Effect of curvature and anisotropy on the finite inflation of a hyperelastic toroidal membrane

The problem of finite inflation of a hyperelastic toroidal membrane under uniform internal pressure is considered in this paper. The work consists of the following two aspects of the inflation problem. Firstly, a formulation for solving the inflation problem efficiently by directly integrating the differential equations of equilibrium without discretization is proposed. The results obtained are compared with those obtained using a discretization method proposed earlier. Secondly, the effects of the geometric and material parameters of the membrane and the internal pressure on the inflation and its stability are studied. The roles of the curvature (specifically, the eigenvalues of the shape operator) of the toroidal geometry and the membrane material parameter on the distortion of the cross-section and occurrence of wrinkling instability are clearly brought out. Based on the Cauchy stress resultants, the limits on the inflation to avoid wrinkling are determined. It is observed that the limit point pressure of the membrane is inversely proportional to the geometric parameter of the torus. The proportionality constant involved is found to vary linearly with the material parameter of the membrane, and involves two universal constants for the toroidal geometry.