Propagation, Reflection and Transmission of Plane Waves in Pre-Stressed Elastic Solids

Hussain, Wasiq (1999) Propagation, Reflection and Transmission of Plane Waves in Pre-Stressed Elastic Solids. PhD thesis, University of Glasgow.

Abstract

This thesis is concerned with the effect of pure homogeneous strain, pre-stress and simple shear deformation on the propagation of homogeneous, surface and interfacial waves in elastic materials. Following a review of previous work regarding the propagation of infinitesimal plane waves in a half-space of incompressible material subject to pure homogeneous strain, the analysis is extended to the influence of pure homogeneous strain on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible isotropic elastic material. In general, the half-spaces consist of different material and are subjected to different deformations. For a certain class of constitutive laws it is shown that a homogeneous plane (SV) wave incident on the boundary from one half-space gives rise to a reflected wave (with angle of reflection equal to the angle of incidence) together with an interfacial wave in the same half-space, while in the other half-space two possibilities arise depending on the angle of incidence, the material properties and the magnitudes of the deformations in the two half-spaces. Either (a) there is a transmitted (homogeneous plane) wave accompanied by an interfacial wave, or (b) there are two interfacial waves with equal speeds of propagation but different rates of (spatial) decay away from the boundary. For a second class of constitutive laws similar behaviour is found for certain combinations of angle of incidence, material properties and deformations, but additional possibilities also arise. In particular, there may be two reflected waves instead of one reflected wave and an interfacial wave, coupled with either possibility (a) or (b) in the second half-space. Equally, there may be two transmitted waves for each of the possible combinations of reflected and interfacial waves in the first half-space. The effect of finite strain principal axis orientation on the reflection from a plane boundary of infinitesimal plane waves propagating in a half-space of incompressible isotropic elastic material is then examined. Attention is focussed on waves propagating in a principal plane of the deformation corresponding to simple shear. For a special class of constitutive laws it is shown that an incident plane harmonic wave propagating in the considered plane gives rise to a surface wave in addition to a reflected wave for every angle of incidence although its amplitude may vanish at certain discrete angles depending on the state of stress and deformation. Unlike the situation in which the underlying deformation is a pure homogeneous strain, however, the amplitude ratio of the reflected (plane harmonic) wave does not in general have unit magnitude, but its magnitude is independent of the pre-stress. Moreover, the angle of reflection differs from the angle of incidence. For materials not in this special class, on the other hand, it is shown that two plane harmonic waves may be reflected when the angle of incidence lies within certain ranges of values (which depend on the shear deformation). Outside this range there is in general a single reflected wave, and a surface wave is generated. This analysis is further used to study the effect of simple shear on the reflection and transmission of plane waves at the boundary between two half-spaces of incompressible elastic material, and, in particular, two half-spaces which form a twin in the sense that equal and opposite simple shears are applied to the two half-spaces. For a special class of constitutive laws it is shown that an incident plane harmonic (shear) wave propagating in the plane of shear in one half-space gives rise to an interfacial wave in each half-space in addition to a reflected and a transmitted plane wave in the respective half-spaces for every angle of incidence, although the amplitudes of the waves may vanish at certain discrete angles (different for each type of wave) depending on the state of deformation. For a specific material not in this special class, corresponding calculations for a particular value of shear show that the nature of the resulting waves is similar to that for the special class of materials. On the other hand, we note that for values of the shear beyond a certain critical value there are certain ranges of angles of incidence for which, instead, two homogeneous plane (shear) waves are reflected and also two transmitted. The dependence of the amplitudes of the reflected, transmitted and interfacial waves on the angle of incidence, the states of deformation and the material properties is illustrated graphically.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Adviser: Ray Ogden
• Keywords: Applied mathematics, Mechanics
• Date of Award: 1999
• Depositing User: Enlighten Team
• Unique ID: glathesis:1999-76468
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 19 Nov 2019 14:18
• Last Modified: 19 Nov 2019 14:18
• URI: https://theses.gla.ac.uk/id/eprint/76468