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Research Papers

Finite Element Modeling of Microcrack Growth in Cortical Bone

[+] Author and Article Information
Susan Mischinski

Department of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085

Ani Ural1

Department of Mechanical Engineering, Villanova University, 800 Lancaster Avenue, Villanova, PA 19085ani.ural@villanova.edu

1

Corresponding author.

J. Appl. Mech 78(4), 041016 (Apr 14, 2011) (9 pages) doi:10.1115/1.4003754 History: Received July 12, 2010; Revised February 14, 2011; Posted March 07, 2011; Published April 14, 2011; Online April 14, 2011

Bone is similar to fiber-reinforced composite materials made up of distinct phases such as osteons (fiber), interstitial bone (matrix), and cement lines (matrix-fiber interface). Microstructural features including osteons and cement lines are considered to play an important role in determining the crack growth behavior in cortical bone. The aim of this study is to elucidate possible mechanisms that affect crack penetration into osteons or deflection into cement lines using fracture mechanics-based finite element modeling. Cohesive finite element simulations were performed on two-dimensional models of a single osteon surrounded by a cement line interface and interstitial bone to determine whether the crack propagated into osteons or deflected into cement lines. The simulations investigated the effect of (i) crack orientation with respect to the loading, (ii) fracture toughness and strength of the cement line, (iii) crack length, and (iv) elastic modulus and fracture properties of the osteon with respect to the interstitial bone. The results of the finite element simulations showed that low cement line strength facilitated crack deflection irrespective of the fracture toughness of the cement line. However, low cement line fracture toughness did not guarantee crack deflection if the cement line had high strength. Long cracks required lower cement line strength and fracture toughness to be deflected into cement lines compared with short cracks. The orientation of the crack affected the crack growth trajectory. Changing the fracture properties of the osteon influenced the crack propagation path whereas varying the elastic modulus of the osteon had almost no effect on crack trajectory. The findings of this study present a computational mechanics approach for evaluating microscale fracture mechanisms in bone and provide additional insight into the role of bone microstructure in controlling the microcrack growth trajectory.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

(a) Schematics of a cortical bone section outlining the orientation of the osteons. Directions 1, 2, and 3 denote radial, circumferential, and longitudinal directions, respectively. (b) A sample microscopy image taken at the transverse plane (AA) showing the microstructure of human cortical bone from a 70-year-old donor. (c) Schematics of the finite element model (IB=interstitial bone, HC=Haversian canal, OS=osteon, and CL=cement line). Note that the black and white dotted lines are possible crack paths where cohesive elements were inserted. Black lines represent the initial crack for three different crack orientations. The arrows indicate the applied loading location and direction.

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Figure 2

(a) Traction-displacement relationship defining the cohesive zone model for both normal and shear modes. (b) 2D solid elements and the compatible 2D cohesive element with four nodes.

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Figure 3

(a) A sample finite element mesh demonstrating crack penetration into an osteon for 0 deg crack. (b) Stress contours showing the different stages of crack propagation for 0 deg crack penetrating into an osteon. (c) A sample finite element mesh demonstrating crack deflection into the cement line for 45 deg crack. (d) Stress contours showing the different stages of crack propagation for 45 deg crack that is deflected by the cement line. Note that the areas defined by the dotted lines in (a) and (c) show the regions for which the stress contours are shown in (b) and (d). In (a) and (c), the initial cracks in the models are represented by completely open surfaces and the new crack growth following the traction-displacement relationship is represented by cohesive elements shown in gray (red in online version). The black dotted lines in (b) and (d) indicate the crack front location and the solid black lines denote the osteon.

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Figure 4

Crack growth behavior for 45 deg and 90 deg crack orientations based on the ratio of the bone to cement line normal fracture properties. Gnc-b/Gnc-cl denotes the ratio of Mode I (opening) fracture toughness of bone to cement line and σnc-b/σnc-cl denotes the ratio of the tensile strength of bone to cement line. The strength and toughness ratios were obtained by keeping the interstitial bone and osteon properties constant (Table 2) while varying the cement line properties. The ranges of cement line properties used to obtain these ratios are marked on the graphs. The solid and hollow circles correspond to individual simulations for penetration and deflection, respectively. The solid lines represent the transition boundary between crack deflection and penetration based on the individual simulation data.

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Figure 5

Crack growth behavior for 45 deg and 90 deg crack orientations based on the ratio of the bone to cement line shear fracture properties. Gsc-b/Gsc-cl denotes the ratio of Mode II (shear) fracture toughness of bone to cement line and σsc-b/σsc-cl denotes the ratio of the shear strength of bone to cement line. The strength and toughness ratios were obtained by keeping the interstitial bone and osteon properties constant (Table 2) while varying the cement line properties. The ranges of cement line properties used to obtain these ratios are marked on the graphs. The solid and hollow circles correspond to individual simulations for penetration and deflection, respectively. The solid lines represent the transition boundary between crack deflection and penetration based on the individual simulation data.

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Figure 6

Crack driving force versus crack extension for 0 deg, 45 deg, and 90 deg cracks. Note that this plot is for Mode I fracture toughness ratio of Gnc-b/Gnc-cl=1 and tensile strength ratio of σnc-b/σnc-cl=1. The same behavior was observed for other strength and fracture toughness ratios.

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Figure 7

Crack driving force versus crack extension for crack penetration into the osteon and crack deflection into the cement line. Note that the crack is oriented at 45 deg.

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Figure 8

Crack growth behavior for three crack lengths oriented at 45 deg based on the ratio of the bone to cement line fracture properties. Gnc-b/Gnc-cl denotes the ratio of Mode I (opening) fracture toughness of bone to cement line and σnc-b/σnc-cl denotes the ratio of the tensile strength of bone to cement line. The strength and toughness ratios were obtained by keeping the interstitial bone and osteon properties constant (Table 2) while varying the cement line properties. The ranges of cement line properties used to obtain these ratios are marked on the graphs. Note that the values reported in the figures denote the amount of crack extension. The trendline represents the average of the deflection and penetration data sets.

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Figure 9

Crack driving force versus crack extension for three crack lengths oriented at 45 deg. Note that this plot is for Mode I fracture toughness ratio of Gnc-b/Gnc-cl=1 and tensile strength ratio of σnc-cl/σnc-cl=3.5. The same behavior was observed for other strength and fracture toughness ratios. Note that the crack extension values shown in the figure (182 μm, 253.5 μm, and 532 μm) correspond to initial crack lengths of 425 μm, 353.5 μm, and 75 μm, respectively.

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Figure 10

Crack growth behavior for 45 deg crack with two different osteon properties. The model with Osteon 1 has the same cohesive properties for both the osteonal and interstitial bones. In the second model, the osteon (Osteon 2) has 40% lower strength and 40% higher toughness than the interstitial bone. Gnc-b/Gnc-cl denotes the ratio of Mode I (opening) fracture toughness of bone to cement line and σnc-b/σnc-cl denotes the ratio of the tensile strength of bone to cement line. The strength and toughness ratios were obtained by keeping the interstitial bone properties constant (Table 2) while varying the cement line properties. The ranges of cement line properties used to obtain these ratios are marked on the graphs. The trendline represents the average of the deflection and penetration data sets.

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