Nonlinear Mechanics of Defects in Solids

该奖项支持对微分几何的现代工具，在固体中的缺陷力学的应用研究。该公式将提高非线性材料，如金属，软材料和生物组织的失效预测的准确性。固体中的缺陷在决定许多工程材料的机械和物理性能方面起着至关重要的作用。然而，在现有的理论工作中，几乎所有的缺陷分析都局限于线弹性。许多物理学理论最自然地用几何概念来表达。本课题的研究成果也将在材料力学中应用微分几何的高级课程的新教材中进行总结。本课题的研究目的是利用现代微分几何的技术，建立缺陷力学的完全非线性几何理论，解决具有分布线缺陷和点缺陷的固体非线性力学中的几个未决问题。在非线性弹性力学的设定中，对缺陷的了解很少。对于具有分布位错、向错、点缺陷及其组合的非线性固体的应力场，将得到几个新的精确解。不太知道的缺陷在大应变的存在下的相互作用，我们的初步结果表明，几何理论将是非常有用的，在这个方向上取得进展。非线性缺陷的动力学和稳定性研究是令人兴奋和智力挑战的问题，将在本项目的过程中进行调查。

This award supports research towards the application of modern tools of differential geometry to the mechanics of defects in solids. The formulation would improve the accuracy of failure predictions for nonlinear materials such as metals, soft materials, and biological tissues. Defects in solids play a crucial role in determining the mechanical and physical properties of many engineering materials. However, the analysis of defects has been restricted to linear elasticity in almost all the existing theoretical works. Many physical theories are most naturally expressed in terms of geometric notions. The research outcomes of this project will also be summarized in a new textbook for an advanced course on differential geometry applied to mechanics of materials.This research program aims to use techniques from modern differential geometry to formulate a fully nonlinear geometric theory of defect mechanics that will enable one to solve several open problems in the nonlinear mechanics of solids with distributed line and point defects. There is very little known about defects in the setting of nonlinear elasticity. Several new exact solutions will be obtained for the stress fields of nonlinear solids with distributed dislocations, disclinations, point defects, and their combinations. Not much is known about the interaction of defects in the presence of large strains and our preliminary results show that the geometric theory will be very useful in making progress in this direction. Dynamics and stability studies of nonlinear defects are exciting and intellectually challenging problems that will be investigated during the course of this project.