Existence and linearization for the Souza-Auricchio model at finite strains

We address the analysis of the Souza-Auricchio model for shape-memory alloys in the finite-strain setting. The model is formulated in variational terms and the existence of quasistatic evolutions is obtained within the classical frame of energetic solvability. The finite-strain model is proved to converge to its small-strain counterpart for small deformations via a variational convergence argument.


1.
Introduction. Shape-memory alloys recover large strains during mechanical or thermomechanical cycling. This results from an abrupt and diffusionless solid-solid phase transformation between different crystallographic configurations, namely the austenite (mostly cubic, predominant at high temperature and low stresses) and the martensites (lower symmetry variants, favored at low temperature or high stresses) [27]. The resulting thermomechanical behavior is at the basis of a variety of innovative applications ranging from sensors and actuators, to Aerospace, Biomedical, and Seismic Engineering [21], among others, and has triggered an intense research activity in the last decades.
Among the many available options, we focus here on a specific phenomenological, internal-variable-type model for polycrystalline materials. The model was originally For any A ∈ R 3×3 sym+ and any real function f : (0, +∞) → R, we define the tensorvalued function A → f (A) ∈ R 3×3 sym+ via diagonalization and without changing notation: if Q ΛQ = A, with Q ∈ SO(3) and Λ diagonal with Λ ij = λ i δ ij (no summation), then In particular, A s for any s ∈ R, log(A), and exp A are uniquely defined on R 3×3 sym+ . For all 3-tensors D ∈ R 3×3×3 we define |D| 2 := D ijk D ijk , the partial transposition (D ) ijk := D jik , and the product with the 2-tensor A as (DA) ijk := D ij A k and (AD) ijk := A i D jk . Note that, along with these definitions, |DA|, |AD| ≤ |A| |D|.
Given any symmetric, positive-definite 4-tensor C ∈ R 3×3×3×3 , the product CA is defined as (CA) ij := C ij k A k . Moreover, we denote by |A| 2 C := A:CA = C ij k A ij A k the corresponding induced (squared) norm on R 3×3 sym . Let ϕ : E → (−∞, ∞] on the general normed space E be either smooth or proper, convex, and lower semicontinuous. We denote the subdifferential of ϕ [14] as ∂ϕ : {ϕ < ∞} → 2 E * with where E * is the topological dual of E.
A final caveat: in the following we use the same symbol c in order to indicate a generic constant, possibly depending on data and varying from line to line.

2.2.
Small strains: constitutive model. We consider a body occupying the reference configuration Ω ⊂ R 3 , which is an open, connected, and bounded subset of R 3 with Lipschitz boundary ∂Ω. We decompose the boundary ∂Ω as ∂Ω = Γ D ∪ Γ tr where Γ D and Γ tr are disjoint and Γ D has positive surface measure. In the following, a Dirichlet boundary condition is imposed on Γ D whereas traction is exerted on Γ tr .
We denote by y : Ω → R 3 the deformation of the body and by u = y−id : Ω → R 3 the corresponding displacement field. The linearized strain ε(u) := (∇u+∇u )/2 is additively decomposed in an elastic part ε el and a transformation part z as The elastic part of the deformation is linearly related to the stress σ ∈ R 3×3 sym via ε el = C −1 σ, where C stands for the isotropic elasticity tensor. The transformation strain z ∈ R 3×3 dev is assumed to be deviatoric, since the martensitic transformations are approximately volume preserving, and can be related to the martensitic content at a given material point. This can be made explicit in the case of a single cristal [41], where to each martensitic variant is associated a (fixed) transformation strain z k , k = 1, . . . , N , determined by crystallography and showing the same elastic response. In a macroscopic description, one assumes the material to be a mixture of the N martensitic variants, described by a mass-fractions vector p := (p 1 , . . . , p N ) ∈ [0, 1] N , with |p| := M i=i p i ≤ 1 (|p| = 0 means that only austenite is present). Accordingly, the transformation strain is assumed to be a function of the massfractions vector p through z(p) = N i=1 p i z i . It follows that the transformation strain z belongs to the N simplex in R 3×3 dev . In the polycrystalline case, it is hence reasonable to assume an isotropic constraint of the form |z| ≤ L , where the scalar L expresses the maximum strain realizable via martensitic transformation.
The equilibrium property of the material are encoded in the free energy functional. The Souza-Auricchio model is characterized by the Helmholtz free energy density where denote respectively the elastic and the inelastic component of the energy. Here and in the following the subscript 0 is used to refer to the small-strain situation. The indicator function I L is defined by and encodes the constraint |z| ≤ L . The function θ → β(θ) is the austenitemartensite transformation stress at temperature θ. It is usually assumed to be vanishing below some critical temperature θ M and to be linearly growing for high temperatures. The original choice β(θ) := β 0 (θ − θ M ) + , where x + := max{x, 0} is the classical positive part and β 0 is a positive material parameter [5,69], leads to some thermodynamical inconsistency and should be replaced by a suitable regularized version [38]. In what follows we however only ask β to be Lipschitz continuous. Note nonetheless that the temperature evolution t → θ(t) is henceforth assumed to be given.
The classical constitutive relations for the conjugated forces σ and X are obtained from the variation of the free energy as The constitutive evolution law at the material point is prescribed through a balance between dissipative and conservative forces as The dissipation potential R 0 is of Von Mises type, namely R 0 (ż) = ρ|ż|, where ρ > 0 denotes some given yield stress. Since R 0 is 1-homogeneous, the constitutive law (2) is rate independent. Given the explicit form of the free energy ψ 0 , the material constitutive relation (2) can be spelled out as dev , at all material points.
2.3. Small strains: quasistatic evolution. By assuming the temperature evolution t → θ(t) to be given, the quasistatic evolution of the medium results from the combination of the constitutive model with the equilibrium system. The state of the system is hence described by the pair q = (u, z). In order to obtain the existence of quasistatic evolutions, we are forced to include in the energy a penalization of phases interfaces in the form [33] V r (z) = µ r Ω |∇z(x)| r dx, r ≥ 1 which sets the problem within the frame of gradient inelastic theories [55,24,25]. This introduces the length scale κ −r for interfaces. Note that r = 1 is admissible. The total energy functional, which includes the internal energy and external power sources, is given by where (t), u := where δ q E := ∂ q E − ∇ · ∂ ∇q E stands for a variational derivative. In particular, for r > 1 we have the system on (0, T ) × Ω, complemented with the boundary conditions where the displacement u D is given and ν = ν(x) denotes the unit (external) normal vector at the boundary point x ∈ ∂Ω, as well as an initial condition for z.

2.4.
Finite strains: constitutive model. A finite-strain version of the original small strains Souza-Auricchio model was proposed in [22,23] and further studied from the analytical viewpoint in [30]. Let F := ∇y = I + ∇u (in components F ij = ∂y i /∂x j ) be the deformation gradient, assumed to be defined almost everywhere on Ω with values in GL + (3). Following the pattern of the classical finite-strain plasticity [40,42], one multiplicatively decomposes [23] F where F e denotes the elastic part of F and F tr the transformation part. The volume preservation constraint amounts to assume det F tr = 1, so that F tr ∈ SL(3). Note that in [60] a further multiplicative splitting of the tensor F tr is introduced, separating indeed the reversible contributions due to phase transitions from plastic contributions (a generalization of the same idea is introduced in [8] in the smallstrain context).
For each deformation gradient we introduce the corresponding (right) Cauchy-Green symmetric tensors C := F F ∈ GL sym+ (3), C e := F e F e ∈ GL sym+ (3), The Green-St. Venant tensor corresponding to the transformation strain plays a central role, since it replaces the infinitesimal transformation strain z. Indeed, the free energy density ψ for the finite strain model is assumed to be where the inelastic free energy density is obtained from the infinitesimal version . This position is readily justified upon noting that E tr = z + O(|z| 2 ).
The expression of the elastic energy as a function of C e rather than F e reflects frame indifference, namely the invariance under change of frame in the physical space. Precisely, frame indifference reads which implies ψ el (F e ) = ψ el (C 1/2 e ) = ψ el (C e ), thanks to the polar decomposition theorem.
The elastic Cauchy Green tensor can be expressed by (7) in terms of the total and transformation strains as C e = F − tr CF −1 tr . We assume isotropic elastic response, that is (3), This implies that the energy depends on C tr , rather than on F tr . In fact, by the polar decomposition, C e = R(C −1/2 tr CC −1/2 tr )R , and we can express the elastic energy solely in terms of the Cauchy-Green tensors [30,34,35] We can therefore define the free energy as a function ψ(C, C tr ) of the state variables C and C tr .
For the sake of definiteness, let us mention the class of Ogden materials [15, Sec. 4.9] whose elastic energy density is complying indeed with all these assumptions. The constitutive evolution equation in the finite-strain model [23] has the form of an associative flow rule forĊ tr . This evolutive equation was brought back to the variational formulation in [30] by means of the following dissipation potential A discussion on the equivalence between the flow rules in terms of F tr and of C tr can be found in [34]. The flow rule (9) can be made more explicit as follows. Let φ : sym → R be the yield function associated with R, namely, let T → φ(C tr , T ) be the Legendre conjugate ofĊ tr → R(C tr ,Ċ tr ) given by [30,34] and compute where we interpret 0/|0| := {F ∈ R 3×3 dev | |F | ≤ 1}. Then, the flow rule (9) can be expressed via classical Kuhn-Tucker complementary conditions as sym is an element of the subdifferential of the free energy, namely 2.5. Quasistatic evolution at finite strains. We now turn to the coupling of the finite strain constitutive material relation with the quasistatic equilibrium system. In analogy with the small-strain model, we shall here consider an interfacial energy contribution of the form which is bounded for C tr ∈ W 1,r (Ω, R 3×3 ) if r > 1. For r = 1, we allow ∇C tr to be measure-valued, so that V 1 (C tr ) actually is proportional to the total variation of C tr for any C tr ∈ BV (Ω, R 3×3 ) (see [2] and Section 3.1). The state variable now reads q = (y, C tr ) and the total energy functional with the external actions is The dissipation is instead given by so that the evolution equation is formally expressed by ∂qR(q,q) + δ q E(t, q) 0.
By computing the variational derivatives [30,34], we can rewrite the latter for r > 1 as in (0, T ) × Ω (compare with (4)- (6)) along with the initial condition (y(0), C tr (0)) = (y 0 , C tr,0 ) on Ω (15) and the boundary conditions where u D is a given boundary datum and (for r > 1) Observe that the tensor S is nothing but the second Piola-Kirchhoff stress tensor, which is defined by where, in comparison with (12), the invariance under rotations of the intermediate configuration is apparent.
3. Energetic solvability. Let us now consider the existence of solutions to (12)- (17). In particular, given suitable external actions and an initial datum (y 0 , C tr,0 ), we aim at finding a solution t → (y(t), C tr (t)) to the system (12)- (17). As strong solvability seems presently unaccessible, we resort to the variational notion of energetic solvability instead [46,47,54]. This formulation is introduced in Subsection 3.1 and existence is proved in Subsection 3.2.
3.1. Energetic formulation. Let us recollect here some basic material on the concept of energetic solutions. All details and motivation can be found in the recent monograph by Mielke & Roubíček [51]. An abstract rate-independent system is specified by the triple (Q, E, R), where Q is the state space, E(t, q) the energy functional, and R(q,q) the dissipation potential, and reads The positive 1-homogeneity condition R(q, λq) = λR(q,q)), for λ > 0, encodes rateindependence. The energetic formulation of the differential problem (18) consists in looking for a trajectory q : where is called the set of stable states at time t. Correspondingly, relation (19) expresses the global stability of the state q at each time. Relation (20) is the energy balance instead. Let us now leave the abstract frame and specify our notion of energetic solutions for the quasistatic evolution at finite strains. Note that existence of energetic solutions of the constitutive model has already been obtained in [30]. We hence extend here those results to the full quasistatic-evolution case.
In order to take nonhomogeneous Dirichlet boundary conditions into account, we perform a classical map-composition splitting of the displacement field [26,45]. The deformation map is expressed as where y D is the prescribed Dirichlet condition on Γ D ⊂ Ω. Note that this allows to have the variable y to be defined on the set where q y > 3. The map y D (·, t) is assumed to be a diffeomorfism. More precisely, by defining H = ∇y D we assume for some c D > 0. The deformation gradient is then written as F = ∇(y D • y) = H ∇y.
The state space Z := Z r for the transformation strain C tr is defined in coordination with the exponent r as follows.
e. in Ω} for r > 1, e. in Ω} for r = 1, so that the nonlocal term V r (C tr ) is finite for all C tr ∈ Z r . The space Y is endowed with its weak topology whereas Z r has the weak topology of W 1,r (Ω; R 3×3 dev ) for r > 1 and the weak star topology of BV (Ω; R 3×3 dev ) for r = 1. Eventually, the full state space is defined as Q := Y × Z.
Note that the regularizing term V in combination with the constraint from I L in the energy E induces strong compactness of the energy sublevels in L p , namely We define the dissipation D as follows. Let R be the dissipation potential defined from (10). We construct the dissipation metric D : SL(3) sym+ × SL(3) sym+ → [0, +∞] associated to R as In particular, the function D defines a distance and can be explicitly evaluated [49,45,34] as It can be proved from (25) The dissipation is then defined as We collect in the following Lemma some properties, useful both in the existence and in the linearization theory. i) D is a distance. ii) D is weakly continuous on bounded sets. iii) Let C n be uniformly bounded with D(C, C n ) → 0. Then, C n C in Z.
Proof. Ad i): the statement follows directly from the analogous statement on D.
Ad ii): let (C n , C n ) (C, C) in Z × Z being bounded. Given the compact embedding Z ⊂⊂ L 1 (Ω, R 3×3 ) we have that (C n , C n ) → (C, C) in L 1 × L 1 so that the local Lipschitz continuity of D entails We conclude by observing via the triangle inequality that Ad iii): owing to the lower bound (26), the convergence D(C, C n ) → 0 implies C n → C in L 1 and the assertion follows as bounded sets in Z are relatively sequential compact.
3.2. Existence. We are now ready to state and prove our existence result.
Assumption (29) expresses the controllability of the so called Kirchhoff stress tensor ∂ Fe ψ el (F e )F e by means of the energy. This condition is compatible with polyconvexity and plays an important role in finite-deformation theories [11,12]. In particular, it implies that ψ el has polynomial growth [12,Prop. 2.7].
The proof of Theorem 3.2 follows the general existence theory for the rateindependent systems [51] and, in particular, [26,45] (see also [35]). Therefore, we shall not provide here a full argument, but rather comment on some specific detail. The classical strategy of the proof consists in the construction of the time-discrete approximate solutions mentioned in the statement and in a limiting procedure, combined with the check that the limit trajectory is actually an energetic solution. Accordingly, we split the proof of Theorem 3.2 into the next two subsections.
3.3. Time-discrete approximate solutions. We start by proving that, for any partition of the time interval, the incremental minimization problems in (30) admit a solution. This follows via the Direct Method by proving lower semicontinuity and coercivity of E(t k i , ·) + D(C tr,i−1 , ·). Lemma 3.1 provides the required lower semicontinuity of the dissipation potential. The following three Lemmas address the energy instead.

Lemma 3.3 (Coercivity of the energy). Under the assumptions of Theorem 3.2,
the energy E is coercive in the following sense Proof. We may assume with no loss of generality that |C tr | ≤ L almost everywhere. Lettingȳ(t, x) := y D (t, y(t, x)), from the coercivity assumption (28)  ≥ c|∇y| qy .
In order to check the lower semicontinuity of the energy, the following Lemma 3.4 on the convergence of minors is needed. According to assumption (27), we write

Lemma 3.4 (Convergence of minors).
Let y k y in W 1,qy (Ω; R 3 ) and P k → P in L p (Ω; SL(3)) with Let H k = H(t, y k ) and ∇ȳ k = H k ∇y k . Then, Proof. Since detP k = 1, we have From the compact embedding W 1,qy (Ω; R 3 ) → C(Ω; R 3 ) we obtain y k → y in L p (Ω; R 3 ) for some p ≥ 1 and, by (23), we conclude that H k → H = H(t, y) in L ∞ (Ω; R 3 ) as well. Therefore In order to establish the desired convergences we will use the basic fact applied to the three minors in (33). The classical weak continuity of the minors of the gradient [10] for q y > 3 and (34) yields Moreover, we clearly have that The assertion hence follows upon checking the following conditions on the indexes The first two are (32) and the last one is a direct consequence of these.
We can now ready to check the weak lower semicontinuity of the energy. Proof. Let (y k , C tr k ) (y, C tr ) in Y × Z. The term C tr → V r (C tr ) is lower semicontinuous for all r ≥ 1. By the compact embedding (24) we have C tr k → C tr in L p , for all p ≥ 1. Therefore, one can extract a subsequence (C tr ) n k converging pointwise almost everywhere to C tr . By the lower semicontinuity of the nonnegative function C tr → ψ in (C tr , θ) and the Fatou lemma, one concludes that, along such subsequence The L p -convergence of (C tr ) n k for any p ≥ 1 implies the L p convergence of (C tr ) 1/2 n k . In fact, the constraints |(C tr ) n k − I| ≤ L and det C tr k = 1 force the eigenvalues of (C tr ) n k to belong to the interval [(1 + L ) −2 , (1 + L )], entailing the Lipschitz continuity of the square root. Hence, Lemma 3.4 yields As ψ el (∇ȳC )) with Ψ convex, the lower semicontinuity of the elastic energy term follows. Eventually, the time-dependent linear term is weakly continuous.

Convergence of time discretizations.
From the solutions to the incremental minimization problems (30) we construct the corresponding backward piecewiseconstant interpolants (y k , C k tr ) on the partition. These can be classically proved [47] to have bounded energy and dissipation, independently of the diameter of the partition τ k , from the coercivity of the energy, the nondegeneracy of the dissipation, and a control of the power of external actions. This last point deserves a specific lemma, for it slightly departs from the standard theory.
where the modulus of continuity ω depend only on data.
As for the remaining two terms in the right-hand side of (37), by observing that namely relation (35). The continuity in time ∂ t E(·, q) follows directly from that oḟ H (see also [45,Thm. 5.3]),θ(t), and˙ .
Owing to the energy and dissipation bound, the interpolants (y k , C k tr ) turn out to admit a not relabeled subsequence which converges to (y, C) weakly in Y × Z. The latter can be proved to be an energetic solution by virtue of the lower semicontinuity of the energy (Lemma 3.5), the continuity of the dissipation (Lemma 3.1.ii) and the continuity of the power of external actions (36). Eventually, convergence of the energies and dissipations can also be checked [51, Thm. 2.1.6, p. 55]. 4. Small-deformation limit for quasistatic evolution. We now turn our attention to small deformation regime. In particular, we prove that in such regime the finite-strain model reduces to the small-deformation one. This consists in a evolution Γ-convergence argument [16,20,51], implying in particular the convergence of the corresponding energetic solutions. In the static case, the seminal contribution in this direction is [17] where a variational justification of linearization in elasticity is provided, see also [1] for successive refinements and [56,57,68] for extensions of the argument have been presented.
In the inelastic, evolutive setting the corresponding small-deformation-limit technique has been presented in [52] in an abstract setting and then applied in the frame of finite plasticity in [53,31,34,35]. We follow here the argument of [35] by adapting it to the different form of the energy functional. The Souza-Auricchio model is here augmented via a gradient term of the inelastic variables, both in the finite and in the infinitesimal-strain case [4]. As a consequence, we take advantage of a strong convergence notion for the inelastic variable. This was not available in the plasticity model considered in [53], where both the finite-strain model and the small-strain limit had no plastic gradient nor in [35], where the gradient of the inelastic variable was included at finite strains only.
The present small-deformation analysis is restricted to the case r = 2 in (11) for the quadratic character of the gradient energy term plays a crucial role. Letting ε > 0 and (y, C tr ) ∈ Q be given, we introduce the rescaled variables and the rescaled Green-Saint Venant strain The rescaling of the energy is performed in a way to obtain the density (1) to first order, namely we define the rescaled energy density as ψ ε (e, z, θ) := ψ el ε (e, z) + ψ in ε (z, θ), The quadratic scaling of the elastic energy is motivated by the assumption of a quadratic behavior of the potential ψ el in the neighborhood of the identity. In particular, we assume (see (43) below) ψ el (I) = 0 and ∂ Fe ψ el (I) = 0. The last condition is equivalent to assuming a stress-free reference configuration. In particular, the elastic energy is supposed to be twice differentiable at I and we define the elasticity tensor C as C := 4∂ 2 Ce ψ el (I) = ∂ 2 Fe ψ el (I) so that the symmetries C ij k = C kij = C ijk hold. In this way the rescaled energy density as a function of the rescaled variables admits the Taylor approximation ψ el ε (e, z) = 1 2 |e−z| 2 C + o(1). which represents the elastic part of (1). The complete rescaled energy functional reads then where its small-strain counterpart (recall (3)) is We define the rescaled dissipation distance as The linear scaling reflects the 1-homogeneity of the dissipation potential R. By exploiting the explicit form of D given in (25) one can compute To first order, the D ε hence reduces to Assume now to be given θ ∈ C 1 [0, T ], ∈ C 1 ([0, T ]; (W 1,qy ΓD (Ω; R 3 )) * ), assume y D = id (for the sake of simplicity, nonhomogeneous conditions can also be considered at the expense of notational intricacies), and initial values z 0ε such that C tr,0 := exp(2εz 0ε ) ∈ S(0) where S(t) denotes stable states at time t ∈ [0, T ] corresponding to (Q, E ε , D ε ). Owing to Theorem 3.2 there exists an energetic solution (y ε , C tr ε ) corresponding to the triple By defining (u ε , z ε ) from (y ε , C tr ε ) via the change of variables (38) we readily find that (u ε , z ε ) is an energetic solution corresponding to (Q 0 , E ε , D ε ) where the space Q 0 is now chosen to be by simply extending trivially the functionals. Note that, for all ε > 0, the trajectory (u ε , z ε ) takes values in the linear, ε-independent state space Q 0 . The weak convergence in Q 0 will hence provide the relevant topology for the Γ-convergence argument. We shall refer to (u ε , z ε ) as finite-strain quasistatic evolutions and denote the corresponding set of stable states at time t ∈ [0, T ] by S ε (t).
We are concerned with the convergence of finite-strain quasistatic evolutions (u ε , z ε ) to a solution (u, z) of the small-strain Souza-Auricchio system corresponding to In particular, (u, z) is en energetic solution of the equilibrium system (4)-(6) with r = 2 along with the homogeneous Neumann condition (∇z) ν = 0 and an initial condition for z [36].
We now state our main convergence result.
Note that this theorem delivers a new proof of the former existence result [4, Thm. 6.1]. The condition (44) is automatically satisfied in the small strain limit for most of the physically meaningful energies. In the case of arbitrary deformations, condition (44) restricts the class of admissible polyconvex energies, see e.g. the discussion in [1]. In the case of Ogden materials (8), condition follows (44) by asking γ i ≥ 2 for some i.
In order to prove of the theorem we apply the general theory of the evolutionary Γ-convergence established in [52]. This relies on (separate) Γ-lim inf inequalities for energy and dissipation: as well as on the existence, for any given (û 0 ,ẑ 0 ) and (u ε , z ε ) (u 0 , z 0 ) with uniformly bounded energies, of a mutual recovery sequence (û ε ,ẑ ε ) such that We shall split the proof in lemmas and start by presenting a coercivity result corresponding to a version of Lemma 3.3 with respect to the rescaled variables.
In particular, we can argue exactly as in [17].
Having established the Γ-lim inf inequalities (45)- (46), the next ingredient of the evolutive-Γ-convergence argument is the specification of a mutual recovery sequence. This is done within the following lemma.
Proof. Note that the definition of the recovery sequence differs from that of [35]. We let The definition of u ε can be rewritten in more intuitive terms as where now y = id + ε u. In the following we will use the shorthand notations C tr,ε := exp(2εz ε ), C tr,ε := exp(2ε z ε ).
Let us summarize some of the relevant properties of the recovery sequence. By the coercivity Lemma 4.2 and the convergence z ε z 0 in H 1 we have the bound From this, for any α ∈ R (in particular, α = 1, −1/2 are relevant to us), by expanding the exponential C α tr,ε = exp(2αεz ε ) we obtain that C α tr,ε = I + 2αεz ε + ε 2 L ε with L ε ∞ ≤ c Passing to gradients in this expansion at α = 1 and using (50) we deduce ∇C tr,ε − 2∇z ε L 2 ≤ cε.
We now establish the inequalities (47)- (48) in subsequent steps, by examining separately different contributions.
Step 1. The lim sup inequality for the dissipation. The explicit expression from (2.4) yields so that the convergence z ε → z in L 2 along with the uniform L ∞ bound on z ε entail (47).
Step 4. The lim sup inequality for the transformation energy. We aim at showing that lim sup ε→0 Ω ψ in ε ( z ε , θ(t ε )) dx − Ω ψ in ε (z ε , θ(t ε )) dx Convergence (49) entails the lim sup inequality for the first two contributions in ψ ε tr , namely the hardening and thermomechanical coupling terms. As for the constraint, it suffices to use (51) for z ε in order to find to check that, since |z 0 | ≤ L almost everywhere, one has I L 1 2ε | exp(2ε z ε ) − I| = 0 a.e., at least definitively for ε → 0. This concludes the proof of the lim sup inequality (54).  [52] can be easily deduced by density. The pointwise strong convergence of (u ε , z ε ) and the convergence of energies and dissipation follow at once from the uniform convexity of the linearized energy E 0 along the same lines as in [53,Cor. 3.8 and Cor. 3.9].