Surface stress and surface elasticity are related to the organization of surface morphology, surface patterns, and surface atomic structures. As the size of the structure approaches the nanometer level, the surface-to-volume ratio increases. Generally the surface energy in deformable solids depends on the surface strain. The surface stress and elasticity influence the distribution of bulk stress near the surface. Interface stress and elasticity also exist at material interfaces and determine the interface properties. In the present study, the singular stress at a wedge corner in an anisotropic two-dimensional joint under tensile loading is analyzed using the molecular dynamic (MD) method and the anisotropic elasticity theory using a boundary condition with interface stress and interface elasticity. Not only the interface stress but also surface stress on the free surface are considered as a special case of an interface. The interface stress and interface elasticity are obtained through the MD analysis. In the case of a two-dimensional joint, the interface stress and elasticity depend on the distance from the wedge corner. In the analysis of anisotropic elasticity, the eigenequation used to determine the order of the stress singularity is newly derived using a boundary condition that considers the interface stress and interface elasticity. The order of the stress singularity varies with distance from the wedge corner. The stress distribution near the wedge corner can be expressed by the relation between the order of the stress singularity and the distance from the wedge corner.