Globally stable quasistatic evolution in plasticity with softening

We study a relaxed formulation of the quasistatic evolution problem in the context of small strain associative elastoplasticity with softening. The relaxation takes place in spaces of generalized Young measures. The notion of solution is characterized by the following properties: global stability at each time and energy balance on each time interval. An example developed in detail compares the solutions obtained by this method with the ones provided by a vanishing viscosity approximation, and shows that only the latter capture a decreasing branch in the stress-strain response.


Introduction
In the study of quasistatic evolution problems for rate independent systems a classical approach is to approximate the continuous time solution by discrete time solutions obtained by solving incremental minimum problems (see the review paper [14] and the references therein).
In this paper we apply this method to the study of a plasticity problem with softening, where the new feature is given by the presence of some nonconvex energy terms. For a general introduction to the mathematical theory of plasticity we refer to [7], [8], [9], [10], and [12]. To focus on the new difficulty, due to the lack of convexity, we consider the simplest relevant model, namely small strain associative elastoplasticity with no applied forces, where the evolution is driven by a time-dependent boundary condition w(t), prescribed on a portion Γ 0 of the boundary of the reference configuration Ω ⊂ R 2 .
The unknowns of the problem are the displacement u : Ω → R 2 , the elastic strain e : Ω → M 2×2 sym (the set of symmetric 2×2 matrices), the plastic strain p : Ω → M 2×2 D (the set of trace free symmetric 2×2 matrices), and the internal variable z : Ω → R. For every given time t ∈ [0, T ] they are related by the kinematic admissibility conditions: Eu = e + p in Ω (additive decomposition) and u = w(t) on Γ 0 . The stress depends only on the elastic part e through the usual linear relation σ := Ce , where C is the elasticity tensor.
Given a sequence of subdivisions of a time interval [0, T ] 0 = t 0 k < t 1 k < · · · < t k−1 k < t k k = T , we assume that an approximate solution (u i−1 k , e i−1 k , p i−1 k , z i−1 k ) is known at time t i−1 k . The approximate solution (u i k , e i k , p i k , z i k ) at time t i k is defined as a solution of the following incremental minimum problem: inf (u,e,p,z)∈A(w(t i k )) where Q is the stored elastic energy, H is the plastic dissipation rate, V is the softening potential, while A(w(t i k )) is the set of functions (u, e, p, z) such that Eu = e + p in Ω, u = w(t i k ) on Γ 0 , and z ∈ L 1 (Ω). The details of the definition of Q, H , V , together with the technical assumptions which are needed for our analysis, are given in Section 2. For the present discussion it is sufficient to know that Q is a quadratic form, H is positively homogeneous of degree one, and V is strictly concave with linear growth.
Due to the nonconvexity of the functional the infimum in (1.1) is not attained, in general. To overcome this difficulty, in this paper we consider a relaxed formulation of this approach (see Proposition 4.11). To preserve the continuity of the energy terms it is convenient to cast the relaxed problem in the language of Young measures. An additional difficulty is due to the linear growth of H and V , which may cause concentration effects. For this reason we formulate the problem in a suitable space of generalized Young measures (see [4,Section 3]).
The next step in our analysis is the study of the convergence of the relaxed approximate solutions as the time step t i k − t i−1 k → 0 as k → ∞ (uniformly with respect to i ). We prove that, up to a subsequence, these solutions converge to a solution of a quasistatic evolution problem formulated in the framework of generalized Young measures. This is characterized by the usual conditions considered in the variational approach to rate independent evolution problems, namely global stability and energy balance (see Definition 4.6), suitably phrased in the language of Young measures. The notion of dissipation required for this purpose is quite delicate and relies on the theory developed in [4].
We also prove that the barycentres of these Young measure solutions define a function (u(t), e(t), p(t), z(t)), where (u(t), e(t), p(t)) is a quasistatic evolution of a perfect plasticity problem (see [3]) corresponding to a relaxed dissipation function, denoted p → H eff (p, 0), which can be computed explicitly in terms of H and V . Some other qualitative properties of the solutions are investigated at the end of Section 4.
This result allows to compare the globally stable solutions obtained in this paper with the solutions delivered by the vanishing viscosity approach of [5]. In particular, we study in Section 5 the globally stable evolution corresponding to the same data considered in [5,Section 7]. The main differences are the following. While the globally stable solution involves generalized Young measures, the vanishing viscosity evolution takes place in spaces of affine functions, since the data in the example are spatially homogeneous. The stress σ(t) corresponding to the vanishing viscosity solution exhibits a decreasing branch, which accounts for the softening phenomenon. On the contrary, the stress of the globally stable solution is nondecreasing and, after a critical time, it becomes constantly equal to the asymptotic value of the stress of the viscosity solution.
2. Notation and preliminary results 2.1. Mathematical preliminaries. We refer to [5] for the standard notation about measures, matrices and functions with bounded deformation. In particular, for every measure µ the symbols µ a and µ s always denote the absolutely continuous and the singular part with respect to Lebesgue measure. The former is always identified with its density. The symbol · 2 denotes norm in L 2 , while · 1 denotes the norm in L 1 , as well as in the space M b of bounded Radon measures. The symbol ·, · denotes a duality pairing depending on the context. Generalized Young measures. We refer to [4] for the definition and properties of generalized Young measures and of time dependent systems of generalized Young measures. The underlying measure λ will always be the two-dimensional Lebesgue measure L 2 . In particular we refer to [4,Section 6] for the definition of barycentre of a generalized Young measure, and to [4,Section 3]  for every f ∈ C hom (U ×Ξ×R). Note that αω ν p does not belong to GY (U ; Ξ) since it does not satisfy the projection property (3.3) of [4].
The reference configuration. Throughout the paper Ω is a bounded connected open set in R 2 with C 2 boundary. Let Γ 0 be a nonempty relatively open subset of ∂Ω with a finite number of connected components, and let Γ 1 := ∂Ω\Γ 0 . On Γ 0 we will prescribe a Dirichlet boundary condition. This will be done by assigning a function w ∈ H 1/2 (Γ 0 ; R 2 ), or, equivalently, a function w ∈ H 1 (Ω; R 2 ), whose trace on Γ 0 (also denoted by w ) is the prescribed boundary value. The set Γ 1 will be the traction free part of the boundary.
Admissible stresses and dissipation. Let K be a closed strictly convex set in M 2×2 D ×R with C 1 boundary. For every value of the internal variable ζ ∈ R, the set is interpreted as the elastic domain and its boundary as the yield surface corresponding to ζ . We assume that there exist two constants A and B , with 0 < A ≤ B < ∞, such that Together with convexity, (2.7) yields Let π R : M 2×2 D ×R → R be the projection onto R. The hypotheses on K imply that there exists a constant a K > 0 such that (2.9) The support function H : will play the role of the dissipation density. It turns out that H is convex and positively homogeneous of degree one on M 2×2 D ×R. In particular it satisfies the triangle inequality H(ξ 1 + ξ 2 , θ 1 + θ 2 ) ≤ H(ξ 1 , θ 1 ) + H(ξ 2 , θ 2 ) . (2.11) Let Φ be the gauge function of K according to [18,Section 4]. Since Φ 2 is strictly convex and differentiable, and 1 2 H 2 = ( 1 2 Φ 2 ) * , by [18,Theorem 26.3] the function H 2 is strictly convex and differentiable, so that the set {(ξ, θ) ∈ M 2×2 D ×R : H(ξ, θ) ≤ 1} is strictly convex with C 1 boundary. The same property holds for the sets for every c ∈ R. From (2.5) it follows that so that H(·, 0) is the support function of K(0) in M 2×2 D . Using the theory of convex functions of measures developed in [6], we introduce the functional H :  where µ > 0 is the shear modulus, so that our assumptions are satisfied. Let Q : M 2×2 sym → [0, +∞) be the quadratic form associated with C, defined by Q(ξ) : It turns out that there exist two constants α C and β C , with 0 < α C ≤ β C < +∞, such that for every ξ ∈ M 2×2 sym . These inequalities imply |Cξ| ≤ 2β C |ξ| . (2.23) The softening potential. Let V : R → R be a function of class C 2 , which will control the evolution of the internal variable ζ , and consequently of the set K(ζ) of admissible stresses. We assume that there exist two constants b V > 0 and M V > 0 such that for every where a K is the constant in (2.9), and V ∞ denotes the recession function of V , defined by for every θ,θ ∈ R with |θ| ≤ R . From (2.14) and (2.26) it follows that there exists a constant C K V > 0 such that for every ξ 1 , ξ 2 ∈ M 2×2 D and every θ 1 , θ 2 ∈ R (see [5,Subsection 2.2]). It is convenient to introduce the function is any measure such that z << λ, and the function V : The definition is extended to M b (Ω) by setting The prescribed boundary displacements. For every t ∈ [0, +∞) we prescribe a boundary displacement w(t) in the space H 1 (Ω; R 2 ). This choice is motivated by the fact that we do not want to impose "discontinuous" boundary data, so that, if the displacement develops sharp discontinuities, this is due to energy minimization.
Elastic and plastic strains. Given a displacement u ∈ BD(Ω) and a boundary datum w ∈ H 1 (Ω; R 2 ), the elastic strain e ∈ L 2 (Ω; M 2×2 sym ) and the plastic strain p ∈ M b (Ω; M 2×2 D ) satisfy the weak kinematic admissibility conditions where n is the outward unit normal, ⊙ denotes the symmetrized tensor product, and H 1 is the one-dimensional Hausdorff measure. The condition on Γ 0 shows, in particular, that the prescribed boundary condition w is not attained on Γ 0 whenever a plastic slip occurs at the boundary. It follows from (2.31) and (2.32) that e = E a u − p a a.e. in Ω and p s = E s u in Ω. Since tr p = 0 , it follows from (2.31) that div u = tr e ∈ L 2 (Ω) and from (2.32) that (w − u) · n = 0 H 1 -a.e. on Γ 0 , where the dot denotes the scalar product in R 2 . Given w ∈ H 1 (Ω; R 2 ), the set A(w) of admissible displacements and strains for the boundary datum w on Γ 0 is defined by 33) The set A reg (w) of regular admissible displacements and strains is defined as Equivalently, (u, e, p) ∈ A reg (w) if and only if u ∈ W 1,1 loc (Ω; R 2 ) ∩ BD(Ω), e ∈ L 2 (Ω; M 2×2 sym ), p ∈ L 1 (Ω; M 2×2 D ), Eu = e + p a.e. on Ω, and u = w H 1 -a.e. on Γ 0 . The stress. The stress σ ∈ L 2 (Ω; M 2×2 sym ) is given by

Relaxation of the incremental problems
In this section we study different forms of relaxation of the incremental minimum problems.
3.1. Convex envelope of the nonelastic part. In this subsection, given (ξ 0 , θ 0 ) ∈ M 2×2 D ×R, we study the convex envelope of the function , it is enough to study the convex envelope of Let G ∞ be the recession function of G, defined by By the homogeneity of H it follows that

4)
where co denotes the convex envelope in M 2×2 D ×R. In particular, co G does not depend on θ 0 and is positively homogeneous of degree 1 in (ξ,θ).
By the previous lemma the convex envelope co F of the function F introduced in (3.1) is given by As H eff is convex and positively homogeneous of degree 1 , it can be written in the form [18,Theorem 13.2]), and (2.28), the function G ∞ can be expressed as the minimum of two convex functions, namely Since H * = χ K , we obtain Using (2.26), (3.11), and the strict convexity of K , it is easy to check that K eff is a bounded closed convex set and that (0, 0) ∈K eff ⊂ K eff ⊂K . (3.12) (3.14) Proof. By (2.10), (3.9), and (3.12) we have (3.2) and (3.3). By (2.27) this implies θ = 0 , so that H eff (ξ, 0) = G(ξ, 0) = H(ξ, 0). By (3.15) we deduce that ξ = 0 . This concludes the proof of (3.13).

16)
where co θ denotes the convex envelope with respect to θ . Moreover there existθ ∈ R and α ∈ [ 1 2 , 1], such that (3.10)). By (2.15) we have Since G ∞ is positively homogeneous of degree 1 , we have co A = {co G ∞ ≤ 1} and co θ A ⊂ {co θ G ∞ ≤ 1} , where co θ A is the smallest set containing A, which is convex with respect to θ , i.e., its intersections with all lines {ξ = const.} are convex. To prove that co A = co θ A, it is enough to show that co θ A is convex. By (3.19) we have that (ξ, θ) ∈ co θ A if and only if there exists θ ⊕ ∈ R such that |θ| ≤ θ ⊕ and (ξ, θ ⊕ ) ∈ A ⊕ . Since A ⊕ is convex, from this property it is easy to deduce that co θ A is convex, hence co A = co θ A. It follows that Since both functions co G ∞ and co θ G ∞ are positively homogeneous of degree 1 , we conclude that co G ∞ = co θ G ∞ . By homogeneity, to prove (3.17) and (3.18) it is not restrictive to assume that H eff (ξ, θ) = 1 , so that (ξ, θ) ∈ co A. From the previous discussion it follows that there exists θ ⊕ ∈ R such that |θ| ≤ θ ⊕ and (ξ, To conclude the proof of (3.17) and (3.18) . Then θ 0 = 0 and the common tangent hyperplane to the graphs of H eff and G ∞ at the point (ξ 0 , θ 0 , G ∞ (ξ 0 , θ 0 )) is the graph of the linear function Proof. The inequality θ 0 = 0 follows from (3.14). Therefore G ∞ is differentiable at (ξ 0 , θ 0 ). Using the convexity of H eff and the inequality H eff ≤ G ∞ , we deduce that H eff is differentiable at (ξ 0 , θ 0 ) and its partial derivatives coincide with those of G ∞ . The formula for the tangent hyperplane follows easily from the Euler identity.
By (2.15) and (2.28) we may suppose θ 0 > 0 . By the homogeneity of the problem it is not restrictive to assume that H eff (ξ 0 , θ 0 ) = G ∞ (ξ 0 , θ 0 ) = 1 . Then the set {L = 1} is the common tangent hyperplane to the hypersurfaces {H eff = 1} and {G ∞ = 1} at the point Therefore, the same argument used for (ξ 0 , θ 0 ) shows that (3.21) Let us prove that ξ 1 = ξ 0 and θ 1 = −θ 0 . Let S be the open segment with endpoints (ξ 0 , θ 0 ) and (ξ 1 , θ 1 ). As L = 1 on the endpoints, it is L = 1 on S . As H eff = 1 on the endpoints, by convexity we have H eff ≤ 1 on S . On the other hand, since the graph of L is tangent to the graph of H eff , by convexity we also have L ≤ H eff . Therefore H eff = 1 on S . By (3.21) we have G ∞ = 1 on S . As H eff ≤ G ∞ , we conclude that H eff = 1 < G ∞ on S .

3.2.
Relaxation with respect to weak convergence. We begin with a result that can be easily deduced from [1]: every (u, e, p) of the admissible set A(w) introduced in (2.33) can be approximated by triples (u k , e k , p k ) in the set A reg (w) introduced in (2.34), so that u k satisfies the boundary condition u k = w H 1 -a.e. on Γ 0 . Theorem 3.6. Let w ∈ H 1 (Ω; R 2 ) and let (u, e, p) ∈ A(w). Then there exists a se- Proof. By [1, Theorem 5.2] for every k there exists a function ψ k ∈ W 1,1 (Ω; R 2 ) such that where · 1,Ω denotes the norm in M b (Ω; M 2×2 D ). By approximation it is clear that we can find a sequence m k → ∞ such that, setting u k := v m k k , we have u k ∈ BD(Ω)∩W 1,1 loc (Ω; R 2 ), u k = w H 1 -a.e. on Γ 0 , u k ⇀ u weakly * in BD(Ω), div u k → div u strongly in L 2 (Ω), and Setting e k := e D + 1 2 div u k I and p k := Eu k − e k , we clearly have that e k → e strongly in L 2 (Ω; M 2×2 sym ) and By lower semicontinuity this implies that p k − p a 1 → p s 1 and p k 1 → p 1 . To deal with the inner variable z we need a technical lemma concerning the approximation of measures on product spaces.
where the norms are computed in the product Hilbert structure of Ξ 1 ×Ξ 2 .
Proof. First of all we observe that |p k for these values of m we also have p km 21 ∈ B R for k large enough. Since the weak * convergence is metrizable on B R , we can construct a sequence m k → ∞ such that p k 21 := p km k 21 satisfies the required properties. Using convolutions it is easy to construct a sequence p k 22 in L 1 (Ω; Ξ 2 ) such that p k 22 ⇀ p 22 weakly * in M b (Ω; Ξ 2 ) and p k 22 . It remains to prove that lim sup By the triangle inequality and by the properties of p k 21 and p k 22 , we have lim sup where the last equality follows from the fact that the measures (p 1 , p 21 ) and (0, p 22 ) are mutually singular.
Moreover, there exist a sequence (u k , e k , p k ) ∈ A reg (w) and a sequence z k ∈ L 1 (Ω) such that u k ⇀ u weakly * in BD(Ω), e k → e strongly in L 2 (Ω; Proof. Owing to the lower semicontinuity of Q and H eff (see the comments after (2.17) and (2.36)), inequality (3.24) follows from the inequality , which is a consequence of (3.4) and (3.7).
We observe that it is enough to prove (3.25) when z 0 belongs to L 1 (Ω) and is piecewise constant on a suitable triangulation. Indeed, there exists a sequence z n 0 of piecewise constant functions which converge to z a 0 strongly in L 1 (Ω). For every n let (u n k , e n k , p n k , z n k ) be a sequence satisfying the second statement of the theorem as k → ∞, with z 0 replaced by z n 0 . Then By (2.25) and by the definition of V we have By a standard double limit procedure it is then easy to construct a sequence (u k , e k , p k , z k ) satisfying the second statement of the theorem. Moreover, we may also assume that (u, e, p) ∈ A reg (w) and z ∈ L 1 (Ω). Indeed, in the general case, combining Theorem 3.6 with Lemma 3.7 we can construct a sequence (u m , e m , p m ) ∈ A reg (w) and a sequence z m ∈ L 1 (Ω) such that u m ⇀ u weakly * in BD(Ω), e m → e strongly in L 2 (Ω; [17,Theorem 3] (see also [11,Appendix]) these properties imply that and the conclusion of the theorem can be obtained by a standard double limit procedure.
Let us fix a piecewise constant function z 0 ∈ L 1 (Ω). Let and let , uniformly with respect to x, it follows that G 1 and G 2 satisfy the same property. Moreover, G 1 and G 2 are piecewise constant in x, uniformly with respect to (ξ, θ). It is easy to see that To conclude the proof, using a standard double limit procedure, it is enough to show that for every i = 1, 2 , (u, e, p) ∈ A reg (w), z ∈ L 1 (Ω), and η > 0 there exist a sequence (u k , e k , p k ) ∈ A reg (w) and a sequence z k ∈ L 1 (Ω) satisfying the properties of the second statement of the theorem and such that Using the approximation argument introduced in [13] we can also assume . Using the Lagrange interpolation on a locally finite grid composed by isosceles right triangles which becomes finer and finer near the boundary, we can replace these functions by new functions u , e , p, and z , with (u, e, p) ∈ A reg (w), such that u is piecewise affine on this triangulation T , while e , p, and z are piecewise constant. Since z 0 is piecewise constant, it is not restrictive to assume that for every x ∈ T and every T ∈ T . We may assume that every triangle T of the triangulation T is relatively compact in Ω.
Let us fix i = 1, 2 and T ∈ T . Then where ξ T is a 2×2 -matrix and c T ∈ R 2 . Moreover, we have Note that this is the only point where the dimension two is crucial. By a standard lamination procedure with interfaces orthogonal to b T we can construct two sequences v k . For every T ∈ T and every δ > 0 let T δ be the triangle similar to T with the same centre and similarity ratio 1 − δ , and let ϕ δ T ∈ C ∞ c (T ) a cut-off function such that ϕ δ T = 1 on T δ and 0 ≤ ϕ δ T ≤ 1 on T . Let us fix a finite subset T ′ ⊂ T , let It is clear that u k ⇀ u weakly * in BD(Ω) and u k = w H 1 -a.e. on Γ 0 . We set We observe that there exists a constant C( Passing to the limit first as δ → 0 , then as ε → 0 , and finally as Ω ′ ր Ω, we can make η arbitrarily small, and this concludes the proof.
Proof. We start by showing that the infimum in (3.31) is less than or equal to the minimum in (3.32). Let (ũ,ẽ,p) ∈ A(w) andz ∈ M b (Ω) be a minimizer of (3.32). By Theorem 3.8 there exist a sequence (ũ m ,ẽ m ,p m ) ∈ A reg (w) and a sequencez m ∈ L 1 (Ω) such thatẽ m →ẽ Indeed, using the definition of T (pm,zm) , we have Since (u − u 0 , e− e 0 , p− p 0 ) ∈ A(w), the minimum (3.32) is less than or equal to the infimum in (3.33).
On the other hand the infimum in (3.31) is greater than or equal to the infimum in (3.33), since for every (ũ,ẽ,p) ∈ A reg (w) and everyz ∈ L 1 (Ω) we can construct a triple (u, e, µ) ∈ B by setting u := u 0 +ũ , e := e 0 +ẽ, and µ t0t1 : . This concludes the proof of the theorem.
3.4. Some structure theorems. We prove now two structure theorems for generalized Young measures whose action on H + {V } equals the relaxed functional H eff evaluated on their barycentres.
By approximation it suffices to prove this equality when f is Lipschitz continuous with respect to ξ 1 , ξ 2 , η with a constant L independent of x (see [4,Lemma 2.4]). In this case we have As π 1 (µ) = δ p , we have which concludes the proof.

Globally stable quasistatic evolution for
where the supremum is taken over all finite families t 0 , t 1 , . . . , t k such that a = t 0 < t 1 < · · · < t k = b . As in the case of the variation Var(µ; a, b) considered in [4, Section 8], we have where the supremum is taken over all finite families t 0 , t 1 , . . . , t k such that a = t 0 < t 1 < · · · < t k = b .
In the following definition we use the homogeneous function {V } defined by (3.30) and the notion of weakly * left-continuous system of generalized Young measures introduced in [4, Definition 7.6]. We are now in a position to state the main theorem of the paper.

4.2.
The incremental minimum problems. The proof of Theorem 4.7 will be obtained by time discretization, using an implicit Euler scheme. Let us fix a sequence of subdivisions (t i k ) i≥0 of the half-line [0, +∞), with 0 = t 0 k < t 1 k < · · · < t i−1 k < t i k → +∞ as i → ∞ , (4.9) We define u i k ∈ BD(Ω), e i k ∈ L 2 (Ω; M 2×2 sym ), and µ i k ∈ SGY ({t 0 k , . . . , t i k }, Ω; M 2×2 D ×R) by induction on i . We set u 0 k := u 0 , e 0 k := e 0 , µ 0 k := δ (p0,z0) , and for i ≥ 1 we define (u i k , e i k , µ i k ) as a minimizer (see Lemma 4.9 below) of the functional over the set A i k of all triplets (u, e, ν) with u ∈ BD(Ω), e ∈ L 2 (Ω; M 2×2 sym ), and ν ∈ SGY ({t 0 k , . . . , t i k }, Ω; M 2×2 D ×R), with the following property: there exist a sequence (ũ m ,ẽ m ,p m ) ∈ A reg (w i k ) and a sequencez m ∈ L 1 (Ω) such that where B i k is the class of all triplets (u, e, ν), with u ∈ BD(Ω), e ∈ L 2 (Ω; and (u, e, p) ∈ A(w i k ), where (p, z) := bar(ν t i k ). The first two equalities follow from the definition of A i k and the continuity properties of the functional (4.11). On the other hand the infimum in the last line is greater than or equal to the infimum in the previous line by Theorem 3.9, and is less than or equal to the infimum in the first line, since A i k ⊂ B i k by (4.12) and (4.13). The existence of a minimizer (u i k , e i k , µ i k ) to (4.11) is guaranteed by the following lemma. Lemma 4.9. For every i the functional (4.11) has a minimizer on A i k , every minimizer for every (u, e, p) ∈ A(w(t i k )) and every z ∈ M b (Ω). Proof. The lemma will be proved by induction on i . Assume that µ i−1 k is defined and . We shall prove that the functional (4.11) has a minimizer (u i k , e i k ,  −1 , η) , hence by the compatibility condition (7.2) of [4] the sequence is bounded uniformly with respect to m. By (4.12) we have ν m it follows that u m is bounded in BD(Ω). Therefore, passing to a subsequence, we may assume that u m ⇀ u weakly * in BD(Ω). By [3, Lemma 2.1] it follows that (u, e, p) ∈ A(w i k ), hence (u, e, ν) ∈ B i k . We claim that e m → e strongly in L 2 (Ω; M 2×2 sym ) .  Since the other term in (4.11) is continuous with respect to the weak * convergence of ν m which contradicts the equalities in Remark 4.8, since (u, e, ν) ∈ B i k . Therefore, (4.17) is proved.
We deduce from (4.17) that (u, e, ν) ∈ A i k and that it is a minimizer of (4.11) in A i k . From now on we set (u i k , e i k , µ i k ) := (u, e, ν). By Remark 4.8 and Theorem 3.9 we obtain min where the last inequality follows from Jensen inequality.
It is enough to take z = z i−1 k in (4.14) and to use the inequality H eff (ξ, θ) ≥ H eff (ξ, 0), which follows from the fact that θ → H eff (ξ, θ) is convex and even.
The following theorem shows that the incremental problems can be considered as a relaxed version of incremental problems defined on functions. For different approaches to the relaxation problem in the context of rate-independent processes we refer to [15] and [16].
Conversely, ifê i andμ i coincide with the function e i k and the measure µ i k obtained in the incremental construction, then there exist two sequences (u m , e m , p m ) ∈ A reg (w(0)), z m ∈ L 1 (Ω) and for every i ≥ 1 two sequences (ũ i,m ,ẽ i,m ,p i,m ) ∈ A reg (w i k ) andz i,m ∈ L 1 (Ω) such that for every i ≥ 0 (4.20) holds with respect to strong convergence and (4.21) holds with respect to weak * convergence, while for every i ≥ 1 .
Proof. Inequality (4.22) follows from (4.20) and (4.21) by the lower semicontinuity of Q in the weak topology of L 2 (Ω; M 2×2 sym ) and the continuity of the duality product in the weak * topology of GY (Ω; (M 2×2 D ×R) 2 ). By Theorem 3.6 and Lemma 3.7 there exist a sequence (u m , e m , p m ) ∈ A reg (w(0)) and a sequence z m ∈ L 1 (Ω) such that u m ⇀ u 0 weakly * in BD(Ω), e m → e 0 strongly in Using [17,Theorem 3] (see also [11,Appendix]) we obtain that Condition (4.20) is trivially satisfied thanks to (4.24). To prove (4.21) we observe that for every i ≥ 1 µ m0...mi = T i (p i,m i ,z i,m i ) (µ m0...mi−1 ) . We now proceed by induction on i . Equality (4.21) for i = 0 is true by construction. Assume that (4.21) holds for i − 1 . Then by Lemma 2.1 The conclusion for i follows from (4.25).

4.3.
Further minimality properties. We now prove that the solutions of the incremental problems satisfy some additional minimality conditions. Lemma 4.12. For every i and k and every t > t i k we have (u, e, p) ∈ A(w(t i k )) , (4.28) where (p, z) := bar(ν t ).
Proof. Let us fix (u, e, ν) as in the statement of the lemma, and letν be the system [4,Remark 7.9]. Since µ i k satisfies (4.12), by (4.27) and (4.28) the triplet (u, e,ν) satisfies (4.12) and (4.13), hence (u, e,ν) belongs to the set B i k defined in Remark 4.8. By minimality we have From the compatibility condition (7.2) of [4] we obtain By the triangle inequality (2.11) we deduce that which gives (4.26) by the compatibility condition (7.2) of [4].
For every i and k we set σ i k := Ce i k and for every t ∈ [0, +∞) we consider the piecewise constant interpolations defined by for t ∈ [t i k , t i+1 k ). We define also µ k as the unique system in SGY ([0, +∞), Ω; M 2×2 D ×R) whose restrictions to the time intervals [0, t i k ] coincide with the piecewise constant inter- for every t ∈ [0, +∞).

4.5.
Proof of the main theorem. Let us fix a sequence of subdivisions (t i k ) i≥0 of the half-line [0, +∞) satisfying (4.9) and (4.10). For every k let (u i k , e i k , µ i k ), i = 1, . . . , k , be defined inductively as minimizers of the functional (4.11) on the sets A i k , with (u 0 k , e 0 k , µ 0 k ) = (u 0 , e 0 , δ (p0,z0) ), and let u k (t), e k (t), σ k (t), w k (t), and [t] k be defined by (4.29) and let µ k be the unique system in SGY ([0, +∞); Ω; M 2×2 D ×R) whose restrictions to the intervals [0, t i k ] coincide with the piecewise constant interpolations of µ i k (see [4,Definition 7.10]). Using Lemma 2.1 and the definition of A i k we can prove by induction on i that (u k , e k , µ k ) ∈ AY ({t 0 k , t 1 k , . . . , t i k }, w k ) for every i and k . This implies that (u k , e k , µ k ) ∈ AY ([0, +∞), w k ) (4.41) for every k . Let us prove that for every T > 0 there exists a constant C T , independent of k , such that sup where the last equality follows from the fact that (µ k ) 0 = δ (p0,z0) . From (2.22), (2.23), (4.33), and (4.43) we deduce that for every k and every t ∈ [0, T ]. The first estimate in (4.42) can be obtained now by using the Cauchy inequality. By (4.33) and the first inequality in (4.42) we have that is bounded uniformly with respect to k and t ∈ [0, T ]. By (4.43) this implies the boundedness of By the compatibility condition (7.2) of [4] and by the equality (µ k ) 0 = δ (p0,z0) we have which, together with the boundedness of (4.45), gives that |ξ| + |θ|, (µ k ) t (x, ξ, θ, η) is bounded. This implies that {V }(θ, η), (µ k ) t (x, ξ, θ, η) is bounded too, so that (2.13) and the boundedness of (4.44) yield the second estimate in for every finite sequence t 1 , . . . , t m in Θ with t 1 < · · · < t m .
By (4.42) and by the weak * lower semicontinuity of the dissipation we can pass to the limit in (4.33) and we obtain  for every T ∈ Θ . By left continuity the same inequality holds for every T ∈ [0, +∞).
Passing to the limit in (4.31), we obtain for every t,t ∈ Θ with t <t. By left continuity the same inequality holds for every t,t ∈ [0, +∞) with t <t.
let F ∞ be the recession function of F with respect to (ξ,θ), and let co F be the convex envelope of F with respect to (ξ,θ). It is easy to see that F ∞ = H + V ∞ . We claim that Indeed, as V is concave, we have V (θ +θ) − V (θ) ≥ V ∞ (θ), which gives where the intermediate equality follows from the fact that µ x,Y t is a probability measure. This implies co F ≥ H eff . The opposite inequality can be obtained arguing as in the proof of (3.4).
By Theorem 3.8 there exist a sequence (ũ k ,ẽ k ,p k ) ∈ A(0) and a sequencez k ∈ L 1 (Ω) such Passing to a subsequence, we may assume that δ (p k ,z k ) converges weakly * to some ν ∈ GY (Ω; M 2×2 D ×R). We note that bar(ν) = (p, 0) and that By Theorem 3.10 we deduce that where λ := |p s | andp λ is the Radon-Nikodym derivative ofp s with respect to |p s |, while z and z λ are two nonnegative functions such that As F ∞ = H + V ∞ , using (4.61) and (4.62) we obtain {F }, ν = H eff (p, 0), hence using also the strong convergence ofẽ k toẽ , we deduce From the definition of F (x,ξ,θ) this is equivalent to saying that converges to Q(e(t) +ẽ) + H eff (p, 0). Since by (ev1), we obtain (4.59) by passing to the limit as k → ∞. Thanks to [3,Theorem 4.7], to conclude the proof of the theorem it is enough to show that Q(e(T )) + D H eff ((p, 0); 0, T ) ≤ Q(e 0 ) + To show this we fix any subdivision (t i ) 0≤i≤k of the interval [0, T ]. From the definition of D H we obtain, using the compatibility condition (7.2) of [4], where the last inequality follows from the Jensen inequality. Recalling that H eff (ξ, θ) ≥ H eff (ξ, 0) for every ξ and θ , from the arbitrariness of the subdivision we obtain (4.64).
Since the functions t → Q(e(t)) and t → Owing to the previous theorem, if (u, e, µ) is a globally stable quasistatic evolution of Young measures, then µ has a weak * derivativeμ t at a.e. time t ∈ [0 + ∞) in the sense of [4,Definition 9.4]. The next theorem deals with the structure ofμ t and shows that the finite part µ Y t of µ t does not evolve.
We first prove that π ∞,a = 0 . We argue by contradiction. Assume that there exists a Borel set A, with L 2 (A) > 0 and λ(A) = 0 , such that π ∞,a (x) > 0 for every x ∈ A.

An example
In this section we assume that C is isotropic, which implies that while the stress σ of the approximable quasistatic evolution satisfies