Nonlinear Problems of Solid Mechanics

提案：DMS-0204505 PI：Stuart S.马里兰州大学帕克分校题目：固体力学的非线性问题。S. Antman提出要处理各种动态和稳态非线性问题的杆，壳，和三维固体。 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，几何精确的理论，包括一般的非线性本构方程被使用。 (Such固体力学中正确表述的非线性问题很少能直接归入现有的数学理论。 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，确定存在性，规律性和适定性如何依赖于材料行为，有助于固体中的冲击和耗散机制理论，发展了固体力学问题的非线性分析和有效计算的新方法。 具体研究领域包括：（i）非线性弹性壳体的外翻，（ii）承受非保守载荷的结构的非线性稳定性，（iii）承受粘滑摩擦的非弹性体的动态稳定性，（iv）全局多参数Hopf分叉，（v）流固相互作用，（vi）参数共振，（vi）强迫问题的吸引子，（vii）不可压缩弹性和粘弹性杆的动力学，（viii）来自非线性弹性的准线性双曲问题，（ix）小惯性的渐近性，（x）耗散机制，（Xi）滞后，和（xii）控制。S. Antman提出了处理各种动态和稳态非线性问题的各种固体组成的各种材料。 控制理论，需要准确地描述的行为，遭受大而快速的变形，可能低于极限的温度和负载的机构，提出了严峻的数学挑战，很少被分析。 该研究将包括（a）物理理论的发展，（B）能够处理控制方程的数学理论的发展，（c）对数值方法发展的贡献，以及（d）具体问题和问题类别的处理。 后者的例子包括：（一）高压下壳体的大屈曲（塌陷）。(ii)非保守载荷作用下可变形结构的动力失稳。 (Such负载可以通过旋转、通过与移动的流体接触或反馈而引起。 (iii)结构在周期力作用下的稳定性。 (iv)流体-结构相互作用，例如，如与运动的流体接触并经历大的和可能破坏性的振荡的船或直升机的螺旋桨或面板的振动。(v)生理学中的流体-固体相互作用，如动脉中的血液流动。 (vi)由可变形体连接的刚体的大运动（如由系绳连接的航天器）。 (vi)分析电磁固体力学的问题，并应用智能材料来控制振动固体的响应，例如，使潜水艇或航天器的振动部分在短时间内停止。

Proposal: DMS-0204505PI: Stuart S. AntmanInstitution: University of Maryland College ParkTitle: Nonlinear Problems of Solid MechanicsABSTRACTS. S. Antman 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. In each case, properly invariant, geometrically exact theories encompassing general nonlinear constitutive equations are to be used. (Such correctly formulated nonlinear problems of solid mechanics can seldom be directly subsumed under available mathematical theories.) The goals of these studies are to discover new nonlinear effects, determine thresholds in constitutive equations separating qualitatively different responses, determine general classes of constitutive equations that are both physically and mathematically natural, examine important kinds of instabilities, determine how existence, regularity, and well-posedness depend on material behavior, 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. Among the specific areas to be studied are (i) eversion of nonlinearly elastic shells, (ii) nonlinear stability of structures subject to nonconservative loads, (iii) dynamic stability of inelastic bodies subject to stick-slip friction, (iv) global multiparameter Hopf bifurcation, (v) fluid-solid interactions, (vi) parametric resonance, (vi) attractors for problems with forcing, (vii) dynamics of incompressible elastic and viscoelastic rods,(viii) quasilinear hyperbolic problems from nonlinear elasticity, (ix) asymptotics of small inertia, (x) dissipative mechanisms, (xi) hysteresis, and (xii) control.S. S. Antman proposes to treat a variety of dynamical and steady-state nonlinear problems for a variety of solid bodies composed of a variety of materials. The governing theories, needed to describe accurately the behavior of bodies suffering large and rapid deformations, possibly underextremes of temperature and loading, present severe mathematical challenges and have seldom been analyzed. The research is to encompass (a) the development of physical theory, (b) the development of mathematical theory capable of handling the governing equations, (c) contribution to the development of numerical methods, and (d) the treatment of specific problems and classes of problems. Examples of the latter include: (i) Large buckling (collapse) of shells under highpressure. (ii) Dynamic instability of deformable structures under nonconservative loads. (Such loads may be induced by rotations, by contact with moving fluids, or feedback.) (iii) Stability of structures subject to periodic forcing. (iv) Fluid-structure interactions, e.g., like that of a propeller or panel of a ship or a helicopter in contact with a moving fluid and undergoing large and possibly destructive oscillations. (v) Fluid-solid interactions in physiology, such as blood flowing in an artery. (vi) Large motions of rigid bodies joined by deformable bodies (like space vehicles joined by a tether). (vi) The analysis of problems of electromagnetic solid mechanics with applications to the use of smart materials to control the response of vibrating solids, e.g., to bring a vibrating part of a submarine or space vehicle to rest in a short time.