The finite circular disc with a central elliptic hole under parabolic pressure

The mechanical response of a finite circular disc with a central elliptic hole is modelled mathematically assuming that the disc is subjected to radial pressure varying according to a parabolic law along two finite arcs of its periphery. The symmetry axis of the pressure forms an arbitrary angle with respect to the major axis of the elliptic hole. Using the complex potentials technique, the displacement and stress fields are determined in the form of infinite series. Assuming then that the short semi-axis of the elliptic hole tends to zero, compact expressions are obtained for the stress intensity factors characterizing the severity of the stress field around the end points of the ellipse's major axis, which now becomes the tips of a discontinuity resembling a mathematic crack. The solution for the stress intensity factors is validated against the respective one proposed by Atkinson et al. in the limiting case of a centrally cracked disc under diametral compressive point forces. The agreement is very good as long as the length of the crack does not exceed the disc's radius. The novelty of the present solution (besides imposing parabolic pressure instead of uniform one or point forces) is that the discontinuity covers the whole range from the circular hole (ring) to the mathematical crack. Moreover, the expressions obtained are exact, complete and easily programmable. From a practical point of view, the solution introduced can be proven a valuable, easy-to-use tool for engineers using the Brazilian-disc test for the determination of fracture toughness of brittle geomaterials. Indeed, the parabolic pressure considered here approaches closely the actual load distribution exerted on the disc during the standardized implementation of the test. In addition, the form of discontinuity considered here is closer to the shape of the cracks actually machined in the specimens of the test which are by no means mathematical cuts but rather they are slits with a finite distance between their lips.