Energy minimization and the formation of microstructure in dynamic anti-plane shear

We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinitedimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite dimensional "chaos". We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3) Using a transformation due to RYBKA, the problem is recast as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in IP spaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular we are unable to obtain analogues of the failure to minimize and non-propagation of strain discontinuity results obtained by BALL, HOLMES, JAMES, PEGO &; SWART [1991] for a one-dimensional model problem.