Mathematical Sciences: "Nonlinear Dynamics in Continuum Mechanics."

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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.