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The peridynamic stress formula proposed by Lehoucq and Silling [1, 2] iscumbersome to be implemented in numerical computations. Here, we show that theperidynamic stress tensor has the exact mathematical expression as that of thefirst Piola-Kirchhoff static Virial stress originated from Irving-Kirkwood-Nollformalism [3, 4] through the Hardy-Murdoch procedure [5, 6], which offers asimple and clear expression for numerical calculations of peridynamic stress.Several numerical verifications have been carried out to validate the accuracyof proposed peridynamic stress formula in predicting the stress states in thevicinity of the crack tip and other sources of stress concentration. Theperidynamic stress is evaluated within the bond-based peridynamics withprototype microelastic brittle (PMB) material model. It is found that the PMBmaterial model may exhibit nonlinear constitutive behaviors at largedeformations. The stress fields calculated through the proposed peridynamicstress formula show good agreements with finite element analysis results,analytical solutions, and experimental data, demonstrating the promisingpotential of derived peridynamic stress formula in simulating the stress statesof problems with discontinuities, especially in the bond-based peridynamics.

Peridynamic stress is the static first Piola-Kirchhoff Virial stress