Surface instabilities of constrained elastomeric layers subject to electro-static stressing

Surface instabilities on the top surface of a soft elastomeric dielectric layer bonded on its bottom to a stiff substrate are investigated for two electro-static conditions: (I) a voltage difference imposed across the top and bottom surfaces of the layer which are both conducting, and (II) a voltage difference imposed across a rigid planar electrode above the layer and the conducting top surface of the layer. In the absence of surface energy, the critical voltage associated with the onset of instability of the planar state is shown to be a surface mode with any wavelength that is short compared to the layer thickness. With no layer pre-stretch and no surface energy, the uniform state in (I) has hydrostatic compression while that of (II) has hydrostatic tension, nevertheless the critical voltage is the same for both problems if the permittivity is the same for both systems. The role of equi-biaxial pre-stretch and surface energy is presented for both problems for neo-Hookean materials and for the Gent generalization. The thrust of the study is the investigation of the post-bifurcation behavior involving nonlinear interactions among the simultaneous surface modes associated with the critical voltage. The post-bifurcation response of Problem I is a crease-like mode with the top surface probing downward towards the lower surface. The opposite occurs for Problem II with the formation of a sharp ridge protruding towards the upper electrode. The bifurcation behavior is highly unstable leading to an extreme sensitivity to small imperfections. The study complements earlier work and highlights unresolved issues.