Nucleation of creases and folds in hyperelastic solids is not a local bifurcation

Abstract

We consider creases and folds in compressed hyperelastic solids from the point of view of bifurcation theory. They refer to highly localized surface deformations that occur at compressive loads significantly below the value of the well-known Biot instability. Much work from the literature attempts to make the case that this phenomenon corresponds to a “local bifurcation” distinct from the Biot instability. A local bifurcation is a path of equilibrium solutions emanating from a (bifurcation) point on the trivial solution branch that exists in all sufficiently small neighborhoods of that bifurcation point. The inference is usually made by first introducing a small surface imperfection; a solution curve is then obtained that is seemingly close to a perfect bifurcation diagram. However, imperfection theory is valid only in some sufficiently small neighborhood of a bifurcation point. Thus, in the absence of an equilibrium path connecting these solutions to the trivial one, there is no justification for concluding that creasing and folding are local bifurcations of the perfect system.In this work, we directly address the nucleation of these solutions in the perfect, imperfection-free case. We demonstrate that surface instabilities in functionally graded and bilayer elastic halfspaces, corresponding to local bifurcations from the homogeneous state, are necessarily smooth and oscillatory; creases/folds eventually do develop along the global bifurcating solution branches, albeit “far” from the trivial solution, as evidenced by the corresponding bifurcation diagrams. In addition, we find that their stable realization occurs at load levels well below that of the initial surface instability. Moreover, we obtain such results for the perfect homogeneous halfspace, by switching the continuation parameter from macroscopic lateral strain to the film-to-substrate shear modulus ratio. When this ratio reaches unity, we obtain the desired localized deformation solution, avoiding the need for analysis near the highly degenerate homogeneous state at the Biot instability.