Mathematical Sciences: Microstructure and its Influence on Composite Material Properties

本方案旨在研究具有新颖或极端弹性行为的微观结构，发展新的变分原理作为分析工具，推导粘弹性复合材料的复合剪切模量的边界，获得平面弹性、反平面弹性和压电性之间的精确关系，寻求耦合方程和非耦合方程之间更普遍的联系。并利用积分方程方法对耦合场问题进行数值求解。复合材料和多晶材料在自然界和工程材料中无处不在。它们在技术上的重要性从地震预测（许多岩石是多晶的），石油工业（含油的岩石），航空航天工业（轻质复合材料），到了解海冰，盐水和冰的复合材料（对气候建模很重要），胶体（流体中悬浮颗粒的复合材料），形状记忆材料（金属相的不同变体的复合材料），到水听器（压电材料的复合材料），到生物应用（骨、壳、肌腱都是复合结构的例子）。这一建议将提高我们对复合材料和多晶材料的认识，从而促进新材料的开发，并改进评估复合材料性能的程序。

DMS-9501025 Milton This proposal aims to investigate microstructures which have novel or extremal elastic behavior, develop new variational principles as a tool for analysis, derive bounds on the complex shear modulus of viscoelastic composites, obtain exact relations between plane elasticity anti-plane elasticity and piezoelectricity, seek a more general link between coupled equations and uncoupled equations, and numerically solve coupled field problems through the use of integral equation methods. Composite and polycrystalline materials are ubiquitous in nature and amongst engineered materials. Their technological importance stretches from applications in earthquake prediction (many rocks are polycrystalline),in the oil industry (rocks containing oil), the aerospace industry (lightweight composites), to understanding sea ice, a composite of brine and ice (important to climate modelling), colloids (a composite of suspended particles in a fluid),to shape memory materials (a composite of different variants of a metallic phase), to hydrophones (composites of piezoelectric materials), to biological applications (bones, shells, tendons are all examples of composite structures). This proposal will enhance our knowledge of composite and polycrystalline materials in ways that will facillitate the development of new materials, and improve procedures for estimating the properties of composites.