Stability and bifurcation of inflation of elastic cylinders

Abstract

International audienceA 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 di¬erential equations. The positive-de­ niteness 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 in®a-tion 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

Rotation, inversion and perversion in anisotropic elastic cylindrical tubes and membranes

“…-the inflation of a spherical (or cylindrical) shell under controlled volume leads to a nonmonotonous response of the pressure [55][56][57]; -the Poynting effect can be considered as an inversion of the normal displacement under pure shear tress (or, equivalently in torsion) [58,59]; -the inversion of the axial strain for pressurized arteries under fixed axial loads [34,60]; -the swelling or shrinking of a rectangular anisotropic elastic tissue with dispersed fibres under uniaxial tension: depending on the degree of dispersion of the fibres such a tissue will either shrink or swell in the direction perpendicular to the fibres (demonstrated numerically in [32]); and -the inversion of rotation for a helical rod under a pure axial load [54].…”

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

“…One interesting approach to the analysis of Hadamard stability is that of [3] in which the integrand function in (4.7) is "suitably" expanded into a series which is then used to develop necessary and sufficient conditions for (4.2). This approach is made reasonably tractable in [3] because of the assumption of incompressibility and the symmetry of the subject boundary value problem.…”

“…This approach is made reasonably tractable in [3] because of the assumption of incompressibility and the symmetry of the subject boundary value problem. In [10], this approach is generalized to compressible materials and three-dimensional perturbations; however, as in [3], the analysis is based on the expandability of the perturbations in suitable series.…”

“…Here, we do not have conditions similar to [3] and [10]. Our aim is to study the form of the planar Couette and twisting Taylor-Couette instabilities, similar to those observed for fluids (see [1,16]).…”

Some solutions for a compressible isotropic elastic material

“…More interestingly, in a comparative study [7] of the stability of a rectangular block under biaxial loadings for the Blatz-Ko, Levinson-Burgess and neo-Hookean materials they showed that 'an incompressible analysis using a realistic material model gives a good approximation to the behaviour of slightly compressible materials in such problems'. The stability properties of two of the (compressible) problems studied in the present article, namely the inflation of a circular cylindrical tube and that of a spherical shell, have recently been examined for incompressible materials by Chen and Haughton [12] and Haughton and Kirkinis [18]. Even though no study of this kind has been published so far in the literature, it seems that for the range of Poisson's ratio and stretches mentioned above, the stability of the solution of a body composed of slightly compressible material, such as the compressible neo-Hookean materials above, could be inferred from an analogous study of the limiting incompressible material.…”