Surface instability of elastic half-spaces by using the energy method

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res. 12, 168–182 (doi:10.1007/BF03184638)) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.