Analytical pressure–deflection curves for the inflation of pre-stretched circular membranes

Abstract

The inflation of circular membranes is a benchmark problem in finite elasticity. Due to its mathematical complexity, so far most efforts have been focused on numerical solutions whereas analytical solutions are still lacking. The present work proposes an analytical formula to compute the pressure–deflection curves of pre-stretched circular membranes. The Mooney–Rivlin hyperelastic model is assumed as constitutive law. Following the semi-inverse method, an analytical model assuming spherical deformed configurations is developed and a pressure–deflection relation is derived. Due to the simplifying hypothesis on the kinematics, the analytical expression for the pressure is not always accurate and requires an adjustment. Two corrective polynomial functions are implemented in the formula to capture the effect of the pre-stretch, which affects the pressure–deflection response depending on the magnitude of deformation and material parameters. The polynomial coefficients are calibrated by fitting numerical solutions of the differential equilibrium equations of the exact theory. The calibration is performed over a wide range of material parameters and pre-stretch, covering all the values of practical interest. As a result, an accurate and ready-to-use pressure–deflection relation for practical applications is obtained. The proposed formula is validated by finite element simulations. Differently from other solutions in literature, our formula holds for both compressible and incompressible materials. Experimental bulge tests on pre-stretched rubber membranes are carried out and the proposed formula is used for the calibration of material parameters. A code developed in the software Mathematica is made available upon request to directly compute the proposed formula.