Solution by quadratures of the problem of a cylindrical crack by the method of matrix factorization

In this paper the axisymmetric problem on a semi-infinite cylindrical crack is considered. On the surfaces of the crack, the normal and tangential components of the traction are prescribed whereas the displacement vector components are unknown and supposed to be discontinuous. The problem is reduced to a 2 × 2 matrix Wiener–Hopf factorization. The solution is found by quadratures. Thus, this is the first example of successful closed-form matrix factorization arisen in the theory of mixed boundary-value problems for elastic bodies with curvilinear spatial defects. In addition, the weight functions for the stressintensity factors are constructed. Numerical results for the stress-intensity factors for two types of loading: (i) the exponential functions and (ii) a point force acting along the axis of symmetry, are reported.