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Applications of symmetries and conservation laws to the study of nonlinear elasticity equations

Date

2014-10-29

Journal Title

Journal ISSN

Volume Title

Publisher

ORCID

Type

Degree Level

Masters

Abstract

Mooney-Rivlin hyperelasticity equations are nonlinear coupled partial differential equations (PDEs) that are used to model various elastic materials. These models have been extended to account for fiber reinforced solids with applications in modeling biological materials. As such, it is important to obtain solutions to these physical systems. One approach is to study the admitted Lie symmetries of the PDE system, which allows one to seek invariant solutions by the invariant form method. Furthermore, knowledge of conservation laws for a PDE provides insight into conserved physical quantities, and can be used in the development of stable numerical methods. The current Thesis is dedicated to presenting the methodology of Lie symmetry and conservation law analysis, as well as applying it to fiber reinforced Mooney-Rivlin models. In particular, an outline of Lie symmetry and conservation law analysis is provided, and the partial differential equations describing the dynamics of a hyperelastic solid are presented. A detailed example of Lie symmetry and conservation law analysis is done for the PDE system describing plane strain in a Mooney-Rivlin solid. Lastly, Lie symmetries and conservation laws are studied in one and two dimensional models of fiber reinforced Mooney-Rivlin materials.

Description

Keywords

Applied Mathematics, Conservation Law, Differential Equations, Elasticity, Hyperelasticity, Lie Symmetry Analysis

Citation

Degree

Master of Science (M.Sc.)

Department

Mathematics and Statistics

Program

Mathematics

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DOI

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