Post-bifurcation equilibria in the plane-strain test of a hyperelastic rectangular block

Of interest here is the bifurcated equilibrium solution of a homogeneous, hyperelastic, rectangular block under finite, plane-strain tension or compression. A general asymptotic analysis of the bifurcated equilibrium path about the principal solution’s lowest critical load is presented using Lagrangian kinematics. The analysis is valid for any compressible hyperelastic material with axes of orthotropy aligned with the block’s axes of symmetry in the reference (stress-free) configuration. The general theory is subsequently applied to blocks of different constitutive laws. Results are presented in the form of bifurcated equilibrium branch’s curvature at the critical load as function of the block’s aspect ratio, since the sign of this curvature determines the branch’s stability. For small aspect ratios there is agreement with existing structural models, while for relatively higher aspect ratios some rather counter-intuitive stability results appear, which strongly depend on the constitutive law.