the space Z := V × H we may define a bilinear form a : Z × Z → R forw = (u1, p1), z = (u2, p2)

a(w, z) =∫

Ω

2µ 〈sym(∇u1 − p1), sym(∇u2 − p2)〉 + λ tr [∇u1 − p1] tr [∇u2 − p2]

+ 2µ H0 〈dev sym p1, dev sym p2〉 + µ L2c 〈Curl p1, Curl p2〉dV

=∫

Ω

2µ 〈sym(∇u1 − p1), sym∇u2〉 + λ tr [∇u1 − p1] tr [∇u2] − 〈σ, p2〉

+ 2µ H0 〈dev sym p1, dev sym p2〉 + µ L2c 〈Curl p1, Curl p2〉dV. (9.20)

Let us also abbreviate j(p) :=∫Ω

R(p) dV =∫Ω

χ∗(p) dV. Thus any strong solution(u, p) of (8.2) having the regularity (8.1) will satisfy the preliminary weak primalproblem of our gradient-plasticity formulation, defined now.

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Definition 9.2. (weak primal problem of gradient plasticity I) Set

W := H10 (Ω, ΓD, R3)) × sym p ∈ L2(Ω, sl(3)), Curl p ∈ L2(Ω), p|ΓD

· τ = 0).

We call

w(t) = (u(t), p(t)) ∈ H1([0, T ]; W ) (9.21)

weak solution of the infinitesimal gradient plasticity model (8.2), if for almost allt ∈ (0, T ) it holds∫

Ω

〈f(x, t), v − u〉dV ≤ a

((u

p

),

(v − u

q − p

))+ j(q) − j(p) ∀ (v, q) ∈ W.

Let us first see in what respect solutions in the sense of Definition 9.2 are relatedto the original strong formulation and what other type of relations they satisfy.

Corollary 9.3. Assume that a smooth solution (u, p) in the sense of Definition 9.2satisfies additionally the insulation condition∫

Ω

3∑i=1

Div(

ddt

pi × (curl p)i

)dV =

∫∂Ω

3∑i=1

⟨ddt


dS ≡ 0. (9.22)

Then (u, p) is a strong solution of (8.2) in the interior of the domain. If the boundaryconditions of (8.2) are satisfied then this solution is a strong solution up to theboundary.

Corollary 9.4. Assume that (u, p) satisfy Definition 9.2. Then the weak form ofbalance of forces and the global energy inequality are satisfied.

Proof. Assume that w = (u, p) is a solution of Definition 9.2. Test the variationalinequality 9.2 with both (v, q) = (ξ + u, p) and (v, q) = (−ξ + u, p). This shows∫

Ω

〈f(x, t), ξ〉dV ≤ a(w, (ξ, 0)),∫

Ω

〈f(x, t),−ξ〉dV ≤ a(w, (−ξ, 0)). (9.23)

Hence from bilinearity, forall ξ ∈ H1([0, T ]; V )∫Ω

〈f(x, t), ξ〉 dV = a(w, (ξ, 0))

=∫

Ω

2µ〈sym(∇u − p), sym(∇ξ)〉 + λtr [∇u − p]tr [∇ξ] dV

=∫

Ω

〈σ,∇ξ〉 dV, (9.24)

i.e. the weak form of balance of forces.

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Assume again that w = (u, p) is a solution of Definition 9.2 and test the vari-ational inequality 9.2 with both (v, q) = (ξ + u, p) and (v, q) = (ξ + u,−p). Thisshows ∫

Ω

〈f(x, t), ξ〉 dV ≤ a(w, (ξ, 0)), no new information∫Ω

〈f(x, t), ξ〉 dV ≤ a(w, (ξ,−2p)). (9.25)

Hence we obtain∫Ω

〈f(x, t), ξ〉 dV ≤ a(w, (ξ, p))

=∫

Ω

2µ〈sym(∇u − p), sym∇ξ〉 + λtr [∇u − p]tr [∇ξ] − 〈σ, p〉

+ 2µH0 〈dev sym p, dev sym p〉 + µ L2c 〈Curl p, Curl p〉dV ,

(9.26)

and further∫Ω

〈f(x, t), ξ〉dV ≤ a(w, (ξ,−2p))

=∫

Ω

2µ 〈sym(∇u − p), sym∇ξ〉 + λ tr [∇u − p] tr [∇ξ] + 2〈σ, p〉

− 4µ H0 〈dev sym p, dev sym p〉 − 2µ L2c 〈Curl p, Curl p〉dV

(9.27)

which shows, using balance of forces

0 ≤∫

Ω

〈σ, p〉 − 2µ H0 〈dev sym p, dev sym p〉 − µ L2c 〈Curl p, Curl p〉dV,

0 ≥∫

Ω

〈σ,−p〉 + 2µ H0 〈dev sym p, dev sym p〉 + µ L2c 〈Curl p, Curl p〉dV

⇒ 0 ≥∫

Ω

ddt

W (∇u(x, t0), p(t), Curl p(t)) dV. (9.28)

The last inequality is sufficient for the global a priori energy inequality. Thus weget the global a priori energy inequality back although we might not be able tospeak about boundary values for Curlp · τ = 0.

9.5. Existence of weak solutions

The approach towards defining weak solutions taken in Definition 9.2 seems to benatural at first sight. However, while the set p ∈ C1(Ω; sl(3)), p|ΓD

· τ = 0 canbe closed to a Hilbert-space under the norm

‖p‖2curl,sym := ‖ sym p‖2

L2(Ω) + ‖Curl p‖2L2(Ω), (9.29)

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the ensuing Hilbert space might not be continuously embedded in L2(Ω) and thecontinuous definition of traces giving p|ΓD

· τ = 0 a precise meaning seems impos-sible.t

Therefore we propose to modify the weak formulation by adding the term‖Curl skew p‖2 in the energy to obtain a modified bilinear form a : H(Ω)×H(Ω) →R for w = (u1, p1), z = (u2, p2) (the space H(Ω) is defined in (9.49))

a(w, z) =∫

Ω

2µ 〈sym(∇u1 − p1), sym(∇u2 − p2)〉 + λ tr [∇u1 − p1] tr [∇u2 − p2]

+ 2µ H0 〈dev sym p1, dev sym p2〉+ µ L2

c (〈Curl p1, Curl p2〉 + 〈Curl skew p1, Curl skew p2〉) dV (9.30)

together with the stronger boundary conditions (9.14) will satisfy the modified weakprimal problem of our gradient-plasticity formulation:

Definition 9.5. (weak primal problem of gradient plasticity II) Set

W := H10 (Ω, ΓD, R3)) ×H.

We call

w(t) = (u(t), p(t)) ∈ H1([0, T ]; W ), p · τ = skew p · τ = 0 on ΓD, (9.31)

weak solution of the modified infinitesimal gradient plasticity model, if for almostall t ∈ (0, T ) it holds∫

Ω

〈f(x, t), v − u〉dV ≤ a

((u

p

),

(v − u

q − p

))+ j(q) − j(p) ∀ (v, q) ∈ H(Ω),

where the Hilbert-space H is defined in Sec. (9.6).

In the following we denote the linear functional of applied loads by and abbre-viate (w) =

∫Ω〈f(x, t), w〉 dV. In order to prove the existence of weak solutions

in the modified sense of Definition 9.5, we use the following abstract result forvariational inequalities.

Theorem 9.6. (Existence and uniqueness for mixed variational inequality) LetH(Ω) be a Hilbert space and W ⊂ H(Ω) be a non-empty closed convex cone. Con-sider the problem of finding w ∈ W such that w(t) ∈ W almost everywhere and

∀ z ∈ W : a(w, z − w) + j(z) − j(w) − (z − w) ≥ 0. (9.32)

Assume that the following hold :

(1) the bilinear form a(·, ·) is symmetric and continuous on H(Ω) and coercive onW, i.e.

a(w, z) ≤ C+‖w‖H(Ω) ‖z‖H(Ω) ∀w, z ∈ H(Ω),

a(z, z) ≥ C+‖z‖2H(Ω) ∀ z ∈ W, (9.33)

tSince p is not symmetric, this is not the space Hcurl which would allow to specify tangentialtraces for p.

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(2) the linear form of external loads is bounded, i.e. ‖ (z)‖ ≤ C+ ‖z‖H(Ω).(3) the functional j is non-negative, convex, positively homogeneous of grade one

and Lipschitz-continuous, i.e.

∀ s ∈ R : j(sw) = |s| j(w), |j(z) − j(w)| ≤ L‖z − w‖H(Ω). (9.34)

Then the problem (9.32) has a unique solution w ∈ H1([0, T ]; W ).

Proof. The proof is given in Theorem 7.3 (p. 166) in Han/Reddy.12

9.6. The Hilbert-space H

To show that the bilinear form a satisfies the assumptions of this theorem, we needto introduce a new Hilbert-space H, which we define and investigate first. Definethe set

X := p ∈ C∞(Ω, M3×3)| skew p(x) · τ = 0, x ∈ ΓD, (9.35)

where ∂Ω is assumed to have C1-boundary. Thus we are led to define aquadratic form

‖p‖2 := ‖ sym p‖2

L2(Ω) + ‖Curl p‖2L2(Ω) + ‖Curl skew p‖2

L2(Ω). (9.36)

Let us show that this defines a norm on X.

Lemma 9.7. ‖p‖ is a norm on X.

Proof. Positivity, homogeneity and the triangle inequality are clear. We need onlyto show that

p ∈ X, ‖p‖ = 0 ⇒ p ≡ 0. (9.37)

Since p is smooth we have

‖p‖ = 0 ⇒ ‖ sym p‖L2(Ω) = 0, and ‖Curl p‖L2(Ω) = 0,

⇒ sym p(x) = 0 ⇒ p(x) = A(x) ∈ so(3), Curl p(x) = Curl A(x) = 0. (9.38)

However, Curl is isomorphic to ∇ on C∞(Ω, so(3)),34 thus the former implies thatA(x) = const. Moreover

[ skew p(x)] · τ1(x) = [skew p(x)] · τ2(x) = 0

⇒ A · τ1 = A · τ2 = 0 ⇒ rank(A) ≤ 1. (9.39)

Since a skew-symmetric matrix A in R3 has either rank two or rank zero (in which

case it is zero) we conclude that A ≡ 0. Thus p = A = 0.

Finally, we take the completion of the encompassing space X with respect tothe norm ‖ · ‖ and define

H := X ‖·‖

, (9.40)

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making H into a Hilbert-space. Thus we have shown that H(Ω, M3×3) is a Hilbert-space with scalar product

〈p1, p2〉 =∫

Ω

〈sym p1, sym p2〉 + 〈Curl p1, Curl p2〉

+ 〈Curl skew p1, Curl skew p2〉dV. (9.41)

Corollary 9.8. ‖p‖2curl,sym is also a norm on X.

The space H is continuously embedded in L2(Ω, M3×3).

Lemma 9.9. (Continuous embedding of H in L2(Ω)) The space H is continuouslyembedded in L2(Ω, M3×3), i.e.

∃C+ ∀ ζ ∈ H : ‖ζ‖L2(Ω) ≤ C+ ‖ζ‖. (9.42)

Proof. We show the inequality for functions in X and pass then to the limit bydensity. Decomposing ζ = sym ζ +skew ζ and using that (pointwise and integrated)

‖ζ‖2 = ‖ sym ζ‖2 + ‖ skew ζ‖2 ⇒ ‖ζ‖2L2(Ω) ≤ ‖ζ‖2

+ ‖ skew ζ‖2L2(Ω), (9.43)

it remains to bound ‖ skew ζ‖2L2(Ω) in terms of ‖ζ‖2

. However,

‖ skew ζ‖2L2(Ω) ≤ ‖ skew ζ‖2

H1(Ω) ≤ C+Poin(Ω, ΓD)‖∇ skew ζ‖L2(Ω)

≤ 2C+Poin(Ω, ΓD) ‖Curl skew ζ‖2

L2(Ω) ≤ 2 C+Poin(Ω, ΓD) ‖ζ‖2

,

(9.44)

where we have used in the first line Poincare’s-inequality for functions ζ ∈ X thatsatisfy automatically skew[ζ]|ΓD

= 0 and in the second line that Curl bounds ∇(pointwise) on so(3). The constant C+

Poin(Ω, ΓD) is the Poincare constant.

Lemma 9.10. The linear space H1(Ω, M3×3)∩skew p(x) = 0, x ∈ ΓD (boundaryterm in the sense of traces) is contained in H.

Proof. Note first that C∞(Ω, M3×3) ∩ skew p(x) = 0, x ∈ ΓD ⊂ X and that

∀ p ∈ X : ‖p‖ ≤ C+‖p‖H1(Ω). (9.45)

Choose φ ∈ H1(Ω, M3×3) ∩ skew p(x) = 0, x ∈ ΓD. Then there is a sequenceφn ∈ C∞(Ω, M3×3) ∩ skew p(x) = 0, x ∈ ΓD such that φn → φ strongly in H1.Since

‖φ − φn‖ ≤ ‖φ − φn‖H1(Ω), (9.46)

this implies ‖φ − φn‖ → 0. Hence φ is contained in the closure of X w.r.t. thenorm ‖ · ‖.

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Remark 9.11. The new space H is not compactly embedded in L2(Ω, M3×3). Tosee this consider a sequence pk = ∇ζk, bounded in H. Because of the gradientstructure it holds that Curl pk ≡ 0 but pk = ∇ζk need not converge.

Lemma 9.12. (H and boundary values) The space H admits continuous tracesin the sense that for tangential vectors τ the following terms can be defined on ∂Ω

p(x, t) · τ ∈ L2(∂Ω), skew p(x, t) · τ ∈ L2(∂Ω). (9.47)

Proof. Since sym p ∈ L2(Ω) and Curl sym p ∈ L2(Ω) the tangential trace sym p · τcan be defined. Similarly, for skew p ∈ L2(Ω) and Curl skew p ∈ L2(Ω) the tangentialtrace skew p · τ can be defined.

Therefore, in this space, the boundary condition p · τ = 0 on ΓD is well definedin the sense of traces. Moreover, we have the characterization

H = sym p ∈ Hcurl(Ω), skew p ∈ H10 (Ω, ΓD, so(3)), (9.48)

which is easily seen by calculating ‖Curl sym p‖2 = ‖Curl(p−skew p)‖2 = ‖Curl p−Curl skew p)‖2.

Having presented our Hilbert-space we are now ready to announce our finalresult.

Theorem 9.13. (Existence and uniqueness of weak solutions) The mixed varia-tional inequality (9.5) together with the strengthened boundary condition (9.14) hasa unique weak solution in Z = H1(0, T ; H(Ω)) with

H(Ω) := H10 (Ω, ΓD, R3) ×H. (9.49)

Proof. In order to apply the abstract framework of Theorem 9.32, we define theclosed, convex subset of the Hilbert-space H

K = p ∈ H, tr [p] = 0, p(x, t) · τ = 0, skew p(x, t) · τ = 0, x ∈ ΓD,and define

H(Ω) := H10 (Ω, ΓD, R3) ×H, W := H1

0 (Ω, ΓD, R3) × K, Z := H1([0, T ]; H(Ω)).

It is easy to see that the bilinear form a, defined in (9.20) is symmetric andcontinuous on H(Ω). Moreover, for z = (u, p) ∈ W we have

a(z, z) =∫

Ω

2µ 〈sym(∇u − p), sym(∇u − p)〉 + λ tr [∇u − p] tr [∇u − p]

+ 2µ H0 〈dev sym p, dev sym p〉+ µ L2

c (〈Curl p, Curl p〉 + 〈Curl skew p, Curl skew p〉) dV

≥∫

Ω

2µ ‖ sym(∇u − p)‖2 + 2µ H0 ‖ dev sym p‖2

+ µ L2c

(‖Curlp‖2 + ‖Curl skew p‖2)

dV

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=∫

Ω

2µ (‖ sym∇u‖2 − 2〈sym∇u, sym p〉 + ‖ sym p‖2)

+ 2µ H0 ‖ dev sym p‖2 + µ L2c

(‖Curl p‖2 + ‖Curl skew p‖2)

dV

≥∫

Ω

2µ(‖ sym∇u‖2 − 2 ‖ sym∇u‖ ‖ symp‖ + ‖ symp‖2

)+ 2µ H0 ‖ dev sym p‖2 + µ L2

c


dV.

(9.50)

Cauchy–Schwarz and Young’s inequalities then imply

a(z, z) ≥∫

Ω

2µ

(‖ sym∇u‖2 − ε ‖ sym∇u‖2 − 1

ε‖ sym p‖2 + ‖ sym p‖2

)+ 2µ H0 ‖ dev sym p‖2 + µ L2

c


dV

=∫

Ω

2µ (1 − ε)‖ sym∇u‖2 + 2µ (1 + H0 − 1ε

) ‖ dev sym p‖2

+ µ L2c

(‖Curlp‖2 + ‖Curl skew p‖2)

dV. (9.51)

Choose ε = (1 + H02 )−1 to get

a(z, z) ≥∫

Ω

µ H0 ‖ sym∇u‖2 + µ H0 ‖ dev sym p‖2

+ µ L2c


dV . (9.52)

Since p ∈ sl(3) it follows

a(z, z) =∫

Ω

µ H0 ‖ sym∇u‖2 + µ H0 ‖ sym p‖2

+ µ L2c (‖Curl p‖2 + ‖Curl skew p‖2) dV

≥ µ(H0 ‖ sym∇u‖2L2(Ω) + min(H0, L

2c) ‖p‖2

). (9.53)

Using Korn’s inequality for the displacement u ∈ H10 (Ω, ΓD, R3) gives finally

a(z, z) ≥ µ(H0 CK ‖u‖2H1(Ω) + min(H0, L

2c) ‖p‖2

).

Thus a is positive definite on W . Moreover, j(q) =∫Ω

R(q) dV is one homogeneousand Lipschitz, due to (9.8). We have shown that within these spaces the problem(9.5) admits a unique weak solution.

10. Discussion

The classical elasto-perfectly plastic Prandtl–Reuss model with kinematic harden-ing has been extended to include a weak nonlocal interaction of the plastic distortionby introducing the dislocation density in the Helmholtz free energy. The evolutionequation for plasticity follows by applying the second law of thermodynamics inthe efficient formulation proposed by Maugin21 together with a sufficient condition

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guaranteeing the insulation condition. This is done in a finite-strain setting basedon the multiplicative decomposition. We apply a strict principle of referential, inter-mediate and spatial covariance. Referential covariance severely restricts the choiceof the hardening contribution. However, full covariance w.r.t. constant rotationsdoes not reduce the gradient plasticity model to a dependence on the plastic metricCp = FT

p Fp, in contrast to the classical case without gradients on the plastic dis-tortion. In this context the gradient plasticity model allows to distinguish in moredetail between form-invariance under all rotations (covariance for local model) andform-invariance under rotations constant over the body (covariance for non-localmodel). The question of the correct invariance requirements to be satisfied in themultiplicative decomposition has been raised many times in the literature. Ourdevelopment shows that this question has a new answer in the non-local setting:for full covariance with respect to rigid rotations of the reference, intermediate andspatial configuration in the multiplicative decomposition the model should satisfythe condition (5.11).

A geometrically linear version of the model is also derived. The proposed gra-dient plasticity model approximates formally the classical model in the large scalelimit Lc = 0. For the rate-independent infinitesimal strain model the following hasbeen obtained: uniqueness of strong solutions with micro-free/hard boundary condi-tions and that a weak formulation can be recast into a mixed variational inequalitywhose well-posedness can be shown in a suitable Hilbert-space.

It remains to see whether this formulation of gradient plasticity is not onlyweakly well-posed but also amenable to an efficient numerical implementation. Theintroduction of a plastic length scale Lc > 0, which acts as a localization limitershould be compared to the Cosserat elasto-plasticity model29–33 which also hasthe ability to remove the mesh-sensitivity. For the Cosserat model it is alreadyshown that this can be implemented in an efficient way. Currently, the dislocationbased plasticity model is being implemented, however, only for the irrotational casewithout plastic spin.35 There, boundary conditions on the symmetric plastic strainεp need not be imposed.

Notation

Let Ω ⊂ R3 be a bounded domain with Lipschitz boundary ∂Ω and let Γ be a

smooth subset of ∂Ω with nonvanishing two-dimensional Hausdorff measure. Wedenote by M

3×3 the set of real 3 × 3 second-order tensors, written with capitalletters. The standard Euclidean scalar product on M

3×3 is given by 〈X, Y 〉M3×3 =

tr[XY T

], and thus the Frobenius tensor norm is ‖X‖2 = 〈X, X〉

M3×3 (we usethese symbols indifferently for tensors and vectors). The identity tensor on M

3×3

will be denoted by 1, so that tr [X ] = 〈X,1〉. We let Sym and PSym denote thesymmetric and positive definite symmetric tensors respectively. We adopt the usualabbreviations of Lie-algebra theory, i.e. so(3) := X ∈ M

3×3 |XT = −X are skewsymmetric second-order tensors and sl(3) := X ∈ M

3×3 |tr [X ] = 0 are traceless

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tensors. We set sym(X) = 12 (XT + X) and skew(X) = 1

2 (X − XT) such thatX = sym(X) + skew(X). For X ∈ M

3×3 we set for the deviatoric part dev X =X − 1

3 tr [X ]1 ∈ sl(3). For a second-order tensor X we let X · ei be the applicationof the tensor X to the column vector ei. The curl of a three by three matrix isdefined to be the vector curl applied on the i.th row, written in the ith row, i.e.

curl

p11 p12 p13

p21 p22 p23

p31 p32 p33

=

curl[p11, p12, p13]

curl[p21, p22, p23]

curl[p31, p32, p33]

.

Acknowledgments

P. Neff is grateful for stimulating discussions with B. Svendsen and B. D. Reddyon models of gradient plasticity and to C. Miehe, A. Bertram and A. Mielke fordiscussions about invariance requirements on the plastic variable. We are grateful toB. D. Reddy for sharing with us the paper36 prior to publication and to K. Hutterfor pointing out Ref. 27. Last but not least we thank two unknown reviewers fortheir detailed remarks which helped to improve this paper. K. Chelminski wassupported by polish government grant: KBN no. 1-P03A-031-27.

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NOTES ON STRAIN GRADIENT PLASTICITY: FINITE STRAIN...

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