Peridynamic stress is the static first Piola-Kirchhoff Virial stress

TL;DR

This paper demonstrates that the peridynamic stress tensor is mathematically equivalent to the first Piola-Kirchhoff Virial stress, simplifying numerical implementation and validated through various numerical tests.

Contribution

It establishes an exact expression for peridynamic stress as the first Piola-Kirchhoff Virial stress, enhancing computational efficiency and accuracy.

Findings

Peridynamic stress matches Virial stress mathematically.

Numerical results agree with finite element and experimental data.

The method effectively captures stress concentrations near cracks.

Abstract

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