The effect of shear on the stability and dynamic properties of elastic bodies.
PhD thesis, University of Glasgow.
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The rôle that shear plays in the dynamical response and associated stability of elastic bodies is investigated within this thesis from two perspectives. Forming the major part of the study is the investigation of infinitesimal wave propagation within elastic material which has been subjected to a static pre-strain corresponding to simple shear.
Initially we consider a prototype problem wherein the theory of incremental motions provides the mechanism for analyzing Rayleigh waves propagating along the surface of an incompressible elastic half-space. This is looked at from a plane strain point of view but, significantly, the direction of propagation is not along a principal axis. Using co-ordinates measured relative to the Eulerian axes in the governing equation and boundary conditions, corresponding to the vanishing of incremental tractions, we derived the secular equation for infinitesimal waves in terms of wavespeed, shear and hydrostatic stress parameters for a particular class of materials. The dependence of the wavespeed on these parameters is illustrated and bifurcation criteria are found through setting the wavespeed to zero, this corresponding to quasi-static incremental displacements. For a general form of incompressible, isotropic strain-energy function we are also able to provide the bifurcation criteria incorporating an additional, material parameter. We also consider the compressible counterpart to this problem and follow the same approach, where possible, in establishing the secular equation for compressible materials.
This approach is also adopted for the next problem in which an infinite layer of incompressible elastic material, having uniform width, is pre-strained and within which infinitesimal waves are propagated along the layer. Owing to the layer width the waves are now dispersive and for three types of incremental boundary conditions we provide dispersion equations involving wavespeed, shear, hydrostatic stress and layer thickness (wavelength) parameters for the same particular class of materials. The interdependence of these parameters is comprehensively detailed for this class along with the bifurcation analysis which is again extended so that it may be applicable to a general incompressible, isotropic material.
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