In this lecture, we discuss the basic issues that arise when generalized plane strain deformations are superimposed on anti-plane shear deformations in isotropic incompressible hyperelastic materials. Attention is confined to a subclass of such materials, namely generalized neo-Hookean materials, for which the strain-energy density depends only on the first invariant of the strain tensor. The governing equations of equilibrium are a coupled system of three nonlinear partial differential equations for three displacement fields. It is shown that, for general plane domains, this system decouples the plane and anti-plane displacements only for the case of a neo-Hookean material. Even in this case, the stress field involves coupling of both deformations. For generalized neo-Hookean materials, universal relations may be used in some situations to uncouple the governing equations.
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