Pitchfork and octopus bifurcations in a hyperelastic tube subjected to compression: Analytical post-bifurcation solutions and imperfection sensitivity

Abstract

This paper studies the axisymmetric deformations of a nonlinearly hyperelastic tube subjected to axial compression. We aim at investigating the critical buckling stresses and modes, deriving the analytical solutions for the post-bifurcation deformations and studying the imperfection sensitivity. For a general isotropic hyperelastic tube, a coupled series-asymptotic method is utilized to derive two simplified model equations with specified constraints on the tube geometry. Then, we specialize to the Blatz–Ko material. With greased end conditions, through linear bifurcation analysis, we obtain the critical stress values and the corresponding mode numbers. The analytical solutions for the post-bifurcation states are constructed by the multiple scales method. By examining the solution behavior in the post-bifurcation regime, it is found that a thick tube could be considerably softer than a thin one. The singularities theory is used to consider the imperfection sensitivity, which reveals the mechanism is the existence of two modes corresponding to the same critical stress. Numerical solutions are also obtained which confirm the validity of the analytical solutions.