Global Bifurcation of Anti-plane Shear Fronts

We consider anti-plane shear deformations of an incompressible elastic solid whose reference configuration is an infinite cylinder with a cross section that is unbounded in one direction. For a class of generalized neo-Hookean strain energy densities and live body forces, we construct unbounded curves of front-type solutions using global bifurcation theory. Some of these curves contain solutions with deformations of arbitrarily large magnitude.Comment: 21 pages, 2 figure

Finite Mechanical Proxies for a Class of Reducible Continuum Systems

We present the exact finite reduction of a class of nonlinearly perturbed wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an inverse eigenvalue problem, we establish an equivalence between the spectral finite description derived from A-C-Z and a discrete mechanical model, a well definite finite spring-mass system. By doing so, we decrypt the abstract information encoded in the finite reduction and obtain a physically sound proxy for the continuous problem.Comment: 15 pages, 3 figure

Global bifurcation of anti-plane shear equilibria

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.Includes bibliographical references

Mathematical foundations of elasticity

[Preface] This book treats parts of the mathematical foundations of three-dimensional elasticity using modern differential geometry and functional analysis. It is intended for mathematicians, engineers, and physicists who wish to see this classical subject in a modern setting and to see some examples of what newer mathematical tools have to contribute