Two-Phase Equilibria in Nonlinear Elasticity
via Global Bifurcation
Timothy J. Healey
We consider two-phase equilibria of elastic bodies. A well known method for such problems is the inference of micro-structure via minimizing sequences of the free energy. In spite of its success, that approach allows for “infinite refinement” of phase mixtures, typically violates equilibrium conditions, ignores local energy minima and is not readily generalized to problems under external loading. In experiments on shape-memory alloys, one typically observes an initial homogeneous state giving rise to more “exotic” states as loading parameters are varied. Accordingly, we propose the use of global bifurcation methods, in the presence of small interfacial energy, to determine paths of equilibria. We mostly present results for a model “two-well” solid in anti-plane shear. We establish the rigorous existence of global bifurcating branches of equilibria, with each branch having the precise symmetry of a certain eigenfunction of the linearization. We then perform “unorthodox” global-numerical path following in appropriate fixed-point spaces; in particular, the orientation of the mesh is prescribed by symmetry. The “unorthodoxy” here refers to the fact that we treat both the capillarity coefficient and the actual loading parameter as “control” variables. We obtain branches of locally stable equilibria exhibiting phase nucleation (and anti-nucleation) and fine layering of phases - all in qualitative agreement with experiment. We also indicate the promise of this approach to elastic models for a class of biomembranes called liposomes - closed vesicles of lipid-bilayer structures (the latter of which mimic the behavior of cellular membranes).