Abstract: In order to solve continuum and discontinuous problems in a uniform way, the numerical manifold method (NMM) introduces two cover systems, i.e., the mathematical cover (MC) and the physical cover (PC). By constructing the MC with the node influence domains in moving least squares interpolation (MLS) as the mathematical cover, the Element Free Galerkin method in the setting of NMM is proposed, named NMM-EFG. The NMM-EFG can easily deal with continuum and discontinuous problems while the pre-processing becomes very easy. A scheme for simulating crack propagation under the small deformation and the large deformation conditions is developed. By the treatment of kinked cracks, the crack can grow at arbitrarily small step without mesh refinement. Compared with results from large deformation, the results from small deformation might be prone to unsafe evaluation.