Analysis of Cavitation in Solids

该奖项支持的研究重点是对固体中某些重要材料失效的数学分析。 这一奋进的目标是定性预测空洞的形成和增长。 为此，principalinvestigator将继续他的研究有关的非线性偏微分方程，这构成了marticalmodel，以确定条件下，这些问题有奇异的解决方案。 基本方程是那些出现在弹性和粘弹性和所需的奇点是点不连续。 将要考虑的问题包括：拟线性椭圆型方程组奇异解的存在性;双曲型方程组奇异解的存在性和可容许准则;某些抛物型方程组奇异解的存在性;变分法中具有奇异性的极小值的存在性;奇异极小值的正则性、优良性质和渐近行为;孤立奇点的最佳位置;以及确定一个拟线性椭圆系统的已知奇异解是否确实是变分法中相应问题的最小值。该基金的研究领域是材料科学中方程的数学分析。 确定材料在外力影响下何时失效的最常用方法是使一块实际材料承受拉伸载荷，直到发生失效，即，“拉它，直到它断裂”。如果一个人只对材料的总性能感兴趣，这是可以的。 然而，如果一个人想了解的原因formaterial失败，那么一个必须求助于数学模型的材料。 对某些橡胶状聚合物（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。 然后这些洞的大小增长，并结合形成裂缝。 在光纤中也观察到类似的现象。 当功率过大时，由于一系列的孔沿着光纤的核心级联而下，可能会发生灾难性的故障。 这些孔洞严重降低了光纤传输信息的能力。 在这项资助中，主要研究者将通过研究弹性和粘弹性理论中的偏微分方程系统，揭示聚合物和玻璃中孔洞产生和生长的机制。

The focus of the research supported by this award is themathematical analysis of certain significant material failures insolids. The goal of this endeavor is the qualitative prediction ofthe formation and growth of voids. Toward this end the principalinvestigator will continue his studies of the relevant nonlinearpartial differential equations, which constitute the mathematicalmodel, in order to determine conditions under which these problemshave singular solutions. The underlying equations are those thatarise in elasticity and viscoelasticity and the desiredsingularities are point discontinuities. Problems that will beconsidered include: the existence of singular solutions toquasilinear elliptic systems; the existence of, and admissibilitycriteria for, singular solutions to hyperbolic systems; theexistence of singular solutions to certain parabolic systems; theexistence of minimizers with singularities for problems in thecalculus of variations; regularity, fine properties, and theasymptotic behavior of singular minimizers; the optimal locationfor an isolated singularity; and, the determination of whetherknown singular solutions to a quasilinear elliptic system areindeed minimizers of the corresponding problem in the calculus ofvariations.The research area of this grant is the mathematical analysis ofequations that arise in Materials Science. The most common way todetermine when a material will fail under the influence of externalforces is to subject a piece of the actual material to tensileloads until failure occurs, i.e., "pull on it until it breaks".This is fine if one is interested only in the gross properties ofthe material. However, if one wants to understand the reasons formaterial failure then one must have recourse to mathematical modelsof the material. Experiments on certain rubbery polymers, calledelastomers, have shown that when one pulls on an elastomer smallholes appear in the material. These holes then grow in size andcombine to form cracks. A similar phenomenon has been observed inoptical fibers. Catastrophic failure, due to a series of holesthat cascade down the core of the fiber, can occur when excessivepower is applied. These holes seriously degrade the ability of thefiber to transmit information. In this grant, the principalinvestigator will uncover the mechanisms that cause the creationand growth of holes in polymers and glasses by examining systems ofpartial differential equations from the theory of elasticity andviscoelasticity.