A VARIATIONAL APPROACH TO A CIRCULAR HYPERELASTIC MEMBRANE PROBLEM

The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight. It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.