Convergent spectral approximations for the thermomechanical processes in shape memory alloys
type: Item
instance of: scholarly article
title: Convergent spectral approximations for the thermomechanical processes in shape memory alloys
zbMATH Open document ID: 0939.35184
DOI: 10.1016/S0362-546X(98)00159-X
author: Pedro Morin
author: R. D. Spies
author: Terry L. Herdman
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2000-02-08T00:00:00Z
review text: In this article, discrete spectral approximations to the nonlinear partial differential equations arising from the conservation laws that model the dynamics of thermomechanical martensitic transformations (solid-solid phase transitions) in one-dimensional shape memory alloys with non-convex Landau-Ginzburg potentials are developed. By using the theories of analytic semigroups and interpolation spaces and a generalization of Gronwall's lemma for singular kernels, the convergence of the approximations is shown to hold not only in the state-space norm but also in the stronger \(\|\cdot \|_\delta\) -norm. The numerical experiments performed using this scheme show that under different initial conditions and distributed external actions the developed model is able to produce solutions whose qualitative behavior is found to be in close agreement with laboratory experiments performed on shape memory alloys under similar conditions. From a practical point of view it would be very important to find the values of the vector parameter \(q\) that ``best fit'' experimental data for a given alloy. This is called the parameter identification problem about which no results are yet known. In this regard the scheme presented here provides a friendly mathematical framework for attacking this problem.
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 74S25
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 74N30
zbMATH DE Number: 1383356
zbMATH Keywords: shape memory alloys
zbMATH Keywords: non-convex potential
zbMATH Keywords: hysteresis
zbMATH Keywords: conservation laws
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: spectral approximations
zbMATH Keywords: analytic semigroups
reviewed by: Leonid B. Chubarov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1383356