Nonlinear Behavior of Trabecular Bone at Small Strains

[+] Author and Article Information
Elise F. Morgan, Oscar C. Yeh, Wesley C. Chang

Orthopædic Biomechanics Laboratory, Department of Mechanical Engineering, University of California, Berkeley, CA 94720

Tony M. Keaveny

Orthopædic Biomechanics Laboratory, Department of Mechanical Engineering, Department of Bioengineering, University of California, Berkeley, CA 94720Department of Orthopædic Surgery, University of California, San Francisco, CA 94143

J Biomech Eng 123(1), 1-9 (Oct 16, 2000) (9 pages) doi:10.1115/1.1338122 History: Received September 30, 1999; Revised October 16, 2000
Copyright © 2001 by ASME
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A tensile stress–strain curve for trabecular bone from the human proximal tibia showing typical nonlinear behavior. The open circle represents the yield point determined by the 0.2 percent offset technique when a strain range of 0.10–0.40 percent is used to define the initial modulus, and the closed circle represents the yield point corresponding to a strain range of 0–0.10 percent. The former strain range was used by Chang et al. 18, and the latter in the current study. As evident, the use of different strain ranges in defining the elastic modulus can result in different yield points. Yield strain can be particularly sensitive to this change in modulus definition.
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For each anatomic site, specimens were machined such that the principal trabecular orientation was aligned with the longitudinal axis of the core. (A) One–two vertebral specimens were cored in the superior–inferior direction from each vertebral body. At each stage of machining the proximal tibiae and femora, contact radiographs were used to identify regions of homogeneous, well-oriented trabecular bone. These regions were then excised in the subsequent machining stage. (B) A 15-mm-thick slab of each proximal femur aligned with the femoral neck axis yielded one specimen from each of the greater trochanteric and neck regions. (C) Medial and lateral portions of each proximal tibia were separated by a sagittal cut (a medial–lateral view of the lateral half of a right tibia is shown here). A 15-mm-thick slab aligned with the trabecular orientation in the sagittal plane was excised from each portion. This slab tended to be in the posterior region and at a small angle (10–20 deg) to the frontal plane. Each slab yielded one specimen; two specimens from each tibia were obtained.
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Percent reductions in tangent modulus compared to the initial tangent modulus for each site in both tension and compression. Error bars represent 1 SD. Reductions in the tangent modulus were calculated at (A) 0.20 percent strain and (B) 0.40 percent strain. Within each graph, groups marked “a” were significantly different (p<0.05) from those marked “b.” The reduction in tangent modulus was greater in tension than in compression (p<0.001) for all sites at 0.40 percent strain and for the bovine tibia at 0.20 percent strain (p<0.001).
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The reduction in tangent modulus computed using strains measured by the 25 mm gage length extensometer attached across the endcaps was compared with that computed using strains simultaneously measured by the 5 mm gage length extensometer attached directly to the specimen surface. Results shown are representative of all anatomic sites and loading modes with respect to the close agreement between the modulus reductions calculated from data from each extensometer. Error bars indicate ±1 SD. Paired t-tests for all groups at each of the seven strains detected no significant differences (p>0.10,power>0.81). These results establish that the observed nonlinearity was not due to any possible slipping at the bone–endcap interface or nonlinear behavior of the cyanoacrylate used to bond the specimens in the endcaps.
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Reductions in modulus as a function of apparent density for all anatomic sites. No significant correlation existed in compression at either (A) 0.20 or (B) 0.40 percent strain. Positive correlations were found in tension at (C) 0.20 and (D) 0.40 percent strain.
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Percent differences in initial modulus resulting from the use of different polynomial orders and strain ranges for each anatomic site in (A) compression and (B) tension. Each difference is computed with respect to the initial modulus determined from a quadratic fit from 0–0.20 percent strain. Each column represents the mean difference for each site; error bars indicate 1 SD. As shown by the small differences corresponding to the nonlinear fits, the initial modulus was not sensitive to either the order of the nonlinear fit or the strain range used in the fit. Much larger differences resulted when moduli from linear and quadratic fits were compared.
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Reductions in tangent modulus in the human vertebra as a function of strain for two different strain rates: 0.50 percent per second and 0.10 percent per second. At each strain, reductions in tangent modulus were greater at the slower strain rate (p<0.001).




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