Elastic solution for a circular disk with a central crack under compressive diametrical load

The splitting tensile strength test (Brazilian test) is widely used because of its simplicity to assess the ultimate tensile strength and fracture behaviour of a variety of brittle materials, with specific reference to ceramics, concrete, and cementitious composites. Indeed, from the linear theory of elasticity, the distribution of the normal stress along any diameter of an uncracked disk subjected to a pair of concentrated diametrical compressive forces is known in closed form. Conversely, a Brazilian disk with a pre-existing central crack of a given length turns out to be a much more challenging problem owing to the mixed boundary conditions to be imposed inside and outside the crack along the crack direction, together with the condition about the stress field along the outer curved boundary of the disk. Few studies deal with such a demanding layout. Among these, based on the weight function method, Dong et al. (2004) evaluated the SIFs varying the angle of the external load with respect the crack plane and the crack length as well. Critical conditions for the achievement of pure mode I and mode II loading have been found also. However, that study is restricted to the neighbouring of the crack tips. In the present study, the full field solution of the Brazilian disk is provided analytically in terms of Airy stress function in bipolar coordinates. The study is handled by examining separately a skew-symmetric and a symmetric loading condition, representative for the mode II and mode I loadings, respectively. Both the situations lead to a Fredholm hypersingular integral equation, whose solution is found through a collocation method by expanding the unknown in series of Chebyshev polynomials. It is pointed out that for mode I loading, a closing or an opening crack may arise. Both these circumstances have been analysed in detail. Conversely to the existing studies, the proposed formulation allows assessing both the displacement and stress fields along the entire diameter of the disk in the direction of the crack for any loading angle.