Two-Phase Equilibria in Nonlinear Elasticity

via Global Bifurcation

Timothy J. Healey

Cornell University

**tjh10@cornell.edu**

** **

** **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).