Lie Symmetry Analysis and Approximate Solutions for Non-linear Radial Oscillations of an Incompressible Mooney–Rivlin Cylindrical Tube

Abstract

The non-linear, second-order differential equation derived by Knowles (1960, Quart. Appl. Math.18, 71–77) which governs the axisymmetric radial oscillations of an infinitely long, hyperelastic cylindrical tube of Mooney–Rivlin material is considered. It is shown that if the boundary conditions are time dependent, then the Knowles equation has no Lie point symmetries, while if the boundary conditions are constant it has one Lie point symmetry corresponding to time translational invariance. The derivation by Knowles (1962, J. Appl. Mech.29, 283–286) of bounds on the period of the oscillation for the heaviside step loading boundary condition is extended to obtain limiting oscillations that exhibit periods that bound the exact period above and below. The Knowles equation for a Mooney–Rivlin material is expanded in powers of a dimensionless parameter, μ, defined in terms of the thickness of the tube wall. To zero order in μ an Ermakov–Pinney equation is obtained which has three Lie point symmetries. It is shown that the differential equation which is correct to first order in μ also has three Lie point symmetries which disappear at second order in μ. For time independent boundary conditions, the three Lie point symmetries of the order μ equation are derived explicitly and the associated first integrals are obtained. The general solution is derived in terms of the three first integrals and it is illustrated for free oscillations and the heaviside step loading boundary condition. The non-autonomous first order in μ equation is transformed to an autonomous Ermakov–Pinney equation and a non-linear superposition principle for the solution to first order in μ is derived and applied to a blast loaded applied pressure that decays linearly with time. The solutions to first order in μ are compared with numerical solutions of the Knowles equation for a thick-walled cylinder and are found to be more accurate than the zero order solutions described by the Ermakov–Pinney equation.