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TECHNICAL PAPERS

A Cellular Solid Model for Modulus Reduction Due to Resorption of Trabeculae in Bone

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

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

A. M. Kraynik

Sandia National Laboratories, Albuquerque, NM 87185-0834

J Biomech Eng 122(5), 511-515 (May 12, 2000) (5 pages) doi:10.1115/1.1289996 History: Received January 23, 2000; Revised May 12, 2000
Copyright © 2000 by ASME
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Figures

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Section through human lumbar vertebrae (L1) showing thinning and resorption of trabeculae: (a) 55-year-old female (volume fraction of bone=0.17), and (b) 86-year-old female (volume fraction of bone=0.07)
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(a) Two-dimensional random Voronoi honeycomb (after Silva et al. 4). (b) Young’s modulus and compressive strength of two-dimensional Voronoi honeycombs plotted against reduction in relative density. The curve labeled “intact” gives the results for reductions in relative density from uniform thinning of the cell walls. The curve labeled “struts removed” gives the results for reductions in relative density from the random removal of cell walls. The Young’s modulus and compressive strengths are normalized by the values for the intact honeycomb (after Silva and Gibson 5).
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Overview of software used to generate the three-dimensional Voronoi structures
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An example of an intact three-dimensional, open-cell Voronoi structure. This structure had 311 struts filling a repeating unit with 3×3×3 cells.
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A three-dimensional, open cell Voronoi structure with 20 percent of the struts removed. In the finite element analysis, struts are removed by deleting one of the seven beam elements in a strut. The figure shows the resulting structure.
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Young’s modulus normalized by that of the intact structure, E*/E0, plotted against percent reduction of relative density. The upper curve is for the intact three-dimensional Voronoi structure, which has had the density reduced by uniform thinning of the cell walls. The lower curve is for the three-dimensional Voronoi structure, which has had the density reduced by removal of struts.
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Comparison of normalized Young’s modulus, E*/E0, plotted against percent density reduction by removal of struts for a random, three-dimensional Voronoi structure, a regular three-dimensional tetrakaidecahedral structure, a random two-dimensional Voronoi structure, and a regular two-dimensional hexagonal structure. In both two-dimensional and three-dimensional structures, the reduction in modulus is similar for the random and the regular structures.

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