Analysis of Cavitation in Solids

该奖项支持的研究重点是固体中某些重大材料失效的数学分析。这一努力的目标是对空洞的形成和增长进行定性预测。为此，首席研究员将继续研究构成数学模型的相关非线性偏微分方程，以确定这些问题具有奇异解的条件。基础方程是弹性和粘弹性中出现的方程，所需的奇点是点不连续点。将要考虑的问题包括：拟线性椭圆系统奇异解的存在性；双曲型系统奇异解的存在性和可容性准则一类抛物型方程组奇异解的存在性变分微积分问题奇异极小值的存在性奇异极小值的正则性、精细性质和渐近性孤立奇点的最优位置；并且，确定拟线性椭圆系统的已知奇异解是否确实是变分学中相应问题的极小解。这项资助的研究领域是材料科学中出现的方程的数学分析。确定材料在外力作用下何时会失效的最常见方法是将一块实际材料施加拉伸载荷直到失效，即“拉它直到它断裂”。如果一个人只对材料的总体特性感兴趣，这是很好的。然而，如果一个人想了解材料失效的原因，那么他必须求助于材料的数学模型。对某些称为弹性体的橡胶聚合物的实验表明，当人们拉弹性体时，材料上会出现小孔。这些孔洞逐渐增大，并结合在一起形成裂缝。在光纤中也观察到类似的现象。当施加过大的功率时，由于一系列的孔沿着光纤核心级联而下，可能会发生灾难性的故障。这些孔严重降低了光纤传输信息的能力。在这项资助中，首席研究员将通过检查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。

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.