Nonlinear Problems of Solid Mechanics

安特曼0708180 S. S. Antman和他的同事处理各种动态和稳态非线性问题的可变形杆，壳和三维固体，可以遭受大的和快速变形。 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。 在每种情况下，适当不变，几何exacttheories包括一般非线性本构方程。 这些研究的目的是（i）发现新的非线性效应，（ii）确定本构方程中分离定性不同响应的阈值，（iii）确定物理上和数学上都自然的本构方程的一般类别，（iv）检查和预测重要类型的不稳定性，（v）确定如何存在，规律性，和适定性的解决方案的governingquasilinear系统的偏微分方程依赖于材料的行为，（vi）有助于理论的冲击和耗散机制的固体，以及（vii）开发固体力学问题的非线性分析和有效计算的新方法。 Antman继续使数学家能够接触到精心制定的物理理论，并使科学家和工程师能够接触到应用分析的现代技术，特别关注研究生。 S. S. Antman和他的同事们开发了强大的新数学方法来分析固体的大而快的变形。 这项工作既有应用于新技术材料，包括“智能”材料和旧的生物材料，如活组织。 （尽管50年前人们就知道如何从第一原理中干净地推导出控制这种变形的方程，而不需要特别的近似，但处理这些方程的数学工具直到现在才被开发和完善。） 所研究的具体问题包括：（1）薄壳状物体在流体流动中的变形，（2）这种壳体在脉动压力下的稳定性损失，（3）鳗鱼的游动，（4）当物体所承受的载荷施加得很慢或载荷很大或物体重量很轻时，（5）壳体的屈曲，（6）旋转物体的奇异效应，（7）激波的发展和消除（激波是速度和变形的大而突然的不连续性），（8）有效计算方法的建立。 Antman正在继续使数学家能够接触到精心制定的物理理论，并使科学家和工程师能够接触到应用数学的现代技术，特别关注研究生。

Antman0708180 S. S. Antman and his coworkers treat a variety of dynamicaland steady-state nonlinear problems for deformable rods, shells,and three-dimensional solid bodies that can suffer large andrapid deformations. The bodies are composed of nonlinearlyelastic, viscoelastic, plastic, viscoplastic, or magnetoelasticmaterials. In each case, properly invariant, geometrically exacttheories encompassing general nonlinear constitutive equationsare used. The goals of these studies are to (i) discover newnonlinear effects, (ii) determine thresholds in constitutiveequations separating qualitatively different responses, (iii)determine general classes of constitutive equations that are bothphysically and mathematically natural, (iv) examine and predictimportant kinds of instabilities, (v) determine how existence,regularity, and well-posedness of solutions of the governingquasilinear systems of partial differential equations depend onmaterial behavior, (vi) contribute to the theory of shocks anddissipative mechanisms in solids, and (vii) develop new methodsof nonlinear analysis and of effective computation for problemsof solid mechanics. Antman is continuing to make carefullyformulated physical theories accessible to mathematicians, and tomake modern techniques of applicable analysis accessible toscientists and engineers, with special attention to graduatestudents. S. S. Antman and his coworkers develop powerful newmathematical methods for analyzing the large and rapiddeformations of solid bodies. This work has applications both tonew technological materials including "smart" materials and toold biological materials, such as living tissue. (Although ithas been known for 50 years how to derive the equations governingsuch deformations cleanly from first principles without ad hocapproximations, the mathematical tools for treating theseequations are only now being developed and refined.) Thespecific problems under study include (1) the deformation of thinshell-like bodies in a fluid flow, (2) the loss of stability ofsuch a shell subjected to pulsating pressures, (3) the swimmingof eels, (4) the rigorous justification of useful approximationsfor the motions of the body when the loads it bears are appliedslowly or when the loads are very large or when the body is oflight weight, (5) the buckling of shells, (6) strange effects forspinning bodies, (7) the development and the preclusion of shocks(which are large and sudden discontinuities in velocity anddeformation), and (8) the construction of effective methods ofcomputation. Antman is continuing to make carefully formulatedphysical theories accessible to mathematicians, and to makemodern techniques of applicable mathematics accessible toscientists and engineers, with special attention to graduatestudents.