Asymptotic bifurcation analysis and post-buckling for uniaxial compression of a thin incompressible hyperelastic rectangle

The problem of uniaxial compression of an incompressible 2D thin rectangle is studied in this paper. We consider the case where the two ends of the rectangle are welded to two rigid bodies. The focus is on the bifurcation analysis and post-buckling solutions. The combined series-asymptotic expansion method is used to derive two coupled non-linear ordinary differential equations (ODEs), which govern the leading-order axial strain and shear strain. Through an analysis on the linearized equations, an algebraic equation for determining the critical stress values of buckling is obtained. For the non-linear coupled ODEs, the numerical solutions of the non-trivial solutions are obtained by providing proper initial guesses. Energy analysis shows that for the first mode of buckling material failure may first happen at the middle point of the bottom surface.