### Variational Methods For Elastic Crystals

University of Minnesota, 1985

Abstract

We discuss stable equilibria of homogeneous crystalline elastic bodies subjected to body forces and dead loading tractions on the boundary. Frequently, austenite and twinned martensite phases occur simultaneously in solid crystals, corresponding to highly nonsmooth deformations. In order to describe this phenomenon, the class of admissible deformations must be enlarged to include discontinuous gradients. This renders impossible the use of techniques of Differential Calculus frequently explored to study equilibrium states in elasticity (e.g., inverse function theorem). Also, the standard methods of the Calculus of Variations cannot be employed due to the failure of growth and convexity conditions which results from the symmetry properties of crystals. Using the material symmetry group and special piecewise homogeneous deformations with discontinuous gradients across planes with particular orientations relative to the lattice vectors, we prove that the total energy is not bounded below unless the body forces and surface tractions vanish identically. By analyzing deformations constructed from violent shearing of slices supported by planes oriented as before for the global problem, we obtain necessary conditions for the metastability of N-phase deformations. One of these conditions is a balance law similar to the balance of angular momenta with the deformation replaced by its gradient. We show that C^{1} deformations with zero traction on the boundary and zero total force must satisfy a first order linear system of PDEs with coefficients depending exclusively on the body force. We also give an example of the application of these necessary conditions. Finally, the molecular theory of solid crystals is used to analyze and construct strain energy functions appropriate for an elastic crystal undergoing a second order cubic-tetragonal phase transition. Differently from the constructions proposed in the current literature, we explore an idea introduced by Ericksen suggesting series representations. Sufficient conditions for stability are obtained by studying the second derivatives of the strain energy function at the critical temperature.