Energy-Minimizing Microstructures in Multiphase Elastic Solids

Citation

Chenchiah, Isaac Vikram (2004) Energy-Minimizing Microstructures in Multiphase Elastic Solids. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/RXE5-9A33. https://resolver.caltech.edu/CaltechETD:etd-05252004-131315

Abstract

This thesis concerns problems of microstructure and its macroscopic consequences in multiphase elastic solids, both single crystals and polycrystals.

The elastic energy of a two-phase solid is a function of its microstructure. Determining the infimum of the energy of such a solid and characterizing the associated extremal microstructures is an important problem that arises in the modeling of the shape memory effect, microstructure evolution (precipitation, coarsening, etc.), homogenization of composites and optimal design. Mathematically, the problem is to determine the relaxation under fixed volume fraction of a two-well energy.

We compute the relaxation under fixed volume fraction for a two-well linearized elastic energy in two dimensions with no restrictions on the elastic moduli and transformation strains; and show that there always exist rank-I or rank-II laminates that are extremal. By minimizing over the volume fraction we obtain the quasiconvex envelope of the energy. We relate these results to experimental observations on the equilibrium morphology and behavior under external loads of precipitates in Nickel superalloys. We also compute the relaxation under fixed volume fraction for a two-well linearized elastic energy in three dimensions when the elastic moduli are isotropic (with no restrictions on the transformation strains) and show that there always exist rank-I, rank-II or rank-III laminates that are extremal.

Shape memory effect is the ability of a solid to recover on heating apparently plastic deformation sustained below a critical temperature. Since utility of shape memory alloys critically depends on their polycrystalline behavior, understanding and predicting the recoverable strains of shape memory polycrystals is a central open problem in the study of shape memory alloys. Our contributions to the solution of this problem are twofold:

We prove a dual variational characterization of the recoverable strains of shape memory polycrystals and show that dual (stress) fields could be signed Radon measures with finite mass supported on sets with Lebesgue measure zero. We also show that for polycrystals made of materials undergoing cubic-tetragonal transformations the strains fields associated with macroscopic recoverable strains are related to the solutions of hyperbolic partial differential equations.

Thesis Files