Contents lists available at SciVerse ScienceDirect

Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 1088–1102

0022-50

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/jmps

A morphing strategy to couple non-local to localcontinuum mechanics

Gilles Lubineau a,n, Yan Azdoud a, Fei Han a, Christian Rey b, Abe Askari c

a King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division, Cohmas Laboratory, Thuwal 23955-6900, Saudi Arabiab LMT Cachan (ENS Cachan / CNRS / UPMC / PRES UniverSud Paris), 61 Avenue du President Wilson, 94235 Cachan Cedex, Francec Applied Math Group, The Boeing Company Seattle, WA, USA

a r t i c l e i n f o

Article history:

Received 5 December 2011

Received in revised form

21 February 2012

Accepted 26 February 2012Available online 3 March 2012

Keywords:

Non-local elasticity

Gluing

Peridynamics

Long-range forces

Multi-scale

96/$ - see front matter & 2012 Elsevier Ltd. A

016/j.jmps.2012.02.009

esponding author. Tel.: þ966 544 700 059.

ail address: gilles.lubineau@kaust.edu.sa (G. L

a b s t r a c t

A method for coupling non-local continuum models with long-range central forces to

local continuum models is proposed. First, a single unified model that encompasses

both local and non-local continuum representations is introduced. This model can be

purely non-local, purely local or a hybrid depending on the constitutive parameters.

Then, the coupling between the non-local and local descriptions is performed through a

transition (morphing) affecting only the constitutive parameters. An important feature

is the definition of the morphing functions, which relies on energy equivalence. This

approach is useful in large-scale modeling of materials that exhibit strong non-local

effects. The computational cost can be reduced while maintaining a reasonable level of

accuracy. Efficiency, robustness and basic properties of the approach are discussed

using one- and two-dimensional examples.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In the accurate simulation of a material’s behavior, strong non-local effects on a low scale may be encountered.Examples are Van der Waals forces in nano-reinforced materials and molecular interactions in dissimilar materialinterfaces (Du et al., 2007). The discrete simulation of these systems at the atomistic scale (molecular mechanics ormolecular dynamics, see Rapaport, 2004) remains out of reach with current computing facilities. An alternative is to usenon-local continuum mechanics, which has the potential to capture the interactions at least partially while reducing thecomputation cost. Here, non-local models include long-range central interactions between material points (Silling, 2000;DiPaola et al., 2009, 2010), in contrast to what we call ‘‘local’’ continuum models in which material points can only interactby means of contact forces. Yet, the simulation of fully non-local continuum models also quickly results in prohibitivecomputational costs for two reasons. First, the assembly operation of the discrete system is more costly as each integrationpoint interacts with multiple neighbors. Second, this leads to non-sparse matrices.

To overcome this issue, non-local models can be reserved for parts of the structure where non-local effects stronglyinfluence the solution, whereas, far from these regions, upscaling to local continuum mechanics is a good compromise toreduce the computational cost. The challenge, then, becomes gluing those models together without introducing large numericalartifacts. The objective of this paper is to present a simple and robust coupling technique between a non-local model, reservedfor the part of the structure where it is necessary, and a compatible local model, used in the rest of the structure.

Gluing techniques in multi-model approaches have been extensively studied in the framework of atomistic-to-continuumcoupling (Belytschko and Xiao, 2003). Curtin and Miller presented a comprehensive review in Curtin and Miller (2003).

ll rights reserved.

ubineau).

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1089

These techniques can be extrapolated to non-local-to-local continuum coupling. One example is the Arlequin approach(BenDhia and Rateau, 2005; Chamoin et al., 2010) based on weighting the energy of the two models in a gluing area byarbitrary blending functions. This approach has been applied recently to non-local-to-local continuum coupling (Han andLubineau, 2012), although it largely increases the size of the system to be solved as the coupling conditions are enforced bythe systematic introduction of Lagrange multipliers. A different approach to atomistic-to-continuum coupling is based on aunified equilibrium equation between the models as introduced by Fish et al. (2007) and Badia et al. (2007). Here, additionalLagrange multipliers are not required, but the blending functions are somewhat arbitrary. We take advantage of couplingmodels through a unified equilibrium equation in this work.

The non-local-to-local continuum coupling method presented relies on the definition of a single model over thestructure, with an equilibrium equation that includes both the local and non-local interactions. The relative weight of eachpart of the model (the local and non-local parts) at any point depends only on the constitutive parameters. The morphingbetween a purely non-local description to a purely local one is the result of the evolution of the material properties relatedto each model. This evolution is constrained by energy equivalence throughout the morphing domain.

The remainder of this paper is organized as follows. Section 2 describes the framework for a non-local model with long-range central forces. We also review some classical results on the definition of an equivalent local continuum underhomogeneous infinitesimal strains. Section 3 introduces the general formulation of the morphing technique. In Section 4,we analyze and validate the precision of the morphing method in unidimensional problems. Finally, Section 5 presentssome numerical benchmarks that validate the approach in bi-dimensional problems.

2. Non-local linear elasticity with long-range central forces

2.1. The non-local continuum with long-range central forces

We start with a brief description of non-local models with long-range central forces. Such models have beenextensively described in the literature. Among others, the reader is referred to the work of Kroner (1967) that introduceda long-range force model, to the work of Eringen (2001) on non-local elasticity models, of Silling (Silling, 2000; Silling andLehoucq, 2010) who described a long-range model in the more general framework of peridynamics, and to the work ofDiPaola et al. (2009, 2010) who presented a three-dimensional, non-local linearized elasticity theory.

We limit ourselves to linear elasticity. We also assume small displacements and infinitesimal transformation. Letmaterial points in a non-local continuum interact through long-range forces (see Fig. 1(a)) such that the equilibriumequation at point x is written for a quasi-static problem asZ

HdðxÞf ðp-xÞ dVpþb ¼ 0, 8x 2 O ð1Þ

where b are the prescribed external body forces and f ðp-xÞ is the specific long-range force that characterizes the action ofpoint p over point x. We denote by HdðxÞ the neighborhood over which long-range forces are neglected, and d theassociated cut-off radius (Fig. 1).

We restrict ourselves to a situation in which the long-range interactions are central, such that

f ðp-xÞ ¼ f ½x�/p�xS�f ½p�/x�pS ð2Þ

where f ½x�/p�xS (respectively f ½p�/x�pS) is the partial interaction due to the action of point p over point x (with respectto point x over point p). The bond interaction is thus described as the superposition of two interactions, each related to oneof the bond extremities (see Fig. 1(b)). An antisymmetric form (with respect to x and p) of the interactions ensures thebalance of the linear momentum (see Silling et al., 2007; Warren et al., 2009) and the central interactions ensure that theyare collinear to the bond vector: x ¼ p�x. If small displacements and linear elasticity are assumed, a possible constitutivemodel is (Silling, 2010; Silling and Lehoucq, 2010)

f ½x�/p�xS¼c½x�ðJxJÞ

2fux ðpÞ�ux ðxÞgex ð3Þ

Ω

bx

p

H (x)

δx

p

f[x](p-x)

e

Hδ (x)

f(p x)^

f[p](x-p)^

Fig. 1. (a) The non-local continuum domain O: HdðxÞ is the horizon of x depending on the characteristic range of long-range forces. (b) Partition of the

interaction, f ðp-xÞ, into partial interactions.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021090

where ex ¼ x=JxJ, and ux ðpÞ is the projection of the displacement at point p over the bond (i.e., ux ðpÞ ¼ uðpÞ � ex ). In thisapproach, the partial bond behaves as a spring with stiffness c½x�ðJxJÞ (a so-called micro-modulus in peridynamics). c is hereintroduced as a general function of the bond length JxJ. Together, Eqs. (2) and (3) lead to

f ðp-xÞ ¼c½x�ðJxJÞþc½p�ðJxJÞ

2fux ðpÞ�ux ðxÞgex ð4Þ

In the case of a homogeneous material (i.e., c½x�ðJxJÞ ¼ c½p�ðJxJÞ ¼ cðJxJÞ), the strain energy density of the non-localmodel at point x is written as

WnlðxÞ ¼

1

4

ZHdðxÞ

cðJxJÞfux ðpÞ�ux ðxÞg2 dVp ð5Þ

2.2. ‘‘Local’’ approximation of the energy

In case the strain is homogeneous in the HdðxÞ neighborhood, it is possible to derive the stiffness operator (the classicalHook tensor) of the equivalent local continuum from the non-local properties. Let us consider an infinitesimalhomogeneous transformation. We have

ux ðpÞ�ux ðxÞCx � e � xJxJ

8p 2 HdðxÞ ð6Þ

with e the infinitesimal strain defined in the local framework (Appendix A) by

e ðxÞ ¼ 12ðruðxÞþ t

ruðxÞÞ ð7Þ

From Eqs. (6) and (5), the strain energy density of an equivalent local model can be written as

WnlðxÞCWl

ðxÞ ¼1

2e : K 0 : e with K 0

¼

ZHdðxÞ

cðJxJÞx � x � x � x

2JxJ2dVp ð8Þ

where ð�Þ is the dyadic product. In Eq. (8), the non-local strain energy density WnlðxÞ is similar to the strain energy density of a

classical local continuum model, WlðxÞ, with a linear elastic material defined by the 4th order stiffness tensor, K 0, evaluated at

point x. Note that in our case, where c is only a function of the bond length, the constitutive model is necessarily isotropic. It iscommon to prove (Appendix B) such that Lame’s parameters of the equivalent local stiffness tensor are given by

l¼ m¼ 2p15

Z d

0cðrÞr4 dr

!ð9Þ

The energy equivalence is used in subsequent sections to design a bridging method between both models.

3. Coupling between the non-local and local models

Let O be the overall domain. We consider a non-local model characterized by a uniform micro-modulus, c0ðJxJÞ, and we

denote by K 0 the elastic tensor of the ‘‘equivalent’’ local model. We propose to derive a surrogate problem that enables us

to couple the non-local model with its equivalent local one to avoid the resolution of the original, fully non-local problem.Then, we assume that far from a specific zone (O2), where the non-local model is used, the deformation is smooth

enough to use the local model in O1 (see Fig. 2). The main problem actually comes from the local/non-local interface. Wepropose to introduce a morphing zone, Om, where both models co-exist.

3.1. A progressive morphing between non-local and local interactions

We assume that @O � @O1. In other words, the non-local subdomain, O2, is totally embedded within the local one. Om isthe morphing zone where a hybrid model is defined in which interactions are both local and non-local and meet thefollowing requirements:

O¼O1 [Om [O2, O1 \Om ¼ |, O2 \Om ¼ | ð10Þ

The coupling is then defined as a simple evolution of the material properties characterizing each model. We build theunified governing equations over O when an hybrid model is used:

�

Kinematic admissibility and compatibility:

e ðxÞ ¼ 12ðr � uðxÞþ � uðxÞÞ 8x 2 O ð11Þ

Fig. 2. The domain O¼O1 [Om [O2.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1091

Zx ðp�xÞ ¼ ux ðpÞ�ux ðxÞ 8ðx,pÞ 2 O ð12Þ

u ¼ u 8x 2 Su ð13Þ

�

Static admissibility:

div sþZ

HdðxÞff ½x�/p�xS�f ½p�/x�pSg dVp ¼ 0 8x 2 O ð14Þ

s � n ¼ T 8x 2 ST ð15Þ

�

Constitutive equations:

s ¼ K ðxÞ : e 8x 2 O ð16Þ

f ½x�/p�xS¼c½x�ðJxJÞ

2Zx ðp�xÞex 8x 2 O ð17Þ

K ðxÞ characterizes the stiffness of the local continuum at any point, x. c½x�ðJxJÞ is the non-local constitutive parameter local

to point x. Using c0ðJxJÞ as references, we define c½x�ðJxJÞ by introducing a weighting scalar function, a, such that

c½x�ðJxJÞ ¼ aðxÞc0ðJxJÞ ð18Þ

Based on Eq. (14), it is important to note that (Fig. 3)

�

a point x strictly belongs to the local model (i.e., x 2 O1) if and only if:

K ðxÞ ¼ K 0 and aðx 0Þ ¼ 0 8x 0 2 HdðxÞ ð19Þ

�

a point x strictly belongs to the non-local model (i.e., x 2 O2) if and only if:

K ðxÞ ¼ 0 and aðx 0Þ ¼ 1 8x 0 2 HdðxÞ ð20Þ

3.2. Defining the morphing: a and K

The function a is defined a priori and is used to select the sub-domains that should be processed with the non-localdomain. The issue is how to calculate the conjugated evolution of K to minimize the coupling artifacts.

3.2.1. Constraining based on the local strain energy density

At any point, x 2 Om, the strain energy density can be written as

WðxÞ ¼1

2s ðxÞ : e ðxÞþ 1

4

ZHdðxÞðf ½x�/p�xS�f ½p�/x�pSÞ � ðuðpÞ�uðxÞÞ dVp ð21Þ

Introducing Eqs. (16)–(18) to (21) leads to

WðxÞ ¼1

2e ðxÞ : K ðxÞ : e ðxÞþ

ZHdðxÞ

c0ðJxJÞaðxÞþaðpÞ

2

fux ðpÞ�ux ðxÞg2

4dVp ð22Þ

Cutting line L

1

0α

u

T

Cutting line L

Ω1 Ω1 Ωm

Ωs

Ω2

Ωm

Ω

δ

Ωs

Ω2

δ

Fig. 3. The general shape of the morphing function is a. Solid lines are initial requirements in O1 and O2 (Eqs. (19) and (20)). The dashed line is a free

evolution of a that needs to be defined. Os is the borderline of Om above which a has to be prescribed as either 0 or 1 (see Eqs. (19) and (20)).

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021092

A general constraint in the following form is now introduced:

WðxÞ ¼Wmðx,u9OÞ ð23Þ

Eq. (23) states that in a general sense, the strain energy density at point x of the morphing zone should depend on aprescribed function, Wm, of the displacement over the whole domain. Various simple constraints can be used. For example:

�

if we define:

Wmðx,u9OÞ ¼

12e ðxÞ : K0

ðxÞ : e ðxÞ ð24Þ

we prescribe the strain energy of the hybrid model to be equal to the strain energy of the homogenized local continuummodel in the local kinematics to the morphing zone at point x.

� if we define:

Wmðx,u9OÞ ¼

ZHdðxÞ

c0fux ðpÞ�ux ðxÞg

2

4dVp ð25Þ

we prescribe the energy of the hybrid model to be equal to the energy of the non-local continuum model in the non-local kinematics of the morphing zone at point x.

The two constraints given by Eqs. (24) and (25) are in general different.

3.2.2. The constraint for smooth coupling

A possible choice for Eq. (23) is based on the assumption that the strain field varies slowly over the morphing area.Indeed, it makes sense that strong gradients should be included within the non-local zone, O2, to take them correctly intoaccount. Then, far from this region of interest, the coupling can be done and the coarse local continuum model can be usedto reduce the computational cost. We assume that the transformation is homogeneous over the neighborhood of point x:

F ðx 0ÞCF ðxÞ ¼ F 8x 0 2 HdðxÞ ð26Þ

In that case, constraints (24) and (25) are obviously equivalent. Using Eq. (23) for any homogeneous strain state naturallyleads to a general relation between the stiffness operators:

K 0¼ K ðxÞþ

ZHdðxÞ

c0ðJxJÞðaðxÞþaðpÞÞ

2

x � x � x � x

2JxJ2dVp ð27Þ

which can be rewritten as

K ðxÞ ¼ ð1�aðxÞÞK 0þ

ZHdðxÞ

c0ðJxJÞðaðxÞ�aðpÞÞ

2

x � x � x � x

2JxJ2dVp ð28Þ

Remark. Eq. (28) is an intrinsic coupling condition directly written on tensors. Provided the morphing function, a, isknown, it automatically defines the stiffness coefficients to be used for the local model at any point in the morphing area.Of course, for practical applications in a chosen orthonormal basis ðe1,e2,e3Þ, the stiffness coefficients can be derived:

KijklðxÞ ¼ ð1�aðxÞÞK0ijklþ

ZHdðxÞ

c0ðJxJÞðaðxÞ�aðpÞÞ

2

xixjxkxl

2JxJ2dVp ð29Þ

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1093

These coefficients satisfy the following symmetries (illustrated in Eq. (32)):

Kijkl ¼ Kijlk ¼ Kjikl ¼ Kklij major and minor symmetries ð30Þ

Kijkl ¼ Kikjl symmetries due to central forces ð31Þ

We have 15 different coefficients for the general situation. An interesting point is that this local stiffness matrix might notsatisfy the symmetries of the original material. Its complexity at a specific point, x, is fully related to the variation of thefunction, a, within the horizon, HdðxÞ:

½K �ðe1 ,e2 ,e3Þ¼

� 0 1 2 &

� � , | �

� � �

� � � � |

� � � � � 1

2666666664

3777777775ðe1 ,e2 ,e3Þ

ð32Þ

4. Discussion of the coupling quality for a one-dimensional problem

Under the assumption of homogeneous transformation, Eq. (28) ensures that the hybrid model is energetically equivalent toboth the purely local and the non-local models. This constraint does not guarantee that spurious effects will not appear, however.

We propose to investigate the ‘‘ghost forces’’ created by the coupling for an unidimensional case. Ghost forces are a well-known concept in atomistic-to-continuum coupling (Tadmor, 1996; Shenoy et al., 1999; Badia et al., 2008; Curtin and Miller,2003; Dobson and Luskin, 2009). They are non-physical (spurious) forces that arise due to approximations made when addressingthe local/non-local interface in the bridge region and can be evaluated by solving an inverse problem in which the strain field isprescribed to be uniform. If ghost forces exist, they will be revealed as the residual of the equilibrium equation.

4.1. Analysis of a one-dimensional problem

Let us study a one-dimensional beam of length l with a prescribed displacement, u0, at its end (Fig. 4).A non-local model is used in O2. A purely local model is used in O1 and O3, with Young’s modulus, E0, consistent with

the non-local model. Om1 and Om2 are the coupling areas where the hybrid model is defined. Different morphing functionsare investigated for the evolution of the morphing parameter, aðxÞ (Fig. 5).

In a one-dimensional problem, it is worth introducing a function, b, that represents the weighting of the local stiffnessparameter:

EðxÞ ¼ bðxÞE0ð33Þ

Then, b is obtained from the constraint on the strain energy density conservation (Eq. (28)), which is rewritten as follows:

bðxÞ ¼ 1�aðxÞþZ xþd

x�dðaðxÞ�aðpÞÞc0ðJp�xJÞ

ðp�xÞ2

4E0dp ð34Þ

Fig. 4. Study of a one-dimensional case: a beam under a prescribed displacement.

The fundamental equations for this problem are:

�

Kinematic admissibility and compatibility:

eðxÞ ¼ duðxÞ

dx8x 2 ½0,l�; uð0Þ ¼ 0; uðlÞ ¼ u0 ð35Þ

�

Static admissibility:

dsdxþ

Z xþd

x�df ðp-xÞ dpþFgðxÞ ¼ 0 8x 2 ½0,l� ð36Þ

Lc

Ωm1Ω1 Ω2

Ωm1Ω1 Ω2

Fig. 5. Possible choices for the morphing function, a: (a) linear and cubic. (b) Heaviside. a may vary over the length Lc defined by Lc ¼ b�a�2d.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021094

�

Constitutive:

sðxÞ ¼ bðxÞE0eðxÞ 8x 2 ½0,l� ð37Þ

f ðp-xÞ ¼aðpÞþaðxÞ

2c0ðJp�xJÞðuðpÞ�uðxÞÞ ð38Þ

Fg(x) represents the linear ghost-force density. To evaluate Fg(x), we prescribe that the solution be one of ahomogeneous beam under tension. The direct solution of this trivial problem is

uðxÞ ¼ xu0

l, s¼ E0 u0

l, e¼ u0

l8x 2 ½0,l� ð39Þ

4.1.1. Ghost force evaluation

Domain x 2 O1 [O3: This model is purely local and, as a consequence, aðxÞ ¼ 0 and bðxÞ ¼ 1. The local equilibriumequation is written as

d

dxðE0eÞþFgðxÞ ¼ 0 ð40Þ

As e is independent on x, FgðxÞ ¼ 0 in this domain.Domain x 2 O2: This model is purely non-local and, as a consequence, aðxÞ ¼ 1 and bðxÞ ¼ 0. The local equilibrium is

given byZ xþd

x�dc0ðJp�xJÞðuðpÞ�uðxÞÞþFgðxÞ ¼ 0 ð41Þ

Since ðuðpÞ�uðxÞÞ is an odd function and c0ðJp�xJÞ is an even function of p�x, the integral is then identically null forevery point in O2. As a result, FgðxÞ ¼ 0 in this domain.

Domain x 2 Om1 [Om2: This model is hybrid and both aðxÞ and bðxÞ vary. The equilibrium equation becomes

u0E0

l

dbðxÞdxþ

1

2

Z xþd

x�dðaðpÞþaðxÞÞc0ðJp�xJÞðuðpÞ�uðxÞÞ dpþFgðxÞ ¼ 0 ð42Þ

Introducing (34) to replace b, we getZ þd�d

aðxþxÞþaðxÞ2

c0ðJxJÞx dx�Z þd�d

1

4

d

dxðaðxþxÞþaðxÞÞc0ðJxJÞx2 dxþ

FgðxÞl

u0¼ 0 ð43Þ

If we choose c0 to be a constant and we use Eq. (8) in one dimension, we get

c0 ¼3E0

d3ð44Þ

Introducing Eq. (44) into Eq. (43) leads to the ghost-force density over the morphing zone, defined as

FgðxÞ ¼3E0u0

2ld3

1

2

Z þd�d

d

dxðaðxþxÞþaðxÞÞx2 dx�

Z þd�dðaðxþxÞþaðxÞÞx dx

( )8x 2 Om1 [Om2 ð45Þ

For comparison, we also evaluate the ghost force that would be obtained by a rough partition, such as the followingclassical choice:

bðxÞ ¼ 1�aðxÞ ð46Þ

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1095

Note that Eq. (46) is a first-order approximation of Eq. (34) as it would be equivalent to neglecting the gradient of a inEq. (34). The ghost-force density related to that choice is

F gðxÞ ¼ E0 u0

l

daðxÞdx�

3

2d3

Z þd�dðaðxþxÞþaðxÞÞx dx

( )8x 2 Om1 [Om2 ð47Þ

We show in the next paragraph that the choice defined by Eq. (46) gives rise to a higher ghost-force density compared toour choice defined by Eq. (28).

4.2. Ghost forces in the one-dimensional case

The morphing is fully defined by the choice of the function, a, and the coupling length, Lc. We propose here to evaluatethe influence of these choices on two quantities that should vanish in the case of perfect coupling: Fg(x), whichcorresponds to the ghost term in Eqs. (36) and (45), and f gðxÞ ¼

R xa FgðsÞ ds, which has the dimension of a stress and which

is directly proportional to the error of the strain solution when solving the surrogate problem. For comparison, we alsogive the quantities F g and f g , which result from the partition constraint (Eq. (46)).

We simplify the reference problem of Fig. 4 by focusing on half of the coupling area, Om1 (it is enough to represent theghost forces in the whole domain as these are symmetric over the coupling area). We also shift the origin to simplify theanalytical evaluation of the ghost forces by defining the shifted abscissa, ~x, defined by ~x ¼ x�ðaþdÞ.

When solving the analytical equation for a different function, four intervals need to be considered and defined as (seeFig. 6)

IA ¼ f ~x= ~xo�dg, IB ¼ ½�d,0�, IC ¼ ½0,d�, ID ¼ d,Lc

2

� �ð48Þ

Case of a linear function: We choose as a morphing function:

að ~xÞ ¼ 0 8 ~xr0, að ~xÞ ¼~x

Lc8 ~x 2 ½0,Lc�, að ~xÞ ¼ 1 8 ~xZLc ð49Þ

Eq. (45) then provides the exact analytical form of Fg and fg:

Fgð ~xÞ

s ¼

0 8 ~x 2 IA,

1

LcF0

~x

d

� �8 ~x 2 IB,

1

LcF0

~x

d

� �þ

1

2

� �8 ~x 2 IC ,

0 8 ~x 2 ID,

8>>>>>>>><>>>>>>>>:

f gð ~xÞ

s ¼

0 8 ~x 2 IA

dLc

f 0

~x

d

� �8 ~x 2 IB

dLc

f 0

~x

d

� �þ

1

2

~x

d

� �8 ~x 2 IC

0 8 ~x 2 ID

8>>>>>>>><>>>>>>>>:

ð50Þ

with

F0ðyÞ ¼14 ð�1�3yþ2y3Þ, f 0ðyÞ ¼

14 ð�y�3

2 y2þ12y4Þ ð51Þ

From this, we get the linear choice of a:

Fgð ~xÞ

s¼ o

1

Lc

� �,

f gð ~xÞ

s¼ o

dLc

� �ð52Þ

Case of a cubic function: We choose as a morphing function:

að ~xÞ ¼ 0 8 ~xr0, að ~xÞ ¼ �2~x

Lc

� �3

þ3~x

Lc

� �2

8 ~x 2 ½0,Lc�, að ~xÞ ¼ 1 8 ~xZLc ð53Þ

Borderof the morphing domain

Transitionpoint

IA IB IC ID

α (linear case)

∼

Fig. 6. Half a coupling zone with a shifted origin. The blue curve is aðxÞ, a linear shape function. (For interpretation of the references to color in this figure

legend, the reader is referred to the web version of this article.)

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021096

Eq. (45) then provides the exact analytical form of Fg and fg:

Fgð ~xÞ

s¼

0 8 ~x 2 IA

dL2

c

F1

~x

d

� �þd2

L3c

F2

~x

d

� �8 ~x 2 IB

dL2

c

F1

~x

d

� �þ3

~x

d

� �þd2

L3c

F2

~x

d

� ��3

~x

d

� �2 !

8 ~x 2 IC

�3

5

d2

L3c

8 ~x 2 ID

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð54Þ

f gð ~xÞ

s ¼

0 8 ~x 2 IA

d2

L2c

f 1

~x

d

� �þd3

L3c

f 2~xd

� �8 ~x 2 IB

d2

L2c

f 1

~x

d

� �þ

3

2

~x

d

� �2 !

þd3

L3c

f 2

~x

d

� ��

~x

d

� �3 !

8 ~x 2 IC

�3

5xdd3

L3c

þ3

10

d2

L2c

8 ~x 2 ID

8>>>>>>>>>>>>><>>>>>>>>>>>>>:

ð55Þ

with

F1ðyÞ ¼�32 y�9

4 y2þ34 y4, F2ðyÞ ¼�

310 þ

32 y2þ3

2 y3� 310y5 ð56Þ

f 1ðyÞ ¼3

20 �34 y2�3

4 y3þ 320 y5, f 2ðyÞ ¼�

18 �

310 yþ1

2 y3þ38 y4� 1

20y6 ð57Þ

From this, we get the cubic choice of a:

Fgð ~xÞ

s ¼ odL2

c

!,

f gð ~xÞ

s ¼ od2

L2c

!ð58Þ

Eqs. (52) and (53) allow us to draw the following conclusions:

�

the ghost-force intensity decreases while smoothening the morphing function; � the intensity of the ghost force is driven by the ratio d=Lc . Then, the morphing area has to be large enough and its

dimensions can be used to control the ghost-force level;

� the ghost forces are vanishing when d tends toward 0. This can be explained easily, as in this case, when the long-range,

non-local central force models degenerate towards a local continuum model.

Illustration of the ghost forces in1D: Exact analytical expressions have been derived. We illustrate here these expressionsin a specific configuration. We fix the parameters to arbitrary dimensions: d¼ 0:5 and Lc ¼ 3¼ 6d (Fig. 7). We compare theghost force, fg(x), and the ghost-force density, FgðsÞds (Fig. 7) for three classical choices of the morphing function(Heaviside, linear or cubic). For comparison, we also give the ghost-force density, F g , and the ghost force, f g , that resultfrom the partition constraint (Eq. (46)).

We note that the ghost forces are localized to the coupling zone and that there is no propagation of the ghost forcesoutside the coupled area for the three choices of alpha. It also appears that the maximum ghost force is always reached inthe vicinity of the points of transition (at x¼1.5 or x¼4.5 for linear and cubic and x¼3 for Heaviside). As stated by theanalytical expressions above, these results show that a higher order of alpha reduces the maximum relative ghost force bysmoothing the transitions. This is confirmed by the two-dimensional benchmark tests (see Section 5). Note that themorphing method using an energy density constraint notably improves the quality of the coupling.

5. Two-dimensional numerical examples of the morphing-based coupling method

The purpose of this section is to illustrate the accuracy and effectiveness of the morphing-based coupling technique ontwo-dimensional problems. Two cases are studied. In the first, plates submitted to either pure traction or pure shearloading are simulated. In the second, a cracked plate is subjected to traction and shear together. All results are compared tothose of a purely non-local model and those of a purely local model.

All computations are implemented under the plane strain assumption. An exponential micro-modulus, c0, is assumed tobe c0ðJxJÞ ¼ cJxJ2e�JxJ=l, where c is a scaling constant and l is a characteristic length. The material parameters are set toc ¼ 2 103 N mm�8 and l¼ 0:1 mm. Both the non-local and local models in the present two-dimensional numericalexamples are implemented using finite element analysis (FEA). Implementations of non-local continuum mechanics usingFEA have already been proposed (Macek and Silling, 2007; Chen and Gunzburger, 2011; Zingales et al., 2011). All elementsapplied in these examples are bilinear quadrilateral elements. The grid size is 0:75 mm, and the radius d of the horizon,

Fig. 7. The influence of aðxÞ (solid) and bðxÞ (dashed) ((a), (d), (g)) on the normalized dimensional ghost-force densities, ðLc=9s9ÞFg ðxÞ (solid) and

ðLc=9s9ÞF g ðxÞ (dashed) ((b), (e), (h)) and on the relative ghost force, f g=9s9 (solid) and f g=9s9 (dashed) ((c), (f), (i)) in functions of x. aðxÞ is tested with a

Heaviside shape ((a)–(c)), a linear shape ((d)–(f)) and a cubic shape ((g)–(i)). Note that the domains O1, Om1 and O2 are depicted as white, light grey and

grey zones, respectively.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1097

HdðxÞ, is 2:625 mm. The equivalent local continuum coefficients, including Young’s modulus and Poisson’s ratio, estimatedby Eq. (8) and applied in two dimensions, are E ¼ 251.3 GPa and n¼ 1

3.

5.1. Example 1: the uniform extension of a plate

This example aims at accurately displaying coupling effects with different weighting functions in two dimensions. Weconsider an homogeneous plate subjected separately to traction and shear conditions. Fig. 8 describes the dimensions ofthe plate. The traction case is shown in Fig. 8(b), with the displacement boundary condition, uy ¼ 48 mm, on the upper edgeof the plate. The middle point of the lower edge is completely fixed and other points of this edge are fixed only in the y

direction. In Fig. 8(c), the plate is under pure shear deformation, with the displacement boundary conditions set to

ux ¼ 0:25y

uy ¼ 0:25x

(

on all edges of the plate.Fig. 9 shows the relative error of the strain components, eyy, obtained by different morphing functions, including

Heaviside, linear and cubic functions. The relative error is defined as ðeyy�eAyyÞ=eA

yy, where eAyy is the average value of the

strain components, eyy. We can see that the perturbations occur near the upper and lower parts of the morphing domain,while all the absolute values are very small (less than 0.5%). The maximum relative error decreases as the weightingfunctions become smoother. The relative error is smallest for the cubic weighting function.

The relative errors in eyy along the longitudinal center line of the plate, marked with the dashed lines in Fig. 8(b)and (c), are shown in Fig. 10 to give us a better understanding of the effects of different weighting functions. As seenin this figure, the Heaviside weighting function results in violent shakes in the middle of the morphing domain

Fig. 8. Dimensions and boundary conditions of a plate (unit of length: mm).

Fig. 9. The relative error on the strain component, eyy , for a plate under traction with different weighting functions: (a) Heaviside, (b) linear and (c) cubic.

Fig. 10. The relative error on strain component eyy along the longitudinal center line of a plate under traction with different weighting functions (unit of

length: mm).

Fig. 11. The relative error on strain component exy for a plate under shear with different weighting functions: (a) Heaviside, (b) linear and (c) cubic.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021098

where the Heaviside function is discontinuous. The linear weighting function has only two peaks on both boun-daries of the morphing domain. The cubic weighting function follows a linear change over the whole morphingdomain. Consequently, the perturbations over the morphing area are dependent on the smoothness of the weightingfunction.

The cases of pure shear are displayed in Fig. 11. The relative error on exy has similar characteristics to that of the tractionexamples in Fig. 9. Additionally, Fig. 11 shows obvious symmetry with respect to the x and y coordinates. We thereforeshow only the error values on the longitudinal center line of the plate in Fig. 12. The shapes of the polylines are the same asthose in Fig. 10, but the peak values are less than those in Fig. 10.

−3 3 9 15 21 27 33 39 45 51 57 63 69 75 81 87 93 99−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

y

Rel

ativ

e er

ror i

n ε

(%)

Fig. 12. The relative error on strain component exy along the longitudinal centerline of the plate under shear with different weighting functions (unit of

length: mm).

Fig. 13. Dimensions and boundary conditions of a cracked plate (unit of length: mm).

εxx0.050

-0.05-0.1-0.15-0.2-0.25-0.3-0.35-0.4-0.45

Fig. 14. The strain component, exx , calculated by (a) the fully local model, (b) the morphing-based coupling model, and (c) the fully non-local model.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1099

5.2. Example 2: a cracked plate

The previous examples illustrate the quality of the proposed technique on benchmark-like trivial solutions. We nowpresent an application example. Let us consider the cracked plate of Fig. 13. It is subjected to both traction and shearconditions on the upper side of the plate, whereas the lower side of the plate is fixed. The non-local continuum model isused in the vicinity of the crack in O2, while the local continuum model is employed in O1. Om, shown as the light greyarea, is the morphing domain between both models.

The strain distributions over the plate are calculated by the fully local continuum model, the morphing-based couplingmodel and the fully non-local continuum model. The contour plots of the strain component, exx, are displayed in Fig. 14. Thestrain result in Fig. 14(a) is obtained by the fully local model. Its crack shape is totally different from the others due to thelack of non-local interactions. Fig. 14(b) shows the solutions of the morphing-based coupling model using the cubicweighting function. It agrees exactly with the strain field in Fig. 14(c) from the fully non-local solution. At the same time, theprecise value curves of the strain component, exx, along the crack line behind the crack tip (at the left endpoints of the curves)are plotted in Fig. 15 in which the morphing-based coupling model and the fully non-local model are compared. They are

15 21 27 33 39 45 51 57 63 69 75 81 87 93 99−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

x

ε

Fully peridynamic modelMorphing−based coupling model

Fig. 15. The strain component, exx, obtained by the fully non-local model and the morphing-based coupling model along the crack line (unit of length: mm).

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021100

very close to each other except for a sudden drop at the right end of the fully non-local solution. The reason is that pointsnear the boundary in the non-local model fail to hold a complete horizon, but this is not related to the coupling technique.

6. Conclusion

We have introduced here a simple and systematic coupling strategy between non-local and local continua. A key pointof the approach is that a single unified model is defined over the whole structure. The coupling is naturally defined as agradient of the material parameters such that the relative weight of the local/non-local interaction can evolve within thestructure. This ensures a very simple implementation of the approach. Coupling multiple non-local areas within a singlelocal continuum remains very simple as this is controlled only by the morphing functions.

Acknowledgments

Funding for this research was provided by KAUST baseline funding and by the BOEING company (agreement ‘‘Analysismethods for multifunctional composite materials in the presence of electromagnetic effect’’). The authors are grateful toKAUST and the BOEING company for their financial support.

Appendix A. Notation for a local continuum

We introduce our notation for the modeling of the equivalent local continuum. Adjacent material points only interactthrough contact forces based on the Cauchy stress tensor, s . The infinitesimal force, dF , applied to an orientedinfinitesimal surface, dS, with a unit normal vector, n, is written as follows: dF ¼ 9dS9s � n.

Let us consider a continuous bounded regular domain, O (see Fig. A1). Its boundary, @O, is divided into two parts: Su

over which the displacements are prescribed to be equal to u , and ST over which surface tractions are prescribed to be

equal to T . We assume that Su [ ST ¼ @O and Su \ ST ¼ |. b is a field of body forces. The material is linear elastic, with K ðxÞ

being the 4th-order stiffness tensor evaluated at point x. This tensor would follow the following equation to guaranteeequivalence with the long-range central force model introduced (see Eq. (8)):

K ðxÞ ¼

ZHdðxÞ

cðJxJÞx � x � x � x

2JxJ2dVp ðA:1Þ

Solving the mechanical problem (i.e., finding the displacement, u, the stress, s , and the strain, e ) requires three sets ofequations that are written as follows for infinitesimal strains:

�

Kinematic admissibility and compatibility:

e ¼ 12ðr � uþ � uÞ 8x 2 O, u ¼ u , 8x 2 Su ðA:2Þ

�

Static admissibility:

div sþb ¼ 0, 8x 2 O; s � n ¼ T , 8x 2 ST ðA:3Þ

Fig. A1. The local continuum model. Domain O and notation.

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–1102 1101

�

Constitutive:

s ¼ K ðxÞ : e 8x 2 O ðA:4Þ

At a point x, the strain energy density is classically obtained as

WlðxÞ ¼ 1

2 s : e ¼ 12e : K ðxÞ : e ðA:5Þ

Appendix B. Equivalence between models for isotropic behavior

We propose to define the equivalence between local and non-local models for an isotropic case. For that purpose, it isinteresting to provide an expression that takes advantage of the invariants of the strain tensor. By introducing ðeI,eII,eIIIÞ asthe principal values of e associated with the principal directions ðU I,U II,U IIIÞ, the spectral decomposition of e becomes

e ¼ eIU I � U IþeIIU II � U IIþeIIIU III � U III ðB:1Þ

We also define the projection of the bond vector, x, on the eigenbasis ðU I,U II,U IIIÞ and the related polar coordinatesystem ðr,y,fÞ such that

xI ¼U I � x ¼ reI with eI ¼ cosðfÞcosðyÞ ðB:2Þ

xII ¼U II � x ¼ reII with eII ¼ cosðfÞsinðyÞ ðB:3Þ

xIII ¼U III � x ¼ reIII with eIII ¼ sinðfÞ ðB:4Þ

From Eqs. (B.2)–(B.4) and (8), we obtain the strain energy density of the non-local model (Einstein summation isassumed):

WnlðxÞ ¼ eiej

ZHdðxÞ

cðJxJÞx2

i x2j

4JxJ2dVp ¼ eiej

Z d

0cðrÞ

r4

4dr

! Z 2p

0

Z p=2

�p=2e2

i e2j cosðfÞ dy df

!ðB:5Þ

The polar integration largely simplifies the result:Z 2p

0

Z p=2

�p=2e2

i e2j cosðfÞ dy df¼

4p5

if i¼ j ðB:6Þ

Z 2p

0

Z p=2

�p=2e2

i e2j cosðfÞ dy df¼

4p15

if iaj ðB:7Þ

Thus, we have

WnlðxÞ ¼

p15

Z d

0cðrÞr4 dr

!ð3ðe2

I þe2IIþe

2IIIÞþ2ðeIeIIþeIIeIIIþeIIIeIÞÞ ðB:8Þ

We now consider the two first invariants of e defined by

I1 ¼ e : 1 ¼ eIþeIIþeIII ðB:9Þ

I2 ¼12 e : e ¼

12ðe

2I þe

2IIþe

2IIIÞ ðB:10Þ

G. Lubineau et al. / J. Mech. Phys. Solids 60 (2012) 1088–11021102

Then, the strain energy density of the non-local model can be rewritten as

WnlðxÞ ¼

p15

Z d

0cðrÞr4 dr

!ðI2

1þ4I2Þ ðB:11Þ

Eq. (B.11) is convenient as it clearly states that the resulting energy density is isotropic depending only on the first andsecond invariants of the strains.

To define the equivalent isotropic local model, we suppose that l and m are the Lame parameters of the equivalentisotropic material in the local model. Then, the strain energy density can be written as

WlðxÞ ¼ 1

2ðlI21þ4mI2Þ ðB:12Þ

Eqs. (B.11) and (B.12) can be made compatible for any homogeneous strain state if and only if l¼ m. We rediscover here aclassical result: a central force long-range model, in which the Cauchy–Born rule is assumed, can only represent anisotropic material with a Poisson’s ratio equal to 1

4 in three dimensions (we can prove that, in two dimensions, Poisson’sratio has to be equal to 1

3). Moreover, by making Eqs. (B.11) and (B.12) equal, we have the following relation:

l¼ m¼ 2p15

Z d

0cðrÞr4 dr

!ðB:13Þ

As is well known, the equivalent continuum is characterized by a single constitutive parameter: L¼ l¼ m.

References

Badia, S., Bochev, P., Fish, J., Gunzburger, M., Lehoucq, R., Nuggehally, M., Parks, M., 2007. A force-based blending model for atomistic-to continuumcoupling. Int. J. Multiscale Comput. Eng. 5, 387–406.

Badia, S., Parks, M., Bochev, P., Gunzburger, M., Lehoucq, R., 2008. On atomistic-to-continuum coupling by blending. Multiscale Modeling Simul. 7,381–406.

Belytschko, T., Xiao, S., 2003. Coupling methods for continuum model with molecular models. Int. J. Multiscale Comput. Eng. 1, 115–126.BenDhia, H., Rateau, G., 2005. The Arlequin method as a flexible engineering design tool. Int. J. Numer. Methods Eng. 62, 1442–1462.Chamoin, L., Prudhomme, S., BenDhia, H., Oden, T., 2010. Ghost forces and spurious effects in atomic-to-continuum coupling methods by the Arlequin

approach. Int. J. Numer. Methods Eng. 83, 1081–1113.Chen, X., Gunzburger, M., 2011. Continuous and discontinuous finite element methods for a peridynamics model of mechanics. Comput. Methods Appl.

Mech. Eng. 200, 1237–1250.Curtin, W., Miller, R., 2003. Atomistic/continuum coupling methods in multi-scale materials modeling. Modeling Simul. Mater. Sci. Eng. 11, R33–R68.DiPaola, M., Failla, G., Zingales, M., 2009. Physically-based approach to the mechanics of strong non-local linear elasticity theory. J. Elasticity 97, 103–130.DiPaola, M., Failla, G., Zingales, M., 2010. The mechanically-based approach to 3d non-local linear elasticity theory. Int. J. Solids Struct. 47, 2347–2358.Dobson, M., Luskin, M., 2009. Analysis of the effect of ghost force oscillation on the quasicontinuum error. Math. Modeling Numer. Anal. 43, 591–604.Du, J., Bai, J., Cheng, H., 2007. The present status and key problems of carbon nanotube based polymers. Express Polym. Lett. 1, 253–273.Eringen, A.C., 2001. Nonlocal Continuum Field Theories. Springer-Verlag, New York, Berlin, Heidelberg.Fish, J., Nuggehally, M., Shepard, M., Picu, C., Badia, S., Park, M., Gunzbunger, M., 2007. Concurrent ATC coupling based on a blend of the continuum stress

and the atomistic force. Comput. Methods Appl. Mech. Eng. 196, 4548–4560.Han, F., Lubineau, G., 2012. Coupling of non-local and local continuum models by the Arlequin approach Int. J. Numer. Methods Eng. 89 (6), 671–685.Kroner, E., 1967. Elasticity theory of materials with long range cohesive forces. Int. J. Solids Struct. 3, 731–742.Macek, R., Silling, S., 2007. Peridynamics via finite elements analysis. Finite Elem. Anal. Des. 43, 1169–1178.Rapaport, D., 2004. The Art of Molecular Dynamics Simulation. Cambridge University Press.Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R., Ortiz, M., 1999. An adaptive methodology for atomic scale mechanics: the quasicontinuum

method. J. Mech. Phys. Solids 47, 611–642.Silling, S., 2000. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209.Silling, S., 2010. Linearized theory of peridynamic states. J. Elasticity 99, 85–111.Silling, S., Epton, M., Weckner, O., Xu, J., Askari, E., 2007. Peridynamic states and constitutive modeling. J. Elasticity 88, 151–184.Silling, S., Lehoucq, R., 2010. Peridynamic theory of solid mechanics. Adv. Appl. Mech. 44, 73–168.Tadmor, E., 1996. The Quasicontinuum Method. Ph.D. Thesis, Brown University.Warren, T., Silling, S., Askari, A., Weckner, O., Epton, M., Xu, J., 2009. A non-ordinary state-based peridynamic method to model solid material deformation

and fracture. Int. J. Solids Struct. 46 (5), 1186–1195.Zingales, M., DiPaola, M., Inzerillo, G., 2011. The finite element method for the mechanically based model of non-local continuum. Int. J. Numer. Methods

Eng. 86, 1558–1576.