Mathematical Sciences: Nonlinear Dynamics in Continuum Mechanics

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The objective of this research is to understand the dynamic response of continuous media via the study of singularities for the governing partial differential equations of motion. Three topics are proposed: (i) Formation of shear bands at high strain rates; (ii) Study of self-similar viscous limits and of radial solutions for hyperbolic systems of conservation laws; (iii) Hydrodynamic limits for discrete velocity models in the case of Riemann data. Shear bands are regions of intensely localized strain that appear during high speed deformations of metals and often precede rupture. For that reason their study is critical for the design of improved materials in situations of high speed plastic flow. The theoretical understanding of structures such as shock waves or shear bands is critical for designing improved algorithms in technological applications where such phenomena are dominant, for instance in ballistic penetration of metals and in various manufacturing processes involving metals.