Mathematical formulation of an analytic approach to the stress field in a flattened Brazilian disc

Abstract

Although more than fifteen years have passed since the flattened Brazilian disc test was introduced, analytic studies on the respective stress field are scarce, mainly due to difficulties related to the shape of its boundary. In this context, an alternative mathematical formulation, providing a full-field solution for the distribution of stresses in the flattened Brazilian disc, is here proposed, based on the configuration of an intact circular disc, under a large number of equal point forces, arranged at equal distances along two symmetric parallel chords of the disc. Considering each point force as the resultant of stresses acting on an eliminated hole, a great number of such forces implies the introduction of a large number of small holes, dividing the uniform disc into three distinct parts; the one between the two chords is to stand as the flattened disc under uniform compression. This particular problem for the uniform disc is here solved with the aid of Muskhelishvili’s complex potential technique, providing a closed-form, full-field solution for the stresses on the as above “truncated” disc, and thus, in a first approximation, on the flattened Brazilian disc. The analysis, apart from highlighting the advantage of the flattened against the classical intact Brazilian disc (regarding the reduction of stress concentration along the disc-loading platen contact region), revealed, also, that the pressure on the flat edges of the flattened disc should be non-uniform in order to better approximate the experimental reality. In this direction, a numerical model was developed, indicating the proper non-uniform distribution of pressure that should be applied along the flat edges of the flattened Brazilian disc, in order for these edges to remain plane after deformation, in accordance to the laboratory implementation of the experiment.