Mechanics of pressurized planar hyperelastic membranes

Abstract

The paper examines the mechanics of inflation of incompressible planar hyperelastic membranes that are rigidly fixed at their boundary and subjected to a uniform pressure. Strain energy functions characterized by the neo-Hookean, Mooney–Rivlin and the Ogden forms are used. Fixity is provided along either circular or elliptical boundaries. The computational results indicate that the strain energy function has a significant influence on the pressure versus inflated volume response of the deformed membrane. When the strain energy function corresponds to a Mooney–Rivlin form, the circular membrane displays no tendency to develop any instability. The equivalent circular membranes composed of both the neo-Hookean and Ogden-type strain energy functions developed an initial ‘ Wrinkling Instability ’. For planar membranes with an elliptical planform, the wrinkling instability is more pronounced; membranes composed of hyperelastic materials with a Mooney–Rivlin form of the strain energy function continue to deform without the development of an initial instability point, whereas membranes composed of both the neo-Hookean and Ogden materials exhibit wrinkling behaviour at critical locations at the interior of the fixed boundary region. The dependency of the strain energy function on the second invariant of the Cauchy–Green strain tensor has an influence in the suppression of hyperelastic effects. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.