Nonlinear Problems of Solid Mechanics

研究人员对各种动态和稳态非线性问题进行了仔细的数学处理，这些问题包括可变形的杆、壳和三维固体，可能与移动流体、变温场和电磁场接触。物体由非线性弹性、塑性、粘塑性或磁（粘）弹性材料组成。在每种情况下，都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。这些研究的目标是发现新的非线性效应和新的不稳定性，确定本构方程的阈值，分离定性不同的响应，确定物理和数学上自然的本构方程的一般类别，确定存在性，规律性和适定性如何依赖于材料行为，为固体中的冲击和耗散机制理论做出贡献。发展固体力学问题的非线性分析和有效计算的新方法。在处理新技术材料和旧生物材料（如活组织）的行为时，基于这种简化假设的小变形和线性响应的传统理论很可能无法捕捉重要的物理现象，实际上可能会给出误导性的信息。例如，对传统理论的特定问题的研究可能表明，当外部施加的载荷或频率超过一定阈值时，固体中会发生危险的不稳定性。对同样问题的分析，如果不采用简化的假设，就可以很好地表明，大量材料在远低于简化理论所预测的阈值时产生不稳定性。由于正确表述的固体力学的精确非线性问题很少能直接归入现有的数学理论来处理控制方程，本项目的目的之一是发展新的数学技术来处理这类问题。这些技术可以一次处理各种材料。研究结果对受荷载的结构（桥梁、建筑物、生物结构）、“智能”结构的控制以及微机电系统（MEMs）都有影响。

AntmanDMS-1008058 The investigator gives careful mathematical treatments of avariety of dynamical and steady-state nonlinear problems fordeformable rods, shells, and three-dimensional solid bodies,possibly in contact with moving fluids, variable temperaturefields, and electromagnetic fields. The bodies are composed ofnonlinearly elastic, plastic, viscoplastic, ormagneto-(visco-)elastic materials. In each case, properlyinvariant, geometrically exact theories encompassing generalnonlinear constitutive equations are used. The goals of thesestudies are to discover new nonlinear effects and new kinds ofinstabilities, determine thresholds in constitutive equationsseparating qualitatively different responses, determine generalclasses of constitutive equations that are both physically andmathematically natural, determine how existence, regularity, andwell-posedness depend on material behavior, contribute to thetheory of shocks and dissipative mechanisms in solids, anddevelop new methods of nonlinear analysis and of effectivecomputation for problems of solid mechanics. For treating the behavior of both new technologicalmaterials and old biological materials, such as living tissue,traditional theories, based on such simplifying assumptions assmall deformations and linear response, may well fail to captureimportant physical phenomena and may, in fact, give misleadinginformation. E.g., the study of a specific problem for atraditional theory may indicate that a dangerous instabilityoccurs in a solid body when an externally applied load orfrequency exceeds a certain threshold. An analysis of the sameproblem without the simplifying assumptions may well show that alarge class of materials give rise to instabilities at thresholdsmuch lower than those predicted by simplified theories. Sincecorrectly formulated exact nonlinear problems of solid mechanicsseldom can be directly subsumed under available mathematicaltheories for treating the governing equations, one purpose ofthis project is to develop new mathematical techniques to handlesuch problems. These techniques can treat whole ranges ofmaterials at one time. The results have consequences forstructures under loads (bridges, buildings, biologicalstructures), for control of "smart" structures, and formicro-electro-mechanical systems (MEMs).