### Achievement

This paper presents a novel analysis explaining a known discrepancy between plane strain and plane stress isotropic bond-based peridynamic models, with respect to Poisson’s ratio, and provides a generalization of that discrepancy to the case of anisotropic material models.

### Significance and Impact

The presented analysis provides a new unifying framework, based on Cauchy’s relations, for the determination of restrictions on plane elasticity bond-based peridynamic models. This analysis sheds light upon the source of discrepancy between plane strain and plane stress isotropic bond-based peridynamic models while generalizing this discrepancy to the anisotropic setting. This work further establishes the relations between the restrictions on these two-dimensional plane elasticity formulations and those on three-dimensional bond-based peridynamic models.

### Research Details

In this work, we explored the limitations of bond-based peridynamics with respect to agreement with classical linear elasticity. We examined the claim posed in [1] for isotropic materials that peridynamic plane strain requires a Poisson’s ratio of 1/4 while peridynamic plane stress requires a Poisson’s ratio of 1/3, and we generalized the analysis to the case of anisotropy. In the general anisotropic setting, we demonstrated that bond-based peridynamics is constrained by Cauchy’s relations. Specifically, we deduced that a two-dimensional bond-based peridynamic model imposes _{1212} _{1122 } for plane strain and _{1212} - (_{3312} )^{2} /_{3333} _{1122} _{1133} _{2233} /_{3333} for plane stress on the three-dimensional elasticity tensor. In particular, we showed that the restrictions posed in [1] are simply consequences of Cauchy’s relations being imposed on the corresponding plane strain or plane stress elasticity tensor in the case of isotropy. This analysis demonstrates that a two-dimensional bond-based peridynamic model describing plane stress cannot approximate a three-dimensional bond-based peridynamic model.

### Overview

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 [1] wherein for isotropic materials a Poisson’s ratio restriction of 1/4 for plane strain and 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 [1] and demonstrates that the Poisson’s ratio restrictions described in [1] 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.

[1] Gerstle W., Sau N., Silling S. (2005) Peridynamic modeling of plain and reinforced concrete structures. In: 18th International Conference on Structural Mechanics in Reactor Technology (SMiRT 18), pp. 54–68.

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Last Updated: May 12, 2020 - 3:30 pm