Large Axisymmetric Deformation of a Nonlinear Viscoelastic Membrane Due to Spinning

Abstract

The problem of a viscoelastic membrane undergoing large planar deformation due to spinning is solved. The membrane, rigidly bonded at its inner boundary and traction-free at the outer, consists of a nonlinear viscoelastic solid whose behavior is modeled by a nonlinear single integral constitutive equation. The model includes the possibility that the relaxation time is influenced by the amount of stretching. Spinning is considered to be sufficiently slow so that shear effects can be neglected. The formulation uses radial and circumferential stretch ratios as dependent variables. These satisfy a system of first-order nonlinear partial-differential integral equations. The numerical procedure at each time step obtains the current spatial derivatives of the stretch ratios in terms of the current and previously determined stretch ratios. This gives essentially a first-order nonlinear system of differential equations, of the same structure as that obtained in the elastic version of the problem [1], which is integrated numerically. Solutions are obtained for both strain-dependent and strain-independent relaxation times.