Compression of a hyperelastic layer-substrate structure: Transitions between buckling and surface modes

This paper studies the bifurcation behaviors of a hyperelastic layer bonded to another hyperelastic substrate of finite thickness subjected to compression. We aim at revealing some interesting transitions between different bifurcation modes as the geometrical parameters vary. A linear bifurcation analysis is carried out for obtaining the bifurcation condition in the framework of exact theory of nonlinear elasticity. This condition, in the form of a determinant with complicated elements, contains a few parameters, and here the task is to analyze it to determine different behaviors. From the critical stretch curves, it is found that there are two mode types for the layer: buckling mode and wrinkling mode. By further considering the eigenfunction, three types of modes for the substrate are identified, including buckling mode, buckling-surface mode and wrinkling-surface mode. A careful analysis is carried out to determine the parameter constraints for each type of modes. In particular, three critical thickness ratios and two critical aspect ratios of the layer are found. As a result, we manage to classify the plane of the aspect ratio of the layer and the thickness ratio into six domains for different mode types and whose boundaries determine where the transitions of mode types take place. Finally, an asymptotic analysis with double expansions for each unknown is carried out to give the explicit formulas for the critical mode number and the critical stretch (which also give an improvement on the existing results for a layer coated to a half-space). Also, simplified relations for those critical thickness ratios and aspect ratios are derived. The asymptotic results also reveal some interesting insights, e.g., why the Poisson’s ratio has little effect and in a wrinkling mode the critical stretch is almost independent of the layer thickness.