Elastic fingering of a bonded soft disc in traction: interplay of geometrical and physical nonlinearities

This work provides a mathematical understanding of the elastic fingering provoked by a large axial extension of a soft solid cylinder bonded between rigid plates. In this prototypical system model, a topological transition from a ground axis-symmetric meniscus is quasi-statically controlled by the applied displacement, which acts as the order parameter of a pitchfork bifurcation. Since the isotropic elastic energy becomes nonconvex under finite strains, geometric nonlinearity is of paramount importance for the loss of uniqueness of the solution of the boundary value problem. Nonetheless, physical nonlinearity in the elastic energy is found to exert an opposite stabilizing effect. It indeed penalizes the local stretching at the free boundary that would arise as a consequence of any change of its Gaussian curvature. The theoretical and numerical results are in agreement with recent experimental observations, showing that elastic fingering is strongly affected by the aspect ratio of the disc and can be even suppressed in soft materials with physical nonlinearity.