Modelling nonlinear dynamics of shape‐memory‐alloys with approximate models of coupled thermoelasticity

Abstract

We present a general methodology for constructing approximate models describing shape memory alloys dynamics. We base our discussion on a general three-dimensional model and the Falk-Konopka representation for the free energy function. By considering a one-dimensional counterpart of that model, we show that with little computational efforts we can reproduce successfully phase transition phenomena with our numerical scheme. The same scheme is applied in our code for the general case. Then, we describe a systematic approach to modelling SMA materials, and demonstrate that approach in deriving a centre-manifold-based low-dimensional model from the general three-dimensional model, preserving all main features of the dynamics. Computer algebra technique allows us to derive such models efficiently and with arbitrary degree of accuracy.

Computational models for multi-scale coupled dynamic problems

“…The behaviour of one-dimensional SMA samples is reasonably well understood, and austenitic-martensitic transformations, as well as the associated hysteresis phenomena, can be reproduced computationally (see [15,16] and references therein). However, the extension of such computational models allowing to describe multidimensional responses is more difficult.…”

“…The technique has been used widely in the context of ODE, including bifurcation analysis and general nonlinear delay-differential equation (e.g., [17] and references therein). From a theoretical point of view the technique is well established (see, e.g., [4,16,18,20] and references therein). Although the technique was conceived for ordinary differential equations, at present time application areas of the centre manifold technique to PDE-based models are growing rapidly.…”

“…Similar to the centre manifold technique the (approximate) inertial manifold methodology allows us to reduce infinite-dimensional PDEs to a finite set of ODEs, but the tools of such a reduction are somewhat different in these two cases. While in the inertial manifold methodology we use asymptotic "slaving rules" (or exponential tracking properties that follow from (2.4)) in an attempt to achieve the asymptotic completeness of the model, in the centre manifold technique the main emphasis is placed on a constructive procedure of obtaining a model, solutions of which are attracted exponentially quickly to the solutions of the original system (see also [16]). …”

Simulation of Nonlinear Thermomechanical Waves with an Empirical Low Dimensional Model

“…This step is equivalent to the one described in [12] where Eq. (2) is rewritten in a general dynamical system form…”

“…The idea of constructing a lower dimensional dynamic system from a given higher dimensional system is to replace the dynamics of the given system by a lower dimensional subspace of the origin state space [7,5,12]. Following the standard procedure, we substitute the approximation Eq.…”