We consider a large deformation plane-strain problem involving a compr
essible orthotropic solid subjected to uniaxial compressive loading al
ong one of the principle directions which is aligned with the boundary
of a half-space. An exact solution for the displacement field is obta
ined and a condition for the smallest compressive load corresponding t
o the onset of a surface instability is determined. It is shown that w
hen the compression occurs along the stiffest direction this condition
is expressible in terms of a cubic polynomial, and that the correspon
ding critical loan is lower than the well-known estimate which determi
nes the critical load to be equal to the inplane shear modulus.