Multiple scale modeling for cortical bonefracture in tension using X-FEM

Élisa Budyn∗,1 – and Thierry Hoc ∗∗

∗ Department of Mechanical and Industrial Engineering, University of Illinois atChicago, 842 West Taylor Street, Chicago IL. 60607, USA.

∗∗ Department of Material Science, LMSSMat, UMR 8579, Ecole CentraleParis, Grande Voie des Vignes, Chatenay Malabry, F-92295.

ABSTRACT. We present a multiple scale method for modeling multiple crack growth in corti-cal bone under tension. The four phase composite Haversian microstructure is discretized bya Finite Element Method. The geometrical and mechanical bone parameters obtained by ex-periments mimic the heterogeneity of bone at the micro scale. The cracks are initiated at themicro scale where a critical elastic-damage strain driven criterion is met and are grown untilcomplete failure in heterogeneous linear elastic media when a critical stress intensity factorcriterion is reached. The cracks are modeled by the eXtendedFinite Element Method. The sim-ulations provide the global response at the macroscopic level and stress and strain fields at themicroscopic level. The model emphasizes the importance of the microstructure on bone failurein assessing the fracture risk.

RÉSUMÉ.Nous proposons une méthode multi-échelle pour modéliser lapropagation de fissuresmultiples dans l’os cortical sous traction. La microstructure Haversienne composite à quatrephases est discrétisée par une méthode éléments finis. Les paramètres du tissu osseux géomé-triques et mécaniques obtenus par l’expérience décrivent l’hétérogénéité à l’échelle microsco-pique. Les fissures sont initiées à l’échelle micro où un critère élastic-endommagement piloté endéformation est atteint et sont propagées jusqu’à rupture complète dans des milieux linéairesélastique hétérogènes. Les fissures sont modélisées par la méthode des éléments finis enrichis(X-FEM). Les simulations montrent la réponse globale à l’échelle macroscopique et les champsde contraintes et de déformations à l’échelle microscopique. Le modèle montre l’importance dela microstructure dans le mécanisme de rupture en vue de déterminer un risque fracturaire.

KEYWORDS:finite element, X-FEM, cortical bone, fracture, composite,multi-scale method.

MOTS-CLÉS :element finis, X-FEM, os cortical, rupture, composite, méthode multi-échelle.

REMN – 16/2007, X-FEM, pages 215 à 238

216 REMN – 16/2007, X-FEM

1. Introduction

We present a multiple scale method for modeling multiple crack growth in corticalbone under tension. The model is decomposed into two parts: the modeling of ran-dom Haversian bone microstructures as a multi phase composite and the modeling ofmultiple crack growth inside this heterogeneous microstructure.

Haversian cortical bone microstructure has been extensively studied as a factor in-fluencing the mechanical behavior of the failure process of bone (Katz, 1976; Aoubizaet al., 1996; Gottesmanet al., 1980; Hogan, 1992). At the micro scale, cortical boneis composed of densely packed concentric lamellar structures (osteons) that are em-bedded in an interstitial matrix (Figure 1). The cement lineis an interface between theosteon and the interstitial matrix. Bone porosity is principally due to Haversian canals.Moreover, bone is a living tissue that can adapt both its structure and its architecture toits mechanical environment (Currey, 2003). This process produces a heterogeneousmaterial with a high variability in osteon bone-mineral density that is related to thelevel of maturation of the newly-formed tissue.

A mechanistic understanding of the role of this complex microstructure is stronglyrelevant to predicting the risk of fracture associated withage and disease. From astatistical point of view, several studies have demonstrated that cortical bone fracturetoughness can be related to osteon density and porosity (Wright et al., 1977; Huietal., 1988; Schaffleret al., 1988; O.Brienet al., 2005). At the microstructural scale,considerable work has been developed to study the initiation and growth of micro-cracks that have been shown to act as a stimulus for bone remodeling. In in vivoconditions, cement lines act as a microstructural barrier for the majority of cracks. Incontrast, in transverse loading the underlying microstructure does not have any signif-icant influence on the crack propagation (Nallaet al., 2003).

Due to the complex microstructural hierarchy, the mechanical behavior and thefracture mechanisms have been very difficult to predict. Haversian cortical bone hasoften been compared to a composite where discontinuities within the material maygenerate mechanical stress concentration sites for crack initiation. Several modelsusing homogenization theory have been developed to predictmacroscopic behav-ior (Gottesmanet al., 1980; Katz, 1971; Aoubizaet al., 1996), but they do not resolvethe spatial distribution of local strain. Donget al. (2005) developed a self-consistentmodel for multi-coated cylinders to obtain more precise bounds for cortical bone.Some attempts to model larger unit cells have been developedby Prendergastet al.(1996), however the geometrical and mechanical local anisotropy were not included.The objective of this work is to develop a model for Haversiancortical bone basedon a multiscale approach in order to study the influence of themicrostructure featureson the strain localization pattern, which is strongly relevant in the mechanism of bonefracture.

To solve this problem the eXtended Finite Element Method (X-FEM) which is anumerical method for treating arbitrary discontinuities without remeshing (Belytschkoet al., 1999; Belytschkoet al., 2001) is an efficient tool. In this method, cracks are

Multiple scale modeling for cortical bone 217

initiated at the micro scale at osteonal locations where a critical elastoplastic straindriven criterion is met. The cracks can therefore be randomly located within the mi-crostructure. In the present paper, the Haversian microstructure that is a four phasecomposite: Haversian canal, osteons, cement line and matrix, is discretized by a fi-nite element method. The geometrical and mechanical bone parameters obtained bynanoindentation measurements are randomly distributed tomimic the nature of boneat the micro scale.

The discontinuities are represented by enriching the standard finite element ap-proximation space. For cracks in the framework of linear elastic fracture mechanics,suitable enrichments are the Heaviside function for the crack and the Westergaardfield for the crack tip. X-FEM is an application of the partition of unity (Melenketal., 1996) and was introduced in (Belytschkoet al., 1999; Dolbowet al., 2001; Moëset al., 1999) and later adapted for multiple cracks in Budynet al. (2004). The cracktopology is represented by vector level sets, a method that was developed by Osheretal. (1988) and Sethian (1999) for problems of interface tracking and later improvedby Burchardet al. (2001) and Osheret al. (2002) for the evolution of curves. Herewe use a narrow banded vector level set method developed by Venturaet al. (2002)and Venturaet al. (2003), because this method simplifies the process of freezing theexisting level sets for cracks (Belytschkoet al., 2001).

The cracks are then grown in heterogeneous linear elastic media when a criticalstress intensity factor criterion is met. The cracks with the maximum stress intensityfactors are grown so that they remain approximately at the critical stress intensity fac-tor by adjusting the load parameter and solving for the corresponding displacement.The solution method follows a "crack length control" algorithm. In the case of com-peting crack tips, a stability analysis is performed to determine the crack configurationpath that leads to the maximum decrease in the potential energy. Stress intensity fac-tors are computed by means of an interaction integral (Moranet al., 1987).

The outline of this paper is as follows. In a first part, the description of the FiniteElement Method to create the random Haversian bone microstructure is given (Sec-tion 2). The eXtended Finite Element Method for initiation and multiple crack growthproblems until coalescence of cracks and percolation is presented (Section 3.3). Sec-tion 4 presents the results and compared with experimental results. Conclusions aregiven in Section 5.

2. Cortical bone microstructure

We consider three dimensional unit cells of cortical bone of0.5 millimeter widthand 0.1 millimeter height. Cortical bone microstructures are represented by a fourphase composite composed by an interstitial matrix and osteonal fibers (hollow discsin two dimensions, hollow tubes in three dimensions) coatedby viscous cement linesand hollowed by Haversian canals considered as free boundaries where normally cy-toplasmic fluid flows freely, see Figure1b.

218 REMN – 16/2007, X-FEM

(a)

ο

l l

l

12

3l 4Γcr

Γu

Γcr

Γcr

Γcr21

43

Ω

A Bx

y

Γt

t = tλ

(b)

Figure 1. (a) Light microscope bone observation. (b) Schematic modelof the corticalunit cell

2.1. Geometrical description

All geometrical parameters are experimentally measured from light microscopepictures of bovine bone samples as in Figure 1a. In our measurements the volumetricfraction of the interstitial matrix is about 34 percent and the average width of thecement lines is close to 5µm. The distributions of the diameters of the osteons andthe Haversian canals are statistically measured from the microscopic observations andregularized by a Gaussian fit (see Figure 2) denotedHo for the osteons andHc for theHaversian canals. The average diameters are 35µm for the Haversian canals and 140µm for the osteons.

The random microstructure is constructed using a hard sphere scheme for whichdetails can be found in Torquato (2000). In this scheme each osteoni is representedin two dimensions as a disc and defined by a positionPi(x, y) and a diameterDi.We assume the osteons are impenetrable. The construction process can be outlinedas follows: the positionPi for one osteon is randomly chosen inside the unit cell. AdiameterDi is randomly chosen within theHo distribution. If the osteon volume isincluded in the unit cell without overlapping another osteon, it is accepted; otherwise,it is rejected. The procedure is repeated until the surface fraction of osteon coversmore than 65 percent. Finally, a diameter for the Haversian canal for each osteon israndomly chosen within theHc distribution in Figure 2. We check that the obtainedporosity is between 5 to 9 percent according to our microscopic observations. Thealgorithm to construct the microstructure is programed in Fortran 90 and then en-

Multiple scale modeling for cortical bone 219

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

100

diameter (µ m)

perc

ent

experiment osteonregularization osteonexperiment Haversregularization Havers

Figure 2. Experimental and regularized distribution of the size of the osteon diametersand the Haversian canal diameters

coded in Python as a script file. This geometry is first discretized using a commercialthree dimensional FEM code (ABAQUS, 2004) and linear tetrahedron elements fortheir efficiency and adaptability to complicated geometries. A preliminary local strainfield analysis is performed to determined the locations where the cracks are initiat-ing. The same geometry is then discretized in a two dimensional X-FEM code usingMaltab (MATLAB7, 2006) and Gmsh (GMSH, 2006) and quadratic triangles. Thelongitudinal axis of the osteons is parallel to the z-direction.

2.2. Mechanical properties

In this study the material properties are assigned in each phase based on experi-mental observations. The four phases of the interstitial matrix, the Haversian canals,the cement lines and the osteons can be described by four different constitutive laws.However, the bone remodeling process produces a heterogeneous material with a highvariability in osteon bone-mineral densities and geometries.Therefore, the local me-chanical properties of the osteons and the cement lines willbe different in each osteon.

The osteons are chosen to be transversally isotropic in the three dimensional FEMAbaqus model used for preliminary strain field analysis and isotropic in the twodimensional XFEM Matlab/Gmsh final model. Hocet al. (2006) have performednanoindentation measurements of the local Young’s modulusin wet conditions forover a hundred osteons in the longitudinal osteonal axis (Fan et al., 2002). It isshown that the local Young’s modulus and the bone-mineral content is reasonablycorrelated (r2 =0.75).We assumed the Young’s moduli to be directly relatedto themineral content in our model. Humid nanoindentation measurement was also per-

220 REMN – 16/2007, X-FEM

5 10 15 20 250

10

20

30

40

50

60

70

80

90

100

E (GPa)

perc

ent

ET exp.

ET regul.

EL exp.

EL regul.

Figure 3. Experimental and regularized distributions of the Young’smoduli of theosteons measured by nanoindentation in the transversal plane and along the axis ofthe osteons

formed in the transversal direction. The obtained experimental distributions of theYoung’s moduli were regularized by a Gaussian fit (see Figure3) denotedHT in thetransversal direction andHL in the longitudinal direction. Gaussian distributions forlocal longitudinal and transversal Poisson’s ratioHνL

andHνTare constructed from

micro-extensometry measurements on millimetric bone samples. The transverse shearmodulusGT is described by a Gaussian distributionHG of mean value and standarddeviation taken from Reillyet al. (1975). The attribution of the material propertiesfor each osteoni can be outlined as follows: a probabilitypi is randomly chosen towhich corresponds a longitudinal Young’s modulus withinHL distribution, a transver-sal Young’s modulus withinHT and the shear modulus withinHG. For the Poisson’sratio, the probability is fixed to1− pi in order to obtain the longitudinal and transver-sal Poisson’s ratioνi

L andνiT . In our model, we imposed a larger Poisson’s ratio to

osteons with lower Young’s moduli considering that less mineralized osteons containmore collagen (Katzet al., 1971). The critical stress intensity factors of the cementlines are linearly correlated to its Young’s modulus by a factor 10−4 to fit in the rangeof 0.7 to 2 MPa (Sheltonet al., 2003; Hazenberget al., 2006). Table 1 summarizesthe material properties of the osteons.

Lakes has shown cement lines present a specific chemical composition which givesan isotropic viscoelastic behavior (Lakeset al., 1979). This composition confers dif-ferent elastic properties compared to the osteons that are encircled by the cement lines.The Young’s modulus is taken 25 percent lower than the Young’s modulus of the os-teon it encircles. The Poisson’s ratio is chosen to be 0.49 (Sabelmanet al., 1997Jul 20-23). Although only the elastic properties of the cement lines are used in thisstudy, The critical stress intensity factors of the cement lines are linearly correlated

Multiple scale modeling for cortical bone 221

Table 1. Microscopic elastic properties

Osteons ET (GPa) EL (GPa) νT νL GT (GPa)Mean value 11.51 20.71 0.17 0.18 3.57

Standard deviation 1.84 3.10 0.06 0.06 0.25Extremum min 7.79 14.33 0 0 2.82Extremum max 15.55 24.71 0.343 0.353 4.3

Matrix 12.66 22.78 0.153 0.162 3.93

to its Young’s modulus by a factor10−4 and increased by 40 percent. The Haversiancanals are modeled by free boundaries and assume no viscous contact between thecytoplasmic fluid and the canal walls.

The interstitial matrix is considered homogeneous and transversally isotropic withYoung’s moduli about 10 percent stronger than the mean values of the osteonal mod-uli. This assumption is consistent with nanoindentation tests performed by Rhoet al.(2002) that reveal higher Young’s moduli in the matrix than in the osteons. The Pois-son’s ratio are chosen 10 percent lower than the mean value ofthe osteonal Poisson’sratio.

3. Multiple crack growth by X-FEM

3.1. Elastic-damage strain driven criterion to initiate cracks

A linear elastic analysis using a FEM discretization with a commercial codeAbaqus is applied to cortical bone microstructures. The transversal Young modu-lus E2 was determined for 4 different microstructure sizes (0.2, 0.3, 0.4 and0.5 mm)and was found to be constant (11.5 GPa) for microstructures of size greater than0.3 mm. Therefore we choose a microstructure of 0.5 mm height and width and 0.1mm thickness that is statistically significant . As many biological materials, failure inbone can be described by an elastic-damage strain driven criterion. In this study wedetermine the initial crack locations in the microstructure by using the critical maxi-mum principal strain. When a critical value of 0.4% (Pattinet al., 1996; Bayraktaret al., 2004) has been reached in the elastic calculation, cracks are initialized for theXFEM calculation.

Tension tests are performed on the microstructure (0.5 mm) given in Figure 4 inthe y-direction to determine regions where the critical yield strain is reached. Notethat the strain localization zone direction depends on the loading direction (Budynetal., 2005). The FEM analysis shows that cracks can be initiated perpendicularly tothe direction of maximum principal stress, mode I in tension. Under vertical tensionloading, initial cracks are placed horizontally starting from the edge of Haversian

222 REMN – 16/2007, X-FEM

Figure 4. 0.5 mm unit cell under0.3% strain tension on the top face where the localregion over0.4% are in light grey near the Haversian canals

canals. Note that we progressively increases the displacement at the top of the faceuntil local zones with a strain above0.4% appear; Figure 4 shows the local maximumprincipal strain fields when the top face of the bone cell is submitted to0.3%.

3.2. Description of the problem

For our X-FEM multiple crack growth problem, we consider a two dimensionalelastic unit cellΩ with boundaryΓ, nc cracks with surfacesΓcr = Γα

cr, α = 1 tonc, andnt crack tips as shown in Figure 1b. The normal to a surface is denoted byn. The cracks are assumed to be traction free. Prescribed tractionst are imposed onthe boundaryΓt and prescribed displacements are imposed onΓu. According to 3.1for the boundary conditions we can prescribe a uniform traction along the top edgeof the cell and fix the displacement in the y-direction along the bottom edge and thedisplacement in the x-direction of node A in Figure 1b. The implementation is limitedto linear elastic fracture mechanics innp multi phase unit cells. The equilibriumequation and boundary conditions are given by:

∇ · σ = 0 in Ω [1]

σ · n = t on Γt [2]

u = u on Γu [3]

whereσ is the stress,u(x) is the displacement field,uT = ux, uy; u is theprescribed displacement onΓu.

Multiple scale modeling for cortical bone 223

Each phase of material is considered elastic governed byC the tensor of elasticmoduli, and in small deformation. The length of each crack isdenotedℓi; the set ofcrack lengths is represented by a matrixℓ = ℓi, i = 1 to nt. The imposed tractiont depends linearly on a scalar parameter called the load factor λ: t = λto, whereto

is a reference traction field.

For a preliminary study of multiple crack growth in corticalbone modeling, thecrack propagation is described by linear elastic fracture mechanics for brittle multiphase body by the following Lagrangian form Budynet al. (2004):

L(ℓ,u) = W(ℓ,u) +

nt∑

i=1

∫

Γicr

Gic dℓi [4]

whereW(ℓ,u) is the potential energy of the system,Gic is the critical energy release

rate at crack tipi. Gic is a constant parameter characteristic of each phase and a

function of (x, y). If we definenact as the number of active crack tips, the secondterm in the right hand side of Equation [4] is the energy dissipated during the growthof thenact active crack tips. The potential energyW(ℓ,u) can be decomposed intothe strain energyWint and the load potentialWext:

W(ℓ,u) = Wint(ℓ,u) −Wext(u) [5]

where

Wint(ℓ,u) =1

2

∫

Ω\Γcr

ǫ(ℓ,u) : C : ǫ(ℓ,u) dΩ [6]

Wext(u) = λ

∫

Γt

to · u dΓ [7]

The equilibrium states of the bodyΩ correspond to the stationary points of [4] orpoints on the boundary of the feasible domain, so the solution u ∈ U corresponds to:

δL = δuW(ℓ,u)δu +

nt∑

i=1

[

∂W(ℓ,u)

∂ℓi

+ Gic

]

δℓi > 0, ∀δu ∈ Uo, ∀δℓi > 0

[8]whereδuW(ℓ,u) is the variation ofW with respect tou, δu is the variation of thedisplacement and dℓi is a crack length differential and:

U = uu ∈ H1 (Ω \ Γcr) , [[u · n]] > 0 onΓαcr,u = u onΓu [9]

Uo = U ∩ uu = 0 onΓu [10]

wheren is the normal to the crack andH1 is the Hilbert space of functions withsquare integrable derivatives. Note that the displacementfield is discontinuousacross the crack and that the jump[[]] in the normal displacement is required to benon-negative.

224 REMN – 16/2007, X-FEM

The above gives:δuW(ℓ,u) = 0 [11]

∂W(ℓ,u)

∂ℓi

+ Gic = −Gi + Gi

c > 0, ∀i ∈ 1, .., nt [12]

where the energy release rate is defined as:

Gi = −∂W(ℓ,u)

∂ℓi

[13]

and [11] is the equilibrium Equation and [12] is the Griffith criterion for each crack tipi. As it is often common to express the crack growth law in termsof stress intensityfactors, the energy release rate at tipi can be rewrittenKi =

√E′Gi whereE′ is the

effective Young’s modulus. At each tipi, we compute the equivalent stress intensityfactor Ki

eq (Pariset al., 1965) expressed as the norm of the contribution in mode Iand mode II of the interaction integrals (Yauet al., 1980; Belytschkoet al., 1999) as asurface integral (Budynet al., 2004):

Kieq =

√

(

KiI

)2+(

KiII

)2[14]

The crack growth law in linear elastic fracture mechanics ofEquation [12] can bewritten:

if 0 < Kieq < Ki

c ∆ℓi = 0 ( no growth )if Ki

eq = Kic ∆ℓi ≥ 0 ( growth )

[15]

where∆ℓi is the crack growth increment of tipi. We do not consider any closure ofthe cracks, though we note that∆ℓi < 0 is not a valid solution. If the crack closes, theinequality[[u · n]] > 0 must be enforced. The cracks are grown in the direction of themaximum hoop stress,θi:

θi = 2 arctan

(

1

4

(

KiI/Ki

II ±√

(

KiI/Ki

II

)2+ 8

))

[16]

The above gives two directions; we choose the angle that corresponds to the posi-tive maximum hoop stress. The stress intensity factorsKi

I andKiII are computed by

an interaction integral. The equivalent stress intensity factor is computed by (14). Theexpressions for theJ-integral and the interaction integral as domain integralscan befound in Moranet al. (1987).

An explicit algorithm is used to satisfy both Equations [11]and [12]. The Equi-librium [11] is discretized and solved for a prescribed traction to and then the loadparameterλ is adjusted to satisfy [12], so that the crack with the maximum stressintensity factor met its critical value. The crack growth increments are set at the be-ginning of each step based on a “crack length control” schemeso that the evolutionis controlled by a monotonically increasing function of thetotal crack length∆ℓtot,i.e. the sum of all active crack lengths (Boccaet al., 1991). In some cases, severalcracks are close to their critical stress intensity factors, the number ofnact active

Multiple scale modeling for cortical bone 225

crack to grow is determined by stability analysis and a set ofcompetitive crack tipsNcomp is determined as in Budynet al. (2004). A stability analysis is conducted byapplying a second variation of the Lagrangian form [4] to determine the most stablecrack configuration evolution corresponding to that with the minimum energy dissi-pation (Budynet al., 2004). The derivatives of the energy release rate with respect tothe crack length (Nguyenet al., 1985; Nguyenet al., 1990), are computed by the gen-eralized X-FEM formulation (Budynet al., 2004) of the FEM formulation developedby Suoet al. (1989), Suoet al. (1992) and Suoet al. (1998), All subdeterminants ofthis matrix[∂Gi/∂ℓj] are computed at timetn−1 and the maximum subdeterminantgives the set of tipsNact that will grow at time steptn determined by:

Nact =

i ∈ Ncomp maxi,j∈Ncomp

det

(

−∂2L(ℓ,u)

∂ℓi∂ℓj

)

= maxi,j∈Ncomp

det

(

∂ (Gi(σo))

∂ℓj

)

≥ 0

[17]

The variation ofGic in Equation [17] vanishes as we considered strong discon-

tinuities in bone material properties andGic constant in each material phase. Note

that when multiple cracks grow at the same time, the Griffith criterion [12] is exactlysatisfied at the crack with the maximum stress intensity factor met its value, and ap-proximately satisfied at the other cracks.

3.3. X-FEM discretization

The eXtended Finite Element Method (X-FEM) is applied to approximate the dis-placement field and incorporate the crack representation with a step function enrich-ment for the discontinuity of the interior of a crack and the asymptotic near-tip dis-placement field enrichment at the tips of the cracks (Belytschkoet al., 1999; Moësetal., 1999; Budynet al., 2004). Figure 5 displays the enrichment scheme. In a meshwith a set of nodesI, all corner nodes of elements crossed by a crack will be enriched.J n contains the set of corner nodes of the elements cut by the crack n enriched bythe step function of crackn (circled nodes in Figure 5).Km contains the set of cornernodes of the element that contains the crack tipm enriched by the branch function(squared nodes in Figure 5).Nc is the set of cracks andNt the set of crack tipsin the entire model. The crack geometry of crackn is described by narrow bandedlevel set functions interpolated by signed distance function fn(x) through (Venturaet al., 2003).

The displacement field is based on modified functions (Staziet al., 2002) adaptedfor multiple crack problems (Budynet al., 2004) is as follows:

uh(x) =

∑

I∈I

NI(x)uI+

nc∑

n=1

∑

J∈Jn

NJ(x)anJ Hn

J (x)+

nt∑

m=1

∑

K∈Km

NK(x)

(

4∑

l=1

bmlK Fm

lK(x)

)

[18]whereNI(x) are the shape functions for the continuous displacement field; NJ(x)are the shape functions applied to the enrichment field. Thischoice of shape functions

226 REMN – 16/2007, X-FEM

(a) (b)

Figure 5. X-FEM representation of a crack. The circled nodes are enriched by thestep function of crackn and the square nodes are enriched with the tip enrichment oftip m. The functionf gives the distance of a point to the crack

for the enrichment field is explained in (Staziet al., 2002; Chessaet al., 2003). anJ

are the additional unknowns for the modified step enrichmentHnJ (x) of crackn and

bmlK are the additional unknowns for the tip enrichment of tipm for the modifiedlth

branch functionFmlK(x); andxJ is the position of nodeJ . The modified enrichment

functionsHnJ andFm

lK are given in Budynet al. (2004). As the enrichment for eachcrack is local, the size of the global stiffness matrix of thesystem is not much largerthan that of the standard stiffness (Budynet al., 2004).

3.4. Crack tip reaching a free boundary or another crack

Within each step, the length of each active crack tip is increased in the direction ofthe maximum hoop stress by:

∆ℓi =∆ℓtot

nact

[19]

wherenact is the number of active cracks that grow at the beginning of stepn; nact isdetermined by a stability analysis at the end of stepn − 1 described in (17).

Three possibilities can arise before percolation: a crack can reach the external freeboundary, reach the free boundary of an Haverse canal or coaslesce with another crack.When this occurs, the crack tip is annihilate or “kill” the crack tip and its near enrich-ment removed and replaced by either a simple step function for free boundaries or a“junction” enrichment for the linked discontinuities derived in Budynet al. (2004).The decision to join boundaries or an another cracks is basedon heuristic consider-ations based on the idea that a very small amount of brittle material subjected to a

Multiple scale modeling for cortical bone 227

tensile stress will break and therefore join the boundary oranother crack (Kienzleret al., 2002; Rubinstein, 1996; Demiret al., 2001; Denget al., 1992; Nemat-Nasseret al., 1987). Note that a special enrichment is developed in this model when a crackjoins another crack in the same element where the tip of the connected crack is lo-cated. In this special case of junction, the near tip enrichment of the connected crackshould be removed as well and changed into a step enrichment to ensure a perfectstrong discontinuity within this entire element that contains the junction.

For each crack tipi that is identified to grow, we consider a virtual increment∆ℓi

virt that is either set, when approaching free boundaries, to∆ℓivirt = ∆ℓtot, where

∆ℓtot is the total crack growth per step, or more restrictively when approaching cracks,to ∆ℓi

virt = max(∆ℓi, ri), where∆ℓi is the increment of growth of [19] andri is theradius of the domain of computation of the interaction integral at tip i; the radiusri being about twice the element length. When the mesh is fine, wehave observedthat when the elements of the annular domain on which theJ-integral is computedare partially truncated by the internal (Haverse canals) orexternal boundary of themodel or contains another crack or another phase, the stressintensity factors still givesthe correct direction of propagation. When another phase encounters the domain ofcalculation of theJ-integral, the correct direction of crack propagation is still obtainedbecause the gradient between the moduli of the phases is not too large. However therestrictive approach criterion is considered when many cracks interact. The length ofthe added increment∆ℓi is also adjusted when the growing crack encounters a freeboundary or another crack so that the tip dies on the boundaryedge.

4. Results

We consider a square of cortical bone of width0.5 mm containing58 cracksinitiated after the analysis of the local strain field shown in Figure 4. The init cell isloaded by tension in thex-direction; the force is prescribed at the top edges and thedisplacement in they-direction is prescribed equal to zero at the bottom edge. Thedisplacement of the bottom left corner node in thex-direction is prescribed equal tozero in Figure 1b. The material properties are random and representative of corticalbone microstructure as described previously and Figure 6 shows the distribution of thelocal transverse Young’s moduli.

The force-deflection curves, expressed as a nominal stress versus nominalstrain curve, are shown for a osteons/cement lines SIF’s contrast of about 6 and 3respectively in Figure 7a and c. The evolution of the nominalstress at each time stepare shown for a osteons/cement lines SIF’s contrast of about6 and 3 respectivelyin Figure 7b and d.In the present study, analysis is static and we consider twocases when the average Stress intensity factor (SIF) of the cement lines are sixtimes or three times tougher as the osteons. In the first case the load deflectioncurve (Figure 7a) shows three phases: from step 1 to 36 the cracks are growinginside the osteons, which produces a strain hardening, fromstep 36 to 68, onedominant crack is able to cross through the cement line of theosteon it originated

228 REMN – 16/2007, X-FEM

−0.3 −0.2 −0.1 0 0.1 0.2 0.3−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

0.9

1

1.1

1.2

1.3

1.4

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Figure 6. (a) Heterogeneous E2 Young’s moduli distribution (MPa). (b) Initial mi-crostructure that is implemented to start the XFEM simulations; the locations of theinitial 58 cracks are determined by the ABAQUS pre-calculation

Table 2. Case when the cement lines are three times tougher than the osteons. Theactive crack’s names are given in Figure 6b. Several events can happen: A: reachesthe outer boundary, B: connects to crack 7, C: reaches an Haversian canal, D: reachescrack 35 but cannot connect 35 again, E: reaches an Haversiancanal, F: reaches theouter boundary, G: reaches the outer boundary, H: reaches the outer boundary

step 2-5 6-10 11 12 13-15 16-18 19 20active crack 7 8 7 36 4 3 4 3

eventstep 21 22-25 26-29 30 31-41 42 43-44 45-46

active crack 35 36 8 36 8 7 8 7event A B C Dstep 47-49 50-51 52-53 54 55

active crack 48 42 10 51 58event E F G H

Multiple scale modeling for cortical bone 229

0 2 4 6 8 10 120

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Figure 7. (a) Load deflection curve (SIF’s contrast of 6), (b) evolution of the loadversus time (SIF’s contrast of 6). (c) load deflection curve (SIF’s contrast of 3), (d)evolution of the load versus time (SIF’s contrast of 3). The ratio of cement line averageSIF’s to osteonal average SIF’s is about 6 (upper curves, thework of separation is1.48 × 10−3 µN.mm ) and about 3 (lower curves, the work of separation is1.73 ×10−4 µN.mm)

from, and damage the matrix. During this period a strain softening is observed asthe matrix looses progressively its structural integrity.Finally from 68 to 88 thecracks grow until complete percolation, but the microstructure had already lost mostof its integrity. In the second case when the cement line are three times tougherthan the osteons, The same patterns are observed in the global load deflectioncurve: from step 1 to 20 the cracks are growing inside the osteons, which producesa strain hardening, from step 20 to 44, one dominant crack crosses through the

230 REMN – 16/2007, X-FEM

(a) (b)

(c) (d)

Figure 8. (a) Initial crack paths, (b) Crack paths ate step 36 (hardening phase), (c)Crack paths at step 68 (softening phase), (d) Crack paths at step 88 (broken mi-crostructure). The ratio of cement line average SIF’s to osteonal average SIF’s isabout 6

cement line of the osteon it originated from, and damages thematrix, which resultsin a strain softening. Finally from 44 to 55 the cracks grow until complete percolation.

We also note the effect of the cement lines over the strain field distribution inFigure 4b and the crack paths in Figure 8 and Table 2 and Figure9d and Table 3.In Budyn et al. (2005), the structural function of the cement lines was studied intransversal compression by comparing the local strain fielddistributions in the sameosteonal geometry in a microstructure including cement lines around the osteons andanother microstructure without cement lines. It was noticed in the absence of cementlines the osteons tend to transfer more deformation to the matrix; with cement lines,

Multiple scale modeling for cortical bone 231

(a) (b)

(c) (d)

Figure 9. (a) Initial crack paths, (b) Crack paths ate step 20 (hardening phase), (c)Crack paths at step 44 (softening phase), (d) Crack paths at step 55 (broken mi-crostructure). Then the ratio of cement line average SIF’s to osteonal average SIF’sis about 3

the osteons remains confined and exhibit a wider range of deformations. This phe-nomenon can partially explain the hardening phase observedin the global behaviordue to primary growth of cracks inside as many osteons as possible before damagingthe matrix. Despite very similar crack paths in Figures 8 and9, by comparing the twoload deflection curves in Figure 7a, we observe that more inhomogeneity makes theresponse richer (Belytschko, 2006). Note that the hardening is more developed whenthe cement lines are tougher and more energy is needed to break the microstructure:the work of separation is1.48 × 10−3 µN.mm when the cement lines are six timestougher than the osteons and1.73 × 10−4 µN.mm when the cement lines are threetimes tougher than the osteons. Therefore the cement lines appear as critical elements

232 REMN – 16/2007, X-FEM

Table 3. Case when the cement lines are six times tougher than the osteons. The activecrack’s names are given in Figure 6b. Several events can happen: A: reaches the outerboundary, B: reaches the outer boundary, C: reaches the outer boundary, D: connectsto crack 7, E: reaches an Haversian canal, F: reaches the outer boundary, G: reachesthe outer boundary, H: reaches the outer boundary, I: reaches the outer boundary, J:connects to crack 6

step 2-5 6-10 11 12 13-15 16-18 19 20active crack 7 8 7 36 4 3 4 3

eventstep 21 22 23 24-26 27-29 30-32 33 34

active crack 35 22 21 5 6 28 27 15event A Bstep 35 36 37-39 40-51 52-53 54-55 56-57 58-62

active crack 16 32 7 8 6 5 6 8event Cstep 63 64-66 67 68 69-70 71 72-73 74-76

active crack 7 36 7 10 7 42 10 21event D E F Gstep 77-78 79-81 82 83-88

active crack 14 13 58 8event H I J

in the protection of bone against fracture and relevant structural elements to study theirevolution in osteoporosis. When the cement lines are less tough, other physical resultschanges: a snap-back is less pronounced, the softening phase is smoothened. Someexperimental observation of the effect of the cement lines as osteonal barriers can beseen in Mohsinet al.(2006). However more experimental data are necessary to inves-tigate the local fracture properties such as the stress intensity factors. Once the matrixis damaged, softening appears at the macroscopic level as the structural integrity ofbone is lost.

5. Conclusion

The results of this preliminary study present a multiple scale approach to describemultiple crack growth in cortical bone under tension. The Haversian microstructure isdiscretized by a statistical finite element model of cortical bone material to investigatethe influence of the local geometrical and mechanical parameters of the microstructureon the mechanical behavior of bone material. Even if in our model majority of geo-metrical and material parameters were based on physical data, some limitations canbe given. For the geometrical description the osteons are modeled by disjoint circu-

Multiple scale modeling for cortical bone 233

lar tubes, which is an idealized geometry and does not allow them to connect to eachother through Volkman canals (Joneset al., 2004). We study bone at the osteonal leveland do not include osteocyte lacunae (Qiuet al., 2005; Prendergastet al., 1996). Wealso modeled interstitial matrix as homogeneous material.In our study we focusedour investigations on the effect of the toughness heterogeneities due to microstruc-ture. Other heterogeneities such as the spatial distribution and shape of the osteons,cement line thickness, the initial defect orientation and distribution ... influence thecrack paths and will be studied in further work.

To implement crack growth the eXtended Finite Element Method is particularlysuitable for multiple crack growth because remeshing is avoided. Higher order ele-ments have been chosen, they are quadratic for the standard displacement field andlinear for the enrichment to give an accurate crack solution. In contrast to boundaryelement methods, the method can easily handle the heterogeneities of the microstruc-ture. The method is explicit, applied to static crack growthand is developed to satisfiesexactly both the equilibrium [11] and the brittle crack growth law [12] at each timestep of the load deflection curve. A stability analysis basedon energy considerationenable to solve the case of competitive crack growth. The response of the unit cellis tracked until complete failure when the cracks have percolated the cell, joined andreached the free boundaries. Some improvement in precisionin the load deflectioncurve will be obtained by using smaller steps and finer meshesto refine the heuris-tic criterion for crack bridging and crack connecting to thefree boundaries. In thepresent study we considered traction loading applied to thetop edge of the cell. Weare currently investigating consistency between the average local stress field when adisplacement is applied on the top edge and the average localstrain field when thecorresponding traction loading to the first displacement boundary case is applied tothe top of cells containing defects. We have already studiedthese issues for cells ofuniform materials containing defects and the global responses was extremely similarin both cases when load BC or corresponding displacement BC were imposed to thetop of the cells. For heterogeneous cells of 0.5 mm without defect it is also possi-ble to recover the same macro field by applying displacement BC or the traction BCcorresponding to the first load BC case.

The model accesses two different scales: a macro scale at theunit cell level anda micro scale for the strain field inside the osteons, in a mimetic osteonal microstruc-ture. At macroscopic scale, the model is able to provide the overall stress and strainresponse.

At the microscopic level, even if the matrix is homogeneous its strain field is not.Moreover, our results show that cement lines have an important role in isolating theosteons from the matrix. This explains why cement lines are implicated with energy infracture processes in the deflection of crack propagation byslowing it down (Advaniet al., 1987) or debonding the osteon (Guoet al., 1995). Cement lines have also beenlinked to energy absorption (Akkuset al., 2001) and viscous damping (Lakes, 2001;Donget al., 2004) in allowing osteon movement inside the interstitialmatrix.

234 REMN – 16/2007, X-FEM

In summary, this model presents preliminary results that characterize the localproperties of cortical bone and emphasizes the importance of the cement lines as anefficient structural component to ensure internal movementat the material level and toprevent the progression of localization damage zones and cracks. Our results suggesthow critical circular porosity (Haversian canals) are sources of localization and frac-ture nucleation for transversal tension loading. More validations of experiments willbe developed for this type of numerical approach, where a dialog between numericalmodeling and experimental investigation can be established. Further investigations ofthe physiological and mechanical properties of the cement lines and the matrix willalso help to refine the modeling of damage and fracture in cortical bone. In the future,introduction in this model of more experimental damage and fracture parameters needto be determined to make the model more physiological and extended to compressionloading. This will lead towards a multiscale study of bone fracture in an entire bone.

Acknowledgments

The authors are grateful for the research support of the University of Illinois atChicago. The first author is also grateful to the CNRS for a Post-doctoral fellowship.We are also grateful to Professor Ted Belytschko for his comments and suggestions.

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1. ARTICLE POUR LA REVUE:

REMN – 16/2007, X-FEM

2. AUTEURS :

Élisa Budyn∗,1 – and Thierry Hoc∗∗

3. TITRE DE L’ ARTICLE :

Multiple scale modeling for cortical bone fracture in tension using X-FEM

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Multiple scale modeling for cortical bone

5. DATE DE CETTE VERSION:

15th February 2007

6. COORDONNÉES DES AUTEURS:

– adresse postale :∗ Department of Mechanical and Industrial Engineering, University ofIllinois at Chicago, 842 West Taylor Street, Chicago IL. 60607, USA.

∗∗ Department of Material Science, LMSSMat, UMR 8579, EcoleCentrale Paris, Grande Voie des Vignes, Chatenay Malabry, F-92295.

– téléphone : (1) 312 996 96 31

– télécopie : (33) 01 41 13 16 16

– e-mail : ebudyn@uic.edu, hoc@ecp.fr

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