Phase transitions in a slender cylinder composed of an incompressible elastic material. I. Asymptotic model equation

Abstract

We present reasons, both experimental and mathematical, for why it is important to consider the radial deformation (thus the lateral traction-free boundary conditions as well) for phase transitions in a slender elastic cylinder. One of the main purposes of this paper is to derive an asymptotic model equation, which takes into account the radial deformation and satisfies the traction-free boundary conditions up to the right order. This is achieved from the three-dimensional field equations through a novel approach that combines a series expansion and an asymptotic expansion. An alternative approach based on Whitham's approach (well used in fluids) is also given. Then, we present some interesting analytic solutions for an infinite cylinder, including those that seem to describe the structure of a phase boundary, the nucleation process and the merge of two-phase boundaries. In particular, by considering the energy distribution based on the nucleation solution, it is revealed that the nucleation process is one of energy localizations, concentrations and separation.