A generalized peridynamic material correspondence formulation using non-spherical influence functions

Peridynamics theory is a nonlocal continuum mechanics theory. The peridynamic material correspondence formulation provides passage for direct incorporation of any material constitutive models from the classical local continuum mechanics theory. The performance of the material correspondence formulation heavily relies on the influence function whose currently used form is spherical and depends only on the bond length. Together with the way how the nonlocal deformation gradient is constructed based on the material point horizon, this results in the well-known issue of material instability in the conventional peridynamic material correspondence formulation. In this paper, a generalized peridynamic material correspondence formulation that enables use both spherical and non-spherical influence functions is developed. A recently proposed class of parameterized non-spherical influence functions that depend on both the bond length and bond direction is used in the generalized formulation. Numerical examples including wave dispersion, small and finite deformation and linear elastic fracture analyses are studied to check the material stability, convergence characteristic, prediction accuracy and applicability to fracture problems without singularity issue of the proposed formulation. It is found that the proposed formulation is inherently stable, possesses linear convergence rate, yields highly accurate predictions for both small and finite deformation problems, and small horizon can be used to improve computational efficiency.