Bond-Based Peridynamics: a Tale of Two Poisson’s Ratios

Jeremy Trageser,Pablo Seleson

doi:10.1007/s42102-019-00021-x

Abstract

This paper explores the restrictions imposed by bond-based peridynamics, particularly with respect to plane strain and plane stress models. We begin with a review of the derivations in Gerstle et al. (2005) wherein for isotropic materials a Poisson’s ratio restriction of $\frac {1}{4}$ for plane strain and $\frac {1}{3}$ for plane stress is deduced. Next, we show Cauchy’s relations are an intrinsic limitation of bond-based peridynamics and specialize these restrictions to plane strain and plane stress models. This generalizes the results from Gerstle et al. (2005) and demonstrates that the Poisson’s ratio restrictions described in Gerstle et al. (2005) are merely a consequence of Cauchy’s relations for isotropic materials. We conclude with a discussion of the validity of peridynamic plane strain and plane stress models formulated from two-dimensional bond-based peridynamic models.

Highlights

Peridynamics was developed as an alternative to classical continuum mechanics for the modeling of material failure and damage [9, 11]
This paper explores the restrictions imposed by bond-based peridynamics, with respect to plane strain and plane stress models
In this work, we focus on peridynamic plane strain and plane stress models formulated from two-dimensional peridynamic models, and we investigate their restrictions

Summary

Introduction

Peridynamics was developed as an alternative to classical continuum mechanics for the modeling of material failure and damage [9, 11]. Under the uniform normal strain (cf Eq 1) in an isotropic material, the strain energy densities for classical plane strain, classical plane stress, and two-dimensional bond-based peridynamics (based on Eq 4) are, respectively (see Appendix A), W1Cε Under the uniform shear strain (cf Eq 2) in an isotropic material, the strain energy densities for classical plane strain, classical plane stress, and two-dimensional bond-based peridynamics (based on Eq 4) are, respectively (see Appendix A), W2Cε. In order to ensure agreement between the classical isotropic plane strain model and the two-dimensional bond-based peridynamic model from Eq 4, we equate W1Cε and W1P in Eq 5 as well as W2Cε and W2P in Eq 6 to find, respectively, c. Model from Eq 4 can only agree with the strain energy density for isotropic classical plane stress when the material has of ν

Constraints Imposed on the Elasticity Tensor by Bond-Based Peridynamics

Conclusions

Strain Energy Densities for Isotropic Classical Plane Strain and Plane Stress

Strain Energy Densities for the Two-Dimensional PMB Peridynamic Model