Structured Deformations and the Microgeometry of Continua

DMS Award AbstractAward #： 0102477 PI： Owen，David机构： 卡内基梅隆大学 应用数学项目经理：Catherine Mavriplis职务：结构化变形和连续体的微观几何结构化变形提供了一个多尺度的几何和运动学，用于描述在宏观水平上发生在亚宏观水平上的光滑和非光滑变形的影响。 拟议的研究重点是（i）应用这种最近开发的多尺度几何来细化和改进在一个以上的晶体系统上经历滑动的单晶的重要连续描述，（二）通过松弛技术从变分法中识别出测量亚宏观空位扫出的体积的几何变量，并理解该变量与不同类型的位错运动，和（iii）获得的关系，完成系统的平衡法律和本构关系时，出现一个连续的身体经历结构化的变形和运动。 这项研究涉及材料科学和工程领域，以及在多个长度和时间尺度上的几何数学研究。 在变形的回形针或其他金属体中，肉眼看起来连续和光滑的几何变化在显微镜下观察时往往显得不稳定和突然。 类似地，塑料夹层袋或其他薄塑料膜的平滑拉伸在宏观尺度和亚宏观尺度上表现不同。 研究物体（回形针、三明治袋或其他物体）因外力或热源而产生的宏观变化的最成功的理论称为场论。 场论的成功首先取决于我们对一种特定物质的物理理解，例如，构成回形针的特定金属或构成夹层袋的特定塑料影响场论的详细形式，其次，影响我们对构成场论的方程的数学理解。 在这个项目中进行的研究是一个正在进行的计划的一部分，以丰富这些领域的理论，允许他们描述和预测的行为，改变形状不同的身体在宏观和亚宏观的长度尺度。 这项研究的预期成果是精炼、改进的场论，可以更精确地模拟技术上重要的材料的行为。

DMS Award AbstractAward #: 0102477PI: Owen, DavidInstitution: Carnegie Mellon University Program: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Structured Deformations and the Microgeometry of ContinuaStructured deformations provide a multiscale geometry and kinematics for describing the effects at the macrolevel of both smooth and non-smooth deformations occurring at submacroscopic levels. The proposed research focuses on (i) applying this recently developed multiscale geometry to refine and improve important continuum descriptions of single crystals undergoing slipping on more than one crystallographic system, (ii) identifying via a relaxation technique from the calculus of variations a geometrical variable that measures the volume swept out by submacroscopic vacancies and understanding the connection between this variable and different types of dislocation movements, and (iii) obtaining relations that complete the system of balance laws and constitutive relations that arise when a continuous body undergoes structured deformations and motions. This research bears on the fields of materials science and engineering and on the mathematical study of geometry at multiple length and time scales. The geometrical changes in a deforming paper clip or other metallic body that appear continuous and smooth to the naked eye often appear to be jerky and abrupt when viewed in a microscope. Similarly, the smooth stretching of a plastic sandwich bag or other thin plastic film appears differently at macroscopic and submacroscopic scales. The most successful theories for studying macroscopic changes in a body - paper clip, sandwich bag, or otherwise -- brought about by applied forces or heat sources are called field theories. The success of field theories rests first on our physical understanding of how a particular substance, e.g., the particular metal comprising the paper clip or the particular plastic constituting the sandwich bag, influences the detailed form of the field theories and, second, on our mathematical understanding of the equations that comprise the field theory. The research undertaken in this project is part of an ongoing program to enrich these field theories by permitting them to describe and predict the behavior of a body that changes shape differently at macroscopic and submacroscopic length scales. The desired outcome of this research is refined, improved field theories that permit more accurate simulations of the behavior of technologically important materials.Date: June 25, 2001