Abstract: Finitecoverbased elementfree method is a numerical method based on finitecover concept and elementfree method. Finitecover is fundamental technique in manifold method which can deal with both continuous and discontinuous problems in a universal form. The prior process of the elementfree method is relatively simple. Finitecoverbased elementfree method can integrates merits of both numerical manifold method and elementfree method. At the same time, it can overcome their individual shortcomings. This method can handle discontinuous problems in a fairly effective way and it is easily implemented. In this paper, fundamental theory of the finitecoverbased elementfree method developed by the authors is briefly presented. The proposed method is successfully applied in the computations of stress intensity factor and numerical simulations of crack expansion. The effectiveness of the present method is verified by some numerical examples.