Mathematical Sciences: Structured Deformations and the Microgeometry of Continua

小行星9703863 拟议的研究是一个正在进行的项目的一部分，以系统的方式纳入连续介质力学的影响，在小长度尺度的几何变化。 这里采用的方法与许多其他方法不同，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“结构化变形”。目前的“前端”方法提供了建模的灵活性，并对各种微结构的屈服、滞后和耗散等重要现象给出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以不同于函数序列的极限的导数。 这两个量之间的差给出了在小长度尺度下由于“无序”引起的变形量的具体量度。 结晶固体、液晶、多晶金属和颗粒材料的现有理论通过“内部变量”或“指向矢”引入了这种变形的测量，例如， 塑性变形、分子取向或空隙率。 虽然这些变量具有自然的物理解释，但在标准方法中，它们必须被接受为理论中的原始对象;对于结构化变形，这些变量的对应物可以直接计算为微观水平上几何变化平均值的极限。 拟议的研究将提供精确的定义和有用的公式运动学量，如“速度和拉伸由于滑移”和“速度和拉伸无滑移”，将实现新的细化一般平衡定律和耗散不等式，并将获得并行细化的具体本构关系，区分一个物质体从另一个。 应用数学家在模拟复杂现象时，如金属棒的弯曲、液晶显示器上出现的对比光场以及沙子通过漏斗的流动，他们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。 在具有技术重要性的个别情况下已经取得了相当大的成功，但缺乏一个系统地包括微观结构的明确和有用的语言。 因此，从研究杆中的变形到研究流过料斗的沙子，几乎需要从建模过程的一开始就开始。 拟议的研究是一个正在进行的计划的一部分，以提供一个新的，相对简单的数学语言，将允许一个更有效和更经济的程序来模拟这些和其他现象。 这个程序的初步成功有助于为一些问题提供简单而有用的答案，例如为什么一个回形针从一堆纸中取出时会弹回复杂的弯曲形状，而不会弹回它在制造早期可能呈现的简单的直线形状。 拟议的研究将继续寻求一种更统一和有效的方法， 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.