This study applies a new scaled boundary element-free Galerkin (EFG) method to determine the coefficients of the Williams’ series expansion for the elastic crack tip asymptotic stress field. Scaled boundary methods can calculate the coefficients of Williams’ series without any special crack tip treatment. Unlike the conventional scaled boundary finite element method, in this paper an EFG approach is used in the circumferential direction of the scaled boundary model. The EFG method offers high accuracy, rapid convergence and smooth stress solutions without post-processing, and no mesh generation is required. The proposed method is verified through numerical examples comparing the new approach with analytical, reference and conventional scaled boundary FEM solutions, and results are obtained which indicate that the new approach converges even more rapidly than the conventional scaled boundary finite element method.