Lie symmetry analysis and approximate solutions for non-linear radial oscillations of an incompressible Mooney-Rivlin cylindrical tube

The non-linear, second-order differential equation derived by Knowles (1960 , Quart. Appl. Math. 18, 71-77) which governs the axisymmetric radial oscil lations of an infinitely long, hyperelastic cylindrical tube of Mooney-Rivl in material is considered. It is shown that if the boundary conditions are Lime dependent, then the Knowles equation has no Lie point symmetries, whil e if the boundary conditions are constant it has one Lie point symmetry cor responding to time translational invariance. The derivation by Knowles (196 2, J. Appl. Mech. 29, 283-286) of bounds on the period of the oscillation f or the heaviside step loading boundary condition is extended to obtain limi ting oscillations that exhibit periods that bound the exact period above an d below. The Knowles equation for a Mooney-Rivlin material is expanded in p owers of a dimensionless parameter, CL, defined in terms of the thickness o f the tube wall. To zero order in mu an Ermakov-Pinney equation is obtained which has three Lie point symmetries. It is shown that the differential eq uation which is correct to first order in mu also has three Lie point symme tries which disappear at second order in mu. For time independent boundary conditions, the three Lie point symmetries of the order mu equation are der ived explicitly and the associated first integrals are obtained. The genera l solution is derived in terms of the three first integrals and it is illus trated for free oscillations and the heaviside step loading boundary condit ion. The non-autonomous first order in mu equation is transformed to an aut onomous Ermakov-Pinney equation and a non-linear superposition principle fo r the solution to first order in mu is derived and applied to a blast loade d applied pressure that decays linearly with time. The solutions to first o rder in mu are compared with numerical solutions of the Knowles equation fo r a thick-walled cylinder and are found to be more accurate than the zero o rder solutions described by the Ermakov-Pinney equation. (C) 2000 Academic Press.