A new approach for analyzing cracked problems in 2D orthotropic materials using the well-known element free Galerkin method and orthotropic enrichment functions is proposed. The element free Galerkin method is a meshfree method which enables discontinuous problems to be modeled efficiently. In this study, element free Galerkin is extrinsically enriched by the recently developed crack-tip orthotropic enrichment functions. Also, a suitable way is applied to select support domains near a crack so that the discontinuity can be modeled without the Heaviside enrichment function. Crack-tip enrichment functions span the possible displacement space that may occur in the analytical solution. For evaluating the mixed-mode stress intensity factors, the interaction integral is applied. Some numerical examples are simulated to investigate the efficacy of the new approach by comparing with other numerical or (semi-) analytical methods.