Nonlinear Homogenization and Applications to Porous and Nematic Elastomers

非线性均质化及其在多孔和向列弹性体中的应用

• 批准号: 0204617
• 负责人: Pedro Ponte Castaneda
• 金额: --
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204617&HistoricalAwards=false
• 关键词: Nonlinear; Homogenization; Applications; Porous; Nematic

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.

提案：DMS-0204617PI：PEDRO PONTE CASTANEDAINSTITITION：宾夕法尼亚大学大学：非线性均质化以及对多孔和列明的弹性弹药的应用提出的，该项目建议开发和应用均化技术，以估算异质材料的量身能源性能，以估算异质材料的量身能源的功能，这些技术是非弹性材料的特征化。 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）。 第一个是具有一个空隙相和一个弹性相的ATWO相复合材料的示例。这些材料在各个行业中广泛用于其绝缘和减震特性。 第二个是涉及由液体晶体弹性相制成的单晶晶粒的多晶骨料的一个例子。它们是表现出“柔软”变形模式的材料，能够通过温度和轻度输入来驱动，并且表现出在足够大的伸展运动中光学透明的显着特性。特别是，Wepropose可以对Hashin进行简单的估计。 - 这些材料的shtrikman和自洽类型，这些材料在其他情况下已经非常有用，因为弹性系统中涉及的有限变形，也有必要使微观结构的演变（例如，这些系统中的孔隙率，质地，空隙或晶粒形状和方向）及其对整体行为的影响。在设备的设计中可能很有用。一般而言，该提案与可能发生弹性（可回收）变形的异质材料系统有关。 这些物质系统的示例包括碳黑色填充弹性体，聚合物泡沫，液晶弹性体，块共聚物和骨骼肌组织。 其机械响应的两个基本特征：1）他们可以经历较大的弹性（可回收）变形； 2）它们表现出非唯一的行为（微弹和其他不稳定性）。 此外，从应用的角度来看，最重要的是，许多这些物质系统的行为可以由外部场（温度，电气，磁性，化学输入）控制。 由于它们具有显着的特性，这些材料通常以复合材料或多晶骨料的形式出现，将继续为许多技术创新提供车辆，从一个多世纪以前的橡胶轮胎到光激活的开关和人工肌肉， 今天。 由于其高度非线性的特性，这些材料系统的表征在数学上也充满挑战。特别是，即使近年来在为这些材料系统开发严格的均质化框架方面取得了很多进展，但在开发“建设性”数学工具方面仍有许多工作要做，以估计该类别中特定系统的本构行为。