Analytical studies on bifurcations of compressions of hyperelastic rectangular

Abstract

In this thesis, we study the bifurcations of two-dimensional rectangular layers under compression. This type of problem is an old one and has been studied widely from different points of view. However, for nonlinearly elastic material, it is very difficult to find analytical post-bifurcation solutions. Our general aim, in this thesis, is to find asymptotic analytical solutions for the compression of nonlinearly elastic rectangular layers. Although the problems of compression of rectangular layers or cylinders have been studied extensively, there are still certain phenomena observed in experiments that have not been explained analytically. The first part of our study is related to an interesting phenomenon that is not solved analytically to the best of our knowledge. This phenomenon is the transition region of the aspect ratio which separates buckling and barrelling found in the experiments by Beatty and co-authors. Friction, which prevents the lateral movement of the end cross-section, might be the cause. Here, we study the compression of a two-dimensional nonlinearly elastic layer under clamped end conditions. Our purpose is to show, under this setting in which the lateral movement of the end cross-section is limited, that there is indeed such a transition region. The second part of our study is concerned with the post-buckling solutions under sliding boundary conditons and welded boundary conditions, respectively. The critical stress values of compression of a rod are given by Eulers buckling formula. However, the post-buckled solutions are seldom obtained. In this part, we consider the approximate solutions and numerical solutions of the post-buckled solutions for two different end conditions. We solve the first problem by constructing asymptotic solutions of the field equations. By using combined series-asymptotic expansions method, we derive two decoupled nonlinear ODEs (ordinary differential equations). One governs the leading-order axial strain and the other governs both the leading-order axial strain and shear strain. By phase plane analysis and the WKB (Wentzel, Kramers & Brillouin) method, it is found that when the aspect ratio is relatively large there is only a bifurcation to barrelling which leads to a corner-like profile on the lateral boundaries of the layer. When the aspect ratio is relatively small there are only bifurcation points which lead to the buckled profiles. A lower bound of the aspect ratio for barrelling and an upper bound for buckling are found, which implies the existence of the above-mentioned transition region. The second problem is studied by a similar approach, i.e., the combined series-asymptotic expansions method. The difference is that we arrive at two coupled ODEs for incompressible hyperelastic materials, which govern the leading-order axial strain and shear strain. Under sliding boundary conditions, we obtain two different explicit approximations of solutions by the method of multiple scales. Numerical computations show that the second approximations are very good approximations. Under welded boundary conditions, we can only obtain the numerical solutions. For both two end conditions, some numerical computations are made to find at which point(s) the material failure occurs first. The main contributions of the thesis are: the transition region of the aspect ratio separating buckling and barrelling observed in experiments is found in our analytical studies and the post-buckling solution and post-barrelling solution are obtained analytically or numerically.