Large Deformation Analysis for a Cylindrical Hyperelastic Membrane of Rubber-Like Material under Internal Pressure

Abstract

Abstract This work studies the large deformation of a cylindrical hyperelastic membrane circumferentially bonded and sealed at each end to a rigid tube. The membrane is subjected to the delivery of an inflating fluid through the tubular channel, causing it to undergo the large, quasi-static axisymmetric deformation. The membrane is made from a rubberlike material and assumed to be isotropic and incompressible. The Lagrangian formalism is employed to develop geometric relations of the deformation field and the system of governing equations in terms of principal stretches and Cauchy stresses. With the material's constitutive laws and proper boundary conditions, the incorporation of the geometric relations and governing equations is made to derive the numerical solution system of the deformation field in the form of two-point boundary-value problem as composed of four first-order ordinary differential equations. Special attention is given to relevant numerical formulation. The Newton-Raphson iterative algorithm, together with the fourth-order Runge—Kutta algorithm, is utilized. A geometric approximation on an inflated bulge is presented to give an initial guess to a deformed membrane profile. In an attempt to obtain convergent behavior of numerical solution along the equilibrium path of deformation, a displacement control strategy is suggested to mimic the quasi-static volume-controlled inflation process. Numerical results are presented. The occurrence of deformation instability is discussed. The effects of various strain energy density functions on inflation kinematics are analyzed.