Study of the Linear Elastic Fracture Mechanics of Cortical ...€¦ · 1 Study of the Linear...

1

Study of the Linear Elastic Fracture Mechanics of

Cortical Bone as a composite material

André Filipe Pinto de Almeida

[email protected]

Instituto Superior Técnico, Universidade de Lisboa, Portugal

November 2016

Abstract— Studying the bone structure is an increasingly

appealing subject for engineers interested in understanding the

properties of this material. The comprehension of the mechanical

behavior of bone is advantageous, because it helps in the

perception of the risk of fracture and it can also serve as a basis

for the development of new and innovative composite materials.

In this work, a microscopic two-dimensional characterization of a

cortical bone is presented using the Abaqus software. The bone

properties required for the development of this work were taken

from literature relatively to bovine bones. Therefore, and in

order to analyze the propagation of preexisting cracks in the

material, an incremental deformation was applied to the bone in

order to calculate the fracture resistance of the model. It is also

studied the importance of the material properties of the osteons,

cement lines and Haversian channels, in the evaluation of how

these bone constituents can cause a disruption in the crack

propagation. In addition, microscopic observations were made to

the cortical bone of a bovine femur, in order to obtain a real

insight of what happens to the cracks on a fractured surface

through a three-point bending test. The results obtained clarify

what makes the cortical bone in a composite material so resistant

to crack propagation, this is because of the observed attraction

that the crack feels for osteons lying in its path.

Keywords— Cortical Bone, Abaqus, Cracks, Osteons, Cement

Lines, Harversian Canals

I. INTRODUCTION

This paper presents an approach to the study of the bovine bone microstructure, structurally identical to human bone, through numerical analysis of 2D models developed using the Abaqus software, where it is implemented a version of the extended finite elements method. The main objective is to understand why the cortical bone is so resistant to fracture.

Several studies of finite elements to single osteons have already been made, in particular by Prendergast and Huiskes, [1], which numerically modeled a single osteon and investigated the relationship between the damage and the applied stress in order to verify if micro lesions can influence local stress fields in the cortical bone microstructure. Dong et al., [2], created a generalized method for estimating the elastic modulus of a composite material reinforced by fibers, this model had its usefulness later on, when it was necessary to determine the dependence of the elastic properties of cortical bone on its porosity. Similarly, Hogan et al., [3], developed a composite model, reinforced by fibers, that was able to represent the Haversian system, considering the osteons as

fibers, the interstitial tissue as a matrix and the cement lines as interface between osteon/matrix. In other studies, carried out by Budyn Hoc, [4], and Abdel-Wahab et al., [5], several multiscale models were developed with extended elements, considering the multiple propagation of cracks in cortical bone when it is subjected to applied stresses. In another experiment, Raeisi Najafi et al., [6], developed a finite model of cortical bone, in microscale, able to relate the interaction between osteons and micro cracks.

All the models implemented have elastic-linear modules, so it was only considered the study of the linear elastic fracture mechanics (LEFM). All of the data used to characterize the constituent materials of the bovine cortical bone was taken from literature. Experimental tests have also been made to bovine cortical bone.

Section II introduces the theoretical concepts. Section III features the case study. In Section IV, the results are presented and discussed. In section V is shown a comparison between the numerical and experimental results. Finally, in Section VI, the conclusions of this work are summarized.

II. TEORIC CONCEPTS

A. Linear Elastic Fracture Mechanics (LEFM)

This theory considers the effect of the stress in front of the crack and assumes approximately the crack as perfectly elastic, with stresses theoretically infinite on its front. The basic principles of these concepts are discussed in this section.

B. The Crack.

When cracks are subjected to loads, they tend to behave as a notch in a material, or as a stress concentrator. The front angle of the crack is extremely small; its radius tends to zero. This geometry causes the appearance of critical stress concentrations in its front, enabling the propagation of the crack. These stresses enhance the appearance of an area where plastic deformation occurs. This plastic zone appears because of the behavior of the material when it is subjected to high stresses, adding complexity to the problem in question. However, using the LEFM theory, the area where plasticity exists is not accounted for, [7], and the stress field resembles an ideal crack, away from what happens in reality, [8] [9] [10] [11]. To overcome this result, there are limits on the plastic zone where the ideal crack has the same behavior of a real

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fissure, [12], however, these limits were not considered in this work.

C. Fracture Modes and the StressIintensiti Factor

The stress intensity factors are used to express the existing stress field in a crack when comparing with other cracks. The critical value is called fracture toughness or 𝐾𝐶 . There are three distinct crack propagation modes corresponding to three different loading modes. Mode I due to normal stresses, enables a severe propagation of the crack through separation surfaces. The Mode II is an inplane shear mode where the crack surfaces slide apart perpendicular to the crack. Mode III refers to torsional loads, enabling the tear of the surfaces of the crack, one under the other. The three modes can be seen in Figure II.1.

Figure II.1: Load Types.

The equation of the total tensile stress field in the crack front area is defined as follows:

Figure II.2 shows the representation of the crack front coordinates.

Figure II.2: Crack front polar coordinates.

The stresses in the crack front are displayed in equations

(II-2) to (II-4):

𝜎𝑥𝑥 = 𝐾𝐼𝑓𝐼(𝑥𝑥)(𝑟, 𝛼) =

=𝐾𝐼

√2×𝜋×𝑟× 𝑐𝑜𝑠 (

𝛼

2)×[1 − 𝑠𝑖𝑛(

𝛼

2)×𝑠𝑖𝑛(

3𝛼

2)]

(II-2)

𝜎𝑦𝑦 = 𝐾𝐼𝑓𝐼(𝑦𝑦)(𝑟, 𝛼) =

=𝐾𝐼

√2×𝜋×𝑟× 𝑐𝑜𝑠 (

𝛼

2)×[1 + 𝑠𝑖𝑛(

𝛼

2)×𝑠𝑖𝑛(

3𝛼

2)]

(II-3)

𝜏𝑥𝑦 = 𝐾𝐼𝑓𝐼(𝑥𝑦)(𝑟, 𝛼) =

=𝐾𝐼

√2×𝜋×𝑟× 𝑐𝑜𝑠 (

𝛼

2)×𝑠𝑖𝑛(

𝛼

2)×𝑐𝑜𝑠(

3𝛼

2)

(II-4)

where:

𝐾𝐼 , 𝐾𝐼𝐼, 𝐾𝐼𝐼𝐼 are the stress intensity factors for Modes I, II and III, respectively;

𝑟 is the distance from the crack tip and α is the angle of the crack tip. This values represent the polar coordinates of the model;

𝜎𝑧𝑧 = 0, for plane stress;

𝜎𝑧𝑧 = ν(𝜎𝑥𝑥 + 𝜎𝑦𝑦), for plane strain, where ν is the

Poisson coefficient;

considering only the mode I, the rest of the components of equation (II-1) are zero.

D. Integral J

The geometry of the contour is made to enclose the crack

front, connecting the two crack surfaces to generate a closed

area. The outline does not need to contract towards the crack,

it only needs to be discriminated where the position of the

crack front is located.

This is due to the fact that the integral J is completely

independent of the chosen contour, assuming that the material

is elastic and in the absence of forces applied to the body or

stresses on the surfaces of the crack, [13]. An example of a

contour is represented in Figure II.3.

Figure II.3: Two-dimensional closed contour.

The full two-dimensional contour can be represented by

equations (II-5) and (II-6), in order to display the closed

contour surrounding the crack front, [13]:

𝐽 = ∮ 𝑚 ∙ 𝐻 ∙ �̅�𝑑Γ

𝐶+𝐶++Γ+𝐶−

(II-5)

and

𝐻 = 𝑊 ∙ 𝐼 − 𝜎 ∙

𝜕𝑢

𝜕𝑥 (II-6)

where:

𝛤 is the contour that contains the crack front;

𝑚 is the normal vector towards the inside of the

contour;

𝑞 is the unity vector in the virtual direction of the

propagation of the crack;

𝐶 is an extension of 𝛤 that enclosures the crack;

𝐶+ e 𝐶− represent, respectively, the upper contour and

the lower contour along the crack;

𝑊 is the energy of elastic deformation;

𝐼 is the identity tensor;

𝜎 is the stress tensor;

𝑢 is the displacements vector.

𝜎(𝑖𝑗)𝑇𝑂𝑇𝐴𝐿 = 𝐾𝐼𝑓𝐼(𝑖𝑗)(𝑟, 𝛼) + 𝐾𝐼𝐼𝑓𝐼𝐼(𝑖𝑗)(𝑟, 𝛼)

+ 𝐾𝐼𝐼𝐼𝑓𝐼𝐼𝐼𝑖𝑗(𝑟, 𝛼)

(II-1)

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E. Classical approach to the Stress Intensity Factor

calculation

For the two-dimensional case studies, in which

quadrilateral elements collapse into triangles, an adjustment of

the nodes of its middle position to position 1/4 is required. The

reason for this amendment is to modify the form of the mesh

element, introducing the correct singularity. The crack front

elements, with the nodes in the 1/4 position, exhibit a 1 √𝑟⁄

singularity corresponding to a deformation of an elastic

member, thereby creating a greater level of accuracy due to

the fact that the analytical solution contains the same term.

The planning of the mesh is depicted in Figure II.4.

Figure II.4: Typical mesh form.

F. eXtended Finite Element Method

The extended finite element method (XFEM) developed by

Ted Belytschko, [14], is an evolutionary advance of the classic

method based on the concept of partition units, where the sum

of shape functions is equal to 1, [15]. This method adds pre-

known information of a given problem to the classic finite

element method, enabling the representation of discontinuities

and singularities regardless of the mesh. This is a powerful

method that enables the case of crack propagation, since it is

not necessary to have a continuous updating mesh as the crack

grows, making possible the free growth of the crack without

having to subject the crack front to a specific mesh.

III. CASE STUDY

The numerical simulations on this paper are intended to confirm two theoretical stages in the process of capturing a crack initiated in interstitial bone. A first phase in which the instability of the crack increases, the propagation of the crack deepens and directs directly to less rigid osteons. The second phase begins as soon as the crack reaches the osteon zone and becomes stuck in the cement line or somewhere inside the osteon. This will be tested for various stiffness values of all the constituents of the cortical bone.

A. Cortical Bone Constitution

In this paper, two-dimensional models of cortical bone with 0.7𝑚𝑚 width and 0.525𝑚𝑚 in height, [5], were considered. The microstructure of the cortical bone area is represented by a composite of four distinct properties: interstitial tissue separating the osteons (hollow discs) from each other; the cement lines, which serve as interface between the interstitial tissue and osteons; and Haversian canals. The models used to represent the microstructure of the cortical bone are in accordance with the state of plane strain. The models are fixed on the lower part of the body, and an incremental deformation ranging between 1% and 2% was applied. All constituents of

the composite were considered homogeneous and isotropic in planes perpendicular to the bone axis, in addition, a perfect connection among all the elements was also considered. The mechanical properties used are shown in the Table 1 and Table 2:

Table 1: Inferior Values, [4], [5].

Symbol 𝑬[𝑮𝑷𝒂] 𝑷𝒐𝒊𝒔𝒔𝒐𝒏

Osteons O1 𝐸𝑜 = 11.510 0.170

Cement line L1 𝐸𝑐 = 8.632 0.490

Table 2: Superior Values, [3], [16], [17].

Symbol 𝑬[𝑮𝑷𝒂] 𝑷𝒐𝒊𝒔𝒔𝒐𝒏

Osteons O2 𝐸𝑜 = 21.1 ± 2 0.3

Interstitial Matrix M2 𝐸𝑚 = 25.1 ± 1.6 0.3

Cement line L2 𝐸𝑐 = 30 0.3

IV. RESULTS

Model 1, presented in Figure IV.1, shows the behavior of the crack, with an initial crack with a length of 0.1𝑚𝑚, considering the material properties shown in Table 3, which were used for analysis 1.

Figure IV.1: Model 1, osteons with the same properties.

Table 3: Material properties of analysis 1.

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sis

1

Case Osteons

[𝑮𝑷𝒂]

Interstitial

Matrix

[𝑮𝑷𝒂]

Cement

lines

[𝑮𝑷𝒂] 1 21.1

25.1

30

2 11.510 30

3 11.510 8.632

4 21.1 8.632

1FI (inclined

crack) 21.1 30

5 21.1 −

Due to the homogeneity of the osteons, regardless of the case, the crack propagates linearly without interruption, both to the left and to the right side of the model. There is no observable attraction of the crack by the osteons; there is no observable bend of the crack path either, even in the case where the crack was inclined at 20° to the horizontal. The changes of the properties in the various cases did not altered anything on the crack propagation behavior.

In Figure IV.2, the observable difference in the value of the maximum stress is noticeable throughout the cases. The points were maximum stress occur are in case 2 (𝜎𝑦 = 96.8𝑀𝑃𝑎)

and 3 (𝜎𝑦 = 97.6𝑀𝑃𝑎). This is because, in the case of case 2,

the osteons have a lower stiffness, giving the model a more flexible structural constitution as a whole, allowing the possibility of a greater elastic deformation in the osteons area, enabling the application of a higher total stress in the model.

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The only difference in case 3 is the Poisson's ratio and modulus of elasticity of the cement line. Reducing the stiffness of the cement line facilitates the deformation of this area, just as happened when it was considered only the osteons with lower stiffness. The gains are minimal but it is perceived that there is a greater resistance of the material to allow a slightly higher stress (𝜎𝑦 = 97.6 MPa). It follows that case 3 is the best

result for crack propagation between osteons.

Figure IV.2: Graphical results for cases 1 to 5.

Knowing that the crack propagates consistently on both sides in the previous analysis, as can be seen in Figure IV.3, the new analysis of the problem aims to disrupt the uniformity of the osteons and cement lines properties, creating a disparity between the right and left side of the model to verify if this imposed imbalance affects the propagation of crack over time.

Figure IV.3: Evolution of the crack in model 1 for osteons with the

same properties.

With this modification, the new model 1, represented in Figure IV.4, becomes more rigid in the left side than on the right side and although the crack has kept its horizontal propagation, this change introduced a discrepancy on the crack, by growing first for the left side until it reaches the end of the model. Only then the crack starts propagating again to the right. The characterization of the components for case 6 is displayed in the following Table 4.

Figure IV.4: Model 1, osteons with different properties

Table 4: Properties of case 6.

Ca

se Osteon.

(left)

[𝑮𝑷𝒂]

Osteon.

(right)

[𝑮𝑷𝒂]

Interstitial

matrix

[𝑮𝑷𝒂]

Cement.

Line

(left)

[𝑮𝑷𝒂]

Cement

Line

(right)

[𝑮𝑷𝒂]

6 21.1 11.510 25.1 30 8.632

It can be seen that from a certain point on, the growth of the crack towards the stiffer side of the model intensifies, and even stops the progression to the other side. The occurrence of this behavior is explained by the reduced elasticity of the osteons on the left side and because the crack always propagates along the same material. Since the osteons on the left side have higher stiffness, they tend to deform less than the less rigid osteons, enabling the occurrence of a larger deformation in the central zone of the model on the left side compared to the right side. The osteons on the right side, by being less stiff, allow the model to stretch more over this side than on the left side. The opening of the crack occurs at lower levels of deformation applied to the left than on the right side of this model.

Examining the graph of Figure IV.5 more closely, it’s relevant that similarly to what happened previously, the stress increased to its peak almost linearly (𝜎𝑦 = 87.7MPa). The

maximum stress point is lower than the one seen in Figure IV.2 for the cases 2 and 3 due to early rupture of the model on the left side before having exhausted the material resistance on the right side.

Figure IV.5: Graphical results of case 6.

0

20

40

60

80

100

0 0.002 0.004 0.006 0.008 0.01

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]

Case 1 Case 2 Case 3Case 4 Case 1FI Case 5

0

10

20

30

40

50

60

70

80

90

0 0.002 0.004 0.006 0.008 0.01

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]

Case 6

5

It is noticed that in the case of having a propagating crack between osteons, without intersecting them, the crack will tend to move to the more rigid side first, only to progress to the less rigid side after the full opening of the crack on the stiff side of the model, as can be seen in Figure IV.6.

Figure IV.6: Evolution of the crack in model 1 for osteons with

different properties.

In model 2, featured in Figure IV.7, the analysis focuses on the opposite case where osteons cross the crack path, serving as a barrier. The distinctive stiffness of each osteon and cement line of this model, as represented in Table 5, causes an imbalance similar to that observed for the model 1.

Figure IV.7: Model 2.


Although the crack has maintained its horizontal propagation, the discrepancy in the model caused the crack to grow firstly to the left, to the less rigid osteon, until reaching the Haversian canal. The crack only advances to the stiffest side, after reaching the Haversian canal of the osteon located on the left side. This is because the crack propagates in three different materials, as can be seen at Figure IV.8 starting its propagation on the interstitial matrix, moving in direction to the cement line and inside the osteon, being obvious that is more energy efficient, in this case the crack to propagate in direction of the least rigid material.

The graphical representation of Figure IV.9 clarifies this idea because, as can be seen, the osteon on left side deforms more to the same stress value as compared to the osteon on the right. The latter, being more rigid, deforms less and is always subject to a higher stress.

After a certain point, as the stress in the osteons continues to increase, the stress at the interstitial matrix decreases, confirming that, after the opening of the crack, it is on the

osteon located at the right side of the model where the maximum stress can be found.

Figure IV.8: Evolution of the crack in model 2.

Figure IV.9: Graphical results for case 7.

In model 3, showed in Figure IV.10, in addition to the osteons aligned with the progression of the crack, there are also other osteons in the model.


The cases made for this model allowed checking the growth of a horizontal crack, centered (Table 6) and decentered (Table 7) relatively to the central osteons, at a vertical distance of 0.0225𝑚𝑚 and 0.0275𝑚𝑚, downwards and 0.0175𝑚𝑚 and 0.0275𝑚𝑚 upwards. The main focus is to verify the crack behavior when passing the cement line and entering the

0

20

40

60

80

100

120

0 0.002 0.004 0.006 0.008 0.01

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]

Intersticia Matrixl Osteon on the Right

Osteon on the Left

An

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sis

2

Case

Osteon

(left)

[𝑮𝑷𝒂]

Osteon

(left)

[𝑮𝑷𝒂]

Interstitial

Matrix

[𝑮𝑷𝒂]

Cement

Line

(left)

[𝑮𝑷𝒂]

Cement

Line

(right)

[𝑮𝑷𝒂]

7 11.510 21.1 25.1 8.632 30

6

Haversian canal. Contrary to what was done until now, it was applied a strain of 2% relative to the height value of the model.


An

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sis

3

Case Osteons

[𝑮𝑷𝒂] Interstitial

Matrix [𝑮𝑷𝒂] Cement Lines

[𝑮𝑷𝒂] 8 21.1

25.1

30

9 11.510 30

10 11.510 8.632

11 21.1 8.632

12 21.1 −

The centered crack propagates horizontally, both to the left and to the right of the model, however it is observable that the crack propagates first towards the osteon on the right, as can been seen in Figure IV.11. Knowing that within the same case all osteons and cement lines have indistinct properties, this outcome proves that there is indeed an attraction visible towards the right side osteon, that is closer than the osteon on the left. The modification of the properties in the various cases did not change anything in the way of how the crack propagates.

Figure IV.11: Crack attraction towards the right osteon.

In the graphical analysis available in Figure IV.12, it is seen that in all the cases the stress increases linearly until reaching the yield stress, bending after this point until it reaches a peak stress. This peak reflects the moment of crack opening after the disruption of osteon on the right side and immediately before the opening of the osteon on the left, i.e., represents the moment before the start of the crack opening after reaching the two osteons. It is found that contrary to what happened in model 1, by reducing the stiffness of the osteons in this model, the overall strength also decreases. It is also observed that the cases where the osteons are type O1, deform more easily at lower stress values, resulting in a more deformable model. After crossing the maximum local point, the stress decreases to its minimum point. This drop in the stress values at increasing deformation values is justified by the crack propagating process. The instants in which the stress values are decreasing corresponds to the time the crack takes to fully propagate, until it reaches its maximum aperture for the stress levels involved.

From the moment that the deformability of the material, in the central part of the interstitial matrix, is exhausted and the existing stress stops being sufficient, it can be verified again an increase of the stress in the model, related with an increasing slope in the graph, associated to the increasing model deformation. The evolution of the crack is represented in Figure IV.13.

Analysis 4, exemplified in

Table 7, portrays the study of decentered cracks, focusing on the crack behavior since its initial length of 0.1𝑚𝑚 until its

final moment. The following Table 7 lists the properties tested in the various cases in analysis of the model 3.


Figure IV.13: Evolution of the centered crack in model 3.


An

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4 Case

Osteon

[𝑮𝑷𝒂]

Interstitial

Matrix

[𝑮𝑷𝒂]

Cement

lines

[𝑮𝑷𝒂]

Vertical

crack

coordinates

[𝒎𝒎] 13

21.12 25.1 30

𝑦 = 0.235

14 𝑦 = 0.24

15 𝑦 = 0.28

16 𝑦 = 0.29

The crack keeps propagating primarily towards the osteon on the right side, but, this time, there are three different situations throughout the cases. In the cases 13 and 16 shown in Figure IV.14 and Figure IV.17 its visible the attraction of the crack by the Haversian canal, however the crack is divided into two when approaching the canal, to the point of cracking its way into its interior, while at the same time, continuing to propagate until the end of the model. The detail of this occurrence is in display in Figure IV.18. This happens in the case 13 for the two sides and in the case 16 only to the left side. To the right of this last case, it is apparent that the crack stops propagating in the cement line and fails to enter into the osteon. In the remaining cases, shown in

0

20

40

60

80

100

0 0.005 0.01 0.015 0.02

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]Case 8 Case 9 Case 10

Case 11 Case 12

7

Figure IV.15 and Figure IV.16 of this analysis, the crack bends directly into the Haversian canal, stopping its propagation there.

Figure IV.14: Final instant of case 13.



Figure IV.17: Final instant of case 16

By investigating the details, it is verified that the curvature of the crack in direction to the Haversian canal is greater as the vertical distance of the crack from the midline of the model increases. This can be seen with higher emphasis on case 16, Figure IV.17, since here the crack passes at a distance of 0.0275𝑚𝑚 in relation to the centered crack. It is noticed by viewing these figures that, beyond the crack being drawn directly to the osteon that its closer located at the initial time, there is also an attraction for the Haversian canal from the moment that the crack crosses the cement line.

In Figure IV.17 and with greater detail in Figure IV.20, it is possible to see that the crack front is stuck in the contact surface between the cement line and the interior of the osteon. This case is explained by the momentary sliding of the crack

between the two materials, separating them, stopping at the instant that it is very difficult to continue the propagation horizontally, to the values of stiffness of these materials. In Figure IV.19 it is showed in detail the stress in the Haversian canal after the crack entry.

Figure IV.18: Detail of case 13.



When looking at the graph of Figure IV.21 there are perceptible distinctions among the cases. On case 13, it is apparent that because there is a fracture on both sides of the model, the stress tends to zero in the end, by ceasing to exist an applied strain. On the other hand, the difference between the cases 14 and 15 is the position of the crack relative to the central line, in that within a given limit of approach of the crack to the osteon, (in this case the limit is established between the line 𝑦 = 0.24𝑚𝑚 and 𝑦 = 0.28𝑚𝑚) this is the predictable result. At the end, it is observed that the crack is stopped by the Haversian canals, and from the point at which the crack starts to widen, the graphical slope in these two cases rises, becoming approximately equal to each other. In case 16, because the crack has been enclosed in the inner periphery of the cement line, the graphical slope has a higher value than for previous cases. To this contributes the fact that there is still enough deformation to be applied and also because the stress concentration in this area is much higher than the verified on the Haversian Canals.

8


After completing the analysis of the previous models, it follows the importance of introducing a random distribution of osteons to understand how this arrangement affects the propagation of the crack, as represented in Figure IV.22.


Both sides of the crack get close to the first osteons

considerably at the same time because the difference between

the distances is small and because they both have the same

properties. No explicit attraction of the crack is observable to

any of these first osteons, there is a slight bend of the crack

when approaching the osteon area, but it is almost

symmetrical at both sides, as can be seen at Figure IV.23, Figure

IV.24 and Figure IV.25. The alterations imposed by the changes

of the properties are only visible after the crack passes the first

osteon. The following Table 8 presents the material properties

of the cases done for this model.

Table 8: Material properties of analysis 5

An

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sis

5

Case Osteon

[𝑮𝑷𝒂] Interstitial Matrix

[𝑮𝑷𝒂] Cement

Lines [𝑮𝑷𝒂] 17 21.1

25.1

30

18 11.510 30

19 21.1 8.632

In the various cases, it is visible the bends that the crack suffers after the first osteon. The crack is drawn from a Haversian canal to the next as it propagates, as can be seen at Figure IV.23, Figure IV.24 and Figure IV.25. However, depending on the case, the crack may be arrested in the first osteon, such is case of case 17, where the crack is secured on the inner surface of the cement line, Figure IV.23.




When comparing the three cases it is perceived that they are quite distinct. While in case 17 the crack stops at cement line, as can be seen more in detail in Figure IV.26, in case 18 the crack propagates vertically to the Haversian canal, as seen in Figure IV.27. In case 19 the fact that the cement line is less rigid, contributes to the crack not getting stuck anywhere, propagating freely between the osteons, culminating in the rupture of the model.


0

20

40

60

80

100

120

140

160

180

0 0.005 0.01 0.015 0.02

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]

Case 13 Case 14 Case 15 Case 16

9


As seen in case 17, the crack is arrested in the cement line, this fact is reflected in the graph of Figure IV.28. In case 18, the crack propagates vertically, something only possible because the osteons of this case have a lower rigidity. The graphical result of this case reflects the instant when the crack reaches the Haversian canal, with a negative slope between 𝜎𝑦 =70𝑀𝑃𝑎 and 𝜎𝑦 = 55𝑀𝑃𝑎. In the case 19, the rigidity of the

cement line is lower and, because of this, it ceases to be a major obstacle to the propagation of the crack. The propagation after the first osteon is irregular, the crack advances for both the right and the left side of the model, but the propagation only depends on which side has an osteon closer. This action is due to the attraction that the crack has to the osteons as it approaches them, such as when the crack arcs to the passage of their surroundings.


V. EXPERIMENTAL RESULTS

The scanning electron microscope (SEM) was used because it

possesses the ability to produce high resolution images for

each sample and thus enables the evaluation of the structure of

each fractured surface. Various fracture surfaces were

photographed to acquire information about the trajectory of

propagation of micro cracks.

The samples were machined from a bovine femur. The cracks

were developed using a three-point bending test.

Figure V.1: Detail of case 18,

model 4.

Figure V.2: Detail of a crack

entering a Haversian canal in a

bovine sample.

Through the analysis of Figure V.1 and Figure V.2, it appears

that in both cases the crack propagated towards the interior of

the Haversian canal, being retained therein. In both cases, it

can be seen the effect that the osteons have on crack

propagation, confirming that the Haversian canals are indeed a

location where the crack propagation ends. It is also noted that

the way the crack reaches the canal depends on the mechanical

properties of the various structural components of the bone in

that location, and because of this reason there isn’t a

completely straight path during the final moments of

propagation.

Another example can be seen from the Figure V.3 and Figure

V.4, in which the crack passes through the interior of the

osteon and continues its propagation. This crack behavior

indicates that there are regions in the cortical bone where the

crack does not stop its propagation within the Haversian canal,

continuing its propagation out of osteon, crossing the cement

line again into the interstitial tissue. The reason for this

occurrence may be associated with the relation between the

mechanical properties of the osteon and the local deformation

that the bone was subjected.

Figure V.3: Detail of case 13,

model 3.

Figure V.4: Crack on the

periphery of an osteon of bovine

bone.

VI. CONCLUSIONS

The bone constitution makes him an extremely durable, relatively light, adaptable and multifunctional material. It is therefore an interesting material to replicate, in order to take advantage of its mechanical performance, in the development of new innovative composite materials.

With this in mind, this paper aimed to analyze the propagation of micro cracks in order to understand what makes the cortical bone so resistant to crack propagation. This was based on the study carried out using Abaqus software in section IV, where there were created four different models in order to

0

10

20

30

40

50

60

70

80

90

100

0 0.002 0.004 0.006 0.008 0.01

𝜎y

[𝑀𝑃𝑎

]

𝜀 [%]

Case 17 Case 18 Case 19

10

analyze the path of the propagation of preconceived cracks, with 0.1𝜇𝑚𝑚 long, It is clear that the objective was achieved.

By analyzing the results for model 1 it was verified that by changing the mechanical properties of bone constituents, specifically by decreasing the stiffness of the osteon and of the cement line, it is possible to achieve higher fracture resistance in the model. It was also possible to see that the path of the crack depends largely on the mechanical properties of the osteons, as well of the properties of the material is which the crack is propagating. As can be seen in section 4, the crack is drawn to the more rigid side of the model first, when passing between osteons without crossing directly with them in any way. Joining this result to that obtained in the analysis of model 2, where it is realized that the crack makes its way towards the less rigid osteon, it is clear that the behavior of the crack can be partly predicted.

When the crack propagates decentered to the Haversian canal, it is clear that because the crack no longer makes contact perpendicularly with the cement line, the latter can completely stop the crack propagation, disregarding the thickness of the cement line, as can be seen in model 3 and 4. This happens because the required conditions are met in terms of the crack approach to osteon, resulting in the crack front being trapped between the inner surfaces of the cement line and the osteon, separating these two materials and stopping at the moment when its position is sufficiently diverted that it is impossible to continue to propagate horizontally, for these levels of stiffness attributed to the osteon.. When the rigidity of osteon ceases to be an impediment to the propagation, the crack tends to propagate vertically breaking the osteon towards Haversian canal, thus stopping its propagation; this example can be seen in the model 4.

It is also possible to observe that once the crack surpasses the cement line and its front enters the inside of the osteon, there is a noticeable bending of the slot towards the Haversian canal. This is particularly noticeable where the crack is decentered relative to the Haversian canal, as in the case of the analysis 4. It is concluded that the crack prioritizes its propagation not only towards the less rigid osteons, but once it enters into these, it tends to bend towards Haversian canal of the osteon that is going through. This bending of the slot during propagation can also be seen in Model 4, where the crack tends to bend towards the osteons, creating bumps in its path.

In the images obtained from the scanning electron microscope in section 5, it is observed the crack propagation to the Haversian canals, as was the case in the numerical results of section 4.

Finally, the conclusion is that the number of osteons present in the crack path is a decisive factor in the ability the crack has to propagate, since with the increasing number of osteons, the smaller the space that the crack has without contacting with an osteon. It is consequently essential to have a balance between the number of osteons and interstitial matrix area, so that the bone is not excessively weak and keeps its mechanical properties, while preventing the free progression of cracks. Therefore, the presence of osteons in the crack path forms the primary barrier to its propagation, either by the presence of the cement line, which can stop the propagation of the crack by itself or the appeal that the crack has for the Haversian canal, which also prevents its propagation.

VII. REFERENCES

[1] P. J. Prendergast e R. Huiskes, “Microdamage and osteocyte-

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[2] X. N. Dong, X. Zhang, Y. Y. Huang e X. E. Guo, “A

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Mechanical Sciences, vol. 47(6), 2005, pp. 922-940.

[3] H. A. Hogan, “Micromechanics modeling of Haversian cortical

bone properties,” em Journal of Biomechanics, vol. 25(5), 1992,

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[4] É. Budyn e T. Hoc, “Multiple scale modeling for cortical bone

fracture in tension using X-FEM,” em Revue européenne de

mécanique numérique, vol. 16, 2007, pp. 213-236.

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[14] T. Belytschko e T. Black, “Elastic crack growth in finite

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