Singular Deformations in Mechanics

建议：DMS-0405646 PI：斯科特J斯佩克特机构：南伊利诺伊大学标题：奇异变形力学摘要首席研究员将分析奇异解某些系统的偏微分方程以及奇异极小相应的能量。 这些问题涉及：奇异解的存在性抛物系统;存在的最小值与奇异性的问题，在变分法;规律性，优良的性能，和渐近行为的奇异极小值;和，确定是否已知的奇异解决方案，以一个准线性椭圆系统确实极小的相应问题，在变分法。 这些问题的动机在于固体力学。 弹性体、玻璃、玻璃状聚合物、薄膜和韧性金属的数学模型产生奇异变形，用于预测这些材料在外力作用下的失效。 感兴趣的失效是由于微观孔的形成和生长以及这些孔随后聚结以在材料中形成裂纹而引起的失效。 当外力施加到弹性体上时，这些孔变得可见，尺寸增大，然后联合收割机组合以破坏材料。 特别感兴趣的问题是：一个孤立的孔的启动和动态增长;孔联合收割机形成裂纹的方式;孔形成的最佳位置;以及温度变化可能导致这种孔形成的方式。 调查的重点是固体中某些重要的材料失效。 其目标是预测材料中空洞和孔洞的形成和增长，这对汽车和国防工业至关重要。 如果一个人想了解的原因，材料失败，然后必须求助于数学模型的材料。 对某些橡胶聚合物（称为弹性体）的实验表明，当人们拉动弹性体时，材料上会出现小孔。 然后这些孔的尺寸变大并联合收割机形成裂纹。 在光纤中也观察到类似的现象。 当施加过大的功率时，由于一系列孔沿着光纤的芯级联而导致的灾难性故障可能发生。 这些孔严重降低了光纤传输信息的能力。 在这项资助中，首席研究员将通过研究弹性和粘弹性理论的偏微分方程系统，揭示导致聚合物和玻璃中孔洞产生和生长的机制。

Proposal: DMS-0405646PI: Scott J SpectorInstitution: Southern Illinois UniversityTitle: Singular Deformations in MechanicsABSTRACTThe Principal Investigator will analyze singular solutions to certain systems of partial differential equations as well as singular minimizers to the corresponding energy. The problems involve: existence of singular solutions to parabolic systems; existence of minimizers with singularities for problems in the calculus of variations; regularity, fine properties, and the asymptotic behavior of singular minimizers; and, the determination of whether known singular solutions to a quasilinear elliptic system are indeed minimizers of the corresponding problem in the calculus of variations. The motivation for these problems lies in the mechanics of solids. Mathematical models for elastomers, glass, glassy polymers, thin films, and ductile metals yield singular deformations that are used to predict the failure of such materials under the application of external forces. The failures of interest are those due to the formation and growth of microscopic holes as well as the subsequent coalescence of such holes to form cracks in a material. When external forces are applied to an elastomer these holes become visible, grow in size, and then combine to break the material. Particular problems of interest are: the initiation and dynamic growth of a solitary hole; the manner in which holes combine to form cracks; the optimal location for hole formation; and, the manner in which a change in temperature can cause the formation of such holes.The research area of this grant is the mathematical analysis of equations that arise in Materials Science. The focus of the investigation is certain significant material failures in solids. The goal is the prediction of the formation and growth of voids and holes in materials that are crucial to the automotive and defense industries. If one wants to understand the reasons that materials fail then one must have recourse to mathematical models of the material. Experiments on certain rubbery polymers, called elastomers, have shown that when one pulls on an elastomer small holes appear in the material. These holes then grow in size and combine to form cracks. A similar phenomenon has been observed in optical fibers. Catastrophic failure, due to a series of holes that cascade down the core of the fiber, can occur when excessive power is applied. These holes seriously degrade the ability of the fiber to transmit information. In this grant, the principal investigator will uncover the mechanisms that cause the creation and growth of holes in polymers and glasses by examining systems of partial differential equations from the theory of elasticity and viscoelasticity.