The remarkable nature of cylindrically anisotropic elastic materials exemplified by an anti-plane deformation

A material is cylindrically anisotropic when its elastic moduli referred to a cylindrical coordinate system are constants. Examples of cylindrically anisotropic materials are tree trunks, carbon fibers [1], certain steel bars, and manufactured composites [2]. Lekhnitskii [3] was the first one to observe that the stress at the axis of a circular rod of cylindrically monoclinic material can be infinite when the rod is subject to a uniform radial pressure (see also [4]). Ting [5] has shown that the stress at the axis of the circular rod can also be infinite under a torsion or a uniform extension. In this paper we first modify the Lekhnitskii formalism for a cylindrical coordinate system. We then consider a wedge of cylindrically monoclinic elastic material under anti-plane deformations. The stress singularity at the wedge apex depends on one material parameter gamma. For a given wedge angle 2 alpha, one can choose a gamma so that the stress at the wedge apex is infinite. The wedge angle 2 alpha can be any angle. It need not be larger than pi, as is the case when the material is homogeneously isotropic or anisotropic. In the special case of a crack (2 alpha = 2 pi) there can be more than one stress singularity, some of them are stronger than the square root singularity. On the other hand, if gamma < 1/2 there is no stress singularity at the wedge apex for any wedge angle, including the special case of a crack. The classical paradox of Levy [6] and Carothers [7] for an isotropic elastic wedge also appears for a cylindrically anisotropic elastic wedge. There can be more than one critical wedge angle and, again, the critical wedge angle can be any angle.