Analysis of Cavitation in Solids

该奖项支持的研究重点是对固体中某些重大材料故障的数学分析。这项工作的目标是对空洞的形成和增长进行定性预测。为此，首席调查员将继续研究构成数学模型的相关非线性偏微分方程式，以确定这些问题有奇异解的条件。基本的方程是那些在弹性和粘弹性中出现的方程，所需的奇点是点不连续。将考虑的问题包括：拟线性椭圆组奇异解的存在性；双曲型系统奇异解的存在性和可容许性准则；某些抛物型系统奇异解的存在性；变分演算中问题的奇异点极小子的存在性；正则性、精细性质和奇异点的渐近行为；孤立奇点的最佳位置；以及确定拟线性椭圆组的已知奇异解是否真的是变分中相应问题的极小值。这项拨款的研究领域是对材料科学中出现的公式的数学分析。确定一种材料何时会在外力的影响下失效的最常见的方法是使一块实际材料承受张拉，直到失效发生，也就是“拉到它断掉”。如果你只对材料的总体性质感兴趣，这是可以接受的。然而，如果一个人想要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，当人们拉动弹性体时，材料中会出现小孔。然后，这些空洞会变大，并结合在一起形成裂缝。在光纤中也观察到了类似的现象。当使用过大的电源时，会发生灾难性的故障，这是由于光纤纤芯上的一系列孔洞造成的。这些洞严重降低了光纤传输信息的能力。在这项拨款中，首席调查员将通过研究弹性理论和粘弹性理论中的偏微分方程组，揭示导致聚合物和玻璃中孔洞产生和增长的机制。

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.