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'.

The Ogden model of rubber mechanics: 50 years of impact on nonlinear elasticity

“…Selvadurai [31] analyses the mechanics of deformation of incompressible planar hyperelastic membranes, rigidly fixed at their boundaries and subject to uniform pressure. He focuses on the neo-Hookean, Mooney–Rivlin and Ogden strain energy forms, and solves the governing equations numerically using the finite element method.…”

On Spencer’s displacement function approach for problems in second-order elasticity theory

“…A second-order elasticity problem of the centrally loaded spherical rigid inclusion analogue of Kelvin's problem was examined by Selvadurai [90]. In order to develop the second-order solution to the localized Kelvin force problem, we utilize the firstorder solution given by equation (35). Avoiding details of lengthy algebraic manipulations, it can be shown that the partial differential equation governing the second-order displacement function takes the form:…”

“…The application of the theory of second-order elasticity to Boussinesq's problem is a natural extension of the study dealing with Kelvin's problem. The first-order solution for Boussinesq's problem is identical in form to Kelvin's solution given by equation (35) and the arbitrary constant corresponding to equation ( 36) is given by:…”

“…The seminal works of RS Rivlin and co-workers presented a comprehensive and systematic approach to the description of hyperelastic materials, commencing with the formulation of constitutive relations by appeal to the theory of invariants, the solution of benchmark problems involving homogeneous strains, inflation and eversion of annular regions, torsion and bending of prismatic bodies, and performing experiments to validate the mathematical approach. References to these accomplishments are also contained in the collected works of RS Rivlin edited by Barenblatt and Joseph [1] and well documented [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] in terms of the contribution of these works to the overall advancement of non-linear continuum mechanics and, in particular, the development of constitutive equations for rubber-like elastic materials [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].…”