Primary and secondary bifurcations of a compressible hyperelastic layer: Asymptotic model equations and solutions

Abstract

In this paper, we study the equilibrium states of a compressible hyperelastic layer under compression after the primary and secondary bifurcations. Starting from the two-dimensional field equations for a compressible hyperelastic material, we use a methodology of coupled series-asymptotic expansions developed earlier to derive two coupled non-linear ordinary differential equations (ODEs) as the model equations. The critical buckling stresses are determined by a linear bifurcation analysis, which are in agreement with the results in the literature. The method of multiple scales is used to solve the model equations to obtain the second-order asymptotic solutions after the primary bifurcations. An analytical formula for the post-buckling amplitudes is derived. Two kinds of numerical solutions are also obtained, the numerical solutions of the model equations by a difference method and those of the two-dimensional field equations by the finite elements method. Comparisons among the analytical solutions, numerical solutions and solutions obtained by the Lyapunov–Schmidt–Koiter (LSK) method in the literature are made and good agreements for the displacements are found. It is also found that at some places the axial strain is tensile, although the layer is under compression. To consider the secondary bifurcation, we superimpose a small deformation on the state after the primary bifurcation. With the analytical solution of the primary bifurcation, we manage to reduce the problem of the secondary bifurcation to one of the first bifurcations governed by a second order variable-coefficient ODE. And, our analysis identifies an explicit function and from the existence/non-existence of its zero one can immediately judge whether a secondary bifurcation can take place or not. The zero corresponds to a turning point of the governing ODE, which leads to non-trivial solutions. Further, by the WKB method the equation (in a very simple form) for determining the critical stress for the secondary bifurcation is derived. We further use AUTO to compute the secondary bifurcation point numerically, which confirms the validity of our analytical results. The numerical solution in the secondary bifurcation branch is also computed by AUTO. It is found that the secondary bifurcation induces a “wave number doubling” phenomenon and also the shape of the layer has a convexity change along the axial direction.