Structured Deformations and the Microgeometry of Continua

DMS奖摘要奖#：0102477PI：Owen，DavidInstitution：Carnegie Mellon大学项目：应用数学项目经理：Catherine Mavriplis标题：结构化变形和连续的微观几何结构变形提供了多尺度几何和运动学，用于描述在宏观水平上发生的光滑和非光滑变形在亚宏观水平上的影响。建议的研究集中在：(I)应用最近发展的多尺度几何来改进和改进对单晶在不止一个晶体系统上滑移的重要的连续统描述，(Ii)通过松弛技术从变分中识别一个几何变量，该几何变量测量亚宏观空位所扫出的体积，并了解该变量与不同类型的位错运动之间的联系，以及(Iii)获得完善的平衡定律和连续物体经历结构化变形和运动时产生的本构关系的系统。这项研究涉及材料科学和工程领域，以及多个长度和时间尺度的几何数学研究。一个变形的回形针或其他金属物体在肉眼看来是连续和光滑的几何变化，在显微镜下看起来往往是陡峭和突然的。同样，塑料夹心袋或其他薄塑料薄膜的平滑拉伸在宏观和亚宏观尺度上表现出不同的表现。在研究物体--回形针、夹心袋或其他东西--由外力或热源带来的宏观变化时，最成功的理论称为场论。场论的成功首先取决于我们对特定物质的物理理解，例如组成回形针的特定金属或构成夹心袋的特定塑料如何影响场论的详细形式，其次取决于我们对构成场论的方程的数学理解。这个项目中进行的研究是一个正在进行的计划的一部分，该计划旨在通过允许这些场论描述和预测在宏观和亚宏观长度尺度上以不同方式改变形状的物体的行为来丰富这些场论。这项研究的预期结果是精炼、改进的领域理论，使之能够更准确地模拟具有重要技术意义的材料的行为。日期：2001年6月25日

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