Rathematical Sciences: Nonlinear Problems of Solid Mechanics

这个项目将涉及固体力学中一些非线性问题的研究。这些问题包括：(1)粘弹性、耗散和冲击的动力学问题（包括非线性粘弹性体和自持振荡、Hopf分岔和流固相互作用的一维问题）；(2)轻柔体的渐近性；(3)非线性弹塑性分岔问题；(4)接触问题；(5)各向异性介质中相的共存；(6)其他问题，如逆分岔问题、非线性磁弹性问题和控制问题。每个问题都将给出一个精确的公式（没有特别的几何简化），每个问题都使用满足标准不变条件和轻微物理限制的一般本构方程。这种本构方程的使用阐明了控制方程的数学结构，并允许检测与材料响应相关的新效应。研究的目标是发现新的现象，特别是那些与不稳定性和阈值的出现有关的现象，处理阐明一般理论的具体问题，并开发用于处理一般问题和特定问题的分析工具。各种动态和稳态非线性问题的杆，壳，和三维固体将被研究。这些物体由非线性弹性、粘弹性、塑性或磁弹性材料组成。在每种情况下，都要使用包含一般非线性本构方程的完全不变的几何精确理论。这些研究的目标是发现新的非线性效应，确定分离定性不同响应的本构方程的阈值，检查重要的不稳定性类型，为固体中的冲击和耗散机制理论做出贡献，并为固体力学问题开发新的非线性分析方法。

This project will involve research on a number of nonlinear problems in solid mechanics. These problems include: (1) dynamical problems of viscoelasticity and dissipation and shocks (including one-dimensional problems for nonlinearly viscoelastic bodies and self-sustained oscillations, Hopf bifurcations, and fluid-solid interactions); (2) asymptotics of light flexible bodies; (3) bifurcation problems of nonlinear elastoplasticity; (4) contact problems; (5) coexistence of phases in anisotropic media; and (6) other problems such as inverse bifurcation problems, nonlinear magnetoelasticity, and control problems. Each problem will be given an exact formulation (without ad hoc geometric simplifications) and each problem employs general constitutive equations satisfying the standard invariance conditions and mild physical restrictions. The use of such constitutive equations illuminates the mathematical structure of the governing equations and permits the detection of new effects associated with material response. The goals of the research are to detect new phenomena, especially those associated with instabilities and the occurrence of thresholds, to treat concrete problems that illuminate the general theories, and to develop analytic tools for the treatment of both general and specific problems. A variety of dynamic and steady-state nonlinear problems for rods, shells, and three-dimensional solid bodies will be studied. The bodies are composed of nonlinearly elastic, viscoelastic, plastic, or magnetoelastic materials. In each case, fully 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 for problems of solid mechanics.