Ulisse Stefanelli

A RATE-INDEPENDENT MODEL FOR THE ISOTHERMAL QUASI-STATIC EVOLUTION OF SHAPE-MEMORY MATERIALS

This note addresses a three-dimensional model for isothermal stress-induced transformation in shape-memory polycrystalline materials. We treat the problem within the framework of the energetic formulation of rate-independent processes and investigate existence and continuous dependence issues at both the constitutive relation and quasi-static evolution level. Moreover, we focus on time and space approximation as well as on regularization and parameter asymptotics.

The Brezis-Ekeland Principle for Doubly Nonlinear Equations

The celebrated Brezis-Ekeland principle [C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), pp. Ai, A1197-A1198, Aii, and A971-A974] characterizes trajectories of nonautonomous gradient flows of convex functionals as solutions to suitable minimization problems. This note extends this characterization to doubly nonlinear evolution equations driven by convex potentials. The characterization is exploited in order to establish approximation results for gradient flows, doubly nonlinear equations, and rate-independent evolutions.

EXISTENCE RESULT FOR A NONLINEAR MODEL RELATED TO IRREVERSIBLE PHASE CHANGES

This paper deals with an initial and boundary value problem for a nonlinear evolution system which may be used to describe irreversible phase transition phenomena. The existence of a global solution is established via a regularization | ap rioriestimate | passage to the limit procedure. Moreover, uniqueness is also discussed under additional regularity assumptions on data.

Existence Result for the One-Dimensional Full Model of Phase Transitions

This note deals with a nonlinear system of partial differential equations accounting for phase transition phenomena. The existence of solutions to a Cauchy-Neumann problem is established in the one-dimensional space setting, using a regularization – a priori estimates – passage to limit procedure.

MAGNETIC SHAPE MEMORY ALLOYS: THREE-DIMENSIONAL MODELING AND ANALYSIS

We present a three-dimensional phenomenological model for the magneto-mechanical behavior of magnetic shape memory materials such as Ni2MnGa. Moving from micromagnetic considerations, we advance to some thermodynamically consistent constitutive relations describing the magnetically-induced cubic-to-tetragonal martensitic transformation in a single crystal. We present an existence analysis for both the constitutive relation problem and the three-dimensional quasi-static evolution problem. Finally, we discuss the reduction of this model to some simpler one by means of a rigorous Γ-convergence analysis.

A discrete variational principle for rate-independent evolution

Abstract We develop a global-in-time variational approach to the time-discretization of rate-independent processes. In particular, we investigate a discrete version of the variational principle based on the weighted energy-dissipation functional introduced in [Mielke and Ortiz, ESAIM Control Optim. Calc. Var. 14: 494–516, 2008]. We prove the conditional convergence of time-discrete approximate minimizers to energetic solutions of the time-continuous problem. Moreover, the convergence result is combined with approximation and relaxation. For a fixed partition the functional is shown to have an asymptotic development by Γ-convergence (cf. [Anzellotti and Baldo, Appl. Math. Optim. 27: 195–123, 1993]) in the limit of vanishing viscosity.

Phase-Field Models with Hysteresis in One-Dimensional Thermoviscoplasticity

This paper introduces a combined one-dimensional model for thermoviscoplastic behavior under solid-solid phase transformations that incorporates the occurrence of hysteresis effects in both the strain-stress law and the phase transition described by the evolution of a phase-field (which is usually closely related to an order parameter of the phase transition). Hysteresis is accounted for using the mathematical theory of hysteresis operators developed in the past thirty years. The model extends recent works of the first two authors on phase-field models with hysteresis to the case when mechanical effects can no longer be ignored or even prevail. It leads to a strongly nonlinear coupled system of partial differential equations in which hysteresis nonlinearities occur at several places, even under time and space derivatives. We show the thermodynamic consistency of the model, and we prove its well-posedness.

THE DE GIORGI CONJECTURE ON ELLIPTIC REGULARIZATION

We prove a conjecture by De Giorgi on the elliptic regularization of semilinear wave equations in the finite-time case.

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.

Linearized plasticity is the evolutionary

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via Γ-convergence for rate-independent processes that energetic solutions of the quasi-static finite-strain elastoplasticity system converge to the unique strong solution of linearized elastoplasticity.