Inflation of a circular elastomeric membrane into a horizontally semi-infinite liquid reservoir of finite vertical depth: Quasi-static deformation model

Similar in both form and function to a classical circular membrane inflation experiment, cell-elastomer composite diaphragm inflation (CDI) experiments have recently been introduced as a methodology for exploring the mechanobiological properties of cell–cell anchoring junctions and cytoskeletal protein networks within a living epithelial sheet. Despite outward similarities, CDI experiments are unique from classical membrane inflation in that the loading associated with the inflation process, though axisymmetric, is inherently non-uniform. In order to better account for the effects of non-uniform loading in a CDI experiment, we introduce 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. A set of coupled nonlinear first-order differential equations is shown to govern the inflation process. Assuming a Mooney–Rivlin (MR) constitutive model, the governing equations are formulated into a boundary value problem, and a procedure is developed for finding numerical solutions that quantify discrete membrane inflation states in terms of deformations, displacements, curvatures, stress resultants, and an inflation pressure distribution. Through iteration of the discrete state solution procedure, we further demonstrate how to generate a polynomial interpolating function that approximates the continuous volume–pressure response of a MR membrane. Select numerical examples are used to simulate the inflation response of a polydimethylsiloxane (PDMS) elastomer membrane used in an actual CDI experiment.