Peridynamics theory is a nonlocal continuum mechanics theory. The peridynamic material correspondence formulation provides passage for direct incorporation of constitutive models from the conventional local continuum mechanics theory. However, the conventional material correspondence formulation suffers from the well-known issue of material instability. The material instability in peridynamics can be understood as the existence of non-unique mapping between deformation states and the resultant force state. This instability poses practical difficulties while using the correspondence model in computational modeling. One consequence of this instability is the oscillation in the predicted displacement field, i.e., existence of zero-energy modes. In addition, the damage correspondence developed based on the same formulation is also problematic, such as inaccurate mapping of damage state from material points to bonds and inaccurate capture of bond-level quantities in the development of correspondence with the conventional local continuum mechanics theory. In this presentation, a recently developed bond-associated peridynamic correspondence framework using non-spherical influence functions will be presented. The use of non-spherical influence functions to develop bond-associated correspondence formulation not only significantly improves the accuracy of calculation of bond-level quantities such as deformation gradient, but also effectively removes the material instability in the peridynamic material correspondence models. More importantly, it shifts the correspondence from material points to bonds, greatly facilitating the discrete fracture modeling by breaking bonds in peridynamics. Consistent with the classical continuum damage mechanics, the damage model is integrated directly into the constitutive relation, allowing a seamless coupling of both the material response and the damage evolution. Numerical examples for the verification of both material correspondence and damage correspondence will be studied. Conclusion and future work will be discussed.