The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone

[+] Author and Article Information
Harun H. Bayraktar, Atul Gupta, Ron Y. Kwon, Panayiotis Papadopoulos, Tony M. Keaveny

Orthopaedic Biomechanics Laboratory, University of California, Berkeley, CA, Department of Mechanical Engineering, University of California, Berkeley, CA, Computational Solid Mechanics Laboratory, University of California, Berkeley, CA, Department of Bioengineering, University of California, Berkeley, CA

J Biomech Eng 126(6), 677-684 (Feb 04, 2005) (8 pages) doi:10.1115/1.1763177 History: Received October 17, 2003; Revised February 06, 2004; Online February 04, 2005
Copyright © 2004 by ASME
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Renderings of the three five-millimeter cube trabecular bone specimens used to develop the multiaxial yield criterion. Age, sex, and volume fraction (Vf) information is shown. See Table 1 for architectural indices.
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Illustration of the method by which the load paths were determined in three-dimensional normal strain space (left) and in normal-shear strain planes (right). For the 3-D normal strain case, after a plane was spanned at an angular increment of θ, the plane itself was rotated about the x-axis by the same angle. A similar approach was taken for the nine normal-shear planes. Note that only one of the nine normal-shear strain combinations is shown.
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Apparent level stress versus normalized strain plot for a sample load path in three-dimensional normal strain space. The 0.2%-offset lines (thin dash) used to determine yield strains along each loading axis are also shown. The normalized strain (the strain along each direction divided by the maximum strain applied in that direction) was used to illustrate chronological yielding in all three directions (marked as points a,b, and c). In this case, yielding first occurred along the x-axis (a), then along the y-axis (b), and finally along the z-axis (c). At the first chronological yield point (a) the strains in the other two directions were calculated to obtain the failure point for the three-dimensional yield surface.
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Yield envelopes in three biaxial normal strain planes: (a) εxx−εyy, (b) εyy−εzz, (c) εxx−εzz. Circles indicate the yield data from all three specimens; solid symbols indicate yielding along the vertical axis; empty symbols indicate yielding along the horizontal axis. Dashed lines shown are quadratic fits to the yield points along each axis. The closed inscribed envelope shown in (a), (b), and (c) is the proposed yield surface (four-parameter modified super-ellipsoid) cross-section in each biaxial normal strain plane.
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Yield envelopes in the nine normal-shear planes: εxx−γxyxx−γyzxx−γxzyy−γxyyy−γyzyy−γxzzz−γxyzz−γyz, and εzz−γxz. Diamond, triangle, and circle indicate the three different specimens. Solid symbols indicate yielding along the shear axis while empty symbols indicate yielding along the normal loading direction. Dashed lines shown are quadratic fits to the yield points along the normal axis. Solid lines are fourth-order polynomial fits to the yield points along the shear axis.
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Histogram of percentage arithmetic error of the yield surface representation for the full (9 coefficients) and reduced (4 coefficients) modified super-ellipsoids versus the finite element data.
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Yield surface (266 points) plotted in 3-D normal strain space for one of the specimens (62-F) (left). The four-parameter modified super-ellipsoid yield surface given by Eq. (6) is shown on the right for comparison.
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Differences in stresses between nonlinear versus linear solutions (Δσ/σlinear) at 0.5% strain for trabecular bone from human femoral neck (FN), proximal tibia (PT), greater trochanter (GT), and vertebral body (VB) decreased with increasing volume fraction. Inclusion of geometrically nonlinear deformations had the least effect in shear loading. For compression and shear, a softening effect was observed, but is shown here using positive values of percentage difference. A stiffening effect was seen in tension.




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