Bifurcation of orthotropic solids

We consider a large deformation plane-strain problem involving a compressible orthotropic solid subjected to uniaxial compressive loading along one of the principle directions which is aligned with the boundary of a half-space. An exact solution for the displacement field is obtained and a condition for the smallest compressive load corresponding to the onset of a surface instability is determined. It is shown that when the compression occurs along the stiffest direction this condition is expressible in terms of a cubic polynomial, and that the corresponding critical load is lower than the well-known estimate which determines the critical load to be equal to the inplane shear modulus.