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This paper deals with error estimates for space-time discretizations in the context of evolutionary variational inequalities of rate-independent type. After introducing a general abstract evolution problem, we address a fully discrete approximation and provide a priori error estimates. The application of the abstract theory to a semilinear case is detailed. In particular, we provide explicit space-time convergence rates for classical strain gradient plasticity and the isothermal Souza-Auricchio model for shape-memory alloys.

This paper focuses on a one-dimensional wave equation being subjected to a unilateral boundary condition. Under appropriate regularity assumptions on the initial data, a new proof of existence and uniqueness results is proposed. The mass redistribution method, which is based on a redistribution of the body mass such that there is no inertia at the contact node, is introduced and its convergence is proved. Finally, some numerical experiments are reported. Mathematics SubjectClassification. 35L85, 35L05, 65N30, 74M15. with Cauchy initial data u(x, 0) = u 0 (x) andu(x, 0) = v 0 (x), x ∈ (0, L), (1.2) and Signorini and Dirichlet boundary conditions at x = 0 and x = L, respectively, 0 ≤ u(0, t) ⊥ u (0, t) ≤ 0 and u(L, t) = 0, t ∈ [0, T ], (1.3)

In this paper we prove the stability of quasi‐static paths of finite dimensional mechanical systems that have an elastic‐plastic behavior with linear hardening. The concept of stability of quasi‐static paths used here is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and to the smallness of the rate of application of the external forces (which plays here the role of the small parameter in singular perturbation problems). The discussion of stability is preceded by the presentation of mathematical formulations (plus existence and uniqueness results) for those dynamic and quasi‐static problems, in a form that is convenient for the subsequent discussion of stability.

This note deals with a three-dimensional model for thermal stress-induced transformations in shape-memory materials. Microstructure, like twined martensites, is described mesoscopically by a vector of internal variables containing the volume fractions of each phase. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and we study space-time discretizations and establish convergence of these approximations. Mathematical formulationWe consider a material with a reference configuration Ω ⊂ R d with d ∈ {2, 3}. This body may undergo displacements u : Ω → R d and phase transformations. The latter will be characterized by a mesoscopic internal variable z : Ω → Z where Z is the Gibbs simplex, associated with the N pure phases e 1 , . . . e N ∈ R N , where e j is the jth unit vector, i.e.,The set of admissible displacements F is chosen as a suitable subspace ofWe consider here the extension of u Dir (t) to Ω, but actually only the trace on Γ Dir would be needed. The internal variable z lives inWe assume also that the material behavior depends on the temperature θ, which will be considered as a time dependent given parameter. Therefore we will not solve an associated heat equation but we will treat θ as an applied load and we denote it by θ appl :This approximation for the temperature is used in engineering models and is justified when the changes of the loading are slow and the body is small in at least one direction: in such a case, excess of heat can be transported very fast to the surface of the body and then radiated into the environment. The linearized strain tensor is given by e(u) def = 1 2 (∇u+∇u T ). The stored-energy potential takes the following formwhere the stored-energy density W (e(u+u Dir (t)), z, θ appl (t)) describes the material behavior. Here σ is a positive coefficient that is expected to measure some nonlocal interaction effect for the internal variable z and l(t) denotes an applied mechanical loading. The total dissipation distance between two internal states z 0 , z 1 ∈ Z is defined viawhere D is a quasi-distance, namelyFinally our problem is assumed to be governed by the energetic formulation of rate independent processes as introduced in [1,2,4,5]. A function (u, z) : [0, T ] → F × Z is called an energetic solution of the rate-independent problem associated with E and D if for all t ∈ [0, T ], the global stability condition (S) and the global energy balance (E) are satisfied, i.e. (S) ∀(ū,z) ∈ F × Z : E(t, u(t), z(t)) ≤ E(t,ū,z) + D(z(t),z), (E) E(t, u(t), z(t)) + Var