Mathematical Sciences: Structured Deformations and the Microgeometry of Continua

9703863欧文提出的研究是一个正在进行的项目的一部分，该项目旨在以系统的方式将小长度尺度上几何变化的影响纳入连续介质力学。这里采用的方法不同于许多其他方法，因为它从一开始就扩大了物体可以经历的变形的集合，包括非经典的“结构变形”。目前，“前端”方法为建模提供了灵活性，并对各种微结构的屈服、滞后和耗散等重要现象进行了简单而直接的预测。所采用的关键数学思想是，函数的导数序列的极限可能不同于函数序列的极限的导数。这两个量之间的差异给出了在小长度尺度上由于“无序”造成的变形量的具体度量。现有的晶体固体、液晶、多晶金属和颗粒材料的理论通过“内部变量”或“指示”引入了这种变形的测量方法，例如塑性变形、分子取向或空隙率。虽然这些变量有自然的物理解释，但在标准方法中，它们必须被接受为理论中的原始对象；对于结构变形，这些变量的对应项可以直接计算为微观几何变化的平均极限。提出的研究将为“速度与滑移拉伸”和“速度与无滑移拉伸”等运动学量提供精确的定义和有用的公式，将实现一般平衡定律和耗散不等式的新细化，并将获得区分不同物质体的特定本构关系的平行细化。应用数学家所面临的一个主要挑战是，如何将每种物质的微观结构的影响纳入数学方程中。应用数学家要对复杂现象进行建模，例如金属棒的弯曲、液晶显示器上对比鲜明的光学场的出现以及沙子在漏斗中的流动。在个别具有重要技术意义的案例中已经取得了相当大的成功，但缺乏一种清晰而有用的语言来系统地包括微观结构。因此，比方说，从研究沙洲的变形转移到研究沙子通过漏斗的流动，几乎需要从建模过程的开始就开始。这项提议的研究是一个正在进行的项目的一部分，该项目旨在提供一种新的、相对简单的数学语言，从而允许更有效、更经济地对这些现象和其他现象进行建模。这个项目的初步成功有助于为一些问题提供简单而有用的答案，比如为什么当从一堆纸中取出回形针时，回形针会弹回一个复杂的弯曲形状，而不是弹回它在制造初期可能假设的简单的直形状。本研究将继续寻求一种更统一、更有效的方法来理解和预测基于微观结构知识的复杂现象。

9703863 Owen The proposed research is part of an ongoing project to incorporate into continuum mechanics in a systematic manner the effects of geometrical changes at small length scales. The approach employed here differs from many others, in that it enlarges from the outset the collection of deformations that a body can undergo to include non-classical "structured deformations." The present, "front-end" approach provides flexibility in modelling and gives simple and direct predictions of important phenomena such as yielding, hysteresis, and dissipation for a variety of microstructures. The key mathematical idea employed is that the limit of a sequence of derivatives of functions may differ from the derivative of the limit of the sequence of func- tions. The difference between these two quantities gives a concrete measure of the amount of deformation due to "disarrangements" at small length scales. Existing theories of crystalline solids, liquid crystals, polycrystalline metals, and granular materials introduce measures of such deformations via "internal variables" or "directors", e.g., plastic deformation, molecular orientation, or void fraction. Although these variables have natural physical interpretations, in standard approaches they must be accepted as primitive objects within a theory; for structured deformations, counterparts of these variables can be calculated directly as limits of averages of geometrical changes at the microlevel. The proposed research will provide precise definitions and useful formulas for kinematical quantitities such as "velocity and stretching due to slip" and "velocity and stretching without slip", will achieve new refinements of general balance laws and dissipation inequalities, and will obtain parallel refinements of specific constitutive relations that distinguish one material body from another. A principal challenge faced by applied mathematicians who model complex phenomena such as the bending of a metal bar, t he appearance of contrasting optical fields on a liquid crystal display, and the flow of sand through a hopper is that of incorporating into the mathematical equations the influence of the microscopic structure of each substance. Considerable success has been achieved in individual cases of technological importance, but there is lacking a clear and useful language for systematically including microstructure. As a result, shifting, say, from studies of deformations in a bar to studies of sand flowing through a hopper requires starting nearly from the beginning of the modelling process. The proposed research is part of an ongoing program to provide a new, relatively simple mathematical language that will permit a more efficient and economical procedure for modelling these and other phenomena. The initial successes of this program help to provide simple and useful answeres to questions such as why a paper clip springs back to a complex, curved shape when removed from a stack of papers and does not spring back to the simple, straight shape that it may have assumed early in its manufacture. The proposed research will continue the quest for a more unified and efficient approach toward understanding and predicting complex phenomena based on knowledge of microstructure.