Biaxial Failure Behavior of Bovine Tibial Trabecular Bone

[+] Author and Article Information
Glen L. Niebur

Department of Aerospace and Mechanical Engineering, The University of Notre Dame, Notre Dame, IN 46556

Michael J. Feldstein, Tony M. Keaveny

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

J Biomech Eng 124(6), 699-705 (Dec 27, 2002) (7 pages) doi:10.1115/1.1517566 History: Received October 01, 2001; Revised June 01, 2002; Online December 27, 2002
Copyright © 2002 by ASME
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Displacement boundary conditions applied to the high-resolution models to achieve pure biaxial strain loading. The filled arrows represent the on-axis applied displacements, and the white arrows the transverse displacements. Displacements in the out-of-plane direction were constrained to zero (rollers), resulting in biaxial plain strain and triaxial stress at the apparent level.
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Typical on-axis and transverse stress-strain curves for equi-biaxial on-axis and transverse applied strain. Discreet 0.2% offset yield points were determined from the on-axis and transverse stress-strain curves. For the multiaxial yield surface, both yield points were plotted along a line emanating from the origin at an angle equal to the arctangent of the applied ratio of transverse to on-axis strain (i.e., along the loading path).
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(a) There was no significant dependence of biaxial yield strain on volume fraction (p>0.25). (b) Both the on-axis and transverse yield stresses were significantly dependent on volume fraction (p<0.05). Linear and power law regressions provided similar predictions of yield stress because of the small range of volume fractions (linear regressions not shown).
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The on-axis apparent yield stress in equi-biaxial plane strain compression was higher than that for unconfined compression for six of seven specimens (unconfined compression data from 17). The transverse yield stresses were always higher for biaxial loading. If the data for plane strain and uniaxial strain are compared statistically, the power law regressions for the on-axis cases are not significantly different (p>0.3) while those for the two transverse cases are (p<0.01).
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The biaxial yield strains for two specimens of bovine tibial trabecular bone. The arrow represents the loading direction in strain space. The angle θ was varied in increments of 22.5° for specimen 1 and 45° degrees for specimen 2. The out-of-plane strains were constrained to zero and there were no shear strains. For each angle, there are two yield points plotted, one from the on-axis stress-strain curve, and the other from the transverse stress-strain curve. Only one point is plotted along some loading directions because loading was stopped before reaching the second 0.2% offset point. The elliptical curves are least squares fits to the data for Specimen 1. The solid curve fits the on-axis yield points, and the dashed curve the transverse yield points. For both curves, the sum of squared errors was less than 0.5% of the total sum of squares (R2=0.99). Yielding occurs when the strain lies outside of either of the ellipses.
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The mean amount of yielded tissue as a percentage of the total tissue in each of equi-biaxial plane strain, and unconfined on-axis and transverse compression quantified at the yield point. Error bars are one standard deviation (n-7). In biaxial loading, where two yield points were calculated for each specimen, the amount of yielded tissue was quantified at the chronologically first to occur, which was always along the transverse loading direction. The amount of tissue that had yielded at the apparent yield point was highest in biaxial loading and lowest for transverse loading for all seven specimens.
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Yielded tissue at the apparent level equi-biaxial compressive yield point, categorized in to portions that yielded in either or both of on-axis or transverse confined compression. Over 80% of the yielded tissue corresponded directly to locations of tissue yielding in on-axis and/or transverse confined compression. Less than 1/2% of the total tissue yields in all three apparent loading modes, accounting for only 4% of the yielded tissue in equi-biaxial compression. Thus, on-axis and transverse failure at the apparent level can be attributed to different locations of yielded tissue within the microstructure.
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Prescribed boundary conditions for axial shear loading. The Z coordinate direction corresponds to the principal trabecular orientation. The (engineering) shear strain is dx/h, and the on-axis normal strain is dz/h. The model was unconstrained in the Y coordinate direction to allow free expansion, while a horizontal strain was applied based on the calculated Poisson’s ratio to simulate free expansion in the X coordinate direction concurrently with the application of shear displacements.
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Axial-shear yield behavior of two specimens determined from high-resolution finite element models (open and closed circles) compared with data reported for mechanical testing in axial-torsional loading (X’s). Experimental data from 6.



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