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An enriched meshless finite volume method for the modeling of material discontinuity problems in 2D elasticity Abstract A 2D formulation for incorporating material discontinuities into the meshless finite volume method is proposed. In the proposed formulation, the moving least squares approximation space is enriched by local continuous functions that contain discontinuity in the first derivative at the location of the material interfaces. The formulation utilizes space-filling Voronoishaped finite volumes in order to more intelligently model irregular geometries. Numerical experiments for elastostatic problems in heterogeneous media are presented. The results are compared with the corresponding solutions obtained using the standard meshless finite volume method and element free Galerkin method in order to highlight the improvements achieved by the proposed formulation. It is demonstrated that the enriched meshless finite volume method could alleviate the expecting oscillations in derivative fields around the material discontinuities. The results have revealed the potential of the proposed method in studying the mechanics of heterogeneous media with complex micro-structures.