Mathematical Sciences: "Nonlinear Dynamics in Continuum Mechanics."

9505342 Tzavaras我们建议研究有关的各种主题的形成，传播和解决的奇异性的动态变形过程中的连续介质。题目是：（i）粘性对双曲守恒律激波传播的影响。(ii)黎曼数据离散速度模型的流体力学极限。(iii)高应变率下剪切带的形成。所提出的技术是从域的偏微分方程定性理论，在必要时使用计算工具来指导理论。所提出的问题各不相同，从具体的模型为特定的现象，一般理论双曲系统的守恒律，发展技术研究从离散过渡到连续理论。 对结构如冲击波和剪切带的理论理解对于在这种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，通常发生在断裂之前。剪切带在金属的弹道穿透和涉及金属的各种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.