On the rectilinear shear of compressible and incompressible elastic slabs

We review some pseudo-planar deformations for the equations of incompressible isotropic nonlinear elasticity first introduced in 1985 by Rajagopal and Wineman. We extend this class of deformations to compressible isotropic and transverse isotropic materials, and also consider the influence of gravity. We consider some new approximate solutions and we discuss the possible relevance of such solutions to the understanding of the complex structure of the fields equations of nonlinear elasticity, using weakly nonlinear theories.