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几何变化的效果纳入连续机械。 此处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人体可以进行的变形收集，以包括非经典的“结构化变形”。 目前的“前端”方法为建模提供了灵活性，并简单而直接地预测了重要现象，例如屈服，滞后和各种微观结构的耗散。 所采用的关键数学思想是，函数衍生物序列的极限可能与功能序列序列的衍生物有所不同。 这两个数量之间的差异给出了由于在小长度上“混乱”而导致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“导管”（例如塑性变形，分子取向或空隙分数）引入了这种变形的度量。 尽管这些变量具有自然的物理解释，但在标准方法中，它们必须被认为是理论中的原始对象。对于结构化变形，可以将这些变量的对应物直接计算为在微观层处的几何变化平均值的限制。 拟议的研究将为运动量定量提供精确的定义和有用的公式，例如“速度和由于滑移引起的延伸”和“速度和伸展，而无需滑动”，将实现一般平衡法律和耗散不平等的新细化，并将获得与另一种物质区与另一种物质主体区分开的特定构型关系的平行精致。 应用数学家面临的主要挑战，他们对复杂现象进行了建模，例如金属棒的弯曲，对比液晶显示器上的光场的外观以及沙子通过料斗的流动是将每种物质微观结构的影响纳入数学方程。 在各个技术重要性的情况下，已经取得了巨大的成功，但是在包括微观结构（包括微观结构）的系统上缺乏清晰且有用的语言。 结果，从钢筋中的变形研究转变为流过料斗的沙子的研究几乎需要从建模过程开始就需要开始。 拟议的研究是一个正在进行的计划的一部分，该计划提供了一种新的，相对简单的数学语言，该语言将允许更有效，更经济的程序来建模这些现象和其他现象。 该程序的最初成功有助于为问题提供简单而有用的答案，例如，当从一堆纸上移除时，纸夹弹簧恢复到复杂的，弯曲的形状，并且不会回到其制造早期可能已经假设的简单，直形的形状上。 拟议的研究将继续寻求一种基于微观结构知识来理解和预测复杂现象的更统一和有效的方法。