Enrichment of three-dimensional numerical manifold method with cover-based contact theory for static and dynamic mechanical response analysis

•The dual cover system, weight function and displacement function of 3D-NMM are proposed.•The element stiffness matrix, inertial force matrix, loading point matrix, etc. are derived.•The contact detection algorithm is realized based on the cover-based contact theory.•The accuracy of the developed code is verified through two continuous and discontinuous models.•The robustness of the developed code is further verified through the low velocity impacting model.