Hamid Said

Type

Text

Type

Dissertation

Advisor

Glimm, James | Jiao, Xiangmin | Li, Xiaolin | Ebin, David

Date

2017-12-01

Keywords

Applied mathematics | Mathematics

Department

Department of Applied Mathematics and Statistics

Language

en_US

Source

This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.

Identifier

http://hdl.handle.net/11401/78229

Publisher

The Graduate School, Stony Brook University: Stony Brook, NY.

Format

application/pdf

Abstract

We present a unified Lagrangian-Hamiltonian formalism for a class of dissipative theories with internal variables based on consistent thermodynamic postulates. We specify two fundamental thermodynamic functions (the free energy and the entropy production rate) that completely determine the state of the system. These functions determine the equations of motion, which are of hyperbolic-parabolic type. A variational principle and Lagrangian theory with dissipation are formulated based on this two function approach. The Lagrangian theory in turn allows for the construction of a novel Hamiltonian theory with dissipation. This Hamiltonian formulation reveals an inner product structure and a refined Kahler structure on the phase space. An existence and uniqueness result is established for all time for the semilinear Hamilton's equations with dissipation as well as local result for the full nonlinear equations. This general two function approach is applied to the problem of failure waves. These waves represent a dynamic mode of brittle fracture that can not be assigned to any of the classical waves. Our theory admits, as special cases, the classical models of Feng and Clifton. Our analysis suggests that the anisotropy of diffusion is related to an observable quantity, the shard size of the comminuted rubble. Furthermore, a Lagrangian-Hamiltonian theory is obtained for failure waves together with mathematical existence. Lastly, the theory is linearized to study the interaction of the reversible processes in the dissipative regime. | 96 pages

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