Wave propagation in residually-stressed materials

The research work included in this thesis concerns the study of wave propagation in elastic materials which are stressed in their initial state. This research is based on the non-linear theory of elasticity. Using the theory of invariants, the general constitutive equation for an isotropic hyperelastic material in the presence of initial stress is derived. These invariants depend on the finite deformation as well as the initial stress. In general, this derivation involves 10 invariants for a compressible material and 9 for an incompressible material. Making use of these invariants, the elasticity tensor is given in its most general form for both the deformed and the undeformed (i.e., the initially stressed reference) configurations. The equations governing infinitesimal motions superimposed on a finite deformations are then used to study the effects of initial stress and finite deformation on wave propagation. For each of the problems carried out in this thesis, the results are specialized for a prototype strain energy function which depends on the initial stress as well as the deformation. The basic theory in each of the problems is formed for the material in the deformed configuration and is later specialized for the undeformed reference configuration. Considering the special case when initial stress is zero, the results are compared with those from the linear theory of elasticity. The problem of homogeneous plane waves in an initially stressed incompressible half-space is considered. The basic theory of the problem is later used to study the reflection of plane waves from the boundary of such a half-space. The reflection coefficients of waves are calculated and graphical representations are given to study the behaviour with reference to the magnitude of initial stress and finite deformation. The study of Rayleigh and Love waves follows thereafter and the basic theory already developed in this thesis is used to study the effect of initial stress on the wave speed of these surface waves. In both cases, the secular equation is analysed in deformed and undeformed configurations and graphs are presented. The problem of wave propagation in a residually stressed inhomogeneous thick-walled incompressible tube which is axially stretched and inflated due to internal pressure, is considered. On the basis of known experimental behaviour, a simple expression for the residual stress is chosen to calculate the internal pressure used to inflate the tube and the axial load to stretch it. The effect of initial stress and stretch on pressure and axial load is studied and graphs are presented. The general theory developed for the deformed configuration for the special model is specialized to the reference configuration and the dispersion relation is analysed numerically