A phenomenological description is given of the non-linear laws of deformation of bonds in the bridged zone of a crack, taking parts of hardening and softening into account. A system of non-linear singular integro-differential equations is obtained for determining the stresses in the bonds in the bridged zone of a crack at the interface of materials. The size of the crack bridged zone is not assumed to be small compared with the size of the crack. A procedure for the numerical solution of the system obtained is considered, based on the method of variable elasticity parameters. Numerical experiments have been conducted to investigate the influence of the parameters of the non-linear part of the bond deformation curve, the size of the crack bridged zone and the magnitude of the external load on the convergence of the iteration process of the solution of the system. The results obtained may be useful (in spite of the limited potential for transferring the solution of non-linear problems to other scales) in developing procedures for solving problems within the framework of the model of a crack with bonds by finite and boundary element methods.