Nonlinear Problems of Solid Mechanics

S. S. Antman将处理杆、壳和三维固体的各种动力学和稳态非线性问题。这些物体由非线性弹性、粘弹性、塑性、粘塑性或磁弹性材料组成。 在每一种情况下，适当不变，几何精确的理论，包括一般非线性constitutive方程被使用。 这些研究的目标是发现新的非线性效应，确定本构方程中分离定性不同响应的阈值，确定物理上和数学上都自然的本构方程的一般类别，检查重要的不稳定性，有助于固体中的冲击和耗散机制理论，并为固体力学问题开发非线性分析和有效计算的新方法。该奖项将支持固体力学的数学研究，如金属，塑料，定性现象，如柱和壳的屈曲，波在杆中的传播，以及可能导致断裂的冲击的形成，将被广泛的数学模型类进行研究。这项工作是跨学科的，结合了工程力学，物理学和数学的现代技术。 现代数学方法的一个重要特点是，它可以同时处理各种材料、各种形状和各种载荷的各种物理问题，这一点在数学界以外并不广为人知。从这个意义上说，这种方法是计算方法的对立面。它一次只能处理一个特定的问题。这项工作的部分成果将是有效的数值方法的发展，这丰富的一类问题，一些结果将被数值说明。

S. S. Antman will treat a variety of dynamical and steady-statenonlinear problems for rods, shells, and three-dimensional solid bodies.The bodies are composed of nonlinearly elastic, viscoelastic, plastic, viscoplastic, or magnetoelastic materials. In each case, properly invariant, geometrically exact theories encompassing general nonlinear constititutive equations are to be used. The goals of these studies are to discover new nonlinear effects, determine thresholds in constititutive equations separating qualitatively different responses, determine general classes of constitutive equations that are both physically and mathematically natural, 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 computationfor problems of solid mechanics.The award will support mathematical research on the mechanics of solids, such as metals, plastics, etc. Qualitative phenomena such as the buckling of columns and shells, the propagation of waves in rods, and the formation of shocks that could lead to fracture will be studied for wide classes of mathematical models. This work is interdisciplinary, combining modern techniquesfrom engineering mechanics, physics, and mathematics. One important featureof the modern mathematical techniques employed, which is not widely known outside of mathematical circles, is that the analytical methods used can handle whole classes of physical problems for different kinds of materials in different kinds of shapes subject subject to different kinds of loadings, all at the same time.In this sense, the approach is the antithesis of computational methods,which can handle only one specific problem at a time. Part of the outcome of this work will be the development of effective numerical methods for this rich class of problems, and some of the results will be illustrated numerically.