Action of normal load on a plane ring-shaped crack inside elastic sphere (Q2753428)

The problem for an elastic sphere containing a ring-shaped crack at its center is considered in the framework of classic elasticity. For its solution, unidirectional spherical coordinate system and a compressed spheroidal coordinate system are introduced. The displacement vector is sought as a sum of two solutions in the two coordinate systems. Fulfillment of boundary conditions results in dual series equations in Legendre polynomials. The resulting integro-algebraic system with respect to coefficients of two series and some auxiliary function is of Fredholm type and can be solved by the method of reduction. Stress concentration factors are listed in the table, and distributions of normal stresses in the middle plane of the sphere are plotted.