Stability of Elastic Crystals

Summary

INTRODUCTION

Twinning, phase transitions and memory shape effects are frequently observed in solid materials with crystalline structure (cf. [3], [19]). As remarked by Wayman & Shimizu [19], it is of great theoretical and technological interest to investigate and predict the onset of such phenomena and to analyse equilibria and stability for this type of material.

The mathematical approach to these problems has recently been addressed by several authors, including Chipot & Kinderlehrer [4], Ericksen [7], [8], [9], Fonseca [10], [11], [12], Fonseca & Tartar [13], James [13], Kinderlehrer [15], Parry [17] and Pitteri [18], within the framework of a continuum theory based on nonlinear thermoelasticity proposed by Ericksen [7], [8]. The thermoelastic behaviour is quite noticeable in certain alloys for some ranges of temperature. As an example, Basinski & Christian [3] found that at room temperature Indium-Thallium alloys are perfectly plastic but for temperatures above 100°C or below 0°C they exhibit rubber like behaviour.

We summarize briefly the notion of an elastic crystal as described by Ericksen [9] and Kinderlehrer [15] (see also Ericksen [7], [8], and Fonseca [10]). In terms of molecular theory, a (pure) crystal is a countable set of identical atoms arranged in a periodic manner. By fixing a cartesian coordinate system with origin at one of the atoms and orthonormal basis [e₁, e₂, e₃]l the position vectors of the atoms of the lattice relative to some choice of linearly independent lattice vectors {a₁, a₂, a₃} may be given by where 114 m_i ∈ Z, i =; 1, 2, 3, and A is the matrix with columns a₁, a₂, a₃.