Asymptotic bifurcation solutions for compressions of a clamped nonlinearly elastic rectangle: Transition region and barrelling to a corner-like profile

In the two experiments by Beatty and his coauthors on the compressions of elastic rectangles/bars, it was found that there is a transition region of the aspect ratio which separates buckling from barrelling. Friction, which prevents the lateral movement of the end cross section, might be the cause. Here, we study the compressions with clamped end conditions. One of the purposes is to show, with this setting in which the lateral movement of the end cross section is limited, that there is indeed such a transition region. By using combined series-asymptotic expansions, we derive two decoupled nonlinear ordinary differential equations (ODEs). By phase plane analysis, the leading-order axial strain can be obtained from one of the ODEs. Then an eigenvalue problem can be formulated from another ODE, which is solved by the WKB (Wentzel-Kramers- Brillouin) method. It is found that when the aspect ratio is relatively large there is only a bifurcation to the 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 a different upper bound for buckling are found, which implies the existence of the above-mentioned transition region. Another finding is that, after the barrelling, no further bifurcation to buckling can occur. The critical buckling loads obtained from our asymptotic solutions are also compared with those obtained from the Euler buckling formula. © 2010 Society for Industrial and Applied Mathematics.