Azimuthal Shear of a Transversely Isotropic Elastic Solid

Abstract

In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either "with" or "against" the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress-strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.

Deformation induced loss of ellipticity in an anisotropic circular cylindrical tube

“…Explicit expressions for σ r θ and σ r z obtained from a given constitutive equation, coupled with (17), can in principle be used to obtain two algebraic formulas (in general implicit and coupled) for γ θ and γ z in terms of the stretches and other parameters of the problem. In general, the solutions for γ θ and γ z may not be unique, as exemplified in the azimuthal shear problem discussed in [5] and [9].…”

“…The azimuthal shear problem for an incompressible isotropic elastic material was studied by Abeyaratne [3] with a focus on loss of ellipticity and the emergence of discontinuous solutions, while for the anti-plane shear problem Silling [4] considered numerically the passage from ellipticity to hyperbolicity of the governing equations resulting from deformation of an incompressible isotropic material containing a crack or a screw dislocation. For an incompressible transversely isotropic elastic material associated with a single family of fibre directions, the problem of loss of strong ellipticity for the azimuthal shear problem was first studied by Kassianidis et al [5], who examined, in particular, the emergence and disappearance of non-uniqueness of solution. This was extended to the case of two symmetrically arranged fibre families by Dorfmann et al [6] and El Hamdaoui and Merodio [7].…”

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

“…It was shown therein that possible continuous solutions are unstable but the discontinuous solution is stable. In a recent paper by Kassianidis et al [4] the azimuthal shear problem has been examined in detail for a transversely isotropic elastic solid. The preferred direction in the reference configuration associated with the transverse isotropy was taken to lie in planes normal to the tube axis and with its tangent making an angle with the radial direction depending only on the radius so as to preserve in-plane radial symmetry during the (isochoric) pure azimuthal shear deformation.…”

On the preservation of fibre direction during axisymmetric hyperelastic mass-growth of a finite fibre-reinforced tube

“…For instance, appropriate use of (5.17a), (5.17b) and (5.17c) converts (5.23) into 24) which is to be solved simultaneously with (5.18). Hence, introduction into (5.24) of the available solution of (5.18), namely (5.19), reveals that the functions f 1 and f 2 must satisfy the differential relationship:…”

“…It is recalled in this connection that the plane strain analysis of an infinitely long transverse isotropic tube offers some analytical simplification and is therefore not rare in conventional hyperelasticity applications (e.g. [23][24][25][26][27][28]). Useful observations stemming from the presented mass-growth modelling, analysis and applications are finally summarised in Sect.…”