### Tuesday, 12 October 2004 - 10:55 AM

### This presentation is part of : Horgan Symposium

### A Global Bifurcation Appraoch to Two-Phase Equilibira in Elastic Solids

**Timothy J Healey**, Cornell University, Cornell University, Theoretical & Applied Mechanics, Kimball Hall, Ithaca, NY 14853
We consider multi-phase equilibria of elastic solids under loading. A well known approach to this class of problems is to infer micro-structure via minimizing sequences of the potential energy, the minimum of which is not attained. In spite of its successes, that method allows “infinite refinement” of phase mixtures, ignores equilibrium conditions and is not easily generalized to problems with external loading. Instead, we propose the use of global bifurcation methods, in the presence of small interfacial energy, to determine paths of equilibria. We present results for “two-well” solids, appropriate for twinning. We establish the rigorous existence of global bifurcating branches of equilibria, with solutions along each branch having the precise symmetry of a certain eigenfunction of the linearization. We also obtain a-priori bounds on solutions, insuring existence in the presence of arbitrarily small interfacial contributions, i.e., "thin" transition layers. We then perform global-numerical path following in appropriate fixed-point spaces (according to the known symmetry of a given branch), i.e., the orientation of our mesh is prescribed by the symmetry. We obtain branches of locally stable equilibria, exhibiting phase nucleation (and anti-nucleation), fine layering of phases, and phase-tip splitting at the boundary - all in qualitative agreement with experiment.

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