Mathematical Sciences: Nonlinear Problems of Solid Mechanics

9623261 Antman The PI建议处理杆、壳和三维实体的各种动态和稳态非线性问题。物体由非线性弹性、粘弹性、塑料、粘塑性或磁弹性材料组成。他还打算研究流固相互作用。在每种情况下，都要使用包含一般非线性本构方程的适当不变的几何精确理论。这些研究的目的是发现新的非线性效应，确定区分不同响应的本构方程的阈值，检验重要的不稳定性，为固体中的激波和耗散机制理论做出贡献，并发展固体力学问题的非线性分析和有效计算的新方法。研究人员建议研究各种经历大动态变形的固体的行为，包括固体与流体相互作用的情况，就像在飞机和船只上发生的那样。通过将现代数学方法应用到具有极大普遍性和精确度的可变形固体理论中，他打算同时处理材料大族的问题。这项工作的目的之一是确定意外的不稳定性，这种不稳定性可能会对技术应用中使用的结构的响应产生不利影响。他还建议为材料行为理论、材料行为控制方法的开发(例如，通过使用涉及电磁效应的智能材料)以及能够处理这些困难和重要应用的新的数学和计算方法的开发做出贡献。***

9623261 Antman The PI proposes to treat a variety of dynamical and steady-state nonlinear problems for rods, shells, and three-dimensional solid bodies. The bodies are composed of nonlinearly elastic, viscoelastic, plastic, viscoplastic, or magnetoelastic materials. He also intends to treat fluid-solid interactions. In each case, properly invariant, geometrically exact theories encompassing general nonlinear constitutive equations are to be used. The goals of these studies are to discover new nonlinear effects, determine thresholds in constitutive equations separating qualitatively different responses, examine important kinds of instabilities, contribute to the theory of shocks and dissipative mechanisms in solids, and develop new methods of nonlinear analysis and of effective computation for problems of solid mechanics. %%% The investigator proposes to study the behavior of various solids undergoing large dynamic deformations, including cases in which the solid is interacting with a fluid, as happens in airplanes and ships. By applying modern mathematical methods to theories of deformable solids formulated with great generality and precision, he intends to treat problems for large families of materials at one time. One aim of this work is to determine unexpected instabilities, which can adversely alter the response of structures used in technological applications. He also proposes to contribute to the theory of material behavior, to the development of methods for the control of material behavior (by using smart materials involving electromagnetic effects, for example), and to the development of new mathematical and computational methods capable of handling these difficult and important applications. ***