The Displacement Field in a Finite Circular Disc with a Central Rectangular Slit

The displacement field in a finite, centrally cracked, circular disc compressed by a pair of radial stress distributions, acting along two anti-symmetric arcs of its periphery, is obtained. This configuration simulates the cracked Brazilian-disc test, used for the determination of fracture toughness of brittle materials. The crack is simulated as a rectangular slit of finite distance between its lips and rounded corners rather than as a mathematical discontinuity of zero distance between its lips. To achieve an analytic, closed-form solution Muskhelishvili's complex potentials’ technique is used, adopting a suitable conformal mapping function described by Savin. Advantage is taken of recently introduced solutions for the finite ring and the finite disc with a central elliptical hole. The formulae obtained permit thorough exploration of the disc's and slit's deformed shapes, enlightening the role of the slit's actual shape. It is concluded that simulating the crack as a mathematical discontinuity may undermine the validity of the test.