On Two-Dimensional Waves of Finite Amplitude in Elastic Materials of Harmonic Type

Author: Jeffers, Robert Humphrey Francis

Year: 1971

Degree: Dissertation (Ph.D.)

Advisor: Knowles, James K.

Committee Member: Unknown, Unknown

Option: Applied Mechanics

DOI: 10.7907/C2ZB-4K70

Abstract

In this thesis, two-dimensional waves of finite amplitude in elastic materials of harmonic type are considered. After specializing the basic equations of finite elasticity to these materials, attention is restricted to plane motions and a new representation theorem (analogous to the theorem of Lamé in classical linear elasticity) for the displacements in terms of two potentials is derived.

The two-dimensional problem of the reflection of an obliquely incident periodic wave from the free surface of a half-space composed of an elastic material of harmonic type is formulated. The incident wave is a member of a special class of exact one-dimensional solutions of the nonlinear equations for elastic materials of harmonic type, and reduces upon linearization to the classical periodic "shear wave" of the linear theory.

A perturbation procedure for the construction of an approximate solution of the reflection problem, for the case where the incident wave is of small but finite amplitude, is constructed. The procedure involves series expansions in powers of the ration of the amplitude to the wavelength of the incident wave and is of the so-called two-variable type. The perturbation scheme is carried far enough to determine the second-order corrections to the linearized theory.

A summary of results for the reflection problem is provided, in which nonlinear effects on the reflection pattern, on the particle displacements at the free surface and on the behavior at large depth in the half-space are detailed.

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