Asymptotic Bifurcation Solutions for Compressions of a Clamped Nonlinearly Elastic Rectangle: Transition Region and Barrelling to a Corner-like Profile

Abstract. Buckling and barrelling instabilities in the uniaxial compressions of an elastic rectangle have been studied by many authors under lubricated end conditions. However, in practice it is very difficult to realize such conditions due to friction. In the two experiments by Beatty and his co-authors, it is found that there is a transition region of the aspect ratio which separates buckling and barrelling. Friction, which prevents the lateral movement of the end cross section, might be the cause. Here, we study the compressions of a two-dimensional nonlinearly elastic rectangle under clamped end conditions. One of the purposes 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. We achieve this by constructing asymptotic solutions of the field equations. By using combined seriesasymptotic expansions, we derive two decoupled nonlinear ordinary differential equations (ODE’s). One governs the leading-order axial strain and another governs both the leading-order axial strain and shear strain. By phase plane analysis, the axial strain can be obtained from one of the ODE’s as the axial resultant force varies. Then an eigenvalue problem can be formulated from another ODE, which is solved by the WKB 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 rectangle. 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 critical buckling loads from our asymptotic solutions are also compared with those obtained from the Euler’s buckling formula.