Mathematical Sciences: Structured Deformations and the Microgeometry of Continua

数学科学：结构变形和康体佳微观几何

• 批准号: 9703863
• 负责人: David Owen
• 金额: $ 7.8万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9703863&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Structured; Deformations; Microgeometry

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.

9703863 Owen 拟议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中。 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能经历的变形集合，以包括非经典的“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直接的预测。 采用的关键数学思想是函数导数序列的极限可能不同于函数序列极限的导数。 这两个量之间的差异给出了小长度尺度下由于“无序”引起的变形量的具体测量。 现有的晶体固体、液晶、多晶金属和颗粒材料理论通过“内部变量”或“导向器”引入了对此类变形的测量，例如塑性变形、分子取向或空隙率。 尽管这些变量具有自然的物理解释，但在标准方法中，它们必须被视为理论中的原始对象；对于结构化变形，这些变量的对应变量可以直接计算为微观水平上几何变化平均值的极限。 拟议的研究将为运动学量（例如“由于滑移而产生的速度和拉伸”和“无滑移的速度和拉伸”）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，并将获得区分一种物质体与另一种物质体的特定本构关系的并行细化。 应用数学家对复杂现象（例如金属棒的弯曲、液晶显示器上对比光场的出现以及沙子通过料斗的流动）进行建模时面临的主要挑战是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案例中已经取得了相当大的成功，但缺乏一种清晰且有用的语言来系统地包括微观结构。 因此，从对棒材变形的研究转变为对流过料斗的沙子的研究几乎需要从建模过程的一开始就开始。 拟议的研究是正在进行的计划的一部分，该计划旨在提供一种新的、相对简单的数学语言，该语言将允许更有效和更经济的程序来建模这些现象和其他现象。 该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，而不会弹回其在制造初期可能呈现的简单直线形状。 拟议的研究将继续寻求一种更加统一和有效的方法来理解和预测基于微观结构知识的复杂现象。