In this thesis one of the mesh-free methods called RKPM is employed to solve the differential equations of strain-gradient elasticity. To this end the corresponding weak form is laid down. Subsequently the relevant stiffness-matrix is obtained by discretization of the weak form. To be sure about the accuracy of the relations, the problem of a plate weakened by a hole under uniform far-field tension, for which the exact solution is available in the literature, is solved. The obtained numerical result is in good agreement with the solution of Eshel and Rosenfeld. Afterwards, a plate containing a crack of finite length subjected to uniform far-field tension (mode I) is considered. This problem...