Analytic studies on bifurcations of a hyperelastic layer-substrate structure under uniaxial compression
Student thesis: Doctoral Thesis
Related Research Unit(s)
This thesis focuses on the bifurcations of a two-dimensional layer-substrate structure under uniaxial compression and its applications to buckled and wrinkled fruits and vegetables. A layer coated to a substrate can be used to model the skin of human beings, fruits and vegetables with exocarp and sarcocarp, the evolution of thin membrane in some biological situations, etc. So it is necessary to study such a structure both mechanically and mathematically due to its widespread applications. In the first part, we focus on the bifurcation behavior of a compressible hyperelastic layer bonded to another compressible hyperelastic substrate. A linear bifurcation analysis is carried out for obtaining the bifurcation condition in the framework of exact theory of nonlinear elasticity. 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. Through a careful analysis, 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. In the second part, we adopt a generalized plane-strain model to establish the geometrical constraint for buckled and wrinkled shapes by modeling a fruit/vegetable with exocarp and sarcocarp as a hyperelastic layer-substrate structure subjected to uniaxial compression. Our point is that there is a critical thickness ratio which separates the buckling and wrinkling modes, independently of the material stiffnesses. More specifically, it is found that if the thickness ratio is smaller than this critical value a fruit/vegetable should be in a buckled shape (under a sufficient stress); if a fruit/vegetable is in a wrinkled shape the thickness ratio is always larger than this critical value. To verify the theoretical prediction, we consider four types of buckled fruits/ vegetables and four types of wrinkled fruits/vegetables with three samples in each type. The geometrical parameters for the 24 samples are measured and it is found that indeed all the data fall into the theoretically predicted buckling or wrinkling domains. In the third part, we revisit the same problem in part one and study the postbifurcation solutions for buckling mode. By specifying the geometrical and material parameters within the buckling mode domains and adopting the combined seriesasymptotic expansions method, two coupled nonlinear ODEs governing the leading order axial strain and shear strain for the upper surface of layer are obtained. At nearcritical loads, a perturbation method is used to derive the amplitude equation by utilizing the solvability condition. Two cases are considered, that is the ratio of Young’s modulus between layer and substrate is of O(1) or large. The two-terms asymptotic solutions and leading order solutions are obtained. It is found that there is a transition between a supercritical bifurcation to a subcritical one when the modulus ratio is increasing. The validity of our results is examined by comparing the analytical solutions with the numerical ones for both cases and good agreements are found. The simple analytical formulas for the deflection, the displacements for both layer and substrate are also obtained, which can give us the insights for how the geometrical and material parameters affect the post-bifurcation states. Especially, the relations of deflection, the critical bifurcation stretch and modulus ratio are non-monotone, which implies there exists the critical modulus ratios such that the deflection and critical stretch attains a minimum value. This suggest a possible way for controlling the deflection amplitude and stability of the structure.
- Elasticity, Bifurcation theory