74C05 Small-strain, rate-independent theories (including rigid-plastic and elasto-plastic materials)
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 the isothermal Souza-Auricchio model for shape-memory alloys.
This paper discusses the convergence of kinetic variational inequalities to rate-independent quasi-static variational inequalities. Mathematical formulations as well as existence and uniqueness results for kinetic and rate-independent quasi-static problems are provided. Sharp a priori estimates for the kinetic problem are derived that imply that the kinetic solutions converge to the rate-independent ones, when the size of initial perturbations and the rate of application of the forces tends to 0. An application to three-dimensional elastic-plastic systems with hardening is given.
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 case of uniform convexity. We study also space-time discretizations and establish convergence of these approximations.
A general abstract approximation scheme for rate-independent processes in the
energetic formulation is proposed and its convergence is proved under various
rather mild data qualifications. The abstract theory is illustrated on several
examples: plasticity with isotropic hardening, damage, debonding,
magnetostriction, and two models of martensitic transformation in
shape-memory alloys.
This paper analyzes a model for phase
transformation in shape-memory alloys
induced by temperature changes and by
mechanical loading. We assume that the temperature is prescribed and
formulate the problem within the framework
of the energetic theory of
rate-independent processes. Existence and uniqueness results are proved.
We study the global spatial regularity of solutions of elasto-plastic models with linear hardening. In order to point out the main idea, we consider a model problem on a cube, where we describe Dirichlet and
Neumann boundary conditions on the top and the bottom, respectively, and periodic boundary conditions on the
remaining faces. Under natural smoothness assumptions on the data we obtain
$u\in L^\infty((0,T);H^{3/2-\delta}(\Omega))$ for the displacements and
$z\in L^\infty((0,T);H^{1/2-\delta}(\Omega))$ for the internal variables.
The proof is based on a difference quotient technique and a reflection argument.
In this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general
three-dimensional system. The plastic behavior is modeled using the classical
three-dimensional von Mises plasticity model. It turns out that this single-yield model leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model,
given by a hysteresis operator of Prandtl-Ishlinskii type whose density
function can be determined explicitly. This result indicates that the use
of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity
is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness
of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.