Abstract: In this paper, the finite cover based element free method is developed for applying to the friction contact problems of tension weak geomaterials. The variational parametric principle of minimum potential energy for plane elastic friction contact problem is presented. Then based on the studies of Simo and Lauren, an effective iterative scheme of the augmented Lagrangian multiplier procedure is established for the tension weak geomaterials and it is numerically implemented within the theoretical framework of the element free method. The slippage problem of an elastic block along a rigid base is analyzed by the proposed algorithm as an example and the numerical results are compared with the conventional element free methods. Discussions are made to address the difference of the proposed finite cover based element free method and conventional element free Galerkin methods in treatment of contact problem of interfaces and potential merits of the proposed method.