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

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 14\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {1}{4}$\end{document} for plane strain and 13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {1}{3}$\end{document} 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.


Introduction
Peridynamics was developed as an alternative to classical continuum mechanics for the modeling of material failure and damage [9,11]. In peridynamics, spatial derivatives are replaced by integral operators which, unlike derivatives, are well-defined at discontinuities. This, in turn, allows material failure and damage to naturally develop within the solution of a peridynamic problem. As a nonlocal theory, peridynamics can be quite computationally expensive [1] and therefore reduced-order models are desirable. In classical linear elasticity, reduced-order models are sometimes employed to reduce computational expenses. Two common reduced-order models are classical plane strain and classical plane stress, which are two-dimensional approximations of three-dimensional models. Peridynamic formulations of plane strain and plane stress have been presented in several works [2-4, 6, 8]. Due to the nonlocality inherent in peridynamics, the computational savings are substantial in these planar formulations compared to their three-dimensional counterparts. However, peridynamic plane strain and plane stress models appearing in these works are not direct approximations of three-dimensional peridynamic models. The recent work in [14] presents novel peridynamic formulations for plane strain and plane stress directly derived from threedimensional peridynamic models. Nevertheless, in this work, we focus on peridynamic plane strain and plane stress models formulated from two-dimensional peridynamic models, and we investigate their restrictions.
Peridynamic models can be classified as state-based [11] or bond-based [9]. In bondbased peridynamics, it is commonly stated that isotropic plane strain models only correspond to materials with Poisson's ratio of 1 4 while isotropic plane stress models only apply to materials with Poisson's ratio of 1 3 . The origin of this claim can be traced to [3]. In this work, we explore the validity of this claim and present a generalization for anisotropic materials.
The organization of this paper is as follows. In Section 2, we review the derivations presented in [3] of a fixed Poisson's ratio in two-dimensional bond-based peridynamic formulations of plane strain and plane stress for isotropic materials. In Section 3, we derive general elasticity constraints in bond-based peridynamics, known as Cauchy's relations. We then specialize these constraints in Section 4 to two-dimensional bond-based peridynamic plane strain and plane stress formulations for anisotropic materials. Conclusions are given in Section 5.

Constraints Imposed on Poisson's Ratio by Bond-Based Peridynamics for Isotropic Plane Strain and Plane Stress
In this section, we explore the assertion presented in [3] of a Poisson's ratio restriction of 1 4 for plane strain and 1 3 for plane stress in isotropic bond-based peridynamics. In order to deduce these restrictions, which we derive below, one equates the strain energy density of an isotropic material in two-dimensional bond-based peridynamics to the strain energy density of an isotropic material in classical planar linear elasticity for two strain states: the uniform normal strain (cf. Fig. 1a), and the uniform shear strain (cf. Fig. 1b), where ε ij are the components of the infinitesimal strain tensor and s 0 is a constant. For an isotropic material (after rotation to principal directions), any other plane state may be considered as a superposition of these two strain states [3]. Similar to the work in [3], we simply state here the classical and peridynamic strain energy densities for these two strain states. However, for the sake of completeness, we additionally provide derivations of the corresponding strain energy densities in Appendix A.
The strain energy density in classical elasticity is where σ ij are the components of the stress tensor and Einstein summation convention is employed for repeated indices. Alternatively for peridynamics, in [3], the case of an isotropic bond-based prototype microelastic brittle (PMB) material model [10] was considered. In that case, the strain energy density is given by where c is a microelastic stiffness constant; s := ξ +η − ξ ξ is the stretch of the bond ξ := x − x; x and x are reference positions of material points with displacements given, respectively, by u(x, t) and u(x , t) at time t; η := u(x , t) − u(x, t) is the relative displacement; and H is a peridynamic neighborhood.
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), Similarly, 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), 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 W Cε 1 and W P 1 in Eq. 5 as well as W Cε 2 and W P 2 in Eq. 6 to find, respectively, Thus, the strain energy density for the isotropic two-dimensional bond-based peridynamic model from Eq. 4 can only agree with the strain energy density for isotropic classical plane strain when the material has Poisson's ratio of ν = 1 4 . Similarly, in order to ensure agreement between the classical isotropic plane stress model and the two-dimensional bondbased peridynamic model from Eq. 4, we equate W Cσ 1 and W P 1 in Eq. 5 as well as W Cσ 2 and W P 2 in Eq. 6 to find, respectively, Thus, the strain energy density for the isotropic two-dimensional bond-based peridynamic model from Eq. 4 can only agree with the strain energy density for isotropic classical plane stress when the material has Poisson's ratio of ν = 1 3 .

Constraints Imposed on the Elasticity Tensor by Bond-Based Peridynamics: Cauchy's Relations
In this section, we develop a generalization of the Poisson's ratio constraints presented in Section 2. Specifically, we do not limit the discussion to two dimensions and we allow anisotropy within the model. Rather than matching constants between strain energy densities of classical and peridynamic models for specific strain states, we perform the comparison for general infinitesimal smooth deformations. To accomplish this, we express the strain energy density of classical linear elasticity in terms of the displacement field. We first recall that in classical linear elasticity the components of the stress and strain tensors are related through a generalized Hooke's Law: where C ij kl are the components of the fourth-order elasticity tensor 1 C. The elasticity tensor has the minor symmetries C ij kl = C jikl = C ij lk and the major symmetry C ij kl = C klij . Substituting Eq. 9 into Eq. 3, we find The last equality in Eq. 10 is obtained from the minor symmetries of the elasticity tensor.
To consider more general bond-based peridynamic models and compare corresponding strain energy densities with those from classical linear elasticity, we employ a linear bondbased peridynamic model with strain energy density given by The function λ(ξ ) is commonly referred to as the micromodulus function and determines the bond response in the linear bond-based peridynamic model. Equation 11 is the most general form of the strain energy density for a microelastic linear bond-based peridynamic model with a pairwise equilibrated reference configuration. 2 As in classical linear elasticity, we assume an infinitesimal smooth deformation so that In this case, the peridynamic strain energy density from Eq. 11 is given by Equating Eqs. 10 and 12, we arrive at 2 The strain energy density in bond-based peridynamics is defined as where w is the pairwise potential function, which for a microelastic peridynamic model is related to the pairwise force function f by [9] f(η, ξ ) = ∂w ∂η (η, ξ ).
Assuming a small deformation, i.e., η 1, we can expand the pairwise potential function in η, provided the required partial derivatives exist, while holding ξ fixed: Taking w(0, ξ ) = 0 and assuming a pairwise equilibrated reference configuration, i.e., f(0, ξ ) = 0 for all ξ = 0, we obtain For the case of a pairwise equilibrated reference configuration, under linearization, the pairwise force function is given by f(η, ξ ) = λ(ξ )(ξ ⊗ ξ )η, where λ is the micromodulus function [9]. Neglecting terms of order O( η 3 ), the pairwise potential function is given by which results in the strain energy density appearing in Eq. 11.
Noting that the right-hand side of Eq. 13 is invariant under any permutation of the indices i, j, k, and l, we immediately deduce that the bond-based peridynamic theory is only applicable to materials whose elasticity tensor C is completely symmetric. Specifically, in addition to the minor and major symmetries that are intrinsic to the elasticity tensor C, the bond-based peridynamic theory imposes the additional symmetry The relations in Eq. 14 are frequently referred to as Cauchy's relations and a historical account of their origin can be found in [7]. These relations are known to occur in classical linear elastic models developed from a molecular theory based on pair potentials between particles [12]. Since bond-based peridynamics employs a pair potential, it is perhaps unsurprising that it is only applicable to materials satisfying Cauchy's relations. This result has been noted in [14]. In order to relate Eq. 14 to the constraints on Poisson's ratio presented in Section 2, we express Eq. 14 in terms of engineering constants [5]. In three dimensions, the expressions are fairly cumbersome for the case of full anisotropy, i.e., triclinic materials. We therefore restrict the discussion to the case of orthotropic symmetry, where we assume the three planes of reflection symmetry coincide with the xy-, xz-, and yz-planes. In this case, there are three relevant Cauchy's relations, 3 In terms of engineering constants, Eq. 15 is given by where E i are Young's moduli, ν ij are Poisson's ratios, G ij are shear moduli, and In two dimensions, there is a single Cauchy's relation, Even in the most general two-dimensional anisotropic case of oblique symmetry, the expression for Eq. 17 in terms of engineering constants is relatively simple: where η ij,kk are coefficients of mutual influence of the second type. 4 When we specialize Eqs. 16a and 18 to the case of isotropy, an interesting development materializes. In the case of isotropy in three dimensions, we have Imposing the restrictions from Eq. 19 on Eq. 16a, we find ν 12 = 1 4 (see also [9]). Alternatively, in the case of isotropy in two dimensions, we have E 1 = E 2 , ν 12 = ν 21 , η 12,11 = η 12,22 = 0, and G 12 = . (20) Imposing the restrictions from Eq. 20 on Eq. 18, we find ν 12 = 1 3 (see also [14]). We observe that with the assumption of isotropy, the constraint on Poisson's ratio obtained in Section 2 for plane strain in bond-based peridynamics is identical to the constraint imposed by Cauchy's relations in three dimensions. Similarly, with the assumption of isotropy, the constraint on Poisson's ratio obtained in Section 2 for plane stress in bondbased peridynamics is identical to the constraint imposed by Cauchy's relation in two dimensions. As we will see in Section 4, these are no mere coincidences.

Implications of Cauchy's Relations in Peridynamics for Anisotropic Plane Strain and Plane Stress
Given the three-dimensional elasticity tensor C 3D , the strain energy density for classical plane strain may be expressed as where, since ε 13 = ε 23 = ε 33 = 0 is assumed for plane strain, the summations are over {1, 2} (cf. Eq. 3). Equation 21 is identical to the two-dimensional formulation of Eq. 10 when one replaces the two-dimensional elasticity tensor components C 2D ij kl with the corresponding three-dimensional components C 3D ij kl . Equating Eq. 21 with the two-dimensional formulation of Eq. 12, we find Thus, in the case of plane strain, two-dimensional bond-based peridynamics imposes a single Cauchy's relation directly on the three-dimensional elasticity tensor. In terms of engineering constants, Eq. 22 is equivalent to Eq. 16a for orthotropic symmetry and simplifies to ν = 1 4 in the case of isotropy (cf. Eq. 19).
Alternatively, given the three-dimensional elasticity tensor C 3D , the strain energy density for classical plane stress may be expressed as 5 where, since σ 13 = σ 23 = σ 33 = 0 is assumed for plane stress, the summations are over {1, 2} (cf. Eq. 3) and are the reduced elastic stiffnesses [13]. Equation 23 is identical to the two-dimensional formulation of Eq. 10 when one replaces the two-dimensional elasticity tensor components C 2D ij kl with the corresponding reduced elastic stiffnesses C ij kl . Equating Eq. 23 with the two-dimensional formulation of Eq. 12, we arrive at C 1212 = C 1122 . From Eq. 24 we deduce In terms of engineering constants C ij kl = C 2D ij kl (cf. [14]) and consequently Eq. 25 is equivalent to Thus, in the case of plane stress, two-dimensional bond-based peridynamics effectively imposes a single Cauchy's relation directly on the two-dimensional elasticity tensor rather than on the three-dimensional elasticity tensor. In terms of engineering constants, Eq. 26 is equivalent to Eq. 18 for oblique symmetry and simplifies to ν = 1 3 in the case of isotropy (cf. Eq. 20).
In classical linear elasticity, plane strain and plane stress models are derived from a three-dimensional model of linear elasticity. In the peridynamic plane strain and plane stress models presented in [3], one begins with a two-dimensional bond-based peridynamic model and informs it with either the classical plane strain or plane stress model. While this generates bond-based peridynamic models which agree with the classical plane strain or plane stress model for infinitesimal smooth deformations, the resulting peridynamic models will not necessarily approximate a three-dimensional bond-based peridynamic model. In fact, as we saw earlier, an isotropic three-dimensional bond-based peridynamic model imposes a Poisson's ratio restriction of ν = 1 4 and therefore the peridynamic plane stress model presented in [3], which requires ν = 1 3 , cannot be a plane stress approximation of an isotropic three-dimensional bond-based peridynamic model.
In general, for anisotropic materials, two-dimensional bond-based peridynamic models for plane stress result in the restriction appearing in Eq. 25, whereas three-dimensional bond-based peridynamic models impose the restrictions appearing in Eq. 14, in particular Eq. 22. Consequently, a two-dimensional bond-based peridynamic model cannot be a 5 Typically, plane stress is applied to thin plate-like structures. This formulation assumes monoclinic symmetry with the plane of reflection symmetry coinciding with the xy-plane. This assumption is essential in keeping the mid-plane of the plate planar under in-plane loading [13]. To obtain Eq. 23, set σ 13 = σ 23 = σ 33 = 0 in Eq. 9 and solve for ε 13 , ε 23 , and ε 33 . Then, substitute the resulting expressions back into the equations for σ 11 , σ 22 , and σ 12 in Eq. 9. Lastly, use the resulting stress-strain relationship in Eq. 3. plane stress approximation of a three-dimensional bond-based peridynamic model. Possibly, the peridynamic plane stress model presented in [3] could be shown to be a plane stress approximation of a three-dimensional state-based peridynamic model, as it is not bound by Cauchy's relations. In [14], it was shown that imposing assumptions similar to those assumed for classical plane stress on a three-dimensional bond-based peridynamic model naturally produces a two-dimensional state-based peridynamic model. The resulting statebased peridynamic model has the same restrictions in terms of engineering constants as the three-dimensional peridynamic model it approximates. Moreover, in [14], it was also shown that imposing similar assumptions to those assumed for classical plane strain on a three-dimensional bond-based peridynamic model does produce a two-dimensional bondbased peridynamic model. As opposed to [3], an important benefit of the plane strain and plane stress peridynamic models presented in [14] is that the original three-dimensional bond-based peridynamic model being approximated is known.

Conclusions
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 [3] 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 twodimensional bond-based peridynamic model imposes C 1212 = C 1122 for plane strain and for plane stress on the three-dimensional elasticity tensor. In particular, we showed that the restrictions posed in [3] 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 bondbased peridynamic model describing plane stress cannot approximate a three-dimensional bond-based peridynamic model.
Substituting the strain states from Eqs. 1 and 2 into Eq. 28, we find, respectively, Alternatively, in classical plane stress, the stress-strain relationship for an isotropic material is given by ⎡ ⎣ σ 11 σ 22 σ 12 Substituting Eq. 30 into Eq. 3, we find the strain energy density for an isotropic material in a state of plane stress is given by Substituting the strain states from Eqs. 1 and 2 into Eq. 31, we find, respectively, we may suppose for infinitesimal deformations that Eq. 36 simplifies to