Axisymmetric necking versus Treloar–Kearsley instability in a hyperelastic sheet under equibiaxial stretching

We consider bifurcations from the homogeneous solution of a circular or square hyperelastic sheet that is subjected to equibiaxial stretching under either force- displacement-controlled edge conditions. derive condition for axisymmetric necking and show, class strain-energy functions considered, critical stretch greater than Treloar–Kearsley (TK) instability less limiting-point instability. An amplitude equation bifurcated derived through weakly nonlinear analysis used show initiation generally sub-critical. Abaqus simulations are conducted verify bifurcation conditions expectation TK should occur first force control, but when displacement controlled, suppressed, it will be observed. It also demonstrated follows growth/propagation process typical all such localization problems.