In this paper, an enriched radial point interpolation method (e‐RPIM) is developed for the determination of crack tip fields. The conventional RBF interpolation is novelly augmented by the suitable trigonometric basis functions to reflect the properties of stresses for the crack tip fields. The performance of the enriched meshfree RBF shape functions is firstly investigated using the surface fitting. The surface fitting results have proven that, comparing with the conventional RBF, the enriched RBF interpolation has: 1) a similar accuracy to fit a polynomial surface; and 2) a much better accuracy to fit a trigonometric surface then the conventional RBF interpolation. It has proven that the enriched RBF shape function will not only possess all advantages of conventional RBF interpolation, but also can accurately reflect the properties of stresses for the crack tip fields. The system of equations for the crack analysis is then derived based on the enriched RBF shape function and the meshfree weak‐form. Crack problems are simulated using this newly developed e‐RPIM method. It has been demonstrated that the present e‐RPIM is very accurate and stable, and it has very good potential to develop a practical simulation tool for fracture mechanics problems.