Mathematical Sciences: Phase Transitions, Defects and Nonconvex Variational Problems

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The general objective of this project is to study material instabilities, such as phase transitions for fluids and elastic solids and metastable equilibrium states for crystals with defects, in a Continuum Mechanics framework. The theoretical tools involved in this program are continuum mechanics, geometric measure theory, partial differential equations, thermoelastodynamics and the calculus of variations. Relaxation and lower semicontinuity properties of nonconvex bulk and interfacial energies will be obtained and singular perturbation methods will be used in nonlinear elasticity to resolve nonuniqueness of equilibria. These results will help us understand and predict the surface structures and geometries of crystals subjected to thermal or mechanical treatments. When the minimum energy is not attained, the role played by the surface tension in stabilizing the oscillations, and the dynamical creation of the microstructure and its evolution within the framework of generalized measure-valued solutions, will be studied. Thermochemical equilibria for coherent two-phase alloys when physical variables, such as composition, are taken into account will be considered. This study will enable predicting the dependence of equilibrium phase composition on the overall composition and volume fraction. Models to analyze defect interaction, dislocations, different types and arrangements of domains and global effects of defects will be addressed. In particular, variational formulations for metastable equilibria of elastoplastic crystals with defects will be analyzed. The mathematical problems addressed in this project -- phase transitions for fluids and alloys, equilibrium states for crystals with defects, defect interaction and global effects of defects -- are areas of intense work in contemporary material science. Only recently they have been addressed mathematically in a systematic way. The questions involved escape the framework of classical mathematical theories. Therefore, accomplishing these goals will require manipulation of some very recent mathematical tools and possibly the introduction of new ones. Addressing such issues may have a significant impact in the industry and technology as it will help to predict the surface structures and geometries of crystals subjected to thermal or mechanical treatments and, in general, it will promote a better understanding of smart materials.