Accepted Manuscript. Well-posedness for the microcurl model in both single and polycrystal gradient plasticity

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1 Accepted Manuscript Well-posedness for the microcurl model in both single and polycrystal gradient plasticity François Ebobisse, Patrizio Neff, Samuel Forest PII: S (16) DOI: /j.ijplas Reference: INTPLA 2146 To appear in: International Journal of Plasticity Received Date: 13 October 2016 Accepted Date: 22 January 2017 Please cite this article as: Ebobisse, F., Neff, P., Forest, S., Well-posedness for the microcurl model in both single and polycrystal gradient plasticity, International Journal of Plasticity (2017), doi: / j.ijplas This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

2 Well-posedness for the microcurl model in both single and polycrystal gradient plasticity François Ebobisse 1 and Patrizio Neff 2 and Samuel Forest 3 February 10, 2017 Abstract We consider the recently introduced microcurl model which is a variant of strain gradient plasticity in which the curl of the plastic distortion is coupled to an additional micromorphictype field. For both single crystal and polycrystal cases, we formulate the model and show its well-posedness in the rate-independent case provided some local hardening (isotropic or linear kinematic) is taken into account. To this end, we use the functional analytical framework developed by Han-Reddy. We also compare the model to the relaxed micromorphic model as well as to a dislocation-based gradient plasticity model. Key words: plasticity, gradient plasticity, variational modeling, dissipation function, micromorphic continuum, defect energy, micro-dislocation. AMS 2010 subject classification: 35D30, 35D35, 74C05, 74C15, 74D10, 35J25. 1 Corresponding author, François Ebobisse, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7700, South Africa, francois.ebobissebille@uct.ac.za 2 Patrizio Neff, Lehrstuhl für Nichtlineare Analysis und Modellierung, Fakultät für Mathematik, Universität Duisburg-Essen, Thea-Leymann Str. 9, Essen, Germany, patrizio.neff@uni-due.de, neff 3 Samuel Forest, MINES ParisTech, Centre des Matériaux, UMR CNRS 7633, BP 87, Evry, France, samuel.forest@ensmp.fr 1

3 Well-posedness for the microcurl model 2 Contents 1 Introduction 2 2 Some notational agreements and definitions 4 3 The microcurl model in single crystal gradient plasticity Kinematics The case with isotropic hardening The balance equations The derivation of the dissipation inequality The flow rule Strong formulation of the model Weak formulation of the model Existence result for the weak formulation Uniqueness of the weak/strong solution The model with linear kinematical hardening The description of the model The weak formulation of the model with linear kinematical hardening Existence and uniqueness for the model with linear kinematic hardening The microcurl model in polycrystalline gradient plasticity The case with isotropic hardening The balance equations The derivation of the dissipation inequality The flow rule Strong formulation of the model Weak formulation of the model Existence result for the weak formulation The model with linear kinematical hardening The relaxed linear micromorphic continuum 31 6 Conclusion Introduction In this paper we consider the so-called microcurl model in plasticity. The model was introduced in 32, 17] to serve the purpose of augmenting classical plasticity with length scale effects while otherwise keeping the algorithmic structure of classical plasticity. We recall that several experimental tests on micron-sized metallic structutres have shown the dependence of the mechanical response on the specimen size (see e.g. 26, 28, 91]). Classtical theories of plasticity are unable to account for those size effects because they do not involve any material length scale. Various models of strain gradient plasticity have been introduced and developped in the past thirty years with the purpose to accommodate the experimentally observed size effects in material behaviour in small scales. The size effects are captured through the introduction of plastic strain gradient

4 Well-posedness for the microcurl model 3 (or its higher derivatives) and sometimes together with higher order stress tensors in the constitutive equations (see e.g. 1, 2, 27, 4, 41, 39, 42, 29, 30, 86] and also the recent contributions in 54, 95, 96]). Models of strain gradient plasticity lead to the flow law involving nonlocal terms. The idea in the microcurl model is simple and straight-forward: the in general non-symmetric plastic distortion p (single crystal plasticity and polycrystal plasticity with plastic spin) with its local in space evolution is energetically coupled to a micromorphic-type additional nonsymmetric tensorial variable χ p via a penalty-like term 1 2 µ H χ p χ p 2. The tensor field χ p is generally assumed to be incompatible i.e., it may not derive from a vector field. The total energy in the model is then augmented by a quadratic contribution acting on the Curl of χ p. The new variable χ p is now determined by free-variation of the energy w.r.t. the displacement field u and the micromorphic field χ p together with corresponding tangential boundary conditions for χ p. This generates the usual equilibrium equation on the one hand and what we will call a micro-balance equation for χ p on the other hand. In the penalty limit H χ, when one expects p = χ p, the variable Curl χ p is then interpreted to be the dislocation density tensor Curl p. The advantage of such a formulation is clear: there is no need for an extended thermodynamic setting, since χ p is not directly taking part in the dissipation. Constitutive laws including dissipative contributions of the microdeformation and microcurl can be proposed, as done in 35] in the general micromorphic case, but they will require additional material parameters whose identification necessitates material specific physical considerations. Thus, also no higher order boundary conditions at interfaces between elastic and plastic parts need be discussed. The resulting model can therefore be described as a pseudo-regularized strain gradient model. The microdeformation variable χ p has at least two different interpretations. First, it can be regarded as a mathematical auxiliary variable used to replace the higher order partial differential equations arising in strain gradient plasticity by a system of two sets of second order partial differential equations for the displacement and microdeformation. This method has computational merits for the implementation of strain gradient plasticity models in finite element codes, see 6]. In that case, H χ is regarded as a mere penalty parameter and should be large enough to enforce the constraint p = χ p. In contrast, the microdeformation variable χ p can also be viewed as a constitutive variable with physical interpretation, for instance based on statistical mechanics, p representing the average plastic distortion over the material volume element and χ p being related to the variance of plastic deformation inside this volume. This interpretation is similar to the microconcentration variable introduced in 94, 34] to solve Cahn Hilliard equations. In this context, H χ must be regarded as a true material higher order modulus to be identified from suitable experimental results. Compared to standard strain gradient plasticity, the microcurl model therefore possesses one additional parameter, H χ, which allows for better description of physical results, as suggested in 18]. An interpretation of the microdeformation χ p was recently proposed in the case of polycrystalline plasticity and damage in 84, 85] where it is related to the grain to grain heterogeneitiy of plastic deformation. Another computational advantage of the microcurl single crystal plasticity model is that the number of independent degrees of freedom (9 tensor components of χ p, or 8 in the case of incompressible microplasticity) is independent of the crystallographic structure of the material and of the number of slip systems. This is in contrast to strain gradient plasticity models involving the directional gradient of the slip variables 40], which require as many degrees of freedom as

5 Well-posedness for the microcurl model 4 slip systems (12 at least in FCC crystals, up to 48 in BCC crystals!). A comparison and discussion of models based on the full dislocation density tensors with models involving densities of geometrically necessary dislocations can be found in 60]. In this paper we will consider the microcurl model in two variants. First, in its original form as a computational approach towards single crystal strain gradient plasticity. We formulate the governing system and show its well-posedness in the rate-independent case. The natural solution space for the micro-variable χ p is the Sobolev-like space H(Curl). Second, we extend the approach formally to polycrystalline plasticity in which the plastic variable ε p := sym p (the plastic strain) is assumed to be symmetric. In this case we still allow for a non-symmetric micro-variable χ p which is now coupled to the plastic variable only via its symmetric part by 1 2 µ H χ sym(p χ p ) 2. This represents an alternative to recently proposed strain gradient plasticity models involving a plastic spin tensor for polycrystals 41, 9, 85]. Again, we show the well-posedness of the formulation. Here, we need recently introduced coercive inequalities generalizing Korn s inequality to incompatible tensor fields 74, 75, 76, 77]. The mathematical analysis (with results of existence and uniqueness) for both variants (single crystal and polycrystalline) is obtained through the machinery developped by Han-Reddy 49, Theorems 6.15 and 6.19] for classical plasticity and recently extended to models of gradient plasticity in 19, 70, 23, 24, 21]. In this approach, the model through the primal form of the flow rule is weakly formulated as a variational inequality and the key issue for its well-posedness is the study of the coercivity of the bilinear form involved on a suitable closed convex subset of some Hilbert space. The polycrystalline variant of the microcurl model bears some superficial resemblence with the recently introduced relaxed micromorphic models 71, 72]. For purpose of clarification, we present the relaxed micromorphic model and clearly point out the differences. In order to put the microcurl modelling framework further on display we finish this introduction with another dislocation based strain gradient plasticity model with plastic spin 23, 24, 21], see Table 1. Here, the microcurl-type regularization would be obtained by considering the microcurl energy 1 2 C iso ε e, ε e µ H χ p χ p µ k 1 sym p µ L2 c Curl χ p 2 (1.1) and for H χ we would recover the model from Table 1. The polycrystalline microcurl variant which we introduce in this paper is, however, based on the energy 1 2 C iso ε e, ε e µ H χ sym(p χ p ) µ k 1 sym p µ L2 c Curl χ p 2. (1.2) This ansatz seems to be appropriate for polycrystalline plasticity without plastic spin. 2. Some notational agreements and definitions Let be a bounded domain in R 3 with Lipschitz continuous boundary, which is occupied by the elastoplastic body in its undeformed configuration. Let Γ D be a smooth subset of with non-vanishing 2-dimensional Hausdorff measure. A material point in is denoted by x and the time domain under consideration is the interval 0, T ].

6 Well-posedness for the microcurl model 5 Additive split of distortion: u = e + p, ε e := sym e, ε p := sym p Equilibrium: Div σ + f = 0 with σ = C isoε e 1 Free energy: 2 Cisoεe, ε e + 1 µ 2 k1 εp µ 2 L2 c Curl p 2 Yield condition: φ(σ E) := dev Σ E σ 0 0 where Σ E := σ + Σ lin curl + Σlin kin, Σ lin curl := µ L2 c Curl Curl p, Σ lin kin := µ k1 εp Dissipation inequality: Σ E, ṗ dx 0 Dissipation function: D(q) := σ 0 q Flow rule in primal form: Σ E D(ṗ) dev ΣE Flow rule in dual form: ṗ = λ dev Σ, E λ = ṗ KKT conditions: λ 0, φ(σ E) 0, λ φ(σ E) = 0 Boundary conditions for p: p n = 0 on Γ D, (Curl p) n = 0 on \ Γ D Function space for p: p(t, ) H(Curl;, R 3 3 ) Table 1: The polycrystalline plasticity model model with linear kinematic hardening and plastic spin studied in 21]. For every a, b R 3, we let a, b R 3 denote the scalar product on R 3 with associated vector norm a 2 = a, a R 3 R 3. We denote by R 3 3 the set of real 3 3 tensors. The standard Euclidean scalar product on R 3 3 is given by A, B R 3 3 = tr AB T ], where B T denotes the transpose tensor of B. Thus, the Frobenius tensor norm is A 2 = A, A R 3 3. In the following we omit the subscripts R 3 and R 3 3. The identity tensor on R 3 3 will be denoted by 1, so that tr(a) = A, 1. The set so(3) := {X R 3 3 X T = X} is the Lie-Algebra of skew-symmetric tensors. We let Sym (3) := {X R 3 3 X T = X} denote the vector space of symmetric tensors and sl(3) := {X R 3 3 tr (X) = 0} be the Lie-Algebra of traceless tensors. For every X R 3 3, we set sym(x) = 2( 1 X + X T ), skew (X) = 1 2( X X T ) and dev(x) = X 1 3tr (X) 1 sl(3) for the symmetric part, the skew-symmetric part and the deviatoric part of X, respectively. Quantities which are constant in space will be denoted with an overbar, e.g., A so(3) for the function A : R 3 so(3) which is constant with constant value A. The body is assumed to undergo infinitesimal deformations. Its behaviour is governed by a set of equations and constitutive relations. Below is a list of variables and parameters used throughout the paper with their significations: u is the displacement of the macroscopic material points; p is the infinitesimal plastic distortion variable which is a non-symmetric second order tensor, incapable of sustaining volumetric changes; that is, p sl(3). The tensor p represents the average plastic slip; p is not a state-variable, while the rate ṗ is an infinitesimal state variable in some suitable sense; e = u p is the infinitesimal elastic distortion which is in general a non-symmetric second

7 Well-posedness for the microcurl model 6 order tensor and is an infinitesimal state-variable; ε p = sym p is the symmetric infinitesimal plastic strain tensor, which is trace free, ε p sl(3); ε p is not a state-variable; the rate ε p is an infinitesimal state-variable; ε e = sym u ε p is the symmetric infinitesimal elastic strain tensor and is an infinitesimal state-variable; χ p R 3 3 is the non-symmetric infinitesimal micro-distortion with sym χ p being the symmetric micro-strain; σ is the Cauchy stress tensor which is a symmetric second order tensor and is an infinitesimal state-variable; σ 0 is the initial yield stress for plastic variables p or ε p := sym p and is an infinitesimal state-variable; f is the body force; Curl p = α is the dislocation density tensor satisfying the so-called Bianchi identities Div α = 0 and is an infinitesimal state-variable; η p = t 0 ε p ds is the accumulated equivalent plastic strain and is an infinitesimal statevariable; γ α is the slip in the α-th slip system in single crystal plasticity while l α is the slip direction and ν α is the normal vector to the slip plane with α = 1,..., n slip. Hence, p = γ α l α ν α. α For isotropic media, the fourth order isotropic elasticity tensor C iso : Sym(3) Sym(3) is given by C iso sym X = 2µ dev sym X + κ tr(x)1 = 2µ sym X + λ tr(x)1 (2.1) for any second-order tensor X, where µ and λ are the Lamé moduli satisfying µ > 0 and 3λ + 2µ > 0, (2.2) and κ > 0 is the bulk modulus. These conditions suffice for pointwise positive definiteness of the elasticity tensor in the sense that there exists a constant m 0 > 0 such that X R 3 3 : sym X, C iso sym X m 0 sym X 2. (2.3) The space of square integrable functions is L 2 (), while the Sobolev spaces used in this paper are: H 1 () = {u L 2 () grad u L 2 ()}, grad =, u 2 H 1 () = u 2 L 2 () + grad u 2 L 2 (), u H1 (), H(curl; ) = {v L 2 () curl v L 2 ()}, curl =, (2.4) v 2 H(curl;) = v 2 L 2 () + curl v 2 L 2 (), v H(curl; ).

8 Well-posedness for the microcurl model 7 For every X C 1 (, R 3 3 ) with rows X 1, X 2, X 3, we use in this paper the definition of Curl X in 70, 92]: curl X 1 Curl X = curl X 2 R 3 3, (2.5) curl X 3 for which Curl v = 0 for every v C 2 (, R 3 ). Notice that the definition of Curl X above is such that (Curl X) T a = curl (X T a) for every a R 3 and this clearly corresponds to the transpose of the Curl of a tensor as defined in 42, 44]. The following function spaces and norms will also be used later. H(Curl;, R 3 3 ) = { X L 2 (, R 3 3 ) } Curl X L 2 (, R 3 3 ), X 2 H(Curl;) = X 2 L 2 () + Curl X 2 L 2 (), X H(Curl;, R3 3 ), (2.6) { H(Curl;, E) = X : E } X H(Curl;, R 3 3 ), for E := sl(3) or Sym (3) sl(3). We also consider the space H 0 (Curl;, Γ D, R 3 3 ) (2.7) as the completion in the norm in (2.6) of the space { X C (, R 3 3 ) X n ΓD = 0 }. Therefore, this space generalizes the tangential Dirichlet boundary condition X n ΓD = 0 to be satisfied by the plastic micro-distortion χ p. The space is defined as in (2.6). H 0 (Curl;, Γ D, E) The divergence operator Div on second order tensor-valued functions is also defined row-wise as div X 1 Div X = div X 2. (2.8) div X 3 3. The microcurl model in single crystal gradient plasticity 3.1. Kinematics Single-crystal plasticity is based on the assumption that the plastic deformation happens through crystallograpic shearing which represents the dislocation motion along specific slip systems, each being characterized by a plane with unit normal ν α and slip direction l α on that plane, and slips

9 Well-posedness for the microcurl model 8 γ α (α = 1,..., n slip ). The flow rule for the plastic distortion p is written at the slip system level by means of the orientation tensor m α defined as Under these conditions the plastic distortion p takes the form so that the plastic strain ε p = sym p is and tr(p) = 0 since l α ν α. n slip m α := l α ν α. (3.1) n slip p = γ α m α (3.2) α=1 ε p = γ α sym(m α ) = 1 γ α (l α ν α + ν α l α ) (3.3) 2 α=1 For the slips γ α (α = 1,..., n slip ) we set Therefore, we get from (3.3) that where m is the third order tensor 1 defined as n slip α=1 γ := (γ 1,..., γ n slip ). p = m γ, (3.4) m ijα := m α ij = l α i ν α j for i, j = 1, 2, 3 and α = 1,..., n slip. (3.5) Let η := (η 1,..., η n slip) with η α being a hardening variable in the α-th slip system The case with isotropic hardening The starting point is the total energy E(u, γ, χ p, η) := Ψ( u, γ, χ p, Curl χ p, η) f, u ] dx (3.6) where the free-energy density Ψ is given in the additively separated form Ψ( u, γ, χ p, Curl χ p, η) : = Ψ lin e (ε e ) elastic energy + Ψ lin curl (Curl χ p ) defect-like energy (GND) + Ψ lin micro(p, χ p ) micro energy + Ψ iso (η) hardening energy (SSD) (3.7), 1 The terminology tensor used here is just intended as matrix since m does not fulfill the rules of change of orthogonal bases for the last index and therefore is not a tensor in the usual sense.

10 Well-posedness for the microcurl model 9 where Ψ lin e (ε e ) := 1 2 ε e, C iso ε e = 1 2 sym( u p), C iso sym( u p), Ψ lin micro(p, χ p ) := 1 2 µ H χ p χ p 2, Ψ lin curl (Curl χ p ) := 1 2 µ L2 c Curl χ p 2, (3.8) Ψ iso (η) := 1 2 µ k 2 η 2 = 1 2 µ k 2 η α 2. Here, L c 0 is an energetic length scale which characterizes the contribution of the defectlike energy density to the system, H χ is a positive nondimensional penalty constant, k 2 is a positive nondimensional isotropic hardening constant. The starting point for the derivation of the equations and inequalities describing the plasticity model is the two-field minimization formulation α E(u, γ, χ p, η) min. w.r.t. (u, χ p ). (3.9) The first variations of the total energy w.r.t. to the variables u and χ p lead to the balance equations in the next section The balance equations The conventional macroscopic force balance leads to the equation of equilibrium div σ + f = 0 (3.10) in which σ is the infinitesimal symmetric Cauchy stress and f is the body force. An additional microscopic balance equation is obtained as follows. Precisely, the first variation of the total energy w.r.t χ p gives for every δχ p C (, R 3 3 ), 0 = d dt E(u, p, χ p + tδχ p, η) t=0 = µ Hχ χ p p, δχ p + µ L 2 c, Curl χ p, Curl δχ p ] dx = µ H χ χ p p, δχ p + µ L 2 c, Curl Curl χ p, δχ p + 3 ( div µl 2 c δχ i p (Curl χ p ) i)] dx i=1 = µ H χ (χ p p) + µ L 2 c Curl Curl χ p, δχ p (3.11) + 3 i=1 which on the one hand gives from the choice δχ p C c µl 2 c δχ i p (Curl χ p ) i, n da, (, R 3 3 ) the micro-balance in strong formulation 2 µ L 2 c Curl Curl χ p = µ H χ (χ p p). (3.12) 2 Here, we have assumed uniform material constants for simplicity.

11 Well-posedness for the microcurl model 10 One the other hand we get 3 i=1 µl 2 c δχ i p (Curl χ p ) i, n da = 0 δχ p C (, R 3 3 ) (3.13) which is satisfied if we choose certain homogeneous boundary conditions on the micro-distortion χ p. Following Gurtin 41] and also Gurtin and Needleman 45] we choose the simple boundary condition χ p n ΓD = 0 and (Curl χ p ) n \ΓD = 0 (3.14) which in the case of models in strain gradient plasticity, where χ p is replaced by p or ε p simply implies that there is no flow of the plastic distortion or plastic strain across the piece Γ of the boundary The derivation of the dissipation inequality. The local free-energy imbalance states that Now we expand the first term, substitute (3.7)-(3.8) and get That is C iso ε e σ, ε e α C iso ε e σ, ε e α Therefore we obtain 0 C iso ε e σ, ε e + α where we set σ, m α γ α α Ψ σ, ε e σ, ṗ 0. (3.15) σ, m α γ α + α Ψ lin micro γ α γ α + α Ψ iso η α ηα 0. (3.16) µ H χ χ p p, m α γ α + µ k 2 η α η α 0. (3.17) (τ α +s α ) γ α +g α η α] = C iso ε e σ, ε e + α α τ α E γ α +g α η α], (3.18) τ α := σ, m α (resolved shear stress for the α-th slip system), (3.19) s α := µ H χ χ p p, m α = µ L 2 c Curl Curl χ p, m α, (3.20) g α := µ k 2 η α (thermodynamic force power-conjugate to η α ), (3.21) τ α E := τ α + s α = Σ E, m α, (3.22) with Σ E being the non-symmetric Eshelby-type stress tensor defined by Σ E := σ + µ H χ (χ p p) = σ µ L 2 c Curl Curl χ p. (3.23)

12 Well-posedness for the microcurl model 11 Since the inequality (3.18) must be satisfied for whatever elastic-plastic deformation mechanism, inlcuding purely elastic ones (for which γ α = 0, η α = 0), equation (3.18) implies the usual infinitesimal elastic stress-strain relation σ = C iso ε e = 2µ sym( u p) + λ tr( u p)1 = 2µ (sym( u) ε p ) + λ tr( u)1 (3.24) and the local reduced dissipation inequality τ α E γ α + g α η α] 0 (3.25) α which can also be written in compact form as Σ α p, Γ α p 0, (3.26) where we define The flow rule α Σ α p := (τ α E, g α ) and Γ α p := (γ α, η α ). (3.27) We consider a yield function on the α-th slip system defined by φ(σ α p ) := τ α E + g α σ 0 for Σ α p = (τ α E, g α ). (3.28) Here, σ 0 is the yield stress of the material, that we assume to be constant on all slip systems and therefore, σ α y := σ 0 g α represents the current yield stress for the α-th slip system 3. So the set of admissible generalized stresses for the α-th slip system is defined as K α := { Σ α p = (τ α E, g α ) φ(σ α p ) 0, g α 0 }, (3.29) with its interior Int(K α ) and its boundary K α being the generalized elastic region and the yield surface for the α-th slip system, respectively. The principle of maximum dissipation 4 associated with the α-th slip system gives us the normality law Γ α p N K α(σ α p ), (3.30) where N K α(σ α p ) denotes the normal cone to K α at Σ α p. That is, Γ α p satisfies Σ α Σ α p, Γ α p 0 for all Σ α K α. (3.31) 3 Note that, for the sake of simplicity, the presented isotropic hardening rule g α does not involve latent hardening and the associated interaction matrix, see 37] for a discussion on uniqueness in the presence of latent hardening. 4 The principle of maximum dissipation (PMD) is shown to be closely related to the so-called minimum principle for the dissipation potential (MPDP) 48, 47, 83], which states that the rate of the internal variables is the minimizer of a functional consisting of the sum of the rate of the free energy and the dissipation function with respect to appropriate boundary conditions. Notice that, as pointed out in 21], both PMD and MPDP are not physical principles but thermodynamically consistent selection rules which turn out to be convenient if no other information is available or if existing flow rules are to be extended to a more general situation.

13 Well-posedness for the microcurl model 12 Notice that N K α = χ K α, where χ K α denotes the indicator function of the set K α and χ K α denotes the subdifferential of the function χ K α. Whenever the yield surface K α is smooth at Σ α p then Γ α p N K α(σ α p ) λ α such that γ α = λ α τ α E τ α E and η α = λ α = γ α with the Karush-Kuhn Tucker conditions: λ α 0, φ(σ α p ) 0 and λ α φ(σ α p ) = 0. Using convex analysis (Legendre-transformation) we find that Γ α p χ K α(σ α p ) flow rule in its dual formulation for the α-th slip system Σ α p χ K α( Γ α p ) flow rule in its primal formulation for the α-th slip system (3.32) (3.33) where χ K α is the Fenchel-Legendre dual of the function χ K α denoted in this context by D α iso, the one-homogeneous dissipation function for the α-th slip system. That is, for every Γ α = (q α, β α ), D α iso(γ α ) = sup { Σ α p, Γ α Σ α p K α} = sup {τe α q α + g α β α φ(σ α E, g α ) 0, g α 0} { σ0 q = α if q α β α, otherwise. We get from the definition of the subdifferential (Σ α p χ K α( Γ α p )) that That is, (3.34) D α iso(γ α ) D α iso( Γ α p ) + Σ α p, Γ α Γ α p for any Γ α. (3.35) D α iso(q α, β α ) D α iso( γ α, η α ) + τ α E (q α γ α ) + g α (β α η α ) for any (q α, β α ). (3.36) In the next sections, we present a complete mathematical analysis of the model including both strong and weak formulations as well as a corresponding existence result Strong formulation of the model To summarize, we have obtained the following strong formulation for the microcurl model in the single crystal infinitesimal gradient plasticity case with isotropic hardening. Given f H 1 (0, T ; L 2 (, R 3 )), the goal is to find: (i) the displacement u H 1 (0, T ; H0 1(, Γ D, R 3 )),

14 Well-posedness for the microcurl model 13 (ii) the infinitesimal plastic slips γ α H 1 (0, T ; L 2 ()) for α = 1,..., n slip, (iii) the hardening variables η α H 1 (0, T ; L 2 ()) for α = 1,..., n slip, (iv) the infinitesimal micro-distortion χ p H 1 (0, T ; H(Curl;, R 3 3 ), with Curl Curl χ p H 1 (0, T ; L 2 (, R 3 3 )) such that the content of Table 2 holds. Additive split of distortion: u = e + p, ε e = sym e, ε p = sym p n silp Plastic distortion in slip system: p = γ α m α with m α = l α ν α, tr(p) = 0 Equilibrium: Div σ + f = 0 with σ = C isoε e = C iso(sym u ε p) Microbalance: µ L 2 c Curl Curl χ p = µ H χ (χ p p), Free energy: α=1 Yield condition in α-th slip system: τe α + g α σ 0 0 where τe α := Σ E, m α with 1 2 Cisoεe, ε e + 1 µ 2 Hχ p χ p µ 2 L2 c Curl χ p µ 2 k2 α ηα 2 Σ E := σ + µ H χ (χ p p) = σ µ L 2 c Curl Curl χ p g α = µ k 2 η α Dissipation inequality in α-th slip system: τe α γ α + g α η α { 0 σ0 q Dissipation function in α-th slip system: Diso(q α α, β α α if q α β α, ) := otherwise Flow rules in primal form: (τ α E, g α ) D α iso ( γα, η α ) Flow rules in dual form: γ α = λ α τ α E τ α E, ηα = λ α = γ α KKT conditions: λ α 0, φ(τe, α g α ) 0, λ α φ(τe α, g α ) = 0 Boundary conditions for χ p: χ p n = 0 on Γ D, (Curl χ p) n = 0 on \ Γ D Function space for χ p: χ p(t, ) H(Curl;, R 3 3 ) Table 2: The microcurl model in single crystal gradient plasticity with isotropic hardening Weak formulation of the model Assume that the problem in Section has a solution (u, γ, χ p, η). We will extensively make use of the identity (3.4). Let v H 1 (, R 3 ) with v ΓD = 0. Multiply the equilibrium equation with v u and integrate in space by parts and use the symmetry of σ and the elasticity relation to get C iso sym( u m γ), sym( v u) dx = f(v u) dx. (3.37)

15 Well-posedness for the microcurl model 14 Now, for any X C (, sl(3)) such that X n = 0 on Γ we integrate (3.12) over, integrate by parts the term with Curl Curl using the boundary conditions and get (X χ p) n = 0 on Γ D, Curl χ p n = 0 on \ Γ D µ L 2 c Curl χ p, Curl X Curl χ p + µ H χ χ p m γ, X χ ] p dx = 0. (3.38) Moreover, for any q = (q 1,..., q n slip) with q α C () and any β = (β 1,..., β n slip) with β α L 2 (), summing (3.36) over α = 1,..., n slip and integrating over, we get where D iso (q, β) dx D iso ( γ, η) dx C iso sym( u m γ), sym(m q m γ) dx ] + µ H χ χ p m γ, m q m γ + µ k 2 η, β η dx 0. (3.39) D iso (q, β) := α D α iso(q α, β α ). (3.40) Now adding up (4.23)-(4.25) we get the following weak formulation of the problem set in Section in the form of a variational inequality: C iso sym( u m γ), sym( v m q) sym( u m γ) + µ L 2 c Curl χ p, Curl X Curl χ p + µ H χ χ p m γ, (X m q) ( χ ] p m γ) + µ k 2 η, β η dx + D iso (q, β) dx D iso ( γ, η) dx f (v u) dx. (3.41) Existence result for the weak formulation To prove the existence result for the weak formulation (3.41), we closely follow the abstract machinery developed by Han and Reddy in 49] for mathematical problems in geometrically linear classical plasticity and used for instance in 19, 88, 70, 23, 24] for models of gradient plasticity. To this aim, equation (3.41) is written as the variational inequality of the second kind: find w = (u, γ, χ p, η) H 1 (0, T ; Z) such that w(0) = 0, ẇ(t) W for a.e. t 0, T ] and a(w, z ẇ) + j(z) j(ẇ) l, z ẇ for every z W and for a.e. t 0, T ], (3.42)

16 Well-posedness for the microcurl model 15 where Z is a suitable Hilbert space and W is some closed, convex subset of Z to be constructed later, a(w, z) = C iso sym( u m γ), sym( v m q) + µ L 2 c Curl χ p, Curl X (3.43) ] + µ H χ χ p m γ, X m q + µ k 2 η, β dx, j(z) = l, z = for w = (u, γ, χ p, η) and z = (v, q, X, β) in Z. D iso (q, β) dx, (3.44) f v dx, (3.45) The Hilbert space Z and the closed convex subset W are constructed in such a way that the functionals a, j and l satisfy the assumptions in the abstract result in 49, Theorem 6.19]. The key issue here is the coercivity of the bilinear form a on the set W, that is, a(z, z) C z 2 Z for every z W and for some C > 0. We let and define the norms V := H 1 0(, Γ D, R 3 ) = { v H 1 (, R 3 ) v ΓD = 0 }, (3.46) P := L 2 (, R n slip ), (3.47) Q := H 0 (Curl;, Γ D, sl(3)), (3.48) Λ := L 2 (, R n slip ), (3.49) Z := V P Q Λ, (3.50) } W := {z = (v, q, X, β) Z q α β α, α = 1,..., n slip, (3.51) v V := v L 2, q 2 P = q α 2 L, β 2 2 Λ = β α 2 L, X 2 Q := X H(Curl;), α α z 2 Z := v 2 V + q 2 P + X 2 Q + β 2 Λ for z = (v, q, X, β) Z. (3.52) Let us show that the bilinear form a is coercive on W. Let therefore z = (v, q, X, β) W. First of all notice that m q L 2 q P β Λ. (3.53) So, a(z, z) m 0 sym( v m q) 2 L (from (2.3)) + µ H 2 χ X m q 2 L + µ L 2 c Curl X 2 2 L 2 +µ k 2 β 2 Λ = m 0 sym v 2 L 2 + sym(m q) sym v, sym(m q) L 2] + µ Hχ X 2 L 2 + m q 2 L 2 X, m q 2 L 2] + µl 2 c Curl X 2 L + µ k 2 2 β 2 Λ

17 Well-posedness for the microcurl model 16 m 0 sym v 2 L 2 + sym(m q) 2 L 2 θ sym( v) 2 L 2 1 θ sym(m q) 2 L 2 ] + µ H χ X 2 L + m q 2 2 L θ X 2 2 L 1 ] 2 θ m q 2 L (Young s inequality) 2 + µ L 2 c Curl X 2 L µ k 2 β 2 Λ µ k 2 q 2 P (using second in (3.53)) ( = m 0 (1 θ) sym v 2 L + m ( ) sym(m q) 2 L + µ H θ 2 χ 1 1 ) q 2 P θ µ k 2 q 2 P + µ H χ (1 θ) X 2 L + µ L 2 c Curl X µ k 2 β 2 Λ ( m 0 (1 θ) sym v 2 L + (m µ H χ ) 1 1 ) + 1 ] θ 2 µ k 2 q 2 P (3.54) + µ H χ (1 θ) X 2 L 2 + µ L 2 c Curl X µ k 2 β 2 Λ (for 0 < θ < 1). So, since the hardening constant k 2 > 0, it is possible to choose θ such that m 0 + µ H χ m 0 + µ H χ µ k 2 < θ < 1, we always are able to find some constant C(θ, m 0, µ, H χ, k 2, L c, ) > 0 such that ] a(z, z) C v 2 V + q 2 P + X 2 H(Curl;) + β 2 Λ = C z 2 Z z = (v, q, X, β) W. (3.55) This shows existence for the microcurl model in single gradient plasticity with isotropic hardening Uniqueness of the weak/strong solution As shown in 21] for a canonical rate-independent model of geometrically linear isotropic gradient plasticity with isotropic hardening and plastic spin, the uniqueness of the solution for our model can be obtained similarly. To this aim, notice that if (u, γ, χ p, η) is a weak solution of the model, then (u, γ, χ p, η) is also a strong solution. In fact, choosing appropriatetly test functions in the variational inequality (3.41), we obtain both equilibrium and microbalance equations on the one hand. The latter which is µ L 2 c Curl Curl χ p = µ H χ (χ p p) is satisfied first in the distributional sense and hence is satisfied also in the L 2 -sense since the right hand side is in L 2 (, R 3 3 ). Therefore, it follows that Curl Curl χ p is also in L 2 (, R 3 3 ). Now going back to (3.41), we also derive the boundary condition (Curl χ p ) n \ΓD = 0 which is now justified because we derived that Curl χ p H(Curl;, sl(3)). On the other hand, we also obtain from (3.41) the following set of inequatlities and hence, (u, γ, χ p, η) is a strong solution. Γ α p, Σ α Σ α p 0 Σ α K α, α = 1,..., n silp (3.56)

18 Well-posedness for the microcurl model 17 Now let us consider two solutions w i := (u i, γ i, χ p i, η i ), i = 1, 2 of (3.41) satisfying the same initial conditions, let Γ α p i = (γi α, ηα i ) and Σα p i := (τ E α i, gi α ) be the corresponding stresses. That is, τ α E i = Σ E i, m α = σ i + µ H χ (χ p i p i ), m α = σ i µ H χ Curl Curl χ p i, m α, (3.57) g α i = µ k 2 η α i so that Γ α p i and Σ α p i satisfy (3.58) Γ α p 1, Σ α Σ α p 1 0 and Γ α p 2, Σ α Σ α p 2 0 Σ α K α. (3.59) Now choose Σ α = Σ α p 2 in (3.59) 1 and Σ α = Σ α p 1 in (3.59) 2 and add up to get That is, Σ α p 2 Σ α p 1, Γ α p 1 Γ α p 2 0. (3.60) σ 2 σ 1, m α γ α 1 m α γ α 2 + µ H χ (χ p 2 χ p 1 ) (p 2 p 1 ), m α γ α 1 m α γ α 2 and adding up over α, we get + (g α 2 g α 1 )( η α 1 η α 2 ) 0 (3.61) σ 2 σ 1, ṗ 1 ṗ 2 + µ H χ (χ p 2 χ p 1 ) (p 2 p 1 ), ṗ 1 ṗ 2 + g 2 g 1, η 1 η 2 0 (3.62) Now, substitute sym p i = sym u i C 1 iso σ i obtained from the elasticity relation, into the expression σ 2 σ 1, ṗ 1 ṗ 2 and ṗ i = χ + L2 c p i Curl Curl χ p H i obtained from the microbalance χ equation into the expression χ p 2 χ p 1, ṗ 1 ṗ 2 and get from (3.62) that σ 2 σ 1, C 1 iso ( σ 2 σ 1 ) + µ H χ p 1 p 2, ṗ 1 ṗ 2 + µ H χ χ p 2 χ p 1, χ p 1 χ p 2 + µ L 2 c χ p 2 χ p 1, Curl Curl χ p 1 Curl Curl χ p 2 + g 2 g 1, η 1 η 2 (3.63) σ 1 σ 2, sym( u 1 ) sym( u 2 ) Now for every t 0, T ], we integrate (3.63) over (0, t) using the boundary conditions on χ p i and using the fact that σ 1 σ 2, sym u 1 sym u 2 dx = 0, we get t d C 1/2 iso (σ 2 (s) σ 1 (s)) 2 L + µ H ds 2 χ p 1 (s) p 2 (s) 2 L µ H 2 χ χ p 2 (s) χ p 1 (s) 2 L 2 0 µ L 2 c Curl χ p 2 (s) Curl χ p 1 (s) 2 L 2 + µ k 2 η 1 (s) η 2 (s) 2 L 2 ] ds 0.

19 Well-posedness for the microcurl model 18 Therefore, we obtain C 1/2 iso (σ 2 σ 1 ) 2 L 2 + µ H χ p 1 p 2 2 L 2 + µ k 2 η 1 η 2 2 L 2 µ H χ χ p 2 χ p 1 2 L 2 + µ L 2 c Curl χ p 2 Curl χ p 1 2 L 2. (3.64) On the other hand, we write the micro-balance equation for p i and χ p i with i = 1, 2, as µ H χ p 1 = µ L 2 c Curl Curl χ p 1 + µ H χ χ p 1, (3.65) µ H χ p 2 = µ L 2 c Curl Curl χ p 2 + µ H χ χ p 2, (3.66) then we subtract, take the scalar product with χ p 1 χ p 2, integrate using the boundary condition and get µ H χ (χ p 1 χ p 2 ) n ΓD = 0 and (Curl χ p 1 Curl χ p 2 ) n \ΓD = 0 p 1 p 2, χ p 1 χ p 2 dx = µ L 2 c Curl χ p 1 Curl χ p 2 2 L 2 + µ H χ χ p 1 χ p 2 2 L 2. (3.67) Therefore, we obtain from (3.64) and (3.67) that which implies that µ H χ p 1 p 2 2 L 2 µ L 2 c Curl χ p 1 Curl χ p 2 2 L 2 + µ H χ χ p 1 χ p 2 2 L 2 (3.68) µ H χ p 1 p 2 L 2 χ p 1 χ p 2 L 2 (3.69) p 1 p 2 L 2 = χ p 1 χ p 2 L 2 and hence, Curl χ p 1 Curl χ p 2 L 2 = 0. (3.70) Now, going back to (3.64), we get Hence, we obtain so far, C 1/2 iso (σ 2 σ 1 ) 2 L 2 + µ k 2 η 1 η 2 2 L 2 0. (3.71) σ 1 σ 2, η 1 = η 2 and Curl χ p 1 = Curl χ p 2 τ α E 1 = τ α E 2. Now, let us prove that γ1 α = γα 2 for every α. In fact, from the definition of the normal cone it follows that Γ α p i = 0 that is, γ i α = η i α = 0 inside the elastic domain Int(K α ) (for the α-slip system), which from the initial conditions imply that γi α = 0 inside Int(K α ). Now, looking at the flow rule in dual form (for the α-slip system) in Table 2, we obtain from τe α 1 = τe α 2 that γ 1 α = γα 2 which implies that γα 1 = γα 2 from the initial conditions. Therefore, we obtain p 1 = p 2 which implies from (3.70) 1 that χ p 1 = χ p 2. Now, it remains to show that u 1 = u 2. This is obtained exactly as in 21]. We repeat the proof here just for the reader s convenience. To this end, we use sym( u i ) = C 1 σ i + sym p i obtained from the elasticity relation and get sym( (u 1 u 2 )) = C 1 (σ 1 σ 2 ) + sym(p 1 p 2 ) = 0, and hence, from the first Korn s inequality (see e.g. 64]), we get (u 1 u 2 ) = 0 which implies that u 1 = u 2. Therefore, we finally obtain u 1 = u 2, σ 1 = σ 2, (γ 1 = γ 2 p 1 = p 2 ), χ p 1 = χ p 2, η 1 = η 2, and thus the uniqueness of a weak/strong solution.

20 Well-posedness for the microcurl model 19 Remark 3.1 It should be stressed that, in the proof, k 2 > 0 is necessary for uniqueness of the displacement field (and of the slip variables). If k 2 = 0, we have perfect plasticity and multiple solutions involving displacement discontinuities along slip lines, as in conventional Hill s plasticity, are possible. The curl operator does not regularize such discontinuities since curl p may vanish in the presence of gradient of slip γ α perpendicularly to the slip planes. It is shown in 31] that gradient models lead to finite width kink bands but still allow for slip band discontinuities, parallel to slip planes, in single crystals The model with linear kinematical hardening Here we consider the model where the isotropic hardening has been replaced with linear kinematical hardening The description of the model Here the free-energy density Ψ is also given in the additively separated form as Ψ( u, p, χ p, Curl χ p ) : = Ψ lin e (ε e ) elastic energy where Ψ lin + Ψ lin curl (Curl χ p ) defect-like energy (GND) + Ψ lin micro(p, χ p ) micro energy + Ψ lin kin (ε p) hardening energy (SSD) e (ε e ) := 1 2 ε e, C iso ε e = 1 2 sym( u p), C iso sym( u p), (3.72) Ψ lin micro(p, χ p ) := 1 2 µ H χ p χ p 2, Ψ lin curl(curl χ p ) := 1 2 µ L2 c Curl χ p 2, (3.73) Ψ lin kin(ε p ) := 1 2 µ k 1 ε p 2 = 1 2 µ k 1 sym p 2. In this case, the equilibrium equation and the microcurl balance are obtained as in (3.10) and in (3.12) respectively. Now, the free-energy imbalance Ψ σ, u = σ, ε e + σ, ṗ and the expansion of Ψ lead to the usual infinitesimal eleastic stress-strain relation σ = 2µ sym( u p) + λ tr( u p)1 = 2µ (sym( u) ε p ) + λ tr( u)1, and the local reduced dissipation inequality Σ E, ṗ 0, (3.74) where the non-symmetric Eshelby-type stress tensor in this case takes the form Σ E := σ + Σ lin micro + Σ lin kin (3.75)

21 Well-posedness for the microcurl model 20 with Σ lin micro := µ H χ (χ p p) = µ L 2 c Curl Curl χ p, (3.76) Σ lin kin := µ k 1 ε p = µ k 1 sym p. (3.77) Two sources of kinematic hardening therefore arise in the model: the size dependent contribution, Σ lin micro, induced by strain gradient plasticity, and conventional size independent linear kinematic hardening Σ lin kin. Following the steps in the derivation of the strong formulation of the microcurl model with isotropic hardening in Section 3.2.4, we get the strong formulation in Table 3 for the model with linear kinematical hardening. Additive split of distortion: u = e + p, ε e = sym e, ε p = sym p Equilibrium: Div σ + f = 0 with σ = C iso ε e = C iso(sym u ε p) Microbalance: µ L 2 c Curl Curl χ p = µ H χ (χ p p), Free energy: 1 Cisoεe, εe + 1 µ 2 2 Hχ p χ p µ 2 L2 c Curl χ p µ k1 sym p 2 2 Yield condition: φ(σ E) := dev Σ E σ 0 0 where Dissipation inequality: Σ E, ṗ 0 Dissipation function: D iso (q) := σ 0 q Flow rule in primal form: Σ E D iso (ṗ) Flow rule in dual form: dev ΣE ṗ = λ dev Σ, E Σ E := σ + µ H χ (χ p p) µ k 1 sym p λ = ṗ KKT conditions: λ 0, φ(σ E, g) 0, λ φ(σ E, g) = 0 Boundary conditions for χ p: χ p n = 0 on Γ D, (Curl χ p) n = 0 on \ Γ D Function space for χ p: χ p(t, ) H(Curl;, R 3 3 ) Table 3: The microcurl model in single crystal gradient plasticity with linear kinematical hardening The weak formulation of the model with linear kinematical hardening The equilibrium and microbalance equations in weak form are C iso (sym u ε p ), sym( v u) dx = f(v u) dx, (3.78) µ L 2 c Curl χ p, Curl X Curl χ p + µ H χ χ p p, X χ ] p dx = 0, (3.79) for every v V and X Q with V and Q defined in (4.31) and in (4.33), respectively. Now, the primal formulation of the flow rule (Σ E D kin (ṗ)) in weak form reads for every

22 Well-posedness for the microcurl model 21 q L 2 (, sl(3)) as D kin (q)dx D kin (ṗ)dx = Σ E, q ṗ dx C iso sym( u p), sym(q ṗ dx (3.80) ] + µ H χ (χ p p) µ k 1 sym p, q ṗ dx. Now adding up (3.78), (3.79) and (3.80) we get the following weak formulation of the microcurl model of single crystal strain gradient plasticity with linear kinematical hardening in the form of a variational inequality: C iso sym( u p), sym( v q) sym( u ṗ) + µ L 2 c Curl χ p, Curl X Curl χ p + µ H χ χ p p, (X q) ( χ ] p ṗ) + µ k 1 sym p, sym(q ṗ) dx + D kin (q) dx D kin (ṗ) dx f (v u) dx. (3.81) That is, setting Z := V P Q with V, P and Q defined in (4.31)-(4.33) and their norms in (4.37), we get the problem of the form: Find w = (u, p, χ p ) H 1 (0, T ; Z) such that w(0) = 0 and where a(ẇ, z w) + j(z) j(ẇ) l, z ẇ for every z Z and for a.e. t 0, T ], (3.82) a(w, z) := j(z) := l, z := for w = (u, p, χ p ) and z = (v, q, Q) in Z. C iso sym( u p), sym( v q) + µ L 2 c Curl χ p, Curl X (3.83) ] + µ H χ χ p p, X q + µ k 1 sym p, sym q dx, D kin (q) dx, (3.84) f v dx, (3.85) Existence and uniqueness for the model with linear kinematic hardening In order to show the existence and uniqueness for the problem in (3.82)-(3.85) using 49, Theorem 6.15], we only need to show here that the bilinear form a is Z-coercive. However, this is obtained following a different approach. We will make use of the following result. Lemma 3.1 The mapping : P Q 0, ) defined by (q, X) 2 := q X 2 L 2 + sym q 2 L 2 + Curl X 2 L 2 (3.86)

23 Well-posedness for the microcurl model 22 is a norm on P Q equivalent to the norm defined by (q, X) 2 P Q = q 2 L 2 + X 2 H(Curl ;). (3.87) Proof. To show that is a norm on P Q, we only check the vanishing property of a norm since the other properties are trivially satisfied. The vanishing property is obtained through the Korn-type inequality for incompatible tensor fields established in 74, 75, 76, 77], namely X 2 L 2 C ( sym X 2 L 2 + Curl X 2 L 2 ) X Q := H 0 (Curl;, Γ D, R 3 3 ). (3.88) In fact, let (q, X) P Q be such that (q, X) = 0, that is, q = X, sym q = 0 and Curl X = 0. Thus we get sym X = 0 and Curl X = 0. From (3.88), we then get X = 0 and hence also q = 0. Now to show that both norms are equivalent, we will first show that (P Q, ) is a Banach space. To this aim, let (q n, X n ) be a Cauchy sequence in (P Q, ). Hence, the sequences (q n X n ), (sym q n ) and (Curl X n ) are all Cauchy in L 2 (, R 3 3 ) and therefore, there exist A, B, C L 2 (, R 3 3 ) such that Thus, q n X n A, sym q n B, and Curl X n C. (3.89) sym X n = sym q n sym(q n X n ) B sym A = sym(b A) and Curl X n C. Hence, it follows from the inequality (3.88) that (X n ) is a Cauchy sequence in (Q, H(Curl;) ). Hence, there exists X Q such that X n X and Curl X n Curl X = C in L 2 (, R 3 3 ). Now q n = X n + (q n X n ) X + A and sym q n sym(x + A) = B. Therefore, (q n, X n ) (X + A, X) 2 = (q n (X + A), X n X) 2 = (q n X n ) A 2 L 2 + sym q n sym(x + A) 2 L 2 + Curl X n Curl X 2 L 2 0. So, the sequence (q n, X n ) converges to (X + A, X) in (P Q, ). Since the two normed spaces (P Q, P Q ) and (P Q, ) are Banach and the identity mapping Id : (P Q, P Q ) (P Q, ) is linear and continuous, then as a consequence of the open mapping theorem, we find that Id : (P Q, ) (P Q, P Q ) is also linear and continuous. Therefore, the two norms and P Q are equivalent and this completes the proof of the lemma. We are now in a position to prove that the bilinear form a in (3.84) is Z-coercive. Lemma 3.2 There exists a positive constant C such that a(z, z) C z 2 Z for every z Z.

24 Well-posedness for the microcurl model 23 Proof. Let z = (v, q, X) Z = V P Q a(z, z) m 0 sym( v q) 2 L 2 (from (2.3)) + µ H χ X q 2 L 2 + µ L 2 c Curl X 2 L 2 + µ k 1 sym q 2 L 2 = m 0 sym v 2 L 2 + sym q sym v, sym q ] + µ H χ X q 2 L 2 + µl 2 c Curl X 2 L + µ k 2 1 sym q 2 L ( 2 m 0 (1 θ) sym v 2 L + m ) ] + µ k 1 sym q µ H χ X q 2 L θ 2 + µl 2 c Curl X 2 L. 2 Now, since the hardening constant k 1 > 0, we choose θ such that m 0 m 0 + µ k 1 < θ < 1, and using Korn s first inequality (see e.g. 64]) and Lemma 3.1, we then get two constants C = C(θ, m 0, µ, H χ, k 1, L c, ) > 0 and C = C (θ, m 0, µ, H χ, k 1, L c, ) > 0 such that a(z, z) C ] ] v 2 L + (q, X) 2 2 C v 2 L + q 2 2 L + X 2 2 H(Curl ;) = C z 2 Z. 4. The microcurl model in polycrystalline gradient plasticity 4.1. The case with isotropic hardening The free-energy density Ψ is given in the additively separated form Ψ( u, ε p, χ p, Curl χ p, η p ) : = Ψ lin e (ε e ) elastic energy where Ψ lin + Ψ lin curl (Curl χ p ) defect-like energy (GND) + Ψ lin micro(ε p, χ p ) micro energy e (ε e ) := 1 2 ε e, C iso ε e = 1 2 sym u ε p, C iso (sym u ε p ), (4.1) + Ψ lin iso(η p ), hardening energy (SSD) Ψ lin micro(ε p, χ p ) := 1 2 µ H χ ε p sym χ p 2, Ψ lin curl (Curl χ p ) := 1 2 µ L2 c Curl χ p 2, (4.2) Ψ lin iso(η p ) := 1 2 µ k 2 η p 2. Here, η p is the isotropic hardening variable. It should be noted that there is no constraint on the skew symmetric part of the microdeformation, skew χ p, in (4.2), due to the fact that no plastic spin is considered in the original plasticity model for skew χ p to be compared with. It will be shown that, in spite of that, no indeterminacy of skew χ p arises in the formulation 5. This represents the most straightforward microcurl extension of a phenomenological polycrystal plasticity model. 5 No simple characterization of skew χ p for H χ is known at present.

25 Well-posedness for the microcurl model The balance equations. As in Section 3.2.1, we have the balance equations: div σ + f = 0 (macroscopic balance), (4.3) µ L 2 c Curl Curl χ p = µ H χ (sym χ p ε p ) Sym(3) (microbalance), (4.4) where (4.4) is supplemented by the boundary conditions χ p n ΓD = 0 and (Curl χ p ) n \ΓD = 0. (4.5) The derivation of the dissipation inequality. The local free-energy imbalance states that Now we expand the first term, substitute (4.1) and get Ψ σ, ε e σ, ε p 0. (4.6) C iso ε e σ, ε e σ, ε p µ H χ sym χ p ε p, ε p + µ k 2 η p η p 0. (4.7) Since the inequality (4.7) must be satisfied for whatever elastic-plastic deformation mechanism, inlcuding purely elastic ones (for which η p = 0, ε p = 0), then it implies the infinitesimal stressstrain relation σ = C iso ε e = 2µ (sym u ε p ) + λ tr(sym u ε p )1 (4.8) and the local reduced dissipation inequality That is, which can also be written in compact form as where σ, ε p µ H χ sym χ p ε p, ε p + µ k 2 η p η p 0. (4.9) σ + µ H χ (sym χ p ε p ), ε p µ k 2 η p η p 0, (4.10) Σ p, Γ p 0 (4.11) Σ p := (Σ E, g) and Γ p = (ε p, η p ) (4.12) with Σ E being a symmetric Eshelby-type stress tensor and g being a thermodynamic force-type variable conjugate to η p and defined as Σ E := σ + µ H χ (sym χ p ε p ) = σ µ L 2 c Curl Curl χ p, (4.13) g := µ k 2 η p. (4.14)

26 Well-posedness for the microcurl model The flow rule We consider a yield function defined by φ(σ p ) := dev Σ E + g σ 0 for Σ p = (Σ E, g). (4.15) So the set of admissible (elastic) generalized stresses is defined as K := {Σ p = (Σ E, g) φ(σ p ) 0, g 0}. (4.16) The principle of maximum dissipation gives the normality law Γ p N K (Σ p ), (4.17) where N K (Σ p ) denotes the normal cone to K at Σ p, which is the set of generalised strain rates Γ p that satisfy Σ Σ p, Γ p 0 for all Σ K. (4.18) Notice that N K = χ K where χ K denotes the indicator function of the set K and χ K denotes the subdifferential of the function χ K. Whenever the yield surface K is smooth at Σ p then Γ p N K (Σ p ) λ such that ε p = λ dev Σ E dev Σ E and η p = λ = ε p with the Karush-Kuhn Tucker conditions: λ 0, φ(σ p ) 0 and λ φ(σ p ) = 0. Using convex analysis (Legendre-transformation) we find that Γ p χ K (Σ p ) flow rule in its dual formulation Σ p χ K( Γ p ) flow rule in its primal formulation (4.19) where χ K is the Fenchel-Legendre dual of the function χ K denoted in this context by D iso, the one-homogeneous dissipation function for rate-independent processes. Γ = (q, β), D iso (Γ) = sup { Σ p, Γ Σ p K} = sup { Σ E, q + gβ φ(σ E, g) 0, g 0} { σ0 q if q β, = otherwise. We get from the definition of the subdifferential (Σ p χ K( η p )) that, That is, for every (4.20) D iso (Γ) D iso ( Γ p ) + Σ p, Γ Γ p for any Γ. (4.21) That is, D iso (q, β) D iso ( ε p, η p ) + Σ E, q ε p + g(β η p ) for any (q, β). (4.22) In the next sections, we present as in the case of signle-crystal gradient plasticity, a complete mathematical analysis of the model including both strong and weak fomrulations as well as a corresponding existence result.