Mathematical Sciences: Analysis of Cavitation in Solids

小行星9703986 首席研究员会分析某些部分 微分方程以及微积分中的某些问题 为了确定这些变化的条件， 问题有单一的解决方案。 基本方程为 那些在弹性和粘弹性中产生的， 奇点是点不连续性。 问题是， 考虑的问题包括：拟线性方程奇异解的存在性 椭圆系统的存在性，和可接受的标准， 双曲型方程组的奇异解;极小解的存在性 变分法中的奇点问题;以及， 确定已知的不连续解是否 拟线性椭圆方程组实际上是 变分法中的相应问题。 %%% 该基金的研究领域是数学分析， 材料科学中出现的方程。 的最常见方式 确定材料在外部环境影响下何时失效 力是使一块实际材料受到拉伸 加载直至发生故障，即，“用力拉”直到它断了 这是好的，如果一个人只感兴趣的总属性， 材料。 但是，如果要了解 材料失效，那么人们必须求助于数学模型 的材料。 对某些橡胶聚合物的实验， 弹性体，已经表明，当一个人拉弹性体小 材料中会出现孔洞。 然后这些洞的尺寸会变大， 联合收割机形成裂纹。 类似的现象发生在光学 纤维 如果一个人试图通过一个光学系统发送太多的能量， 然后，一系列孔可以形成在光纤的芯中。 这些孔严重降低了光纤的传输能力 信息. 在这项资助中，首席研究员将发现 导致聚合物中空穴产生和生长的机制 通过研究偏微分方程组 从弹性和粘弹性的理论。 ***

9703986 Spector The principal investigator will analyze certain partial differential equations as well as certain problems in the calculus of variations in order to determine conditions under which these problems have singular solutions. The underlying equations are those that arise in elasticity and viscoelasticity and the desired singularities are point discontinuities. Problems that will be considered include: the existence of singular solutions to quasilinear elliptic systems; the existence of, and admissibility criteria for, singular solutions to hyperbolic systems; the existence of minimizers with singularities for problems in the calculus of variations; and, the determination of whether known discontinuous solutions to quasilinear elliptic systems are in fact minimizers of the corresponding problem in the calculus of variations. %%% The research area of this grant is the mathematical analysis of equations that arise in Materials Science. The most common way to determine when a material will fail under the influence of external forces is to subject a piece of the actual material to tensile loading until failure occurs, i.e., "pull on it" until it breaks. This is fine if one is interested only in the gross properties of the material. However, if one wants to understand the reasons for material failure 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 occurs in optical fibers. If one tries to send too much power through an optical fiber then a series of holes can form in the core of the fiber. 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. ***