A Phenomenological Model for Predicting Fatigue Life in Bovine Trabecular Bone

[+] Author and Article Information
P. Ganguly, T. L. A. Moore, L. J. Gibson

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139Exponent Failure Analysis Associates, Inc. Philadelphia, PA 19104

J Biomech Eng 126(3), 330-339 (Jun 24, 2004) (10 pages) doi:10.1115/1.1762893 History: Received August 01, 2003; Revised October 27, 2003; Online June 24, 2004
Copyright © 2004 by ASME
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A characteristic stress-strain curve for bovine trabecular bone under monotonic compression. The initial modulus (E0m) was calculated by the best-fit line in the strain range 0.001–0.004. The secant modulus (Esecm) at higher strains were calculated by the ratio of the stress and strain.
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Calculation of the secant modulus for a given cycle for a sample tested at a maximum compressive normalized stress (σ/E0f) of 0.0066.
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Variation of the normalized secant modulus with increasing strain for representative specimens under monotonic compression. The secant modulus variation for each sample has been represented by the best-fit linear plot.
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Variation of secant moduli with increasing strain for the trabecular bone samples at normalized stress (Δσ/E0f) ranges of (a) (0.005, 0.006], (b) (0.006, 0.007] (c) (0.007, 0.008] and (d) (0.008, 0.009]. The experimental points have been identified by the sample numbers. The upper and lower bounds from the monotonic tests have been included for comparison, and shown in the plots by the solid lines (upper and lower).
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Plots showing the accumulation of total residual strain with increasing number of cycles, for typical samples in the (0.005, 0.006], (0.006, 0.007], (0.007, 0.008] and (0.008, 0.009] normalized stress ranges. (a) The initial part of the residual strain plots, showing the primary and the secondary phases. (b) The secondary residual strain evolution at large number of cycles. The plot for Δσ/E0=(0.008,0.009] has been omitted from (b), the fatigue life at that stress level being comparable to the number of cycles shown in (a). Note that the residual strain analysis for the same normalized stress range shown in (a) and (b) corresponds to different specimens.
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Experimentally observed variation in the secondary residual strain accumulation rate with changes in normalized stress. The best-fit line was obtained by fitting a power-law curve on the entire experimental data (open, solid and gray squares). The points that appear to be bounding the data were identified (upper bound: solid squares, lower bound: gray squares) and separate power-law curves were fit to obtain the upper and lower bound variation equations.
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Upper and lower bounds of the decrease of the normalized secant modulus predicted by the numerical model, for normalized stress ranges (a) (0.005, 0.006], (b) (0.006, 0.007], (c) (0.007, 0.008] and (d) (0.008, 0.009]. Note that upper and lower bound responses for both f1 and f2 have been considered in determining the model responses. The experimental points, identified by their sample numbers, have been included for comparison.
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Numerical model prediction of the upper and lower bounds of the normalized secant modulus reduction, at normalized stress ranges (a) (0.005, 0.006] and (b) (0.008, 0.009]. Note that for the model bound constructions, upper and lower bound responses for f1 and a median response for f2 (see Table 1) was assumed. The experimental points, identified by their sample numbers, have been included for comparison.
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The predicted S-N curve for bovine trabecular bone. The dotted line signifies the number of cycles corresponding to the endurance limit. The lower and upper bounds for the endurance limit are shown in the figure. The experimental data reflect cyclic compressive loadings of waisted samples of bovine trabecular bone 10.




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