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TECHNICAL PAPERS

Analysis of Crack Growth in a 3D Voronoi Structure: A Model for Fatigue in Low Density Trabecular Bone

[+] Author and Article Information
A. M. Makiyama, S. Vajjhala, L. J. Gibson

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

J Biomech Eng 124(5), 512-520 (Sep 30, 2002) (9 pages) doi:10.1115/1.1503792 History: Received December 01, 2001; Revised May 01, 2002; Online September 30, 2002
Copyright © 2002 by ASME
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Figures

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(a) Construction of a two-dimensional Voronoi structure. (b) A Voronoi honeycomb with about 500 cells (Reproduced from Silva et al. (1995), with the kind permission of Elsevier Science.)
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(a) A three-dimensional open-cell Voronoi structure with 27 cells and roughly 300 struts. In our models, the strut diameter was 150μm and the average length of the struts was 588 μm. (b) A micro-computed tomography image of trabecular bone with a relative density of 9.4%. (Reproduced with the kind permission of Dr. Ralph Muller, ETH, Zurich.)
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Flow chart giving an overview of the fatigue analysis.
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The three crack shapes used in our analysis. The crack shape is characterized by the parameter α.
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Normalized stress intensity factors, ΔK/σπa plotted against relative crack length, a/R, for different crack shapes, for (a) tension (b) bending and (c) torsion. (Reproduced from Levan and Royer, 1993 (Reproduced from Int. J. Fracture with the kind permission of Kluwer Academic Publishers.)
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(a) Young’s modulus, normalized by the initial modulus, E/Eo, plotted against number of cycles of fatigue loading, N on a linear scale. The data shown are for Δσ/Eo=0.006 and α=0.5, for structure A. (b) Young’s modulus, normalized by the initial modulus, E/Eo, plotted against number of cycles of fatigue loading, N on a semi-log scale. Open symbols represent experimental results 11, closed symbols represent model results.
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Average of the normalized Young’s modulus, E/Eo, for the three Voronoi structures plotted against number of cycles of loading, N. The error bars represent one standard deviation. All results for α=0.5.
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Normalized stress range, Δσ/Eo, plotted against the number of cycles to failure, Nf. Failure is defined as the number of cycles to give E/Eo=0.90. (a) For relative densities, ρ*s=1, 5 and 10%, for ao=33 μ,α=0 for Voronoi structure A. (b) For initial crack length, ao=16, 33, 66 μ, for ρ*s=5% and α=0 for Voronoi structure A. (c) For crack shapes, α=0, 0.5, 1.0, for ρ*s=5% and ao=33 μ for Voronoi structure A. (d) For three different Voronoi structures, for ρ*s=5%,ao=33 μ and α=0.
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Stress range plotted against number of cycles to failure, for ρ*s=1, 5 and 10%, for ao=33 μ and α=0 for Voronoi structure A
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(a) Number of cycles to failure, Nf plotted against relative density, ρ*s for Δσ/Eo=0.4%, 0.6%, 1.0%, for ao=33 μ,α=0 for Voronoi structure A. (b) Number of cycles to failure, Nf plotted against normalized stress range, Δσ/Eo, for α=0, 0.5, 1.0, for ao=16 μ, and ρ*s=5% for Voronoi structure A. (c) Number of cycles to failure, Nf, plotted against α for Δσ/Eo=0.4%, 0.6%, 1.0%, for ρ*s=5%,ao=33 μ, for Voronoi structure A. (d) Number of cycles to failure, Nf, plotted against crack size, ao, for Δσ/Eo=0.4%, 0.6%, 1.0%, for ρ*s=5%, and α=0 for Voronoi structure A.
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Normalized stress range, Δσ/Eo plotted against number of cycles to failure, Nf comparing numerical model for open cell Voronoi structure and experimental data for bovine trabecular bone 1011. Finite element model with ao=33 μ and α=0 for Voronoi structure A.

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