D.M. Haughton

Stability and bifurcation of inflation of elastic cylinders

A method of obtaining a full (two–dimensional) nonlinear stability analysis of inhomogeneous deformations of arbitrary incompressible hyperelastic materials is presented. The analysis that we develop replaces the second variation condition expressed as an integral involving two arbitrary perturbations, with an equivalent (third–order) system of ordinary differential equations. The positive–definiteness condition is thereby reduced to the simple numerical evaluation of zeros of a well–behaved function. The general theory is illustrated by applying it to the problem of the inflation of axially stretched thick–walled tubes. The bifurcation theory of such deformations is well known and we compare the bifurcation results with the new stability analysis.

Flexure and compression of incompressible elastic plates

Abstract We consider the bifurcation problem for three-dimensional isotropic, incompressible elastic plates subjected to a combined flexure and axial compression. The compressive loading being along the axis of the right circular cylinder formed by the flexure. Some limited analytical results and more comprehensive numerical results are presented for Neo–Hookean materials. Firstly, we give basic results for a plate subjected to pure flexure and then we show how a pure bending mode and a buckling mode due to axial compression can interact.

On the eversion of incompressible elastic cylinders

Abstract The eversion of incompressible isotropic elastic cylinders is considered. The main problem is formulated as a bifurcation problem with the undeformed thickness ratio as a parameter. Numerical results are presented for several different material models and, in all cases, we find that there is a critical thickness for the tube. Thinner tubes can be everted to a cylindrical shape, thicker tubes will undergo a bifurcation upon eversion and so the cylindrical shape will not be attained. We find that the length to radius ratio of the tube only plays a significant role for one particular bifurcation mode.

Cavitation in compressible elastic membranes

Abstract A thin disk of compressible elastic material is subjected to a uniform radial tension. For the Blatz-Ko material it is shown numerically that solutions containing a cavity exist provided that the tension exceeds some modest value. Analytical solutions are obtained for another material and non-existence is demonstrated.

The Effect of Elastic Surface Coating on the Finite Deformation and Bifurcation of a Pressurized Circular Annulus

A plane-strain theory of an elastic solid coated with a thin elastic film on part or all of its boundary was developed recently by Steigmann and Ogden (1997a). In this paper the theory is applied to the (plane-strain) problem of a thick-walled circular cylindrical tube which is subject to both internal and external pressure and which has an elastic coating on one or both of its circular cylindrical boundaries. The effect of the coating on the symmetrical response of the annular cross-section of the tube is determined first. It is noted, in particular, that while the pressure may exhibit a maximum followed by a minimum during inflation for an uncoated tube it may be a monotonic increasing function of the radius for a coated tube with coating elastic modulus sufficiently large. Next, the possibility of bifurcation from a symmetrical configuration is examined and again the influence of the coating is analysed. The effect of a coating on the outer boundary is compared with that on the inner boundary. Specifically, during compression, coating on the outer boundary delays bifurcation compared with the uncoated case. On the other hand, when the coating is on the inner boundary, bifurcation is either delayed or advanced relative to the uncoated situation depending on the values of the bending stiffness and tube thickness parameters. Generally, bifurcation is delayed by an increase in the magnitude of the bending stiffness of the coating at fixed values of the other parameters.

Wrinkling of annular discs subjected to radial displacements

Abstract We consider an annular elastic membrane subjected to radial displacements or specified radial stresses on the inner and outer edges. The aim is to provide a simple demonstration of tension field theory combined with the use of a relaxed strain-energy function to model wrinkling. Both incompressible and compressible Varge strain-energy functions are considered and exact solutions are found for both the smooth and wrinkled states.

A Comparison of Stability and Bifurcation Criteria for Inflated Spherical Elastic Shells

The nonlinear stability analysis introduced by Chen and Haughton [1] is employed to study the full nonlinear stability of the non-homogeneous spherically symmetric deformation of an elastic thick-walled sphere. The shell is composed of an arbitrary homogeneous, incompressible elastic material. The stability criterion ultimately requires the solution of a third-order nonlinear ordinary differential equation. Numerical calculations performed for a wide variety of well-known incompressible materials are then compared with existing bifurcation results and are found to be identical. Further analysis and comparison between stability and bifurcation are conducted for the case of thin shells and we prove by direct calculation that the two criteria are identical for all modes and all materials.

Existence of Exact Solutions for the Eversion of Elastic Cylinders

The existence of eversion deformations of an elastic cylinder into another right circular cylinder is studied. It is found that such deformations are associated with strain-energy functions of separable form W(λ1,λ2,λ3) =φ(λ1)+φ(λ2)+φ (λ3), and thus can serve as a test for materials of this kind. Under certain constitutive assumptions, there always exists a cylindrical eversion deformation that satisfies exact pointwise traction free boundary conditions over the entire surface of the cylinder. For such solutions the cavity must remain open upon eversion.

Wrinkling of inflated elastic cylindrical membranes under flexure

We investigate the bending of inflated cylindrical elastic membrane tubes with particular reference to the initiation of wrinkling and the subsequent deformation. While the equilibrium equations reduce to quadratures we still require a numerical evaluation. Both incompressible and compressible materials are considered and we look at different ways of constructing the (post wrinkling) shape of the membrane. The effect of changing volume with increased bending is also investigated.

Further Results for the Eversion of Highly Compressible Elastic Cylinders

We consider the eversion problem for highly compressible hyperelastic thick-walled cylinders. We focus attention on two features of such problems that are not adequately described by standard analysis. We investigate, first, closure of the cavity for sufficiently thick tubes and, second, the instability of relatively thin tubes. We find that the closure of the cavity can be ascribed to the fact that actual foam cylinders behave differently in tension and compression. However, the instability of thinner cylinders seems to be governed by some other mechanism. In the course of the analysis, we show how exact solutions to the equilibrium equation for the eversion problem can be generated. Unfortunately such solutions are not compatible with the boundary conditions.