A note on arbitrarily loaded penny-shaped cracks in hexagonal crystals

Abstract

It is shown that a general penny-shaped crack situated in a basal plane of a hexagonal crystal can be studied in a straightforward way using the corresponding results appropriate to an isotropic medium. The stress intensity factors are derived, and discussed for the case in which the crack is subjected to a unidirectional shear traction.We shall consider an infinite body composed of linearly elastic material of hexagonal symmetry containing a penny-shaped crack situated in a basal plane. Taking axes in such a way that the basal planes are normal to the z-axis, we shall suppose that the crack occupies the region z= 0, 0 < r < c, where (r, 0, z) £orm a system of cylindrical polar coordinates. We consider the problem in which the crack is loaded by means of tractions applied to its two faces, so that arz(r, 0, 0), aoz(r, 0, 0) and o-zz(r, 0, 0) are prescribed for 0 < r < c, where we use aij (r, O, z) to denote the/j-component of stress at the point (r, 0, z). No loading is applied at infinity, so that o-ij~0 as r2+z2~oc.This general penny-shaped crack problem, but in an isotropic body, has been solved by Guidera and Lardner [1] by regarding the crack as a Somigliana dislocation. If we denote by Au~(r, O) the components of the displacement discontinuity across the crack--i.e. Aui(r, O) = ui(r, O, O+)-ui(r, 0, 0-) (r<c)--then by means of the Somigliana formula an expression can be written down for the stress at a general point in terms of the Au~, and equating this to the prescribed traction on the crack provides a system of three integral equations for the three quantities Aur, Au o and Auz. These integral equations can then be solved.The stress field of an arbitrary dislocation loop lying in a basal plane of a hexagonal material has been evaluated by Tupholme [2]. In particular it was shown in this paper that the field fulfils an observation previously made by Hirth and Lothe [3] to the effect that certain of the stress components on the plane of the loop can be obtained directly from the corresponding isotropic results by means of appropriate replacements of the * Permanent address:

The indentation of a half-space of hexagonal elastic material by a circular punch of arbitrary end-profile

“…The integral equation approach to contact problems is similar to that used by Guidera and Lardner 1-3] to solve the problem of an arbitrarily loaded penny-shaped crack in an isotropic medium. It has been pointed out in a note by Lardner and Tupholme [4] that, by virtue of certain results concerning the properties of dislocation loops in hexagonal crystals derived by Tupholme [5], the solutions for penny-shaped cracks in isotropic media can readily be extended to solve the analogous problems in hexagonal media. In the present paper we shall take advantage of this observation to solve the indentation problem for a hexagonal half-space by extension of Guidera's integral equation method.…”

“…Indentation of a half-space of hexagonal elastic material 79 [3] enables us to write down integral formulae for the stresses of a Somigliana dislocation in the basal plane of a hexagonal material [4].…”

“…For the present contact problem, we are interested in the stress component %~(r, 0, 0), and the appropriate integral formula for this quantity is as follows [3][4][5]. …”