Nonconvex Homogenization and Applications

该奖项将支持非线性均匀化领域中一类具有挑战性的数学问题的研究。具体而言，该项目涉及有限弹性复合材料（如弹性泡沫或颗粒增强橡胶）的有效或“均匀化”行为的研究。这些都是异质材料系统，其特征是非线性的，事实上非凸的局部能量函数。其关键思想是利用适当选择的“线性比较复合”的可用均匀化估计，以生成相应的非线性复合的估计。这是PI在具有凸能量函数的非线性问题（例如，塑性、非线性导电性等）。通过适当地扩大“试验”线性比较介质的类，将开发一个广义版本的方法，将适用于nonconvex problems.本研究的目的是因此开发的数学方法估计的宏观行为的非线性弹性复合材料，受到有限的变形。这项工作将直接关系到技术上重要的材料系统的范围，从普通的，如弹性泡沫，到更复杂的，如形状记忆合金。特别是，这项工作将侧重于颗粒增强弹性体的模型（例如，炭黑填充橡胶）和多孔弹性固体（例如，因此，将产生对这些材料的响应的有效估计，这些估计易于工程界使用，并且可以与实验研究进行比较。

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.