Stresses and Displacements in an Elliptically Perforated Circular Disc Under Radial Pressure

The complex potentials governing the elastic equilibrium of a finite circular disc, elliptically perforated at its center, are obtained using Muskhelishvili’s formulation. The disc is subjected to non-uniform distribution of pressure along two symmetric finite arcs of its periphery. Given the complex potentials, the stress- and displacement-fields can be determined everywhere on the disc by introducing a novel flexible interpretation of the conformal mapping, suitably adjusted to the computational process. The results of this procedure are given for various strategic loci and are critically discussed. The length of the loaded arc is considered similar to that obtained from the solution of the intact disc-circular jaw elastic contact problem assuming that the disc is compressed between the steel jaws of the device suggested by the International Society for Rock Mechanics for the implementation of the Brazilian-disc test. Concerning the distribution of the externally induced pressure along the loaded arcs, it is proven that for the general asymmetric configuration (i.e. the axes of the elliptical hole are neither parallel nor normal to the loading axis) the induced asymmetric displacement field does not permit maintenance of equilibrium of the disc as a whole in case the disc is considered exclusively under a distribution of radial pressure: Additional tractions must be exerted along the loaded arcs, obviously in the form of frictional stresses. Besides, providing full-field, analytic expressions for stresses and displacements, the main advantage of the present solution is that, by properly choosing the ratio of the ellipse’s semi-axes, the solution of three additional configurations, of major importance in engineering praxis, are obtained: These of the intact disc, the circular ring and the cracked disc.