This paper explores the restrictions imposed by bond-based peri- dynamics, particularly with respect to plane strain and plane stress models. We begin with a review of the derivations in [3] wherein for isotropic mate- rials a Poisson's ratio restriction of1 4for plane strain and1 3for plane stress This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05- 00OR22725 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to pub- lish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan). Sandia National Laboratories is a multimission laboratory managed and operated by Na- tional Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical re- sults and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Jeremy Trageser Computer Science and Mathematics Division Oak Ridge National Laboratory One Bethel Valley Road, P.O. Box 2008, MS-6211, Oak Ridge, TN 37831-6211 E-mail: trageserje@ornl.gov Present address: Center for Computing Research Sandia National Laboratories P.O. Box 5800, MS-1322, Albuquerque, NM, 87185-1322 E-mail: jtrages@sandia.gov Pablo Seleson Computer Science and Mathematics Division Oak Ridge National Laboratory One Bethel Valley Road, P.O. Box 2008, MS-6211, Oak Ridge, TN 37831-6211 E-mail: selesonpd@ornl.gov 2 Jeremy Trageser, Pablo Seleson 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 [3] and demonstrates that the Poisson's ratio restrictions described in [3] 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. Keywords bond-based peridynamics Cauchy's relations plane strain plane stress 1 Intoduction Peridynamics was developed as an alternative to classical continuum mechan- ics for the modeling of material failure and damage [9,11]. In peridynamics, spatial derivatives are replaced by integral operators which, unlike derivatives, are well-de ned 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 elastic- ity, reduced-order models are sometimes employed to reduce computational expenses. Two common such 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 [2,3,4,6,8]. Due to the nonlocality inher- ent in peridynamics, the computational savings are substantial in these pla- nar 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 three-dimensional 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 classi ed as state-based [11] or bond-based [9]. In bond-based peridynamics, it is commonly stated that isotropic plane strain models only correspond to materials with a Poisson's ratio of1 4while isotropic plane stress models only apply to materials with a Poisson's ratio of1 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 xed 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 con- straints in Section 4 to two-dimensional bond-based peridynamic plane strain Bond-based peridynamics: A tale of two Poisson's ratios 3 "22=s0 "11=s0 (a) Uniform normal strain."22=s0 "11=s0 (b) Uniform shear strain. Fig. 1: Uniform strain states used to relate elastic and microelastic constants. and plane stress formulations for anisotropic materials. Conclusions are given in Section 5. 2 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 of1 4for plane strain and1 3for 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.Figure 1a), "11="22=s0and"12= 0; (1) and the uniform shear strain ( cf.Figure 1b), "11="22=s0and"12= 0; (2) where"ijare the components of the in nitesimal strain tensor and s0is 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]. Similarly 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 WC=1 2ij"ij; (3) whereijare the components of the stress tensor and Einstein summation convention is employed for repeated indices. 4 Jeremy Trageser, Pablo Seleson 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 WP=1 4Z Hcs2kkd; (4) wherecis a microelastic sti ness constant, s:=k+kkk kkis the stretch of the bond :=x0x,xandx0are reference positions of material points with displacements given, respectively, by u(x;t) and u(x0;t) at timet,:= u(x0;t)u(x;t) is the relative displacement, and His a peridynamic neigh- borhood. Under the uniform normal strain (1) in an isotropic material, the strain energy densities for classical plane strain, classical plane stress, and two- dimensional bond-based peridynamics (based on (4)) are, respectively (see Appendix A), WC" 1=Es2 0 (1 +)(12); WC 1=Es2 0 1;andWP 1=c3s2 0 6: (5) Similarly, under the uniform shear strain (2) in an isotropic material, the strain energy densities for classical plane strain, classical plane stress, and two-dimensional bond-based peridynamics (based on (4)) are, respectively (see Appendix A), WC" 2=Es2 0 1 +; WC 2=Es2 0 1 +;andWP 2=c3s2 0 12: (6) In order to ensure agreement between the classical isotropic plane strain model and the two-dimensional bond-based peridynamic model (4), we equate WC" 1andWP 1in (5) as well as WC" 2andWP 2in (6) to nd, respectively, c=6E (1 +)(12)3andc=12E (1 +)3)=1 4: (7) Thus, the strain energy density for the isotropic two-dimensional bond-based peridynamic model (4) can only agree with the strain energy density for isotropic classical plane strain when the material has a Poisson's ratio of =1 4. Similarly, in order to ensure agreement between the classical isotropic plane stress model and the two-dimensional bond-based peridynamic model (4), we equateWC 1andWP 1in (5) as well as WC 2andWP 2in (6) to nd, respectively, c=6E (1)3andc=12E (1 +)3)=1 3: (8) Thus, the strain energy density for the isotropic two-dimensional bond-based peridynamic model (4) can only agree with the strain energy density for isotropic classical plane stress when the material has a Poisson's ratio of =1 3. Bond-based peridynamics: A tale of two Poisson's ratios 5 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. Speci cally, 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 clas- sical and peridynamic models for speci c strain states, we consider general in nitesimal smooth deformations. To accomplish this, we express the strain energy density of classical linear elasticity in terms of the displacement eld. We rst recall that in classical linear elasticity the components of the stress and strain tensors are related through a generalized Hooke's Law: ij=Cijkl"kl; (9) whereCijklare the components of the fourth-order elasticity tensor1C. The elasticity tensor has the minor symmetries Cijkl=Cjikl=Cijlkand the major symmetryCijkl=Cklij. Substituting (9) into (3), we nd WC=1 2Cijkl"ij"kl =1 8Cijkl@ui @xj+@uj @xi@uk @xl+@ul @xk =1 2Cijkl@ui @xj@uk @xl:(10) The last equality in (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 bond-based peridynamic model with strain energy density given by WP=1 4Z H()()2d: (11) The function () 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 mi- croelastic linear bond-based peridynamic model with a pairwise equilibrated reference con guration2. As in classical linear elasticity, we assume an in nites- imal smooth deformation so that i@ui @xj(x;t)j: 1To avoid confusion, in later arguments we add the superscript 2 Dor 3Dto the elasticity tensor Cand its components to refer to two or three spatial dimensions, respectively. 2The strain energy density in bond-based peridynamics is de ned as WP:=1 2Z Hw(;)d; 6 Jeremy Trageser, Pablo Seleson In this case, the peridynamic strain energy density (11) is given by WP=1 4Z H()iikkd 1 4Z H()ij@ui @xjkl@uk @xld =1 21 2Z H()ijkld@ui @xj@uk @xl:(12) Equating (10) and (12), we arrive at Cijkl=1 2Z H()ijkld: (13) Noting that the right-hand side of (13) is invariant under any permutation of the indices i;j;k; andl, we immediately deduce that the bond-based peridy- namic theory is only applicable to materials whose elasticity tensor Cis com- pletely symmetric. Speci cally, 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 Cijkl=Cikjl: (14) The relations (14) are frequently referred to as Cauchy's relations and a his- torical 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 wherewis the pairwise potential function, which for a microelastic peridynamic model is related to the pairwise force function fby [9] f(;) =@w @(;): Assuming a small deformation, i.e., kk 1, we can expand the pairwise potential function in, provided the required partial derivatives exist, while holding  xed: w(;) =w(0;) +@w @i(0;)i+1 2@2w @i@k(0;)ik+O(kk3): Takingw(0;) = 0 and assuming a pairwise equilibrated reference con guration, i.e., f(0;) =0for all 6=0, we obtain w(;) =1 2@fi @k(0;)ik+O(kk3): For the case of a pairwise equilibrated reference con guration, under linearization, the pair- wise force function is given by f(;) =()( ), whereis the micromodulus func- tion [9]. Neglecting terms of order O(kk3), the pairwise potential function is given by w(;) =1 2()ikik=1 2()()2; which results in the strain energy density (11). Bond-based peridynamics: A tale of two Poisson's ratios 7 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 (14) to the constraints on Poisson's ratio presented in Section 2, we express (14) in terms of engineering constants [5]. In three di- mensions the expressions are fairly cumbersome for the case of full anisotropy, i.e., triclinic materials. We therefore restrict the discussion to the case of or- thotropic symmetry, where we assume the three planes of re ection symmetry coincide with the xy-,xz-, andyz-planes. In this case, there are three relevant Cauchy's relations3, C3D 1212=C3D 1122;C3D 1313=C3D 1133;andC3D 2323=C3D 2233: (15) In terms of engineering constants, (15) is given by G12=E1E2 (E31323+E212); (16a) G13=E1E2E3 (1223+13); (16b) G23=E2E3 (E21213+E123); (16c) whereEiare Young's moduli, ijare Poisson's ratios, Gijare shear moduli, and :=E1E22E2E3121323E1E32 23E2 22 12E2E32 13: In two dimensions there is a single Cauchy's relation, C2D 1122=C2D 1212: (17) Even in the most general two-dimensional case of oblique symmetry, the ex- pression for (17) in terms of engineering constants is relatively simple: G12=E212 1122112;1112;22; (18) whereij;kk are coecients of mutual in uence of the second type4. When we specialize (16) and (18) to the case of isotropy, an interesting development materializes. In the case of isotropy in three dimensions, we have E1=E2=E3; 12=13=23;andG12=G13=G23=E1 2(1 +):(19) 3The other relations, C3D 1123=C3D 1312;C3D 2213=C3D 2312;andC3D 3312=C3D 2313; are trivially satis ed as each term is zero for orthotropic symmetry. 4Note that to retain standard notation for peridynamics and engineering constants, we abuse notation by using ias the relative displacement components and ij;kk as the co- ecients of mutual in uence of the second kind. Throughout this work the subscripts will dictate which quantity is referenced. 8 Jeremy Trageser, Pablo Seleson Imposing (19) on (16), we nd 12=1 4(see also [9]). Alternatively, in the case of isotropy in two dimensions, we have E1=E2; 12=21; 12;11=12;22= 0;andG12=E1 2(1 +): (20) Imposing (20) on (18), we nd 12=1 3(see also [14]). We observe that with the assumption of isotropy, the constraint on Pois- son'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 bond-based 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. 4 Implications of Cauchy's relations in peridynamics for anisotropic plane strain and plane stress Given the three-dimensional elasticity tensor C3D, the strain energy density for classical plane strain may be expressed as WC"=1 2C3D ijkl"ij"kl; (21) where, since "13="23="33= 0 is assumed for plane strain, the summations are overf1;2g(cf.(3)). Equation (21) is identical to the two-dimensional formulation of (10) when one replaces the two-dimensional elasticity tensor components C2D ijklwith the corresponding three-dimensional components C3D ijkl. Equating (21) with the two-dimensional formulation of (12), we nd C3D 1212=C3D 1122: (22) Thus, in the case of plane strain, two-dimensional bond-based peridynamics imposes a single Cauchy's relation directly on the three-dimensional elastic- ity tensor. In terms of engineering constants, (22) is equivalent to (16a) for orthotropic symmetry and simpli es to =1 4in the case of isotropy ( cf.(19)). Alternatively, given the three-dimensional elasticity tensor C3D, the strain energy density for classical plane stress may be expressed as5 WC=1 2C0 ijkl"ij"kl; (23) 5Typically, plane stress is applied to thin plate-like structures. This formulation assumes monoclinic symmetry with the plane of re ection 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 (23), set 13=23=33= 0 in (9) and solve for "13;"23;and "33. Then, substitute the resulting expressions back into the equations for 11;22;and12 in (9). Lastly, use the resulting stress-strain relationship in (3). Bond-based peridynamics: A tale of two Poisson's ratios 9 where, since 13=23=33= 0 is assumed for plane stress, the summations are overf1;2g(cf.(3)) and C0 ijkl:=C3D ijklC3D ij33C3D 33kl C3D 3333(24) are the reduced elastic sti nesses [13]. Equation (23) is identical to the two- dimensional formulation of (10) when one replaces the two-dimensional elastic- ity tensor components C2D ijklwith the corresponding reduced elastic sti nesses C0 ijkl. Equating (23) with the two-dimensional formulation of (12), we arrive atC0 1212=C0 1122. From (24) we deduce C3D 1212(C3D 3312)2 C3D 3333=C3D 1122C3D 1133C3D 2233 C3D 3333: (25) In terms of engineering constants C0 ijkl=C2D ijkland consequently (25) is equiv- alent to C2D 1212=C2D 1122: (26) Thus, in the case of plane stress, two-dimensional bond-based peridynamics e ectively 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, (26) is equivalent to (18) for oblique symmetry and simpli es to =1 3in the case of isotropy ( cf.(20)). In classical linear elasticity, plane strain and plane stress models are de- rived 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 in nitesimal 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 =1 4and there- fore the peridynamic plane stress model presented in [3], which requires =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 peridy- namic models for plane stress result in the restriction (25), whereas three- dimensional bond-based peridynamic models impose the restrictions (14), in particular (22). Consequently, a two-dimensional bond-based peridynamic model cannot be a plane stress approximation of a three-dimensional bond-based peri- dynamic 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 10 Jeremy Trageser, Pablo Seleson produces a two-dimensional state-based peridynamic model. The resulting state-based peridynamic model has the same restrictions in terms of engineer- ing 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 peridy- namic model does produce a two-dimensional bond-based peridynamic model. As opposed to [3], an important bene t 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. 5 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 of1 4while peridynamic plane stress requires a Poisson's ratio of1 3, and we generalized the analysis to the case of anisotropy. In the general anisotropic setting, we demonstrated that bond-based peridynamics is con- strained by Cauchy's relations. Speci cally, we deduced that a two-dimensional bond-based peridynamic model imposes C1212=C1122for plane strain and C1212(C3312 )2 C3333=C1122C1133C2233 C3333for 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 correspond- ing 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. A Derivations of strain energy density results from Section 2 In this section, we present derivations of the strain energy density results utilized in Section 2 to derive the Poisson's ratio restrictions in isotropic bond-based peridynamics. A.1 Strain energy densities for isotropic classical plane strain and plane stress In classical plane strain, the stress-strain relationship for an isotropic material is given by 2 411 22 123 5=E (1 +)(12)2 41  0 1 0 0 01 2(12)3 52 4"11 "22 2"123 5: (27) Substituting (27) into (3), we nd the strain energy density for an isotropic material in a state of plane strain is given by WC"=E 2(1 +)(12) (1)("2 11+"2 22) + 2"11"22+ 2(12)"2 12 : (28) Bond-based peridynamics: A tale of two Poisson's ratios 11 Substituting the strain states (1) and (2) into (28), we nd, respectively, WC" 1=Es2 0 (1 +)(12)andWC" 2=Es2 0 1 +: (29) Alternatively, in classical plane stress, the stress-strain relationship for an isotropic material is given by2 411 22 123 5=E 122 41 0 1 0 0 01 2(1)3 52 4"11 "22 2"123 5: (30) Substituting (30) into (3), we nd the strain energy density for an isotropic material in a state of plane stress is given by WC=E 2(12) "2 11+"2 22+ 2"11"22+ 2(1)"2 12 : (31) Substituting the strain states (1) and (2) into (31), we nd, respectively, WC 1=Es2 0 1andWC 2=Es2 0 1 +: (32) A.2 Strain energy densities for the two-dimensional PMB peridynamic model We begin by expressing the relative displacement for the two strain states (1) and (2). Under the strain state (1), we nd =s0h1;2i: (33) Under the strain state (2), we nd =s0h1;2i: (34) We only consider the strain energy density at material points within the bulk of the body, so that one may suppose the peridynamic neighborhood H=B2D (0), i.e., the two-dimensional disk of radius centered at the origin. To obtain the strain energy density for the two- dimensional PMB peridynamic model under the strain state (1), we substitute (33) into (4) to nd WP 1=c 4Z B2D (0)(k+kk k)2 kkd=c 4Z B2D (0)s2 0kkd=c3s2 0 6: (35) To obtain the strain energy density for the two-dimensional PMB peridynamic model under the strain state (2), we substitute (34) into (4) to nd WP 2=c 4Z B2D (0)(k+kk k)2 kkd=c 4Z B2D (0)q (1 +s0)22 1+ (1s0)22 2kk2 kkd =c3 6 s2 0+ 24(1 +s0)E 1;2ps0 s0+ 1 : (36) Here,E(x;k) :=Rx 0p 1k2t2p 1t2dtis the incomplete elliptic integral of the second kind. In linear elasticity, we are concerned with in nitesimal deformations. Consequently, by noticing the limit lim s0!0WP 2 s2 0= lim s0!0c3 6 s2 0+ 24(1 +s0)E 1;2ps0 s0+ 11 s2 0=c3 12; (37) 12 Jeremy Trageser, Pablo Seleson we may suppose for in nitesimal deformations that (36) simpli es to WP 2=c3s2 0 12: (38) Acknowledgements Research sponsored by the Laboratory Directed Research and De- velopment Program of Oak Ridge National Laboratory, managed by UT-Battelle, LLC, for the U. S. Department of Energy.