Nonlinear Homogenization and Applications to Porous and Nematic Elastomers

提案：DMS-0204617PI: Pedro Ponte castanedinstitute: University of宾夕法尼亚大学标题：非线性均匀化及其在多孔和向列弹性体中的应用摘要本项目旨在开发和应用均匀化技术来估计非凸能量函数表征的非均质超弹性材料系统的宏观行为。将详细分析两个典型的例子：多孔弹性体和“多域”液晶弹性体（LCEs）。第一个例子是两相复合材料，一个空洞相和一个弹性相。这些材料因其绝缘和减震性能而广泛应用于各个行业。第二种是涉及由液晶弹性体相制成的单晶颗粒的多晶聚集体的示例。它们是表现出“软”变形模式的材料，能够被温度和光输入驱动，并且还表现出在足够大的拉伸下变得光学透明的显著特性。特别地，我们建议为这些材料开发Hashin-Shtrikman和自洽类型的简单估计，这已经被发现在其他情况下非常有用。然而，由于弹性体系统中涉及的有限变形，因此有必要描述这些系统中微观结构（例如孔隙率、织构、空洞或晶粒形状和取向）的演变及其对整体行为的影响。由于相关能量函数的非凸性，还必须考虑到不稳定的可能发展，特别是因为这种不稳定模式可能在器件设计中有用。这个研究项目将涉及数学分析中几种令人兴奋的工具的组合，包括变分法、凸分析、微分方程和优化，并可能影响我们对复杂材料本构理论的理解。这一建议是关于异质材料系统，可以经历大的弹性（可恢复）变形。这些材料系统的例子包括，除其他外，炭黑填充弹性体，聚合物泡沫，液晶弹性体，嵌段共聚物和骨骼肌组织。它们的力学响应有两个基本特征：1)它们可以承受较大的弹性（可恢复）变形；2)它们表现出非独特的行为（微屈曲和其他不稳定性）。此外，从应用的角度来看，最重要的是，许多这些材料系统的行为可以由外部场（温度，电，磁，化学输入）控制。由于其卓越的性能，这些材料通常以复合材料或多晶聚集体的形式出现，将继续为许多技术创新提供载体，从一个多世纪前的橡胶轮胎到今天的光激活开关和人造肌肉。由于它们的高度非线性特性，这些材料系统的表征在数学上也具有挑战性。特别是，尽管近年来在为这些材料系统开发严格的均质框架方面取得了很大进展，但在开发“建设性”数学工具以估计这一类特定系统的本构行为方面仍有许多工作要做。

Proposal: DMS-0204617PI: Pedro Ponte CastanedaInstitution: University of PennsylvaniaTitle: Nonlinear homogenization and applications to porous and nematic elastomersABSTRACTThis 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 atwo-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, wepropose 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 alsocharacterize 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 betaken 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 complexmaterials 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.