In this paper a new formulation of the stiffness matrix for a joint element in element free Galerkin method (EFGM) is presented. The meshless methods are new techniques in numerical analysis, particularly on rock mechanics. These methods are suitable for calculation of large deformation happened in rock mechanics. New formulations of the stiffness matrix for a joint element in element free Galerkin method; as one of meshless methods, is presented in this paper. This formulation was performed in local coordinate system and also was used to model rock joints and soil-soil and soil-structure interfaces. Moreover the formulation is presented for both elastic and elastoplastic state. The new solution of the joint stiffness matrix was obtained in this research and also it was observed that this method can be easily used in an EFG code. The results show good agreement between EFGM calculation and finite difference method.
The joint element was originally developed by Goodman et al.  in plane strain and plane stress state. They analyzed, jointed rock mass that it has been applied to soil-structure and rockstructure interaction studies. Subsequently, it was determined that to more accurately model the behavior of geological structures. Zienkiewicz et al.  presented the use of continuous isoparametric elements with a simple nonlinear material property for shear and normal stresses, assuming uniform strain in the thickness direction. Ghaboussi et al.  presented an axisymmetric slip surface element, which they reported to be singular under certain conditions. For a continuous mortar joint of thickness e obeying to a Masars damage law , defined by a Young modulus E, a Poisson coefficient υ and a damage threshold εdo. Faruque  presented an axisymmetric interface element that involves an iterative scheme to obtain convergence. Factors affecting this rate convergence were also discussed. The sand-structure interface developed by Zeghal and Edil . They were modeled based on physical observations and surface of slippage was idealized to be sinusoidal based on an intensive numerical simulation program. Wang et al. a constitutive model based on limit concept is proposed to predict the behavior of rock interfaces and joints.
The moving least squares approximation was developed by Lancaster and Salkauskas  for data fitting and surface construction. Nayroles et al.  has used MLS to construct shape function and named it diffuse element method (DEM). DEM was modified by Belytschko  and who named it the element free Galerkin method.