Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Estimation of material parameters from volume–pressure data

A method is introduced for estimating values of the hyperelastic material parameters and the residual membrane tension (prestretch) that best describe a set of volume-pressure inflation response data acquired during a cell-elastomer composite diaphragm inflation (CDI) experiment. Based on a model for the quasi-static inflation of a prestretched clamped circular isotropic incompressible hyperelastic membrane into a horizontally semi-infinite incompressible liquid reservoir of finite vertical depth, the fit methodology presented here is developed explicitly for a Mooney-Rivlin (MR) strain energy function. The parameter fit is formulated as a nonlinear least-squares optimization problem (NLSOP), and two empirical constraints are postulated to restrict the domain of model fit parameters to physically relevant values. A numerical least-squares regression analysis utilizing the Nelder-Mead simplex algorithm is proposed for solving the constrained NLSOP. To demonstrate the capabilities of the method, the fit procedure is utilized to estimate the residual membrane tensions and the MR constitutive parameters C1 and C2 that characterize the pseudoelastic loading and unloading responses of a polydimethylsiloxane elastomer membrane as observed in the context of a CDI experiment. The material parameter estimates are verified as being both accurate and repeatable, and the MR non-uniform inflation model is shown to produce better fits (in a least-squares sense) than direct polynomial fitting of the original data set. Extension of the fit methodology to volume-pressure data measured for cell-elastomer composite diaphragm specimens, though possible, requires the development of new constitutive models that can better account for the complex rheological behavior of a living epithelial sheet.