Mathematical Sciences: "Nonlinear Dynamics in Continuum Mechanics."

9505342 Tzavaras我们提出研究连续介质动态变形过程中奇点的形成、传播和解析等相关课题。主题是：(i)黏度对双曲守恒律激波传播的影响。（ii） Riemann数据离散速度模型的流体力学极限。（iii）在高应变速率下形成剪切带。所提出的技术来自于偏微分方程的定性理论领域，在必要时使用计算工具来指导理论。提出的问题从具体现象的具体模型，到守恒定律的一般理论双曲系统，到研究从离散理论到连续理论过渡的发展技术。对激波和剪切带等结构的理论理解对于在这些现象占主导地位的技术应用中设计改进的算法至关重要。剪切带是在金属高速变形过程中出现的强烈局部应变区域，通常在破裂之前出现。剪切带在金属的弹道穿透和各种涉及金属的制造过程中起着重要作用。

9505342 Tzavaras We propose to study various topics related to the formation, propagation and resolution of singularities during the dynamic deformation of continuous media. The topics are: (i) Effect of viscosity on shock propagation for hyperbolic conservation laws. (ii) Fluid-mechanic limits for discrete velocity models for Riemann data. (iii) Formation of shear bands at high strain rates. The proposed techniques are from the domain of qualitative theory of partial differential equations, with the use of computational tools when necessary to guide the theory. The proposed problems vary from concrete models for specific phenomena, to general theory hyperbolic systems of conservation laws, to developing techniques for studying the transition from discrete to continuum theories. The theoretical understanding of structures such as shock waves and shear bands is critical for designing improved algorithms in technological applications where such phenomena are dominant. Shear bands are regions of intensely localized strain that appear during high speed deformations of metals and often precede rupture. Shear bands play a major role in ballistic penetration of metals and in various manufacturing processes involving metals.