Research Papers

Bovine Cortical Bone Stiffness and Local Strain are Affected by Mineralization and Morphology

[+] Author and Article Information
É. Budyn1

Department of Mechanical Engineering,  University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607ebudyn@uic.edu

J. Jonvaux

Department of Mechanical Engineering,  University of Illinois at Chicago, 842 W. Taylor Street, Chicago, IL 60607jjonva2@uic.edu

C. Funfschilling

 Innovation and Research SNCF, 45 rue de Londres, 75379 Paris Cedex 08, Francechristine.funfschilling@sncf.fr

T. Hoc

Department of Mechanical Engineering,  LTDS UMR 5513/MSSMAT UMR 8579, Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully Cedex, Francethierry.hoc@ec-lyon.fr


Corresponding author.

J. Appl. Mech 79(1), 011008 (Dec 08, 2011) (12 pages) doi:10.1115/1.4004644 History: Received March 03, 2010; Revised July 12, 2011; Published December 08, 2011; Online December 08, 2011

A multiscale analysis of the mechanical behavior of bovine Haversian cortical bone is presented in the frame-work of linear elasticity. Cortical bone displays a complex microstructure that includes four phases: Haversian canals, osteons, cement lines, and interstitial bone. Based on close experimental observations, a Monte Carlo algorithm is implemented to build the natural bone composite microstructure. To represent the hierarchical nature of bone, the algorithm incorporates macroscopic morphological components, such as its porosity and osteonal volume fraction, as well as microscopic parameters, such as the characterized distributions of the osteons diameters. Bone local mechanical properties are measured by nanoindentation and microextensometry. The three-dimensional microstructures are discretized by a finite element method in order to evaluate the representative volume element of bovine cortical bone. The numerical model calculates the macroscopic bulk and material Young’s moduli and describes the local stress and strain. How geometrical or mechanical factors affect bone failure is investigated through a comparison of the macroscopic anisotropy and local strain to experimental data.

Copyright © 2012 by American Society of Mechanical Engineers
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Figure 1

Definitions of morphometric parameters in light microscopy (LM) observations of a transverse section of bovine Haversian cortical bone: (a) phase denominations, (b) phase contours and representative measurements.

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Figure 2

(a) Back-scattered electron microscopy (BSEM) observation of a transverse section of bovine cortical bone. (b) Schematic of a transverse section of bovine cortical bone describing the morphometric model parameters.

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Figure 3

(a) Experimental distributions of osteon and Haversian canal diameters where fso and flo denote the proportions of small and large osteons, respectively, and fsc and flc are the proportions of small and large Haversian canals, respectively. (b) Experimental distributions of osteon Young’s moduli measured by nanoindentation in the transverse plane ET and along the axis of the osteons EL .

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Figure 4

Algorithm for porosity adjustment. The temporary Haversian canals (related to P′) are in light blue; the final Haversian canals (related to P) are in dark blue.

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Figure 5

(a) Convergence study of (a) transverse stiffness values and (b) longitudinal stiffness values with respect to the mesh size

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Figure 6

Correlation coefficients between the experimental diameter distribution and the numerical diameter distribution for all sets of microstructures of different sizes L. The osteonal and Haversian diameters are presented.

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Figure 8

(a) Macroscopic transverse Young’s moduli and (b) macroscopic longitudinal Young’s moduli versus L. The mean and standard deviation are calculated for ETu¯,S and ELu¯,S. Note ETt¯,S and ELt¯,S are calculated for few microstructures of mean macroscopic moduli when Dirichlet boundary conditions are applied.

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Figure 9

Definition of the external normal vector n to the Haversian canal surface Γc and the contours Cc to calculate the Young’s modulus (a) under transverse compression, (b) under longitudinal compression

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Figure 10

Comparison of (a) transverse and (b) longitudinal Young’s moduli based on surface and volume calculations

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Figure 11

(a) Experimental local transverse strain along the x1 -axis in the real bone. (b) Average Green–Lagrange compressive strain of the osteons versus transverse Young’s modulus ET . “RVE” and “Leq ” denote the osteonal strain distributions in a microstructure of RVE size and of a microstructure of equivalent size, respectively. “Inside real” and “outside real” denote the strain distributions of the osteons inside the observation window and of the osteons that have been excluded because partially cut, passed through by a Volkmann canal or eaten by another osteon.

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Figure 12

Local E22 strain distributions in the different constitutive bone phases: interstitial bone, osteons, cement lines. The strains are presented for two MC microstructures of RVE size and identical geometrical arrangements that either contain cement lines or are deprived of them.

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Figure 13

Normal strain in the loading direction (a) three-dimensional model under transverse compression in the x2 -axis direction. (b) Three-dimensional model under longitudinal compression in the x3 -axis direction.

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Figure 7

(a) Transverse boundary conditions, u¯2 is negative nonzero on the top,u¯2 = 0 on the bottom,u¯3 = 0 on the back lines (x2=x3=0) and (x3  = 0 and x2  = L), u¯1=0 on the mid-points (black dots). (b) Longitudinal boundary conditions, u¯3 is negative nonzero on the front,u¯3 = 0 on the back,u¯2=0 on the bottom lines (x2  = x3  = 0) and (x3  = t and x2  = 0), u¯1=0 on the mid-points (black dots).




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