A thin plate with a curvilinear crack under the action of in-plane stresses is considered. A linear elasticity model is assumed to model the behavior of the material of the plate in the bulk. A non-linear boundary condition which includes a surface tension depending on the curvature of the crack is given on the crack surface. Using Muskhelishvili’s formulas for stresses and displacements and Savruk’s integral representations of complex potentials the problem is reduced to a complex singular integro-differential equation. The regularization of this equation by reduction to the system of two real Fredholm equations is discussed. The equation is solved numerically and the results are presented.