On the shape of the free surface of a rubber-like material in proximity to a rigid contactor

Motivated by recent experimental research on thin elastic films by Monch and Herminghaus and by Shenoy and Sharma, we build on Andreussi and Curtin's pioneering work on the wrinkling of a free surface with residual stress and investigate the wrinkling of residually stressed solid films subjected to van der Waals or other types of interaction from a proximate contactor. We consider an incompressible rubber-like film of infinite depth, account for the influence of the contactor and both residual surface stress and curvature on the surface free-energy density, and impose force balance both in the bulk and on the surface. From the resulting linearized bulk and superficial equations, we derive a quintic dimensionless dispersion relation and perform a parametric study to see when linearly stable or unstable behavior of the film surface is manifested. Compared with the quadratic dispersion relation of Andreussi and Curtin, the present model always yields more linearly stable wrinkled configurations possible for the film. Thus, for experimental or design applications, the addition of the contactor and inclusion of surface curvature effects may be advantageous.