Adrien Petrov

On Existence and Approximation for a 3D Model of Thermally Induced Phase Transformations in Shape-Memory Alloys

This paper 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. We assume that the temperature variations are prescribed. The problem is formulated mathematically within the energetic framework of rate-independent processes. An existence result is proved and temporal regularity is obtained in the case of uniform convexity. We also study space-time discretizations and establish convergence of these approximations.

Error Estimates for Space-Time Discretizations of a Rate-Independent Variational Inequality

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.

MATHEMATICAL RESULTS ON EXISTENCE FOR VISCOELASTODYNAMIC PROBLEMS WITH UNILATERAL CONSTRAINTS

This paper focuses on a damped wave equation and the evolution of a Kelvin–Voigt viscoelastic material, both problems being subject to unilateral boundary conditions. Under appropriate regularity assumptions on the initial data, both problems possess a weak solution which is obtained as the limit of a sequence of solutions of penalized problems; the functional properties of all the traces are precisely identified through Fourier analysis, and this enables us to infer the existence of a strong solution, i.e., a solution satisfying almost everywhere the unilateral conditions.

Error Bounds for Space-Time Discretizations of a 3D Model for Shape-Memory Materials

This paper deals with error estimates for space-time discretizations of a three-dimensional model for isothermal stress-induced transformations in shapememory materials. After recalling existence and uniqueness results, a fully-discrete approximation is presented and an explicit space-time convergence rate of order \(h^{\alpha/2} + \tau^{1/2}\) for some α ∈ (0,1] is derived.

A weighted finite element mass redistribution method for dynamic contact problems

This paper deals with a one-dimensional wave equation being subjected to a unilateral boundary condition. An approximation of this problem combining the finite element and mass redistribution methods is proposed. The mass redistribution method is based on a redistribution of the body mass such that there is no inertia at the contact node and the mass of the contact node is redistributed on the other nodes. The convergence as well as an error estimate in time are proved. The analytical solution associated with a benchmark problem is introduced and it is compared to approximate solutions for different choices of mass redistribution. However some oscillations for the energy associated with approximate solutions obtained for the second order schemes can be observed after the impact. To overcome this difficulty, an new unconditionally stable and a very lightly dissipative scheme is proposed.

A mathematical model for the third-body concept

The third-body concept is a pragmatic tool used to understand the friction and wear of sliding materials. The wear particles play a crucial role in this approach and constitute the main part of the third-body. This paper aims to introduce a mathematical model for the motion of a third-body interface separating two surfaces in contact. This model is written in accordance with the formalism of hysteresis operators as solution operators of the underlying variational inequalities. The existence result for this dynamical problem is obtained by using a priori estimates established for Faedo–Galerkin approximations, and some more specific techniques such as anisotropic Sobolev embedding theory.

Solvability of a pseudodifferential linear complementarity problem related to a viscoelastodynamic contact model

Abstract This paper focus on a simplified viscoelastic problem subjected to unilateral boundary conditions. The problem is reduced to a pseudodifferential linear complementarity problem using Fourier techniques. Under appropriate regularity assumptions on the data, the diagonal process is employed to prove the existence result to this pseudodifferential linear complementarity problem.

MATHEMATICAL RESULTS ON THE STABILITY OF QUASI-STATIC PATHS OF ELASTIC-PLASTIC SYSTEMS WITH HARDENING

In this paper, existence and uniqueness results for a class of dynamic and quasistatic problems with elastic-plastic systems are recalled, and a stability result is obtained for the quasi-static paths of those systems. The studied elasticplastic systems are continuum 1D (bar) systems that have linear hardening, and the concept of stability of quasi-static paths used here takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the system. That concept is essentially a continuity property relatively to the size of the initial perturbations (as in Lyapunov stability) and relatively to the smallness of the rate of application of the forces (which plays here the role of the small parameter in singular perturbation problems).

Some results on the stability of quasi-static paths of elastic-plastic systems with hardening

In this paper, a concept of stability of quasi-static paths is discussed. This takes into account the existence of fast (dynamic) and slow (quasi-static) times scales in the mechanical systems that have an elastic-plastic behavior with linear hardening. The proposed concept is essentially a continuity property relative to the size of the initial perturbations (as in Lyapunov stability) and relative to the smallness of the rate of application of the forces (which here plays the role of the small parameter in singular perturbation problems). Existence and uniqueness results for the dynamic and quasi-static problems are recalled and the stability of quasi-static paths for elastic-plastic systems with hardening is obtained.