B ifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers. Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.

https://hal.archives-ouvertes.fr/hal-01302310/document

Bifurcation of inflated circular cylinders of elastic material under axial loading—II. Exact theory for thick-walled tubes

Bifurcations in axially compressed thick elastic tubes

In this paper, the problem of finite deformation of a cylindrical rubber tube with an internal rigid tube is examined, where the rubber tube is subjected to axial loads on its two ends. A reasonable mathematic model is formulated by using the nonlinear field theory. Then, the implicit solutions that describe the cases of tension and compression are derived. The influence of end loads, thickness and length on the finite deformation is discussed in detail. Numerical simulations are given simultaneously. It also shows that for a large domain of the middle part of the tube, the deformation is very close to a uniform case, but near the two ends of the tube, the change is very fast, which coincides with the mechanical backgrounds.

Finite Deformation of a Class of Rubber Cylindrical Tubes Subjected to Axial Loads

In this paper, the deformation of a thick-walled circular cylindrical shell of incompressible isotropic elastic
material is considered. The shell, which is made of Three-Term strain energy function is subjected to the
combined external and axial loading pressure. In order to obtain the relevant eigenvalues, which is the main
objective of the work, the incremental equilibrium equations are solved with two numerical, i.e. Adams-Moulton
and Compound matrix methods. Finally the bifurcation behavior is investigated by plotting the radius changes
with respect to the changes of the length of the cylinder.

https://ijsts.shirazu.ac.ir/article_2432_a0e85c47b16bd83821a243d1f7592b3f.pdf

The axisymmetric bifurcation analysis of an elastic cylindrical shell subjected to external pressure and axial loading

In this paper, the problem of a rubber-like hollow circular cylinder subjected to a finite uniform initial stress in the axial direction is considered. An infinitesimal axial-symmetric deformation is then superposed on the prestressed hollow cylinder and the axial-symmetric instability of the cylinder caused by the initial stress is investigated. The analysis is based on the general theory of mechanics of incremental deformations established by Biot[ll]. It is shown that introducing a displacement potential function, φ, the field equation previously derived by Sun[15] is obtained. The solution of this field equation is shown to be given in terms of modified Bessel functions of the first and the second kind in r and trigonometric functions of z . By satisfying the homogeneous boundary conditions, the characteristic equation governing stability of the hollow cylinder is obtained. Numerical results examining the critical initial stress as a function of the slenderness ratio and the geometry of the cylinder are presented and discussed.

C.T. Sun

Papers

Instability of a hollow rubber-like cylinder under initial stress