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Research Papers

Scale and Boundary Conditions Effects on the Apparent Elastic Moduli of Trabecular Bone Modeled as a Periodic Cellular Solid

[+] Author and Article Information
Congyu Wang

Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, H3G 1M8, Canadawangcongyu@hotmail.com

Liang Feng

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801lfeng4@illinois.edu

Iwona Jasiuk

Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801ijasiuk@illinois.edu

J Biomech Eng 131(12), 121008 (Nov 17, 2009) (11 pages) doi:10.1115/1.4000192 History: Received August 13, 2008; Revised August 26, 2009; Posted September 10, 2009; Published November 17, 2009; Online November 17, 2009

We study apparent elastic moduli of trabecular bone, which is represented, for simplicity, by a two- or three-dimensional periodic cellular network. The term “apparent” refers to the case when the region used in calculations (or specimen size) is smaller than a representative volume element and the moduli depend on the size of that region and boundary conditions. Both the bone tissue forming the network and the pores (represented by a very soft material) are assumed, for simplicity, as homogeneous, linear elastic, and isotropic. In order to investigate the effects of scale and boundary conditions on the moduli of these networks we vary the specimen size and apply four different boundary conditions: displacement, traction, mixed, and periodic. The analysis using periodic boundary conditions gives the effective moduli, while the displacement, traction, and mixed boundary conditions give apparent moduli. The apparent moduli calculated using displacement and traction boundary conditions bound the effective moduli from above and below, respectively. The larger is the size of the region used in our calculations, the closer are the bounds. Our choice of mixed boundary conditions gives results that are very close to those obtained using periodic boundary conditions. We conduct this analysis computationally using a finite element method. We also investigate the effect of mismatch in elastic moduli of bone tissue and soft fill, trabecular bone structure geometry, and bone tissue volume fraction on the apparent elastic moduli of idealized periodic models of trabecular bone. This study gives guidance on how the size of the specimen and boundary conditions (used in experiments or simulations) influence elastic moduli of cellular materials. This approach is applicable to heterogeneous materials in general.

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Copyright © 2009 by American Society of Mechanical Engineers
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Figures

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Figure 10

Comparison of the apparent elastic stiffness components (a) C1111app and (b) C1212app for two dimensions and three dimensions obtained using displacement and traction boundary conditions and Unit cell 1

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Figure 1

(a) SEM image of trabecular bone; (b) idealized 2D periodic model of trabecular bone

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Figure 4

Apparent stiffness components as functions of Eb/Es (Eb=13GPa, νb=νs=0.3, and bone tissue volume fraction 20%) (a) C1111d, C1212d, and C1122d; (b) C1111t, C1212t, and C1122t

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Figure 5

Comparison of the apparent elastic stiffness components (a) C1111app, (b) C1212app, and C1122app obtained using displacement, traction, and mixed boundary conditions, and the effective elastic stiffness components obtained using periodic boundary conditions (Unit cell 1 was used)

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Figure 6

Comparison of the apparent elastic stiffness components (a) C1111app, (b) C1212app, and (c) C1122app obtained using Unit cells 1 and 2 and displacement and traction boundary conditions

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Figure 7

Unit cell of idealized periodic model of trabecular bone without sharp corners

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Figure 8

Comparison of the apparent elastic stiffness components C1111app, C1122app and C1212app, obtained using three different volume fractions (5%, 10%, and 20%) and displacement and traction boundary conditions, for four different window sizes (δ=δ0, 2δ0, 3δ0, and 6δ0) using Unit cell 1

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Figure 9

The 3D periodic model of trabecular bone with 3×3×3 unit cells for 20% bone tissue volume fraction

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Figure 2

Two different units cells: (a) Unit cell 1 and (b) Unit cell 2

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Figure 3

Three different window sizes δ0, 2δ0 and 3δ0

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