Inhomogeneous non-unidirectional deformations of an elastic wedge

We consider the deformation of a nonlinearly elastic wedge in the case of a stored energy function that is given by a power law. This model, introduced by Knowles, reduces to the classical neo-Hookean model when the power-law exponent is unity, and for certain values of the exponent the equations of anti-plane strain lose ellipticity allowing one to carry out interesting mathematical analyses of the governing equations. Reminiscent of the situation in Jeffery–Hamel flows of the Navier–Stokes fluid (see Jeffery, Phil. Mag. Series 6. 29 (1915); Hamel, Deutsch. Math. Ver.25 (1916)) we find that depending on the wedge angle, in addition to solutions that are unidirectional, that is either towards or away from the apex of the wedge, solutions that are not unidirectional, in that the displacement is towards the apex in certain regions of the wedge and away from the apex in other regions, are also obtained. Solutions that are not unidirectional have been shown to exist by McLeod and Rajagopal (Arch. Rational Mech. Anal.147 (1999)). We are able to provide numerical solutions which confirm these findings not only for the case where we consider the boundary conditions employed in their work, but also for traction boundary conditions. The case where $n<0.5$ corresponding to the domain where the equation loses ellipticity, is also investigated and solutions obtained.