Nonconvex Homogenization and Applications

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This award will support research on a challenging class of mathematical problems in the field of nonlinear homogenization. Specifically, the project is concerned with the study of the effective or "homogenized" behavior of finite elastic composites such as elastic foams or particle-reinforced rubber. These are heterogeneous material systems that are characterized by nonlinearand in fact nonconvex local energy functions. The key idea is to make use of available homogenization estimates for suitably chosen "linear comparison composites" to generate corresponding estimates for the nonlinear composites. This is an idea that has been pioneered with success by the PI in the context of nonlinear problems with convex energy functions (e.g., plasticity, nonlinear conductivity, etc.). By suitably enlarging the class of "trial" linear comparison media, a generalized version of the method will be developed that will be suitable for nonconvex problems.The objective of this research is thus to develop mathematical methods for estimating the macroscopic behavior of nonlinear elastic composite materials that are subjected to finite deformations. The work will be directly relevant to arange of technologically important material systems, from the commonplace,such as elastomeric foams, to the more sophisticated, such as shape-memoryalloys. In particular, the work will focus on models for particle-reinforced elastomers (e.g., carbon-black filled rubbers) and porous elastic solids (e.g., polymeric foams).As a result, effective estimates for the response of such materials will be producedthat are easy to use by the engineering community and that can be compared to experimental studies.