Large amplitude oscillations of a tube of incompressible elastic material

Abstract

Introduction.In recent years a number of problems have been solved in the nonlinear theory of elasticity for incompressible bodies (see [1] and [2] for discussion and references). The work of Rivlin and others has shown that the assumption of incompressibility makes possible the solution of these problems without any special assumptions as to the form of the strain energy function characteristic of the material. Because of the severe non-linearity involved, the special problems considered have been static ones and the methods of attack have been essentially inverse. The question of waves in incompressible materials has been considered by Ericksen [7], The present paper treats the dynamic problem of axially symmetric oscillations of an infinitely long circular cylindrical tube of incompressible elastic material. The symmetry of the motion and the condition of incompressibility combine to permit the reduction of the problem to one to which the methods of the theory of non-linear vibrations of single-degree-of-freedom systems can be applied. The approach is semi-inverse in the sense that the determination of all physical quantities is made to depend on one unknown function which satisfies a non-linear ordinary differential equation.The specific problem considered here is that of arbitrary amplitude free oscillations of the tube when it is set in motion under given initial conditions and subject to no surface pressures. It is shown that periodic motions are possible for a large class of strain energy functions, and an expression for the period of oscillation is given in terms of the strain energy function. The results are shown to take a particularly simple form in the limiting case of a thin tube. Explicit results are given for the strain energy corresponding to a material of Mooney-Rivlin type.The static problem of the symmetrical deformation of a tube of incompressible material by uniform internal and external pressures is a special case of a more general problem solved by Rivlin [3] and discussed in [1]. It may be remarked that results similar to those contained in this paper can be obtained for the radial oscillations of a spherical shell of incompressible material. The corresponding static problem of the symmetrical expansion of a spherical shell under pressure was solved by Green and Shield [4] and is also discussed in [1].2. Formulation of the problem. Consider an infinitely long circular cylindrical tube of homogeneous, isotropic, elastic, incompressible material which in its undeformed state has inner radius rt and outer radius r2 . A point in the tube which at time t has cylindrical coordinates R, 8, z is assumed to have been at the point r, 6, z in the undeformed state. The motion is thus completely described by the function R = R(r, t).

Finite amplitute oscillations in curvilinearly aeolotropic elastic sylinders

“…4, the differential equation governing the motion of the cylindrical tube is derived and is reduced to a form identical to that of [6], [7] for isotropic materials. 3) X" is the material co-ordinate system with the metric tensor Gaf and x' is the spatial co-ordinate system with the metric tensor .…”

“…4 and the other two follow from substitution into (3.6) and (3.7). As Knowles [6] remarked, (3.7) implies that at infinity a normal force is required over the ends of the cylinder. Finally, the invariants are: Thus, using Truesdell's theorem [11], [12], the acceleration potential is:…”

“…Following the procedure in [12] and evaluating the stresses at r = rj and r -r2 in (4.3) and subtracting the second from the first, and using which is the form derived by Knowles [6], [7]. Next, define…”

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“…Introduction. Radial motion of a non-linear elastic tube is well understood [1], [2]. In considering the corresponding motion of a non-linear viscoelastic tube an equation describing the behaviour can be derived in a straight forward manner.…”

“…On the other hand given AP and JHt is a more difficult problem to find $(0> and unlike the elastic case [1] it does not seem possible to proceed in general.…”