Hertzian Fracture in Single Crystals with the Diamond Structure

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Abstract
Extension of an earlier theory of Hertzian fracture in brittle isotropic materials is here made to include the case of brittle single crystals, with particular reference to crystals having the diamond structure. A detailed description is first given of the inhomogeneous stress field in a flat, elastic specimen loaded normally with a hard sphere. The geometry of cracks growing in such a stress field is then discussed, taking into account the anisotropy in surface energy relevant to the diamond-structure crystals. Analyses of the mechanics of crack growth into the crystals subsequently indicate that for a certain range of indenter size, the Hertzian crack passes through four equilibrium stages, as it does in glass, before reaching its fully developed length. As a result Auerbach's law, which states that the critical load on a spherical indenter necessary to produce a fully developed Hertzian fracture is proportional to the radius of the indenter, holds within this certain range of indenter size. This law is experimentally confirmed Hertzian fracture tests on single crystals of silicon. The mechanism of crack initiation and growth outlined in this paper is then discussed in terms of conclusions made by previous workers on the nature of the Hertzian cracks in the diamond structure crystals. Finally, possible application of the Hertzian test to the study of some mechanical properties related to the fracture surface energy in brittle single crystals is indicated.