In this paper, the linearly conforming enriched radial basis point interpolation method is implemented for the elasto-plastic analysis of discontinuous medium. The linear conformability of the method is satisfied by the application of stabilized nodal integration and the enrichment of radial basis functions is achieved by the addition of linear polynomial terms. To implement the method for the analysis of a discontinuous medium, an interface layer is assumed between different materials. Interfaces are simulated by the concept of linkage element and there is no need of node or element in the traditional sense. The stiffness of each interface layer has been taken into account by defining normal and tangential stiffness coefficients along the layer. The displacement of each point across the interface layer is tied to the displacement of surrounding material nodes. The final system of equations is derived by the combination of equations for different parts of a discontinuous medium in the global coordinate. Based on the derived equations a computer code has been developed and the results of analysis with the mesh-free method are compared with the results of the finite element analysis and experimental tests.