General solutions for plane extensible elasticae having nonlinear stress-strain laws

Abstract

A general finite deformation solution is obtained for the equilibrium of hydrostatically loaded elasticae whose underformed shapes are circular arcs. The nonlinear stress-strain relations employed give the bending moment and the axial force as derivatives of a strain energy function with respect to suitable strain measures. The representation of the solution involves arbitrary constants of integration which can accommodate any end conditions consistent with equilibrium. Examples are given. The constraint of inextensibility is examined and a perturbation procedure for small extension is developed. In an appendix, the stress-strain laws are derived by an appropriate reduction of the equations for three-dimensional hyperelasticity.