Exact solutions of the mixed axisymmetric problem of the torsion of an elastic space containing a spherical crack

The problem of the torsion of an elastic space, weakened by a spherical crack, is reduced to a system of paired summation equations in first-order associated Legendre functions. It is assumed that the load, applied to the crack surface, can also be represented in the form of a series in associated Legendre functions. Using special differential operators, this system is reduced to permitting an exact elementary solution of a system of equations in Legendre polynomials. Two examples are given. The solution is compared with a known result in the literature. The problem of the effect of curvature of the surface on the stress intensity factor is investigated.