Nonlinear Problems of Solid Mechanics

固体力学非线性问题

• 批准号: 0204505
• 负责人: Stuart Antman
• 金额: $ 36.82万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204505&HistoricalAwards=false
• 关键词: Nonlinear; Problems; Solid; Mechanics

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.

提案：DMS-0204505PI：Stuart S. Antmanininstitution：马里兰州大学公园的公园：固体机械学的非线性问题。 S. Antman提议治疗杆，壳和三维固体的各种动力学和稳态非线性问题。 这些身体由非线性弹性，粘弹性，塑料，粘塑料或磁弹性材料组成。 在每种情况下，都应使用涵盖一般非线性本构方程的几何理论不变，几何精确的理论。 （在可用数学理论下，很少会直接将这种正确提出的固体机制的非线性问题直接归为。）这些研究的目标是发现新的非线性效应，确定本构方程中阈值分隔质量不同的响应，确定本构方程的一般性等方程的一般性，这些方程在物理上和数学上的重要性依赖性，并确定了对实质性的依赖性，并确定了对不存在的依赖性，并且依赖性依赖性，并确定了对构成的依赖性，并确定了对构成的依赖性，并且是在实质性方程式中的依赖性，并确定了范围的依赖性，并确定了构成方程的依赖性，并确定了构成方程的依赖性，并确定了构成方程的依赖性，并确定了构成方程的依赖性，并且可以确定构成方程的依赖性。固体中的冲击和耗散机制的理论，并为固体力学问题开发了非线性分析和有效计算的新方法。 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强迫，（vii）不可压缩的弹性和粘弹性杆的动力学，（（viii）非线性弹性弹性问题，（ix）小惯性的渐近性，（x）耗散机制，（xi）肌病和（xii）控制。 S. Antman提议治疗由各种材料组成的各种固体的各种动力和稳态非线性问题。 需要准确描述遭受巨大和快速变形的身体的行为，可能是温度和负载的不足，呈现出严重的数学挑战，并且很少被分析。 该研究是包括（a）物理理论的发展，（b）能够处理管理方程的数学理论的发展，（c）对数值方法发展的贡献，以及（d）对特定问题和问题类别的处理。 后者的例子包括：（i）高压下壳的大屈曲（倒塌）。 （ii）在非保守载荷下可变形结构的动态不稳定性。 （这种载荷可以通过旋转，与移动的流体接触或反馈来诱导。）（iii）结构的稳定性受到周期性强迫。 （iv）流体结构相互作用，例如，就像船舶或船体面板的相互作用，或与移动的流体接触的直升机，并经历了大型且可能具有破坏性的振荡。 （v）生理学中的流体固定相互作用，例如在动脉中流动的血液。 （vi）刚体的大动作由可变形的身体（例如由系绳连接的太空车辆）加入。 （vi）分析电磁固体力学问题，应用于使用智能材料来控制振动固体的响应，例如，将海底或空间车辆的振动部分带到短时间内。