The determination of limiting pressure in simultaneous elongation and inflation of nonlinear elastic tubes

&WI&-A nonlinear elastic tub&r membrane bonded at its ends to rigid plates is subjected to internal pressure and elongation. For a fixed elongation, pressure initially increases with radius, reaches a local ~ximum and then decreases. The purpose of this work is to determine this limiting pressure for each prescribed elongation. The usual boundary value problem formulation is such that a two dimensional search must be conducted. This can be computationally very costly. A method is presented which reduces the determination of the limiting pressure to a one-dimensional search. A numerical example is presented. An interesting p~enomenun in non~n~ elasticity arises in the problem of the inflation of a sph&d me,mbrane by internal pressure. In studying this problem for an incompressible Mooney model, Green and Shield111 showed that the inflating pressure d-need not montonicahy increase with deformed radius r. In particular, p can increase montonically. to a local maximum, decrease to a local minimum and then increase once more. This local maximum represents a limiting pressure. If a higher internal pressure is applied, the membrane will either respond dynamically or assume a much larger equ~b~um state. The same phenomenon appears in other membrane problems, such as the inflation by lateral pressure of a flat circular membrane clamped along a boundary[2] or the inflation of a torus by internal pressure[3]. In each of these cases, the pressure has a local maximum when considered as a function of an appropriate deformation parameter. Again for the reason discussed above, it is useful to know the magnitude and deformation at this local m~imum. In the above examples there is only one load parameter, the internal pressure. Now consider a nonlinear, elastic, cylindrical membrane which is bonded at its ends to rigid plates. The membrane is to be subjected to simultaneous inflation and elongation. Two load parameters must now be specified, the internal pressure and either the elongation or force applied to the end plates. This problem was first solved an~~ic~fy by Kydoniefs and Spencer[4] for the special case of zero end forces. They presented deformed profiles for a number of internal pressures. Their results indicate that the relation between pressure and deformed radius at mid-length has a local maximum. However, this case is governed by only one load parameter, the pressure. The present work is concerned with determining the local maximum in the more general case, when two parameters are varied. For the cases of the pressurized spherical and toroidal membranes, and the clamped membrane, it is possible to construct the pressure-geometry relation by an inverse procedure. The problem formulations are such that the local maximum can be found by a one parameter search. On the other hand, the formulation of the tube problem is such that a two parameter search must be conducted. Since this can be very time consuming, often prohibitive in computing cost, an alternate method is desirable The one presented here is an optimization scheme using a projected gradient. By means of this method, the search for the maximum pressure is confined to a path (a one dimensional space) instead of a two dimensional space, allowing a significant saving in computer time to be realized. It is applied to the tube problem to obtain results for this interesting case as well as to demonstrate the method for more complicated problems.