A morphing approach to couple state-based peridynamics with classical continuum mechanics

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1 A morphing approach to couple state-based peridynamics with classical continuum mechanics Item Type Article Authors Han, Fei; Lubineau, Gilles; Azdoud, Yan; Askari, Abe Citation A morphing approach to couple state-based peridynamics with classical continuum mechanics 01 Computer Methods in Applied Mechanics and Engineering Eprint version Post-print DOI.1/j.cma Publisher Elsevier BV Journal Computer Methods in Applied Mechanics and Engineering Rights NOTICE: this is the author s version of a work that was accepted for publication in Computer Methods in Applied Mechanics and Engineering. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Computer Methods in Applied Mechanics and Engineering, January 01. DOI:.1/ j.cma Download date 1//01 1::0 Link to Item

2 Accepted Manuscript A morphing approach to couple state-based peridynamics with classical continuum mechanics Fei Han, Gilles Lubineau, Yan Azdoud, Abe Askari PII: S00-(1)000- DOI: Reference: CMA 0 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: August 01 Revised date: 0 December 01 Accepted date: December 01 Please cite this article as: F. Han, G. Lubineau, Y. Azdoud, A. Askari, A morphing approach to couple state-based peridynamics with classical continuum mechanics, Comput. Methods Appl. Mech. Engrg. (01), This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

3 *Manuscript Click here to download Manuscript: Morphing_stated_based.pdf Click here to view linked Referenc A morphing approach to couple state-based peridynamics with classical continuum mechanics Abstract Fei Han a,, Gilles Lubineau a,, Yan Azdoud a, Abe Askari b a King Abdullah University of Science and Technology (KAUST) Physical Science and Engineering Division, COHMAS Laboratory, Thuwal -00 Saudi Arabia b Propulsion System Engineering, Boeing Commercial Airplane, WA, USA A local/nonlocal coupling technique called the morphing method is developed to couple classical continuum mechanics with state-based peridynamics. State-based peridynamics, which enables the description of cracks that appear and propagate spontaneously, is applied to the key domain of a structure, where damage and fracture are considered to have non-negligible effects. In the rest of the structure, classical continuum mechanics is used to reduce computational costs and to simultaneously satisfy solution accuracy and boundary conditions. Both models are glued by the proposed morphing method in the transition region. The morphing method creates a balance between the stiffness tensors of classical continuum mechanics and the weighted coefficients of state-based peridynamics through the equivalent energy density of both models. Linearization of state-based peridynamics is derived by Taylor approximations based on vector operations. The discrete formulation of coupled models is also described. Two-dimensional numerical examples illustrate the validity and accuracy of the proposed technique. It is shown that the morphing method, originally developed for bond-based peridynamics, can be successfully extended to state-based peridynamics through the original developments presented here. Keywords: Morphing method, State-based peridynamics, Classical continuum mechanics, Coupling method Corresponding authors. Tel.: +. address: gilles.lubineau@kaust.edu.sa (G. Lubineau). fei.han@kaust.edu.sa (F. Han) Preprint submitted to CMAME December 0, 01

4 Introduction Recently, Silling [0] developed peridynamics, a nonlocal theory of solid mechanics, based on integral equilibrium equations instead of on the classical use of the partial differential equation. It is assumed that the equilibrium of a material point is attained by an integral of internal forces exerted by nonadjacent points across a finite distance. This nonlocal model is mathematically compatible with crack initiation and propagation, as integral equations can naturally handle discontinuities. These advantages have attracted considerable attention to peridynamics in recent years [1, ]. Peridynamics has been successfully applied to crack propagation [, 1], investigating impact on a brittle solid [], failure analyses of composites [, ] and nanotube reinforced composites []. The first peridynamic formulation that most researchers have applied from the literature is the bond-based model. A bond represents interaction forces between pair-wise points. In the bond-based peridynamic (BPD) model, the forces within each bond are central forces that are determined independently of each other. As a result, this model is restricted to the specific constitutive behavior of isotropic materials with a Poisson ratio of 1/ in dimensions or 1/ in dimensions [0, ]. To break the restriction, Silling et al. reformulated the peridynamic theory called the state-based model, in which bond forces remain. However, each bond force is determined depending on the collective deformation of all the bonds in the neighborhood of each endpoint. Thus, the state-based peridynamic (SPD) model overcomes the limitation of Poisson ratios in the BPD model. The SPD model has been employed to study plasticity [, ], damage and fracture [, ] of materials. However, the peridynamic theory, in particular the SPD model comes with an important computational consumption that limits its application. Additionally, peridynamics is characterized by volume-like boundary conditions, rather than the conventional traction-like boundary conditions, increasing the inconvenience of application for engineers. As a result, a reasonable strategy is to preserve peridynamics for fine-scale descriptions where key mechanisms are considered to strongly impact the solution such as damage or fracture, and to use continuum mechanics for the other parts of the structure, where the conventional continuum model saves considerable computational costs and satisfies boundary conditions on the premise of solution accuracy. In this ways, researchers can couple the SPD and classical continuum mechanics (CCM) models efficiently. Recently, some coupling schemes have been proposed to combine CCM and BPD models; for example, the variable horizon method [1, 1], the force-based coupling method [, 1], the Arlequin coupling method [] and the morphing method [1]. The variable horizon method blends the local and nonlocal equations by reduction of the peri-

5 dynamic horizon in the vicinity of the nonlocal model interface where the mathematical incompatibility is greatly reduced. The force-based coupling method blends BPD and CCM models into a coupled force equilibrium equation by a weighting function and a Taylor approximation. On the other hand, the Arlequin coupling method employs a partition of unity approach to couple energy equations of both models in an overlapping domain. Furthermore, the morphing method constructs a balanced relationship between stiffness tensors of the CCM model and the weighted nonlocal parameters of the BPD model through the equivalent energy density of both models. As a result, the morphing method implements the transition between both models by means of only this simple and unified balance over the whole structure, making it a versatile and powerful technique. Some developments in morphing-based coupling between BPD and CCM models have already been achieved [, ]. In this paper, we further develop the morphingbased coupling strategy for coupling SPD and CCM models. This is an important step towards the application of coupled formulations, because, to date, no technique has been capable of coupling SPD and CCM models. The fundamental concepts that make morphing-based coupling successful are detailed here. The remainder of this paper is organized as follows: Section reviews the key formulation of the SPD model; based on the SPD formulation, we develop the linearization of the SPD model based on vector operations in Section ; Section is devoted to the morphing method between linearized SPD and CCM models; Section presents the finite element discretization of the morphing method; and benchmark examples are shown in Section. In addition, conversion formulas between SPD and CCM parameters for homogeneous, isotropic materials are derived and principles of virtual work and minimum potential energy of the SPD model are proved in the Appendices.. State-based peridynamic formulation We list basic definitions in the SPD theory below; readers can find most from [] with the exception of the different expressions of strain energy density and force state field. We rewrite those expressions as the functions of a full extension of a bond rather than the deviatoric extension [1]. In Section, we can then approximate the full extension to get a linearized SPD model. In this work, we focus on ordinary and elastic materials []. Let H δ (x) be a spherical neighborhood centered at an arbitrary point x in R. Its radius, the horizon, is δ, δ > 0. Definition 1. Define a reference vector state, X, under which the image of a vector ξ H δ (x) remains itself. That is X ξ = ξ. (1)

6 Note that a vector ξ H δ (x) henceforth represents ξ = p x, p H δ (x). Definition. Define a deformation vector state, Y, which is the image of a vector ξ H δ (x) with respect to the point x under the deformation, such that where u denotes a displacement field. Y [x] ξ = ξ + u(x + ξ) u(x), () Definition. Define the direction of the deformation vector state Y to be the vectorvalued state M given by M (Y) ξ = Y ξ Y ξ where the notation calculates the length of a vector. Definition. Let e be the extension scalar state such that ξ H δ (x) / Y ξ 0, () e ξ = y ξ x ξ, y ξ = Y ξ, x ξ = ξ, ξ H δ (x). () In practice, e ξ is the change in length of the bond ξ during the transformation. Definition. Suppose a scalar state ω satisfies the following properties: { ω ξ 0, ξ Hδ (x), H H δ (x) with nonzero volume such that ω ζ > 0, ζ H. Then, ω is called an influence function []. Furthermore, if an influence function ω depends only on the value ξ, then ω is said to be spherical and ω ξ = ω s ξ. Definition. The weighted normalization factor m is a scalar value defined by () m = (ωx) x, () (note that this quantity is called a weighted volume by Silling et al. in []; however we prefer not to call it volume as its physical dimension is different from m ) where scalar states x and ω have been defined above, and the dot product of two states A and B is defined by A B = A ξ B ξ dv ξ. () H δ (x)

7 Definition. The dilatation, θ, is a scalar-valued function defined by: θ (e) = (ωx) e. () m It should be noted (see []) that, for small deformations, θ (e) is directly equal to the change in volume over the neighborhood of point x, H δ (x). Definition. The macro-elastic energy density W p of an ordinary, elastic peridynamic solid is given by W p (e) = λ θ (e) + τ (ωe) e, () m where λ and τ are two parameters and ω is the influence function. Since the concept of strain largely differs in peridynamics compared to classical solid mechanics, we name the strain energy density at every point in the peridynamic solid the macro-elastic energy density, which is the density of elastic energy when the strain is assumed to be uniform over H δ (x). Eq. () can be derived from the definition of strain energy density in [] (see Appendix A for details). For a homogeneous, isotropic, linear-elastic material, the conversion formulas between the Young s modulus, E, the Poisson ratio, ν, and the Lamé parameters Λ and µ in the CCM and the corresponding parameters λ and τ in SPDs are derived in Appendix B. Definition. The equilibrium equation at point x is defined for a quasi-static problem as {T [x] p x T [p] x p } dv p = b (x), () H δ (x) where b denotes body forces, and T is called a force vector state [] whose state value is the force vector (per unit volume squared) that the point p H δ (x) exerts on the point x or vice-versa. For an ordinary material [], we have where t denotes the scalar force state field. T = t M, () Definition. For an ordinary and elastic material, the macro-elastic energy density can be written by W p (e) (see Eq. ()). Because W p is the function of a scalar state, e, the following equation holds (see Appendix C for details), t = W p e. (1)

8 From Eq. (), the force scalar state for an ordinary and elastic material is given by. A linearized model of state-based peridynamics t = λ θ m ωx + τ ωe. (1) m The definitions mentioned above are a general formulation of the SPDs for an ordinary and elastic material; they are widely known and used within the peridynamic community []. In this work, we restrict ourselves to coupling the SPD with CCM models. This CCM model is in the framework of small perturbations [], in which the displacements and the displacement gradients are relatively small compared to unity (i.e., u 1, and u 1). The linearization of the CCM model is well known and is based on the introduction of the linearized strain tensor ε. Here, a similar linearization step needs to be introduced into the SPDs based on small perturbations, in which the displacement and the difference between displacements of two points are relatively small compared to the horizon [1] (i.e., u δ, and u (x + ξ) u (x) δ, ξ H δ (x)). Once we adopt the above assumptions, the SPD model will consequently be approximated to its linearized model. The linearization of this SPD model for an ordinary and elastic material is derived as follows. For any point x on the peridynamic solid, let l = sup u(p) u(x), p H δ (x). (1) p x δ From the assumption of the infinitesimal deformation for the SPD model, we know l δ [1]. We define a vector, η, as the difference between displacements of two points, x and p (i.e., η = u(p) u(x)). Then, we define φ to be a scalar-valued function of vectors which is given by φ(ξ + η) = ξ + η. (1) From Eq. (1) we know that η is an infinitesimal value (i.e., η l δ). Thus, we can expand φ at ξ using the first-order Taylor approximation, which is written as φ(ξ + η) = φ(ξ) + φ(ξ) ξ η + o( η ) ξ + ξ (1) ξ (u(p) u(x)) + o(l).

9 We define the displacement state, U, as: Let p = x + ξ, Eq. () can be recast as U [x] ξ = u(x + ξ) u(x), ξ H δ (x). () U [x] p x = u(p) u(x), p H δ (x). (1) Additionally, the direction state of a bond ξ is defined by ˆM ξ = ξ ξ. (1) According to Eqs () and (), we can rewrite the extension scalar state using ξ and η as e ξ = ξ + η ξ. (0) Substituting Eq. (1)-(1) into Eq. (0) and ignoring the little-o of l, it yields e ˆM U. (1) Then, we introduce Eq. (1) into Eq. (). The new dilatation ˆθ can be rewritten as ˆθ (U) = (ωx) U. () m The macro-elastic energy density Ŵ p of an ordinary and elastic peridynamic solid is given by Ŵ p (U) = λ ˆθ (U) + τ ωu m ˆM ˆMUdV. () H δ (x) And the force scalar state, ˆt, for an ordinary and elastic material is given by ˆt = λ ˆθ m ωx + τ m ω ˆMU. () Let the deformation vector be defined by υ = ξ + η; the extension scalar state can be rewritten as e ξ = υ υ η. () Similar to Eq. (1), we can expand υ η at υ, and then we substitute the expansion into Eq. () and ignore the higher-order terms. Consequently, the extension scalar state

10 satisfies another approximate expression under the assumption of small perturbation in SPD material: e υ υ U () = M (Y) U. Comparing Eq. (1) with Eq. (), we know that ˆM ξ M (Y) ξ. () This is simply a consequence of the small perturbation assumption and indicates that bond directions are similar at the first-order approximation between the reference and current configuration. Therefore, according to Eq. (), we can write the new force vector state, ˆT, for an ordinary and elastic material using Eqs. () and (): ˆT ξ = ˆt ξ ˆM ξ = λ m ˆθ (U ζ ) ω ξ X ξ + τ m ω ξ ˆM ξ ˆM ξ U ξ = λ ω ξ ω ζ ξ ζ U ζ dv m ζ + τ m ω ξ ˆM ξ ˆM ξ U ξ. H δ (x) In fact, this linearized force vector state for an isotropic material can also be calculated by the dot product between the displacement state and the modulus state [1], which has been derived in. Example : Linear Isotropic Solid of [1]. However, we linearized the SPD model by vector operations rather than by the state operations shown in [1]. The linearization process described above is based on the assumption of a small deformation and on the continuities of the function φ on the vector fields of ξ and η. However, continuities in displacement are not necessary for this linearization, which preserves the ability of the SPD model to deal with discontinuities in displacement.. A morphing-based coupling strategy between SPDs and CCM.1. A hybrid CCM and SPD model with morphing parameters We consider a complete domain,, which is composed of three sub-domains: 1, and m, such that = 1 m, 1 =, 1 m = and m =. Let the sub-domains 1 and be treated by a CCM model and a SPD model, respectively. We mainly focus on the finite morphing domain, m, where both models co-exist and work cooperatively. The displacement ū is imposed on the part Γ u of, and the surface forces f are imposed on the complementary part Γ f of. n denotes the outward unit ()

11 normal to Γ f. In addition, the whole domain is subjected to body forces denoted by b. For clarity and with no major restrictions, we assume that (Γ u Γ f ) 1 and that the sub-domains and m are totally embedded within 1. We also assume is embedded within m such that m becomes a transition domain between 1 and (see Figure 1). We consider the SPD model, which is characterized by constants λ 0, τ 0 and by a constant function, ω ξ, of material point (i.e., ω [x] ξ = ω ξ, x \ 1 ), which will henceforth be assumed. We also denote by E 0 the fourth-order elasticity tensor of the equivalent CCM model. Then, we propose a hybrid model for coupling the SPD and CCM models. SPD model CCM model f u u f b m b n Hybrid model Figure 1: The whole body is composed of 1, and m. The coupling is defined as a simple evolution of the material properties characterizing each model. Here we use the morphing concept introduced in [1]: both material models virtually co-exist at every point. However, the weight of each model in the constitutive equation is tuned to ensure a smooth transition from a local to a nonlocal continuum description and vice-versa. We build the unified governing equations over when a hybrid model is used:

12 Kinematic admissibility and compatibility H δ (x) ε = 1 ( u(x) +t u(x)), x \, () U [x] p x = u(p) u(x), x, p \ 1, (0) u(x) = ū(x), x Γ u. (1) Static admissibility divσ(x) + {T [x] p x T [p] x p } dv p = b(x), x, () σ(x) n = f(x), x Γ f. () Constitutive equations σ(x) = E(x) : ε(x), x \, () T [x] p x = λ (x) m + H δ (x) ω p x ω q x (p x) (q x) U q x dv q τ (x) m ω p x ˆM p x ˆM p x U p x, x, p \ 1. () Note that E(x) characterizes the stiffness tensor of CCM at any point, x. The influence function ω was assumed before to be constant for any point, x \ 1. λ(x) and τ(x) are constitutive parameters of the SPD model at point, x. Using λ 0 and τ 0 as references, we define λ(x) and τ(x) by introducing a morphing scalar function, β (0 β(x) 1, x ) such that λ(x) = β(x)λ 0 and () τ(x) = β(x)τ 0. () Remark: We can also define different morphing functions for parameters λ(x) and τ(x). The morphing method presented below continues to work for different functions. Thus, we only consider the single morphing function, β, in this paper.

13 Specific models and their deformation energy densities From the equilibrium Eq. () and the constitutive Eqs. () and (), we note that the parameters, β(x) and E(x), completely determine at any material point, x, the relative weight of each model (CCM or SPD): For a point x, if and only if E(x) = E 0 and β(x ) = 0, x H δ (x), () then this point x strictly belongs to the CCM model. Consequently, the strain energy density at this point can be written as For a point x, if and only if W (x) = 1 ε(x) : E0 : ε(x). () E(x) = 0 and β(x ) = 1, x H δ (x), (0) then this point x strictly belongs to the SPD model. According to Eq. (), the macro-elastic energy density at this point can be written as where ξ = p x. W (x) = λ0 ˆθ (U [x]) + τ 0 ω ξ U [x] ξ m ˆM ξ ˆM ξ U [x] ξ dv ξ, H δ (x) For a point x, if and only if (1) E(x) 0 and x H δ (x) such that 0 < β(x ) < 1, () then we say this point x belongs to the hybrid model. From Appendix D, we know that the hybrid strain/macro-elastic energy density at this point can be written as W (x) = 1 σ (x) : ε (x) + 1 () {T [x] p x T [p] x p } U [x] p x dv p. H δ (x)

14 We will assume that the influence function ω is spherical (i.e., ω ξ = ω s ξ ) for the remainder of this paper to describe the coupling method with clarity and without a loss of generality. It means that the material we consider below is isotropic []. Then, the Eq. () can be expanded and simplified as follows: W (x) = 1 ε (x) : E (x) : ε (x) + 1 [ ] {λ 0 β(x)ˆθ (U [x]) + β(p)ˆθ (U [p]) ω H δ (x) m s ξ X ξ U [x] ξ +τ 0 [β(x) + β(p)] ω s ξ U [x] ξ ˆM ξ ˆM } ξ U [x] ξ dv p, where ξ = p x. Note that U ξ = U ξ and X ξ = X ξ are applied in the derivation of Eq. ()... Defining the conjugated stiffness tensor E based on the a priori morphing function β The morphing function β is user-defined a priori to delimit the SPD sub-domains. The challenge is then to properly calculate the conjugated evolution of E to minimize the coupling artifacts...1. An assumption of smooth strain field over the morphing domain We assume that the strain field varies slowly over the morphing area. Indeed, it makes sense that strong gradients should be included within the SPD domain,, to correctly account for them. Then, far from this domain of interest, the coupling can be done and the CCM model can be used to reduce computational costs. We assume that the strain is homogeneous over the neighborhood of the point, x, with a horizon, δ, in the morphing domain: () ε(q) ε(x) = ε, q H δ (x). () Note that the horizon here is δ rather than δ, so that we can deduce the following approximations for the displacement state, U, and the volumetric deformation, ˆθ. That is, we have a straight-forward relationship between strains and displacements: And we have ˆM ξ U [p] ξ ξ ε ξ, p H δ (x) and ξ H δ (p). () ξ ˆθ(U [p]) ˆθ(U [x]), p H δ (x), () 1

15 where Eq. () was used in this derivation. Introducing Eq. () into Eq. (), the hybrid strain/macro-elastic energy density can be further simplified as follows: where ξ = p x. W (x) = 1 ε (x) : E (x) : ε (x) + 1 β(x) + β(p) { λ 0ˆθ (U [x]) ωs ξ X ξ U [x] ξ H δ (x) m +τ 0 ω s ξ U [x] ξ ˆM ξ ˆM } ξ U [x] ξ dv p,... Equivalences between deformation energy densities Let us consider a homogeneous material in, that is under homogeneous deformation; then the strain energy density should be independent of the morphing function, β. This means that the strain (or macro-elastic) energy density is the same at a point, x, whatever the model is. In this case, we have that the strain energy density (i.e., Eq. ()) is equal to the macro-elastic energy density (i.e., Eq. (1)). It yields 1 ε(x) : E0 : ε(x) = λ0 ˆθ (U [x]) + τ 0 m H δ (x) ω ξ U [x] ξ ˆM ξ ˆM ξ U [x] ξ dv ξ. By applying Eq. (), one can obtain E 0 = λ0 ω m s ζ ω s ξ ζ ζ ξ ξdv ζ dv ξ H δ (x) H δ (x) + τ 0 ω m s ξ ξ ξ ξ ξ dv ξ ξ, H δ (x) where ω ξ = ω s ξ is applied, ζ = q x and ξ = p x, q, p H δ (x). Similarly, an equivalence also exists between the strain energy density (i.e., Eq. ()) and the hybrid energy density (i.e., Eq. ()). Thus, we have 1 ε(x) : E0 : ε(x) = 1 ε (x) : E (x) : ε (x) + 1 β(x) + β(p) { λ 0ˆθ (U [x]) ωs ξ X ξ U [x] ξ H δ (x) m +τ 0 ω s ξ U [x] ξ ˆM ξ ˆM } ξ U [x] ξ dv p. 1 () () (0) (1)

16 By reapplying Eq. (), Eq. (1) can be recast as E 0 = E(x) + λ0 m + τ 0 m H δ (x) H δ (x) H δ (x) β(x) + β(p) ω s ζ ω s ξ ζ ζ ξ ξdv ζ dv p β(x) + β(p) ω s ξ ξ ξ ξ ξ dv ξ p, where ζ = q x and ξ = p x, q, p H δ (x). Moreover, by substituting Eq. (0) into Eq. (), we can calculate the conjugated stiffness tensor, E(x), at any point, x, as E(x) = (1 β(x)) E 0 + λ0 m H δ (x) + τ 0 m H δ (x) H δ (x) β(x) β(p) ω s ζ ω s ξ ζ ζ ξ ξdv ζ dv p β(x) β(p) ω s ξ ξ ξ ξ ξ ξ dv p. Eq. () provides a basic coupling constraint between constitutive parameters. When the a priori morphing function, β, is provided, it automatically defines the stiffness coefficients to be used for the CCM model at any point in the morphing area. Remark: A detailled analysis of ghost forces and how they are related to the choice of the morphing function is not provided here. Yet, the reader can refer to a previous discussion on ghost forces related to the one-dimensional morphing method for bondbased models in [1].. Finite element discretization We now focus on numerically solving the hybrid model (i.e., Eqs. ()-()) with the constitutive constraint (i.e., Eq. ()). One of the ways of doing this is to discretize the equation of minimum potential energy (i.e., Eq. (D.)). From the resulting discrete energy equation, we can then derive the linear algebraic equation for finite element computations. For the derivation method and some of the notations we used below, refer to those in [0]. We divide the whole domain,, by a finite number of elements, V i, i = 1,,, n, where n is the number of elements. These elements are nonoverlapping, but common 1 () ()

17 vertices, called mesh nodes, are shared between adjacent elements. Thus, we can write that = V 1 V V n. Because = 1 m is defined in Section.1, we assume for definiteness and without a loss of generality that 1 = V 1 V V n, m = V n +1 V n + V n and = V n +1 V n + V n, for 1 < n < n < n. In addition, we define the divisions of the neighborhood of any point x (i.e., H δ (x) ). Indeed, there exists a minimal set of elements, A x, which is defined by A x = {Vx 1, Vx,, Vx h(x) } {V 1, V,, V n }, such that (H δ (x) ) B x, x \ 1, where B x = Vx 1 Vx Vx h(x). By extending the influence function ω ξ to the domain B x (refer to Def. (D.)), the equation of the potential energy can also be rewritten by replacing the subdomain (H δ (x) ) with the subdomain B x. That is Π(u) = W c (ε) d + W s (η) d Q b (u) d Q f (u) ds, () Γ f where ε and η are functions of u, and W c (ε) = 1 ε (x) : E (x) : ε (x), W s (η) = λ0 β (x) ω m s ξ ω s ζ η x, p ξ ζη x, q dv q dv p B x B x + τ 0 β (x) ω m s ξ η x, p ξ ξ B x ξ η x, p dv p + λ0 β (p) ω m s ξ ω s γ η x, p ξ γη p, q dv q dv p B x B p + τ 0 β (p) ω m s ξ η p, x ξ ξ B x ξ η p, x dv p, Q b (u) = b(x) u(x) and Q f (u) = f(x) u(x), where ζ = q x and ξ = p x, q, p H δ (x) and γ = q p, q H δ (p). Note that η p, x ( ξ) is replaced with η x, p ξ in the third term on the right-hand side of W s (η). The displacement solution can be approximately expressed in the finite element scheme using piecewise interpolation techniques. Let u i denote the displacement solution over the element V i, which is given by u i (x) = N i (x)d i, i = 1,,, n, () 1

18 where N i is the matrix of shape function and d i is the nodal displacement. Substituting Eq. () into the Eq. (), an approximate potential energy function can be rewritten as n n Π(u) W i i c (ε i ) dv x + W s ( η) dv x i=1 V i i=1 V i n n () Q i b (u i ) dv x Q i f (u i ) ds x, V i S i i=1 where S i is the boundary of element V i and i=1 W i c (ε i ) = 1 [GN i(x)d i ] T [ Ē (x) ] [GN i (x)d i ], W s i ( η) = λ0 h(x) m h(x) j=1 k=1 V j x V k x β (x) ω s ξ ω s ζ [N j (p)d j N i (x)d i ] T [ξ ζ] [N k (q)d k N i (x)d i ] dv q dv p + λ0 h(x) h(p) β (p) ω m s ξ ω s γ [N j (p)d j N i (x)d i ] T j=1 k=1 V j x V k p [ξ γ] [N k (q)d k N j (p)d j ] dv q dv p + τ 0 h(x) β (x) + β (p) ω m j=1 Vx j s ξ [N j (p)d j N i (x)d i ] T [ ] ξ ξ [N ξ j (p)d j N i (x)d i ] dv p, Q i b (u i ) = [N i (x)d i ] T {b(x)} and Q i f (u i ) = [N i (x)d i ] T { f(x)}, where G denotes a matrix of differential operators, η represents a vector defined by the discretized displacement field, the notations [ ] and { } denote a matrix and a vector, respectively. Let the number of mesh nodes in the global discretized domain be N, then the whole nodal displacements can be defined as d T = {d 1, d,, d N }. Thus, for the nodal displacements d i in any element V i, we know that d i d, i = 1,,, n. Moreover, one can define that d i = C i d, i = 1,,, n, () 1

19 where C i is a diagonal matrix in which the diagonal entries may be 0 or 1, depending on the nodes of element V i. Substituting Eq. () into Eq. (), the potential energy Π yields the function of d such that Π(d) = 1 dt Kd d T F, () where K = F = n [GN i (x)c i ] T [ Ē (x) ] [GN i (x)c i ] dv x i=1 V i + λ0 h(x) h(x) n β (x) ω m s ξ ω s ζ [N j (p)c j N i (x)c i ] T i=1 j=1 k=1 V i V j x V k x [ξ ζ] [N k (q)c k N i (x)c i ] dv q dv p dv x + λ0 h(x) h(p) n β (p) ω m s ξ ω s γ [N j (p)c j N i (x)c i ] T ξ n i=1 j=1 k=1 V i [ξ γ] [N k (q)c k N j (p)c j ] dv q dv p dv x + τ 0 h(x) n m i=1 j=1 V i Vx j [ ] ξ ξ [N j (p)c j N i (x)c i ] dv p dv x and i=1 V j x V k p V i [N i (x)c i ] T {b(x)}dv x + β (x) + β (p) ω s ξ [N j (p)c j N i (x)c i ] T n [N i (x)c i ] T { f(x)}ds x. S i i=1 Furthermore, a linear system including the solution of the nodal displacement vector d can be derived from Eq. () using a condition similar to Eq. (D.). That is which yields. Numerical examples Π(d) δd = 0, δd, () d Kd F = 0. (0) The accuracy and effectiveness of this morphing method are illustrated in this section. First, we consider benchmark examples, where a -dimensional (D) plate is prescribed

20 under either pure traction or pure shear boundary conditions (see Figure ). Next, we subject a cracked plate to shear conditions. The SPD model is adopted around the crack and the CCM mode is applied close to the boundary where the displacement conditions are prescribed. Both models are coupled by the morphing method in the transition region, which is far from the crack (see Figure ). y m x Center 1. (a) 1.0 Figure : Dimensions and boundary conditions of a plate under uniform deformations (unit of length:µm). y m x u x u 1. Crack y u y Fixed Point (b) Zoom In Figure : Dimensions and boundary conditions of a cracked plate (unit of length:µm). Here all examples are considered to satisfy the plane strain assumption, so that the component of displacement, u z, is constant (without a loss of generality, let u z = 0), 1 Center u x 0. o u y (c)

21 and so that the components of displacement, u x and u y, are functions of x and y. Therefore, the SPD model, which is built upon the displacements (see Eq. ()), is strictly implemented on the D plate. A spherical influence function, ω s ξ, is assumed to be ω s ξ = ξ e ξ /l, where l is a characteristic length that is assumed to be l = 0.00µm. The influence function, ω s ξ, here includes a square of ξ, in order to ensure the accurate integral in Eq. () when ξ approaches zero. The horizon, δ, is 0.1µm. The material parameters of the SPD model, λ and τ, are GPa and 0GPa, respectively. The equivalent stiffness parameters in the CCM model applied in the D examples, including Young s modulus and Poisson ratio, are E = 1GPa and ν = 0., respectively. The D conversion formulas between λ, τ and E, ν can be derived similarly to the process described in Appendix B for -dimensional (D) examples. All numerical examples are implemented using the finite element analysis detailed in Section. All elements applied in these examples are bilinear quadrilateral grids; the size of grids is 0.0µm..1. Example the uniform deformation of a plate To accurately show the effects of the morphing method, we simulate a homogeneous plate under uniform extensions: pure traction deformation and pure shear deformation. The dimensions of this plate are shown in Figure (a) and the traction conditions are shown in Figure (b), where the bottom of the plate is fixed in the y direction and a completely fixed point occurs in the middle of the x direction. Another displacement boundary condition, ū y = 0.1µm, is imposed on the top of this plate. In Figure (c), the shear boundary conditions are imposed around the plate, such that { ūx = 0.1y, (1) ū y = 0.1x. Figure shows the relative error of the strain component, ε yy, in the whole deformed plate. The results in Figure (a) and (b) are calculated using different morphing functions (i.e., β(x), x ) including linear and cubic functions [, 1]. More discussions about morphing functions can be found in paper [1]. The relative error is defined as (ε yy ε yy )/ ε yy, where ε yy is the analytical solution of the uniform strain components, ε yy. From Figure, we can see that the perturbations occur close to the boundaries of the morphing domain. The absolute value of the maximum error in Figure (a) is greater than %, which is several times greater than those in Figure (b). Moreover, the majority of the plate shows homogeneous strain fields because the range of errors for the majority of Figure (a) is [ 0.%, 0.%], while in Figure (b) the range is only 1

22 from 0.1% to 0.1%. Our conclusions are similar to those of bond-based coupling [1], for which the influence of smoothness of morphing functions was studied in detail. The cases of pure shear deformation are displayed in Figure, where the relative error of the strain component, εxy, is shown, which is calculated by (εxy ε xy )/ε xy, where ε xy denotes the analytical solution of the strain. The images of the perturbations in Figure have similar characteristics to those of the traction examples in Figure. Additionally, the shapes of the perturbation areas in Figures and have symmetries. However, the axes of symmetry are obviously inconsistent with those axes of loading and structure. We think the main reason is the complex anisotropy that arises in the morphing domain. Indeed, even if the original material is isotropic, the morphing technique makes it anisotropic in the morphing domain for its local part, which can change the principal directions of the error. Figure : The relative error on the strain component, εyy, for a plate under traction conditions with different weighting functions: (a) linear and (b) cubic... Example a cracked plate By comparing with benchmark-like solutions in the previous examples, we demonstrate the quality of the proposed morphing method for the SPD model. Here, we present an application example: a cracked plate subjected to both traction and shear conditions along its top. The bottom of the plate is completely fixed. The dimensions of the plate, the shape and position of the crack as well as all boundary conditions are shown in Figure. From the figure, we can see that the SPD model covers the crack in Ω. The boundary conditions are imposed on the boundary of Ω1, where the CCM model is adopted. The morphing method is applied to both models in Ωm shown as the light grey domain. 0

23 Figure : The relative error on the strain component, εxy, for a plate under shear conditions with different morphing functions: (a) linear and (b) cubic. Figure shows the strain component, εxy, where strain results are calculated by the pure CCM model (a,d); the morphing model with a linear morphing function (b,e); and the pure SPD model (c,f). The top figures show the results as whole deformed configurations and the bottom figures are close-ups of the top figures formed around the deformed crack. The top figures (a-c) have nearly the same strain fields with the exception of very thin areas at top and bottom boundaries of Figure (c). The reason is that points near the boundary in the pure SPD model fail to hold a complete horizon such that they lose a part of their material properties; however, this is not related to the morphing technique. Figures (e) illustrates the results such as contour plots and shapes of crack tip calculated by the morphing method are the same as those in Figure (f), which is solved by the pure SPD model. However, the results in Figure (e,f) are different from those in Figure (d), which are calculated by the pure CCM model.. Conclusions We have proposed a morphing approach for coupling the SPD and CCM models. It is essentially a development of the morphing techniques that have been previously presented for coupling bond-based peridynamics with classical continuum mechanics. The proposed coupling method enables and simplifies failure simulation of materials, such as damage and fracture, without any limitation to the Poisson ratio. The proposed morphing technique constructs a single unified balance equation between constitutive parameters of the SPD model and the CCM model. This equation determines a gradient of material parameters over the whole structure through a mor1

24 Figure : The strain component, εxy, calculated by (a,d) the pure CCM model, (b,e) the morphing model using a linear morphing function and (c,f) the pure SPD model. phing function. Consequently, the morphing method achieves a smooth transition from the SPD model to the CCM model or vice versa. From the numerical examples, we conclude that the morphing method does not only preserve the accuracy of the solutions and capture the nonlocal effects at the key domain, such as crack tips, but also avoids the artificial factors near the boundaries and reduces computational costs compared with the pure SPD model. Acknowledgements The authors gratefully acknowledge the financial support received from KAUST baseline and the Boeing Company for the completion of this work.

25 Appendix A. Macro-elastic energy density of an ordinary, elastic peridynamic material Following the derivations proposed by [], the extension scalar state, e, is divided into the isotropic part, e i, and the deviatoric part, e d, such that e = e i + e d and e i = θ(e)x (A.1) From [], the macro-elastic energy density (i.e., the strain energy density in []) of the elastic and ordinary peridynamic model is given by W p = kθ (e) + α ( ωe d ) e d, (A.) where k and α are positive constants. According Eqs. () and (A.1), Eq. (A.) can be recast by W p = kθ (e) + α { ( ω e θ(e)x )} ( e θ(e)x ) = 1 (k m ) α θ (e) + α (A.) (ωe) e, where the definitions of m and θ (i.e., Eqs. () and ()) are applied in the derivation. In addition, a similar expression with the spherical influence function can be found in [1]. From Eq. (A.), we find that the units of two coefficients, k and α, are inconsistent with each other, so we introduce two new parameters to replace k and α such that λ = k m α and τ = m α. (A.) (A.) Then, the macro-elastic energy density can be written as Eq. () with two new coefficients: λ and τ. These two parameters do not only have the same units, but are also similar to the Lamé parameters in the classical elasticity. More studies of these SPD parameters can be found in Appendix B. Appendix B. Conversion formulas between SPD and CCM parameters We aim to create the relations between SPD and CCM parameters for a D homogeneous, isotropic, linear-elastic material. These SPD parameters are either (k and α)

26 in Eq. (A.) or (λ and τ) in Eq. (). The CCM parameters include the Young s modulus, E, the Poisson ratio, ν, and the Lamé parameters Λ and µ. Here, let s consider a small homogeneous deformation with deviatoric strain tensor, ε d ij, i, j = 1,,, such that ε d kk = 0, k = 1,,, where the Einstein summation convention on repeated index is adopted. For a pure distortion of the material (i.e., without dilatation), Eq. () in [] provides a relation between α and µ: α = 1 m µ, (B.1) where µ is named shear modulus or Lamé s second parameter. Now, we consider a pure dilatation of the material (i.e., without distortion). For the classical elasticity, in this case, the deviatoric strain tensor ε d ij = 0. On the other hand, we have the approximation between the deviatoric extension state, e d, and the deviatoric strain tensor, ε d ij form Eq. () in []. That is e d ξ εd ijξ i ξ j. (B.) ξ Thus, in this case, we know e d = 0. Then the macro-elastic energy density from Eq. (A.) can be rewritten as H δ (x) W p = k θ (e). (B.) Substituting the approximate expression of θ (i.e., Eq. ()) into Eq. (B.) and then applying Eq. (), Eq. (B.) can be recast as W p = k ( ) ε H δ (x) m ω s ξ ξ ξdv ξ H δ (x) m ω s ξ ξ ξdv ξ ε = k ( ) (B.) ε ij m ω s ξ ξ i ξ j dv ξ m ω s ξ ξ k ξ l dv ξ ε kl. H δ (x) The evaluation of this integral can be largely simplified by accounting for the spherical symmetry of ω s. As ω s is spherical, any index, i or j (and k or l), with non-repeated indices (such as {1, }) integrates to 0. Only integrals with repeated indices need to be considered (i.e., three integrals). Here we use the spherical coordinate system ξ 1 = r sin φ cos ψ, ξ = r sin φ sin ψ, ξ = r cos ψ: δ π π ω s ξ ξ1dv ξ = ω s ξ r sin φ cos ψr sin φdφdψdr = m, (B.) H δ (x) 0 0 0

27 where, according to Eq. (), m = π δ ω 0 s ξ r dr. Similarly, ω s ξ ξdv ξ = m, H δ (x) H δ (x) ω s ξ ξ dv ξ = m. According to Eqs. (B.), (B.) and (B.), Eq. (B.) can be rewritten as W p = 1 kε ijε kl δ ij δ kl = 1 kθ, (B.) (B.) (B.) where δ ij is the Kronecker s delta operator and Θ = ε ii is the trace of the strain tensor in the CCM. For the homogeneous, isotropic, linear-elastic material, we assume that the macro-elastic energy density in the SPDs should be equal to the strain energy density in the CCM for all pure dilatation deformations. The strain energy density, W c, in the CCM for pure dilatation can be written as follows []: W c = 1 (Λ + ) µ Θ. (B.) Thus, comparing Eq. (B.) with Eq. (B.), one can obtain k = Λ + µ. Substituting Eqs. (B.1) and (B.) into Eqs. (A.) and (A.), they yield λ = Λ µ and τ = µ. (B.) (B.) (B.1) For a D isotropic material, the well-know conversion formulas [] in the CCM between Lamé parameters and stiffness coefficients (i.e., Young s modulus, E, and the Poisson ratio, ν) are Eν Λ = and (1 + ν) (1 ν) (B.1) E µ = (1 + ν). (B.1)

28 Substituting Eqs. (B.1) and (B.1) into Eqs. (B.) and (B.1), it follows that E (ν 1) λ = and (1 + ν) (1 ν) (B.1) E τ = (1 + ν). (B.1) Young s modulus, E, must be positive. The range of the Poisson ratio, ν, is 0 < ν < 1 for most D materials. Thus, τ must be positive according to Eq. (B.1); however, λ can be negative via Eq. (B.1). Appendix C. A definition of scalar force state In this Appendix, the definition of scalar force state (i.e., Eq. (1)) will be derived. To do this, we define an extension vector state, Z, which is given by Z = M(Y)e. According to the definition of a direction vector state (i.e., Eq. ()), we have M(Z) = M(Y)e M(Y)e = M(Y) M(Y) = M(Y), where M = 1 is used in the last step. Thus, the following equation yields And we also have Z = M(Z)e. Z = M(Z)e = e, (C.1) (C.) (C.) (C.) where the property M = 1 is applied again. Then, we replace e with Z in Eqs. () and (). The dilatation θ can be recast as θ(e) = θ(z) = (ωx) Z. m (C.) The macro-elastic energy density can be rewritten as W p (e) = W p (Z) = λ θ (Z) + τ (ω Z ) Z, m (C.) Because of limited space, here we omit a part of the proof that can be written according to the proof in Proposition.1 of [], as long as Y and y are replaced

29 with Z and e, respectively. Then, one can get the following equation according to Eq. () of []: t = W p (Z) e which is the definition of the scalar force state in Eq. (1). = W p (e), (C.) e Appendix D. Principles of virtual work and minimum potential energy The principle of virtual work for peridynamics has been treated in the references [1, ]. Here, we derive the principle of virtual work for the hybrid CCM and SPD model. Then, we propose a proof of the equivalence between the principle of minimum potential energy and the principle of virtual work. Then, the formula of the hybrid strain/macro-elastic energy density (i.e., Eq. ()) is derived form the proved principle of minimum potential energy for the hybrid model. For convenience, let T (x) be the integral of all peridynamic forces to which a point, x, is subjected by all points, p H δ (x). That is T (x) = {T [x] p x T [p] x p } dv p, x. (D.1) H δ (x) Then, the equilibrium equation, Eq. (), can be rewritten as divσ(x) = b(x) T (x), x. (D.) We consider an arbitrary displacement field, δu(x), which satisfies δu(x) = 0, x Γ u. From the equilibrium equation, Eq. (), and the traction boundary condition, Eq. (), one can create their equivalent integral equation: δu(x) (divσ(x) + b(x) + T (x)) d δu(x) (σ(x) n f(x) ) ds = 0. (D.) Γ f Considering the Green identity, the symmetry of stress tensor and the strain-displacement equation (i.e., Eq. ()), the first term in the volume-integration of Eq. (D.) yields δu(x) divσ(x)d = δε (x) : σ (x) d + δu(x) σ(x) nds. (D.) Γ f If we substitute Eq. (D.) into Eq. (D.), it can be rewritten as (δε (x) : σ (x) δu(x) b(x) δu(x) T (x)) d δu(x) f(x)ds = 0. (D.) Γ f

30 Additionally, we derive below the work done by the peridynamic force field, T (x). For clarity and with no major restrictions, we extend the influence function ω ξ to the whole domain, and note that ω is a constant function for any point, x, such that ω η = 0, η / H δ (x), ω ξ 0, ξ H δ (x), (D.) H H δ (x) with a nonzero volume such that ω ζ > 0, ζ H. Note that this extension does not change the displacement solution of the original equations. By using the definition, Eq. (D.), and Eq. (), the Eq. (D.1) can be rewritten as T (x) = {T [x] p x T [p] x p } d p, x. (D.) Then, the third term of volume-integration in Eq. (D.) can be rewritten based on Eq. (D.) and Eq. (1). That is, δu(x) T (x)d = {T [x] p x T [p] x p } X δu(x) d p d x. (D.) By exchanging variables, we can see that {T [x] p x T [p] x p } X δu(x) d p d x (D.) = {T [x] p x T [p] x p } X δu(p) d p d x Then from Eq. (D.), Eq. (D.) yields δu(x) T (x)d = 1 {T [x] p x T [p] x p } {X δu(p) X δu(x) } d p d x. (D.) Based on Eq. (D.) and the symmetry of the elastic tensor, the principle of virtual work (i.e., Eq. (D.)) can then be rewritten as f(x) δu(x)ds + b(x) δu(x)d = σ (x) : δε (x) d Γ f + 1 (D.) {T [x] p x T [p] x p } {X δu(p) X δu(x) } d p d x.

31 The left- and right-hand sides of Eq. (D.) represent the virtual work done by external forces and internal forces, respectively. In an elastic system and in the absence of temperature variations, it is commonly believed that this principle of virtual work is equivalent to the principle of minimum potential energy [1]. We will prove below that it still works for this elastic hybrid local/nonlocal system. Here, we write the principle of minimum potential energy for this hybrid model as the following equation: Π(u) = 1 σ (x) : ε (x) d + 1 {T [x] p x T [p] x p } U [x] p x d p d x (D.1) f(x) u(x)ds b(x) u(x)d = Minimum Value, Γ f where ε and U are functions of u. Note that U is a vector state. According to its definition, however, we can also replace U by a vector, η, which is given by η(u) x, p = u(p) u(x). By substituting these vectors, η, and the constitutive Eqs. () and () into Eq. (D.1) and further extending the influence function ω ξ to the whole domain, it can be rewritten as Π(u) = W c (ε) + W s1 (η) + W s (η) F Γ (u) F (u) = Minimum Value, (D.1)

32 where W c (ε) = 1 ε (x) : E (x) : ε (x) d x, λ(x) W s1 (η) = ω p x ω q x η x, p (p x) m (q x) η x, q d q d p d x λ(p) + ω x p ω q p η p, x (x p) m (q p) η p, q d q d x d p λ(x) = ω p x ω q x η x, p (p x) m (q x) η x, q d q d p d x ( λ(x) = ω p x (p x) η x, p d m p ) ω q x (q x) η x, q d q d x ( ) λ(x) = ω p x (p x) η x, p d m p d x, τ(x) (p x) (p x) W s (η) = ω p x η x, p η x, p d m p x p d x τ(p) (x p) (x p) + ω x p η p, x η p, x d m x p p d x τ(x) (p x) (p x) = ω p x η x, p η x, p d m p x p d x, F Γ (u) = f(x) u(x)ds and F (u) = Γ f b(x) u(x)d. Here, ε and η are functions of u, and the variable interchange between p and x is applied in the equations of W s1 (η) and W s (η). Π(u) denotes the total potential energy of the solid body, which is the sum of the elastic deformation energy stored in the deformed body and the potential energy of the applied forces. For an arbitrary displacement δu, Eq. (D.1) means that there is a diagonal matrix 0

33 Ψ written as ψ Ψ = 0 ψ ψ where ψ i, i = 1,, satisfy 0 ψ i 1, such that where Π(u) u (D.1) Π(u + δu) = Π(u) + Π(u) u δu + 1 δut H [Π] (u + Ψδu)δu Π(u), (D.1) is the gradient of the scalar-valued function, Π(u), with respect to u, and H [Π] (u+ψδu) denotes the Hessian matrix of the function, Π, at the vector, u+ψδu. Note that Eq. (D.1) is the second-order Taylor expansion with the Lagrange form of the remainder. Applying the chain rule, 1 δut H [Π] (u + Ψδu)δu = 1 ( ) T ( ε(u + Ψδu) Wc (ε) ε(u + Ψδu) δu u ε u + 1 ( ) T ( η(u + Ψδu) Ws1 (η) η(u + Ψδu) δu u η u + 1 ( ) T ( η(u + Ψδu) Ws (η) η(u + Ψδu) δu u η u ) δu ) δu ) δu. (D.1) It is worth noting that above ε η and actually denote the Jacobian matrices of ε and u u of η, respectively. For simplicity s sake, we define δε and δη such that ε(u + Ψδu) δε = δu and u η(u + Ψδu) δη = δu. u 1 (D.)