Nonlinear Problems of Solid Mechanics

固体力学非线性问题

• 批准号: 1008058
• 负责人: Stuart Antman
• 金额: $ 38.3万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1008058&HistoricalAwards=false
• 关键词: Nonlinear; Problems; Solid; Mechanics; Nonlinear; Problems; Solid; Mechanics

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).

Antmandms-1008058研究者对动态和稳态的非线性问题进行了仔细的数学处理，可伪造的杆，壳和三维固体，可能与移动的流体，可变温度场和电磁场接触。 这些身体由非线性弹性，塑料，粘膜，ormagneto-（visco-）弹性材料组成。 在每种情况下，都使用适当的，几何精确的理论，其中包含通用的构成方程。 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非线性分析和固体力学问题的有效计算。 为了治疗新技术材料和旧生物学材料的行为，例如生物组织，传统理论，基于这种简化的假设和线性响应，很可能无法捕获重要的物理现象，实际上，实际上可能会产生误导性信息。 例如，对临界理论的特定问题的研究可能表明，当外部施加的载荷频率超过一定阈值时，固体体内的危险不稳定。 在没有简化假设的情况下对同一问题进行的分析可能表明，材料类别的材料类别在阈值下引起的不稳定性低于简化理论所预测的材料。 可以直接将固体力学的确切非线性问题直接包含在可用的数学理论中，以处理管理方程，这是一个项目的一个目的是开发新的数学技术来处理问题。 这些技术可以一次处理整个材料。 结果在负载（桥梁，建筑物，生物结构）下具有后果的堡垒，以控制“智能”结构和emicro-Electro机械系统（MEMS）。