The Response of Composites and Quasiconvexity

提案：DMS-041103PI: Graeme W. milton机构：犹他大学标题：复合材料和准面蠕变的响应摘要这项工作将通过建立总蠕变边界和识别表现出最大或最小可能蠕变的微观结构来提高我们对复合材料蠕变的理解。它还将确定在线弹性中最适合引导应力的微观结构。这些复合材料对于分配或集中结构中的力是有用的。此外，微结构将与异常霍尔系数相识别，当存在磁场时，这可能有助于减少电路设计中的杂散电场。在工作的第二部分，将研究三个基本问题：在张量空间中描述与层次层合几何的有效张量集相关的曲面；探索秩一凸性与拟凸性的关系并开发了新的工具来限定复合函数的有效模和限定函数的拟共面化。这些问题对于在复合材料的什么响应是可能的或不可能的问题上取得进展是重要的。他们的解决方案可以，在各种各样的情况下，使数值易于处理的搜索可能的宏观响应的复合材料。更好地理解材料的宏观响应在技术上具有重要意义。从理解工程材料的宏观反应（对国防、汽车和航空航天工业至关重要），到理解多晶和多孔岩石的宏观反应（与地震预测和石油工业相关），再到理解海冰的宏观反应（对气候建模很重要），这种重要性是全面延伸的。了解生物材料（如组织、骨骼、外壳和肌腱）的宏观反应。在过去，许多有用的复合材料是通过试错、模仿自然界中发现的复合材料或凭直觉获得的。在本世纪，我们将有更大的灵活性来生产定制的复合材料，以满足特定的需求。提高我们对复合材料响应的理解将有助于这些设计复合材料的构建。

Proposal: DMS-041103PI: Graeme W. MiltonInstitution: University of UtahTitle: The Response of Composites and QuasiconvexityABSTRACTThis work will improve our understanding of creep in composite materials, through establishing bounds on the total creep and identifying microstructures that exhibit the maximum or minimum possible creep. It will also identify microstructures which are best for guiding stress in linear elasticity. These composites should be useful for distributing or concentrating the force in a structure. Additionally, microstructures will be identified with an anomalous Hall coefficient, which could potentially be useful in reducing stray electric fields in circuit design when there is a magnetic field present. In a second part of the work three fundamental problems will be studied: characterizing the surfaces in tensor space associated with sets of effective tensors of hierarchical laminate geometries; exploring the connection between rank-one convexity and quasiconvexity; and developing new tools for bounding the effective moduli of composites and for bounding the quasiconvexification of a function. These problems are important for making progress on the question of what responses of composites are or are not possible. Their solution could, in a wide variety of cases, make numerically tractable the search for the possible macroscopic responses of composites.A better understanding of the macroscopic response of materials is of central technological importance. This importance stretches across the board, from understanding the macroscopic response of engineered materials (of critical importance to the defence, automative, and aerospace industries), to understanding the macroscopic response of polycrystalline and porous rocks (relevant to earthquake prediction and to the oil industry), to understanding the macroscopic response of sea ice (important to climate modelling), to understanding the macroscopic response of biological materials (such as tissues, bones, shells and tendons). In the past many useful composites were obtained by trial and error, or by mimicking composites found in nature, or by using intuition. In this century we will have greater flexibility to produce designer composites tailor made to meet specific needs. Improving our understanding of the response of composites will facilitate the construction of these designer composites.