Department of Mechanical Engineering,Villanova University,
800 Lancaster Avenue,Villanova, PA 19085
Finite Element Modeling ofMicrocrack Growth in CorticalBoneBone is similar to fiber-reinforced composite materials made up of distinct phases such asosteons (fiber), interstitial bone (matrix), and cement lines (matrix-fiber interface). Mi-crostructural features including osteons and cement lines are considered to play animportant role in determining the crack growth behavior in cortical bone. The aim of thisstudy is to elucidate possible mechanisms that affect crack penetration into osteons ordeflection into cement lines using fracture mechanics-based finite element modeling. Co-hesive finite element simulations were performed on two-dimensional models of a singleosteon surrounded by a cement line interface and interstitial bone to determine whetherthe crack propagated into osteons or deflected into cement lines. The simulations inves-tigated the effect of (i) crack orientation with respect to the loading, (ii) fracture tough-ness and strength of the cement line, (iii) crack length, and (iv) elastic modulus andfracture properties of the osteon with respect to the interstitial bone. The results of thefinite element simulations showed that low cement line strength facilitated crack deflec-tion irrespective of the fracture toughness of the cement line. However, low cement linefracture toughness did not guarantee crack deflection if the cement line had highstrength. Long cracks required lower cement line strength and fracture toughness to bedeflected into cement lines compared with short cracks. The orientation of the crackaffected the crack growth trajectory. Changing the fracture properties of the osteoninfluenced the crack propagation path whereas varying the elastic modulus of the osteonhad almost no effect on crack trajectory. The findings of this study present a computa-tional mechanics approach for evaluating microscale fracture mechanisms in bone andprovide additional insight into the role of bone microstructure in controlling the micro-crack growth trajectory. �DOI: 10.1115/1.4003754�
IntroductionBone is a hierarchical material that has different structural com-
onents in various length scales. At the microstructural level, bones similar to a fiber-reinforced composite material made up ofistinct phases including osteons �cylindrical structures formed byayers of lamellae as the fibers�, interstitial bone �remnants of oldsteons as the matrix�, cement lines �interface between osteonsnd interstitial bone as the matrix-fiber interface�, and pores �in-luding Haversian canals that are located within osteons, resorp-ion spaces, and Volkmann’s canals� �Figs. 1�a� and 1�b��. The
icrostructure of bone contributes significantly to its fracture re-istance. Microstructural features such as osteons and cementines are considered to play an important role in determining therack growth trajectory in cortical bone �1–10�. Cracks that pen-trate osteons may lead to complete failure of the bone. On thether hand, crack deflection into cement lines slows down therack propagation and increases the crack resistance of the bone.espite these observations, the underlying mechanisms and fac-
ors that control the crack growth trajectory in cortical bone haveot been completely understood.
In engineering literature crack deflection and penetration at annterface between two solids, similar to the case in bone, haveeen studied in detail starting with the work of Cook and Gordon11�. Subsequent research in this area focused on the analysis ofhe crack penetration/deflection phenomenon at an interface be-ween two distinct isotropic materials �12–17� and orthotropic ma-
1Corresponding author.Contributed by the Applied Mechanics Division of ASME for publication in the
OURNAL OF APPLIED MECHANICS. Manuscript received July 12, 2010; final manuscripteceived February 14, 2011; accepted manuscript posted March 7, 2011; published
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terials �18� without incorporating the competition between crackdeflection and penetration. Later works focused on understandingthe conditions under which crack deflection and penetration occurat interfaces using energy-based �19–21� and strength-based crite-ria �22�. More recently, finite element modeling was utilized toincorporate both the strength- and energy-based approaches toevaluate crack deflection/penetration in engineering materials�23–26�. The literature on crack deflection and penetration forcomposite materials in engineering applications provide funda-mental information that can help understand the crack growthmechanisms in bone. As a result, the aim of this study is to de-velop a computational mechanics approach to evaluate microscalefracture mechanisms in bone and to provide further insight intothe role of osteons and cement lines in crack propagation behaviorof cortical bone.
The studies on transverse histological sections of cortical bonetest specimens showed that the majority of the microcracks werein the interstitial bone or were stopped by the cement lines�1,8,27�. Cracks that penetrate into osteons formed a smaller per-centage of overall microcracks. Furthermore, experimental obser-vations revealed that long cracks were more likely to penetrateosteons than to deflect into cement lines, whereas short crackswere arrested at the cement lines �7�. It was also shown that newlyformed osteons stopped long cracks more frequently comparedwith older osteons �5�.
Although the material properties of the osteons and interstitialbone have been studied extensively �28–32�, only a few studiesinvestigated the cement line properties �33–35�. These studies fo-cused on the evaluation of the shear strength of the cement lineinterface via osteon push-out tests. Burr et al. �2� proposed that
cement lines may promote crack deflection due to their low min-
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ralization compared with osteonal and interstitial bones. How-ver, a more recent study suggested that the cement lines may beore mineralized �36�.There have been limited computational studies that investigated
he effect of the cortical bone microstructure on microcrackrowth behavior. Advani et al. �37� and Najafi et al. �38� em-loyed linear elastic fracture mechanics to determine how crackrowth direction and crack velocity interacted with osteons. Bu-yn et al. �39,40� applied an elastic-damage criterion for microc-ack initiation and propagation to a microstructural model of theortical bone using extended finite element method �FEM�. Inddition, microstructural finite element models were employed toetermine the influence of porosity, osteonal bone, and orientationf the Haversian system on the macroscopic elastic moduli andoisson’s ratio of the cortical bone �41,42�. Prendergast anduiskes �43� used gap elements to model cement line debonding
nd evaluated its effect on the strains induced in lacunae. In ad-ition to the computational simulations, analytical calculationsere also performed to determine the influence of cement lines on
he fracture toughness of cortical bone �44�, to investigate thenteraction between microcracks and osteons �45�, to understandhe role of cement line fracture toughness in crack growth �46�,nd to predict the effect of the elastic mismatch between the os-eonal and interstitial bones in promoting or retarding the crackrowth toward osteons �47�.
Recently, cohesive finite element modeling has been introduceds an effective computational modeling technique for simulating
ig. 1 „a… Schematics of a cortical bone section outlining therientation of the osteons. Directions 1, 2, and 3 denote radial,ircumferential, and longitudinal directions, respectively. „b… Aample microscopy image taken at the transverse plane „AA…
howing the microstructure of human cortical bone from a 70-ear-old donor. „c… Schematics of the finite element model „IBinterstitial bone, HC=Haversian canal, OS=osteon, and CLcement line…. Note that the black and white dotted lines areossible crack paths where cohesive elements were inserted.lack lines represent the initial crack for three different crackrientations. The arrows indicate the applied loading locationnd direction.
ortical bone fracture. Cohesive finite element modeling has been
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used in many engineering applications �48–54� including investi-gation of the influence of fracture toughness and cohesive strengthon crack deflection at interfaces for engineering materials such asceramics �23–25�. However, until recently, it has not been appliedto bone fracture. The applications of cohesive finite element mod-eling in bone predicted the age-related fracture toughness loss�55,56�, and the effect of porosity and material properties on crackgrowth in cortical bone �57�. Cohesive modeling was also used tosimulate the load displacement response in cortical bone testing�58–60�, and to determine the whole bone fracture load in humanradius bone �61–64�. In addition, it has been utilized to assesstrabecular bone fracture under quasistatic �65� and dynamic load-ing conditions �66,67�. Furthermore, the influence of collagencross-linking on the stress-strain behavior of bone tissue �68� andthe feasibility of using a double-cantilever beam test specimen formeasuring bone fracture toughness �69� was evaluated by cohe-sive finite element modeling. This study will expand the applica-tion of cohesive finite element modeling to the investigation ofmicroscale bone fracture processes.
In summary, the review of literature indicates that the mecha-nisms and factors that control the microcrack growth trajectory inbone have not been identified in detail via experimental measure-ments or computational simulations. The aim of this study is toelucidate possible mechanisms and factors that affect crack pen-etration into osteons or deflection into cement lines using fracturemechanics-based cohesive finite element modeling. The simula-tions performed in this study investigate the effect of the follow-ing factors on crack growth trajectory: �i� crack orientation withrespect to the loading direction, �ii� fracture toughness andstrength of the cement line, �iii� crack length, and �iv� elasticmodulus and fracture properties of the osteon with respect to theinterstitial bone. The results of these simulations will provide ad-ditional insight into the role of cortical bone microstructure inmicrocrack growth using a finite element modeling approach thathas not been applied to bone fracture at the microscale.
2 Methods
2.1 Finite Element Model. A two-dimensional finite elementmodel of a single osteon surrounded by a cement line interfaceand interstitial bone was created using ABAQUS �version 6.8, 2007,Simulia, Providence, RI� to simulate crack propagation into os-teons or crack deflection into cement lines �Fig. 1�c��. The modelrepresents a bone area of 1�1 mm2. The osteon was located atthe center of the bone area with a diameter of 200 �m and isconcentric with the Haversian canal of diameter 50 �m. Thesedimensions are based on average values reported in literature �70�.The interaction of multiple osteons was not considered in the cur-rent work. An initial crack was introduced into the model withorientations varying from 0 deg �perpendicular to the loading di-rection� to 90 deg �parallel to the loading direction� and lengthsfrom 75 �m to 425 �m. An incremental displacement loadingwas applied at the upper and lower corners of the geometry thatpromoted crack propagation �Fig. 1�c��. Fixed boundary condi-tions were applied at the corners where no loading was applied.The loading for the model was chosen to facilitate crack growthand induce stress fields similar to local stress states experiencedby the bone. The models were meshed using plane strain three-noded linear triangular finite elements. A mesh convergence studywas performed and the converged solid and cohesive elementsizes were used in the simulations. Possible crack paths in theinterstitial bone, the osteon, and the cement line were representedby cohesive finite elements. The interstitial bone and osteons wereassumed to be homogeneous and were modeled as orthotropicmaterials using elastic properties obtained from experimentalmeasurements �28� and analytical calculations �71� �Table 1�.
2.2 Computational Method. In this study, crack propagationis simulated by cohesive finite element modeling that captures the
nonlinear behavior of the process zone, which cannot be modeled
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y linear elastic fracture mechanics. The cohesive model repre-ents the physical processes occurring in the vicinity of a propa-ating crack by a simplified traction-displacement relationshipnd isolates the fracture process from the surrounding continuumonstitutive model. In this paper, a short summary of the compu-ational method will be presented. Additional details of the devel-pment of cohesive finite element modeling in engineering mate-ials and in cortical bone can be found in Refs. �52,54,55,61�.
Cohesive models are defined by a traction-displacement rela-ionship that captures the material softening via the descendingart of the curve where the traction transferred between the ma-erial surfaces decreases as the crack opening displacement in-reases. The decreasing part of the traction-displacement curveepresents the fracture process zone in bone including effects suchs crack bridging and microcracking. There are various types ofraction-displacement relationships proposed in literature to modelhe fracture process including linear and exponential relationships53,54,72�. Previous studies showed that the most influential co-esive model parameters that affect the fracture behavior are theracture toughness and the strength whereas the shape of the co-esive relationship has a little effect on the resulting fracture be-avior �73�. Therefore, in this study, a bilinear traction-isplacement curve was employed �Fig. 2�a��.
ti =��ic
�c�i, �i � �ic
�ic
��iu − �i���iu − �c�
, �iu � �i � �ic
0, �i � �iu
� �1�
here subscript i for all variables denotes normal �n� or shear �s�odes, ti and �i are the traction and opening displacement, �ic is
able 1 Orthotropic elastic properties of the osteonal and in-erstitial bones used in the simulations. Note that these valuesre based on experimental measurements †28‡ and analyticalalculations †71‡. Directions 1, 2, and 3 denote radial, circum-erential, and longitudinal directions, respectively. Note alsohat the shear modulus and Poisson’s ratios are assigned theame values for osteonal and interstitial bones due to the lackf separate shear modulus and Poisson’s ratio information in
iterature for interstitial bone and osteons.
Osteonal bone Interstitial bone
1 �GPa� 16.6 19.7
2 �GPa� 17.0 18.5
3 �GPa� 25.1 27.1
12 �GPa� 7.2 7.2
23 �GPa� 8.4 8.4
31 �GPa� 7.1 7.1
13 0.237 0.237
23 0.247 0.247
12 0.334 0.334
Fig. 2 „a… Traction-displacement remodel for both normal and shear m
compatible 2D cohesive element with fo
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the peak traction representing the maximum strength of the cohe-sive model, �c is the critical displacement at which the crackinitiates, and �iu is the failure displacement when a completecrack forms. The area under the traction-displacement curve de-fines the fracture toughness of the material and can be calculatedas
Gic =�0
�iu
tid�i �2�
where Gic is the fracture toughness in normal and shear modes. Ina mixed mode loading case, assuming that the shear behavior isthe same in all directions, the parameters required to define thecohesive zone are the critical normal and shear stresses ��nc ,�sc�,and the fracture toughness in normal and shear modes �Gnc ,Gsc��Fig. 2�a��. The initial ascending slope of the curve is a penaltystiffness in the numerical formulation and is generally chosen tobe as high as possible in order to obtain a very small �c value,satisfying numerical convergence that ensures a close approxima-tion to the exact solution of the nonlinear equations �52�. Eachcohesive element follows the traction-crack opening profile toform a crack. Damage initiates following the interaction functionthat involves the current stress state �tn , ts� and critical stresses��nc ,�sc� �52�.
� tn
�nc2
+ � ts
�sc2
= 1 �3�
Furthermore, the crack is formed following a mixed mode fracturecriterion:
Gn
Gnc+
Gs
Gsc= 1 �4�
where Gn and Gs are the current values of energy release rate.In the finite element context, cohesive models are represented
as surface elements with zero thickness and are compatible withsolid elements �Fig. 2�b��. In two dimensions, the cohesive inter-face elements are composed of two line elements with zero thick-ness, which coincide in the reference configuration �Fig. 2�b��.Each line element has n nodes, which gives 2n nodes for thecohesive element.
2.3 Finite Element Simulations. Using the FEM model out-lined in Fig. 1�c�, the following simulations were performed.
�i� Crack orientation. Simulations with initial crack orienta-tions ranging from 0 deg to 90 deg were run while varyingnormal and shear cohesive model properties ��nc, �sc, Gnc,and Gsc� of the cement line. The variation in the crackorientation allows for the investigation of mixed modecrack growth behavior. The normal and shear ratios werevaried independently while keeping the other group ofvariables constant. The cohesive properties of the intersti-tial bone and the osteon were the same and were heldconstant for all simulations �Table 2� based on the experi-
onship defining the cohesive zonees. „b… 2D solid elements and the
latiod
ur nodes.
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mental measurements reported in literature �9,74,75�. Theinitial crack length was 353.5 �m for all simulations.
�ii� Crack length. For 45 deg crack orientation, three initialcrack lengths, 75 �m, 353.5 �m, and 425 �m, were in-vestigated. These initial crack lengths lead to a total crackpropagation of 532 �m, 253.5 �m, and 182 �m, respec-tively. For each crack length, normal cohesive model prop-erties ��nc and Gnc� of the cement line were varied. Theinterstitial bone and osteon were assigned the same cohe-sive properties that were held constant for all simulations�Table 2�.
�iii� Elastic modulus of the osteon. For 45 deg crack orienta-tion, the elastic modulus of the osteon was increased by100% and decreased by 50% compared with the valuesreported in Table 1 for osteonal bone. Similar to the otherinvestigations, the normal cohesive model parameters ofthe cement line were varied with respect to the osteonaland interstitial bones. The interstitial bone and osteon wereassigned the same cohesive properties that were held con-stant for all simulations �Table 2�. The initial crack lengthwas 353.5 �m for all simulations.
able 2 Cohesive model parameters for the osteonal and in-erstitial bones †9,74,75‡. Note that �nc=normal strength, �scshear strength, Gnc=fracture toughness for opening mode,nd Gsc=fracture toughness for shear mode.
Cohesive properties for cortical bone
nc 43.5 MPa
sc 68 MPa
nc 1.16 N/mm
sc 2.97 N/mm
Fig. 3 „a… A sample finite element mesh demonstrating ccontours showing the different stages of crack propagationelement mesh demonstrating crack deflection into the cemdifferent stages of crack propagation for 45 deg crack that isdotted lines in „a… and „c… show the regions for which the strcracks in the models are represented by completely opendisplacement relationship is represented by cohesive elemen
in „b… and „d… indicate the crack front location and the solid bla
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�iv� Fracture properties of the osteon. For 45 deg crack orien-tation, the normal cohesive strength of the osteon was de-creased by 40% and the normal fracture toughness wasincreased by 40% compared with the fracture propertieslisted in Table 2 to represent the properties of a newlyformed osteon based on experimental measurements �76�.The interstitial bone properties were kept at their originalvalues as reported in Table 2. The normal cohesive modelparameters of the cement line were varied with respect tothe interstitial bone. The initial crack length was353.5 �m for all simulations.
The results of the simulations outlined above were evaluated todetermine whether the crack propagated through the osteon ordeflected into the cement line. Using these results, the crack de-flection and penetration regimes were plotted for each case as afunction of bone to cement line strength and fracture toughnessratio �Gnc−b /Gnc−cl versus �nc−b /�nc−cl, and Gsc−b /Gsc−cl versus�sc−b /�sc−cl�. In addition, the changes in the crack driving forcewith crack orientation, crack length, elastic properties, and cementline properties were assessed.
3 ResultsThe finite element simulations showed that the crack growth
pattern in the bone depended on the crack orientation with respectto the loading direction, the ratio of the shear and normal fractureproperties of the cement line to the bone matrix, the crack length,and the fracture properties of the osteon relative to the interstitialbone. The difference between osteonal and interstitial bone elasticmoduli did not affect the crack growth path. The strength andfracture toughness of the cement line both played a role in deter-mining the crack propagation trajectory.
Cracks oriented at 0–30 deg penetrated the osteon for all frac-ture toughness and strength ratios �Fig. 3�a��. On the other hand,
k penetration into an osteon for 0 deg crack. „b… Stress0 deg crack penetrating into an osteon. „c… A sample finitet line for 45 deg crack. „d… Stress contours showing theected by the cement line. Note that the areas defined by thecontours are shown in „b… and „d…. In „a… and „c…, the initial
urfaces and the new crack growth following the traction-shown in gray „red in online version…. The black dotted lines
racforen
defless
sts
ck lines denote the osteon.
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racks from 37.5 deg to 90 deg deflected into the cement line forertain strength and fracture toughness ratios �Fig. 3�c��. Thetress contours for penetration and deflection cases are shown inigs. 3�b� and 3�d� for various stages of crack growth. The firsttress contours in Figs. 3�b� and 3�d� show the stress state whenohesive elements start opening marking the crack initiation. Fol-owing the initiation, cohesive elements continue to open and therack front reaches the osteon. At that stage, the crack either pen-trates the osteon or is deflected by the cement line depending onhe properties of the cement line as well as the crack orientation.n Fig. 3�b� the crack penetrates into the osteon whereas in Fig.�d� the crack deflects into the cement line and continues to growithin the cement line.A boundary separating the crack deflection and penetration re-
imes was determined for different crack orientations based onormal and shear properties �Figs. 4 and 5�. These results indicatehat the bone microstructure may impede crack growth based onhe relative fracture properties of its matrix and interface phasesith the exception of the most detrimental modes.Varying the angle between the initial crack and the direction of
he applied load from 0 deg to 90 deg increased the load requiredo initiate and propagate the crack �Fig. 6�. The load required toropagate a crack under an opening mode was significantly lower.urthermore, a crack that penetrated the osteon continued to growith increasing load, whereas a crack that was arrested at the
ement line showed no crack growth while the load continued toncrease. A sample loading case is shown for 45 deg crack withement line properties leading to crack penetration and deflectionFig. 7�. The crack propagation load is the same up to the pointhere the crack encounters the cement line after which the crackropagation continues for the penetration case and stops for theeflection case. This behavior indicates that once the microcracks deflected into the cement line, it may completely be arrestednd not become a macroscale crack that can lead to fracture.
Altering the fracture toughness and strength of the cement linen the normal and shear directions had almost no effect on therack initiation load but influenced the crack growth trajectory.ecreasing the strength and fracture toughness ratios reduced the
ement line’s ability to deflect the crack, indicating that stronger,
ig. 4 Crack growth behavior for 45 deg and 90 deg crackrientations based on the ratio of the bone to cement line nor-al fracture properties. Gnc-b /Gnc-cl denotes the ratio of Mode I
opening… fracture toughness of bone to cement line andnc-b /�nc-cl denotes the ratio of the tensile strength of bone toement line. The strength and toughness ratios were obtainedy keeping the interstitial bone and osteon properties constantTable 2… while varying the cement line properties. The rangesf cement line properties used to obtain these ratios arearked on the graphs. The solid and hollow circles correspond
o individual simulations for penetration and deflection, re-pectively. The solid lines represent the transition boundaryetween crack deflection and penetration based on the indi-idual simulation data.
ougher cement lines facilitated crack propagation into the osteon.
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Although both the fracture toughness and strength ratio of thecement line contributed to determining the crack path, the cementline strength was more critical for promoting crack deflection.Low cement line shear and normal strengths facilitated crack de-flection irrespective of the fracture toughness of the cement lineindicating that low cement line interface strength relative to thebone matrix induces microcrack deflection for any fracture tough-ness ratio �Figs. 4 and 5�. The fracture toughness ratio influencedthe crack trajectory only for low bone to cement line strengthratios. However, low cement line fracture toughness did not guar-antee crack deflection if the cement line had high strength. Theresults showed that for crack deflection to occur the cement lineneeded to have less than one-third of the strength of the matrix�Figs. 4 and 5�.
For a given fracture toughness ratio in the normal direction, the90 deg crack required a weaker cement line to deflect comparedwith the 45 deg crack �Fig. 4�. For the shear direction, the oppo-
Fig. 5 Crack growth behavior for 45 deg and 90 deg crackorientations based on the ratio of the bone to cement line shearfracture properties. Gsc-b /Gsc-cl denotes the ratio of Mode II„shear… fracture toughness of bone to cement line and�sc-b /�sc-cl denotes the ratio of the shear strength of bone tocement line. The strength and toughness ratios were obtainedby keeping the interstitial bone and osteon properties constant„Table 2… while varying the cement line properties. The rangesof cement line properties used to obtain these ratios aremarked on the graphs. The solid and hollow circles correspondto individual simulations for penetration and deflection, re-spectively. The solid lines represent the transition boundarybetween crack deflection and penetration based on the indi-vidual simulation data.
Fig. 6 Crack driving force versus crack extension for 0 deg, 45deg, and 90 deg cracks. Note that this plot is for Mode I fracturetoughness ratio of Gnc-b /Gnc-cl=1 and tensile strength ratio of�nc-b /�nc-cl=1. The same behavior was observed for other
strength and fracture toughness ratios.
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ite behavior was observed �Fig. 5�. When both shear and normalroperties of the cement line were varied simultaneously thearger strength ratio dominated the crack growth trajectory. Forxample, when the normal property ratios were within the pen-tration regime �Fig. 4�, if the shear property ratios attained aalue that was in the deflection regime �Fig. 5�, the crack de-ected into the cement line.The simulations showed that the crack trajectory was influenced
y the crack length. Long cracks required cement lines with lowtrength and fracture toughness to be deflected into the cementine �Fig. 8�. These results are in line with experimental investi-ations, which demonstrated that short cracks were stopped at theement line, and long cracks were able to penetrate the osteon �7�.n addition, the initial crack length affected the initiation load butid not significantly change the load at which the crack encoun-ered the osteon �Fig. 9�.
When the osteon fracture properties were varied with respect tohe interstitial bone, representing newly formed osteons, theoundaries defining the crack penetration/deflection regimes
ig. 7 Crack driving force versus crack extension for crackenetration into the osteon and crack deflection into the ce-ent line. Note that the crack is oriented at 45 deg.
ig. 8 Crack growth behavior for three crack lengths orientedt 45 deg based on the ratio of the bone to cement line fractureroperties. Gnc-b /Gnc-cl denotes the ratio of Mode I „opening…
racture toughness of bone to cement line and �nc-b /�nc-cl de-otes the ratio of the tensile strength of bone to cement line.he strength and toughness ratios were obtained by keeping
he interstitial bone and osteon properties constant „Table 2…hile varying the cement line properties. The ranges of cement
ine properties used to obtain these ratios are marked on theraphs. Note that the values reported in the figures denote themount of crack extension. The trendline represents the aver-
ge of the deflection and penetration data sets.
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shifted to the right �Fig. 10�. The ratio between the interstitialbone and the cement line properties had to be higher both instrength and toughness for the crack to deflect into the cement linewhen the osteon was less brittle than the interstitial bone. Chang-ing the elastic modulus of the osteon had almost no effect oncrack trajectory or the initiation and propagation loads. This studyonly evaluated the crack behavior once it encountered the osteonand did not investigate how the crack propagation toward an os-teon may be affected by the modulus mismatch as in previousstudies �38,47�.
4 Discussion and ConclusionsThe current study focused on the microstructural constituents of
bone and their effects on crack growth using cohesive finite ele-
Fig. 9 Crack driving force versus crack extension for threecrack lengths oriented at 45 deg. Note that this plot is for ModeI fracture toughness ratio of Gnc-b /Gnc-cl=1 and tensile strengthratio of �nc-cl /�nc-cl=3.5. The same behavior was observed forother strength and fracture toughness ratios. Note that thecrack extension values shown in the figure „182 �m,253.5 �m, and 532 �m… correspond to initial crack lengths of425 �m, 353.5 �m, and 75 �m, respectively.
Fig. 10 Crack growth behavior for 45 deg crack with two dif-ferent osteon properties. The model with Osteon 1 has thesame cohesive properties for both the osteonal and interstitialbones. In the second model, the osteon „Osteon 2… has 40%lower strength and 40% higher toughness than the interstitialbone. Gnc-b /Gnc-cl denotes the ratio of Mode I „opening… fracturetoughness of bone to cement line and �nc-b /�nc-cl denotes theratio of the tensile strength of bone to cement line. Thestrength and toughness ratios were obtained by keeping theinterstitial bone properties constant „Table 2… while varying thecement line properties. The ranges of cement line propertiesused to obtain these ratios are marked on the graphs. Thetrendline represents the average of the deflection and penetra-
tion data sets.
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ent modeling. The simulation results showed that the crack tra-ectory was influenced by the strength and fracture toughness ofhe cement line relative to the bone matrix, crack length, crackrientation, and fracture properties of the osteon relative to thenterstitial bone. FEM simulations performed in this study utilizedohesive zone modeling, which incorporated both strength andracture toughness of the bone and the cement line to determinehe crack growth trajectory. The studies to date on cortical boneonsidered the influence of either only fracture toughness �46� ortrength �38,47�. Therefore, the current study provides additionalnsight into the crack propagation behavior in bone at the micro-cale by incorporating both the strength and fracture toughness ofhe bone in the simulations.
The results presented in this study are in agreement with thetudies performed on other engineering materials �23,24� usinghe cohesive modeling approach. In previous studies �23,24�,rack deflection was found to be promoted by lower interfaceoughness and strength as found in this study. Furthermore, ouronclusion on the importance of the cement line strength for crackeflection is supported by the findings in previous studies thathowed a strong influence of interface strength on the crack de-ection behavior compared with interface toughness �23,24�. Thenergy-based crack deflection/penetration criteria developed bye and Hutchinson �20�, which was later applied to bone �46�,
uggested that an interface with low toughness will be sufficient toeflect a crack. However, these results are based on pre-existinginks and do not take into account possible multiple paths in theame analysis as is the case in the current study. The previoustudies that employed cohesive modeling demonstrated a nonlin-ar boundary between the crack penetration and deflection re-imes �23,24� whereas our study found a linear boundary. Thisiscrepancy may be due to the wide range of toughness ratiosnvestigated in previous studies including values less than 1 �24�or which the nonlinear behavior was more pronounced. In theurrent study, cement line was assumed to be a weak interfaceased on experimental observations �2� leading to strength andoughness ratios larger than 1. In combination with the resultsresented for other engineering materials, our simulations demon-trate the strength of cohesive modeling in improving the under-tanding of crack growth mechanisms in biological materials.
The osteon push-out tests done by Dong et al. �34� found inter-ace shear strength values �5.38–13.65 MPa�, which were signifi-antly less than the continuum shear strength of the cortical bone68 MPa� �75�, or bone lamellae strength values obtained by push-ut tests �45.97–93.74 MPa� �34�. Based on these measurements,he bone matrix to cement line shear strength ratios vary between.4–6.9 and 8.5–17.4 for strong and weak cement lines, respec-ively. Comparison of the experimental results with the currentimulation results shows that there is a wide variation in the rela-ive strength of bone and cement line, which can significantly alterhe microcrack propagation behavior. Low strength cement lineshat were identified in the experiments will always promote crackeflection whereas cement lines, which have high strength, mayot be able to function as a crack arrest mechanism depending onhe properties of the surrounding bone. In addition, the range ofhear strength values for cement line that defines the crackenetration/deflection boundaries in Fig. 5 is within the range ofxperimentally measured values outlined above. This indicateshat the cement line properties that correspond to deflection orenetration in the simulations are within the range of experimen-ally measured values.
The finite element simulations performed in this study wereimed to evaluate the experimentally observed crack deflection/enetration behavior in the bone through a computationally effi-ient 2D model. The 2D models used in this study are capable ofapturing the crack propagation in the radial/circumferential planehat was observed in experimental studies �5,7,77�; however, theyo not allow for simulation of crack deflection in the longitudinal
irection. Despite this limitation, our simulations are significant in
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providing insight into the crack growth process in bone and arecapable of modeling the physical phenomenon and identifying thematerial properties of the cement line that influence crackdeflection/penetration behavior that was observed in the experi-ments.
Another limitation of the model used in this study is related tothe cohesive model parameter selection. The cohesive parametersused in the finite element models were based on the experimentalmeasurements performed on bone specimens that include bothosteons and interstitial bone. These experimental measurementsare not capable of measuring individual fracture properties fordifferent components of the bone. Although these tests do notdirectly measure the cohesive zone properties, they provide infor-mation on the overall fracture toughness and the strength of thebone. Direct experimental data on the cohesive properties of boneare not available due to the difficulty of such measurements. How-ever, the selection of the absolute value of these parameters is notexpected to affect the main conclusions of this study since theresults are based on the ratios between the cement line, osteons,and interstitial bone properties. As a result, the ratios between thecohesive model parameters are more significant than their abso-lute values. An additional point is that the current study did notemploy periodic arrangement of osteons and boundary conditions,which can be of importance for a multiscale study of bone frac-ture.
The incidents of bone fracture increases with age indicating thereduced fracture resistance in older bone. It has been shown that ifa crack penetrates an osteon it is more likely to propagate intoother osteons leading to catastrophic failure �7�. Age-relatedchanges in the bone are anticipated to affect the bone matrix aswell as the cement line properties. Therefore, crack deflection orpenetration behavior may change from young to old bone as aresult of this change. Experimental studies in this area showeddiminished effectiveness of fracture toughening mechanisms withincreasing age �78–80�. Furthermore, majority of the osteons inyounger human cortical bone were found to deflect the crackswhereas in older bone crack penetration into osteons was thedominant mechanism �77�. These experimental results indicatethat the crack deflection/stopping mechanism in older bones iscompromised most likely due to the material level changes insmaller length scales. It has been shown that aging and diseaseresult in material level changes in bone such as adverse changes inthe collagen network �81–83� and increase in mineral content ofbone �84�. However, the influence of aging on the material prop-erties of the cement line has not been clearly identified due to thedifficulty in experimental measurement of cement line properties.Therefore, the results of the current study provide new informa-tion that can be applied to understanding of how the crack growthbehavior is compromised based on age-related changes in theproperties of interstitial bone, osteon, and cement line. In addition,experimental observations demonstrated that recently formed os-teons arrested the cracks at the cement line more frequently com-pared with older osteons �5�. The experimental results combinedwith our simulation results may indicate that the cement line prop-erties in newly formed osteons are lower compared with the olderosteons facilitating crack deflection into cement lines.
Age-related changes in the bone matrix may lead to changes inthe material level properties affecting the process zone size in thebone. Previous studies have shown that if the process zone sizeincreases then the strength and toughness ratios for attaining crackdeflection also increase, requiring much weaker interfaces �24�. Ifaging affects the toughness, strength, and elastic modulus of bone,process zone size in bone may increase. As a result, the crackdeflection behavior may be adversely affected from these age-related changes and the crack deflection/penetration boundariesfound in this study may shift toward the left incorporating thiseffect.
Bone constantly renews itself through the bone remodeling pro-
cess leading to the formation of osteons and cement lines. Bone
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emodeling is believed to have evolved to repair damage in theone by targeting microcracks �85,86�. Some of the studies thatocused on the structure-function relationship of osteons sug-ested that osteonal bone may contribute to the bone remodelingrocess by altering the strain fields in the bone through cementine debonding. The change in strain fields due to debonding mayct as a trigger at the cellular level for remodeling to repair dam-ge �43�. In addition, the formation of osteons through bone re-odeling may provide resistance against fracture �87� and slow
own crack growth �2� by preventing the microcracks from attain-ng a catastrophic size before they can be replaced with the re-
odeling process. As a result, the osteonal bone structure and therack deflection/penetration behavior investigated in this studyay be an integral part of the bone remodeling process and the
one repair mechanism while providing additional mechanicalenefits.
In summary, this study demonstrated a computational approachor evaluating the influence of bone’s microstructure on crackrowth trajectory. The findings presented here are significant inetermining the role of bone microstructure in crack growth, instablishing a computational method that aids the evaluation ofechanical response of bone, and in providing insight into micro-
cale mechanisms in bone through finite element modeling thatannot be directly evaluated via experiments.
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