Global bifurcation of anti-plane shear equilibria
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Bifurcation theoretic methods are used to construct families of solutions for two problems arising in non-linear elasticity. These solution curves are shown to exhibit interesting phenomena that are both mathematically challenging and physically relevant. In the first part, we consider an unbounded elastic slab subjected to anti-plane shear deformation and under the influence of a body force. For one class of materials and forces, there is shown to be a loss of ellipticity at the terminal end of the bifurcation curve. More specifically, the ellipticity of the governing equations degenerates as the strain reaches a critical value determined by the material in question. For another class of materials and forces, we prove that broadening occurs; that is, the displacements within our family of equilibria remain uniformly bounded, but their effective supports become arbitrarily large. In the next part, we investigate anti-plane shear deformations on a semiinfinite slab with a non-linear mixed traction displacement boundary condition. Energy estimates are used to show that broadening cannot occur in this setting. Once more we apply global bifurcation theory and deduce extreme behavior at the terminal end of the curve. It is shown that arbitrarily large strains are encountered for a class of idealized materials that do not allow for a loss of ellipticity. We also consider degenerate materials, prove that ellipticity breaks down, and most importantly show that a concurrent blow-up in the second derivative occurs.
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Ph. D.