This project proposes to develop and apply homogenization techniques for estimating the macroscopic behavior of heterogeneous hyperelastic material systems that are characterized by nonconvex energy functions. Two prototypical examples will be analyzed in detail: porous elastomers and "polydomain" liquid crystal elastomers (LCEs). The first is an example of a two-phase composite with one void phase and one elastic phase. These materials are used extensively in various industries for their insulation and shock absorption properties. The second is an example of a polycrystalline aggregate involving single-crystal grains made of a liquid-crystal elastomeric phase. They are materials that exhibit "soft'' modes of deformation, are capable of being actuated by temperature and light inputs, and also exhibit the remarkable property of becoming optically transparent at sufficiently large stretches. In particular, we propose to develop simple estimates of the Hashin-Shtrikman and self-consistent type for these materials, which have already been found to be extremely useful in other contexts. However, because of the finite deformations involved in elastomeric systems, it will be necessary to also characterize the evolution of the microstructure (e.g., porosity, texture, void or grain shape and orientation) in these systems and its implications on the overall behavior. Because of the non-convexity of the relevant energy functions, the possible development of instabilities must also be taken into account, especially because such unstable modes may be useful in the design of devices. This program of research will involve an exciting combination of several tools in mathematical analysis, including calculus of variations, convex analysis, differential equations, and optimization, and is likely to impact our understanding of constitutive theory of complex materials in general.
This proposal is concerned with heterogeneous material systems that can undergo large elastic (recoverable) deformations. Examples of these material systems include, among others, carbon-black-filled elastomers, polymeric foams, liquid crystalline elastomers, block copolymers and skeletal muscle tissue. Two essential features characterize their mechanical response: 1) they can undergo large elastic (recoverable) deformations; and 2) they exhibit non-unique behavior (micro-buckling and other instabilities). In addition, most importantly from the applications point of view, the behavior of many of these material systems can be controlled by external fields (temperature, electric, magnetic, chemical inputs). Because of their remarkable properties, these materials, usually appearing in the form of composites or polycrystalline aggregates, will continue to provide the vehicles for many technological innovations, ranging from rubber tires, more than a century ago, to light-activated switches and artificial muscles, today. Because of their highly nonlinear properties, the characterization of these material systems is also mathematically challenging. In particular, even though much progress has been made in recent years in developing rigorous homogenization frameworks for these material systems, much work remains to be done in terms of developing "constructive" mathematical tools to estimate the constitutive behavior of specific systems within this class.
