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
Your Session has timed out. Please sign back in to continue.


Lotz,  J. C., Cheal,  E. J., and Hayes,  W. C., 1991, “Fracture prediction for the proximal femur using finite-element models: Part I-Linear analysis,” J. Biomech. Eng., 113, pp. 353–360.
Lotz,  J. C., Cheal,  E. J., and Hayes,  W. C., 1991, “Fracture prediction for the proximal femur using finite-element models: Part II-Nonlinear analysis,” J. Biomech. Eng., 113, pp. 361–365.
Cheal,  E. J., Hayes,  W. C., Lee,  C. H., Snyder,  B. D., and Miller,  J., 1985, “Stress analysis of a condylar knee tibial component: influence of metaphyseal shell properties and cement injection depth,” J. Orthop. Res., 3, pp. 424–434.
Keyak,  J. H., and Rossi,  S. A., 2000, “Prediction of femoral fracture load using finite element models: an examination of stress- and strain-based failure theories,” J. Biomech., 33, pp. 209–214.
Cody,  D. D., Gross,  G. J., Hou,  F. J., Spencer,  H. J., Goldstein,  S. A., and Fyhrie,  D. P., 1999, “Femoral strength is better predicted by finite-element models than QCT and DXA,” J. Biomech., 32, pp. 1013–1020.
Ford,  C. M., Keaveny,  T. M., and Hayes,  W. C., 1996, “The effect of impact direction on the structural capacity of the proximal femur during falls,” J. Bone Miner. Res., 11, pp. 377–383.
Liebschner,  M. A. K., Rosenberg,  W. S., and Keaveny,  T. M., 2001, “Effects of bone cement volume and distribution on vertebral stiffness after vertebroplasty,” Spine, 26, pp. 1547–1554.
Oden,  Z. M., Selvitelli,  D. M., and Bouxsein,  M. L., 1999, “Effect of local density changes on the failure load of the proximal femur,” J. Orthop. Res., 17, pp. 661–667.
Keaveny,  T. M., Wachtel,  E. F., Ford,  C. M., and Hayes,  W. C., 1994, “Differences between the tensile and compressive strengths of bovine tibial trabecular bone depend on modulus,” J. Biomech., 27, pp. 1137–1146.
Morgan,  E. F., and Keaveny,  T. M., 2001, “Dependence of yield strain of human trabecular bone on anatomic site,” J. Biomech., 34, pp. 569–577.
Cowin,  S. C., 1986, “Fabric dependence of an anisotropic strength criterion,” Mech. Mater., 5, pp. 251–260.
Tsai,  S., and Wu,  E., 1971, “A general theory for strength of anisotropic materials,” J. Comp. Mat., 5, pp. 58–80.
Wu, E., 1974, “Phenomenological anisotropic failure criterion.,” Mechanics of Composite Materials, G. Sendecky, ed., Academic Press, New York, pp. 353–431.
Keaveny,  T. M., Wachtel,  E. F., Zadesky,  S. P., and Arramon,  Y. P., 1999, “Application of the Tsai-Wu quadratic multiaxial failure criterion to bovine trabecular bone,” J. Biomech. Eng., 121, pp. 99–107.
Niebur,  G. L., Feldstein,  M. J., and Keaveny,  T. M., 2002, “Biaxial failure behavior of bovine tibial trabecular bone,” J. Biomech. Eng., 124, pp. 699–705.
Patel, M. R., 1969, “The deformation and fracture of rigid cellular plastics under multiaxial stress,” p. 196. Berkeley, CA: University of California, Berkeley, CA.
Zaslawsky,  M., 1973, “Multiaxial-stress studies on rigid polyurethane foam,” Exp. Mech., 2, pp. 70–76.
Gibson,  L. J., Ashby,  M. F., Zhang,  J., and Triantafillou,  T. C., 1989, “Failure surfaces for cellular materials under multiaxial loads-I. Modelling,” Int. J. Mech. Sci., 31, pp. 635–663.
Triantafillou,  T. C., Zhang,  J., Shercliff,  T. L., Gibson,  L. J., and Ashby,  M. F., 1989, “Failure surfaces for cellular materials under multiaxial loads-II. Comparison of models with experiment,” Int. J. Mech. Sci., 31, pp. 665–678.
Fenech,  C. M., and Keaveny,  T. M., 1999, “A cellular solid criterion for predicting the axial-shear failure properties of trabecular bone.,” J. Biomech. Eng., 121, pp. 414–422.
Bayraktar, H. H., and Keaveny, T. M., 2004, “Mechanisms of uniformity of yield strains for trabecular bone,” J. Biomech., In Press.
Morgan,  E. F., Bayraktar,  H. H., Yeh,  O. C., and Keaveny,  T. M., 2004, “Contribution of inter-site variations in architecture to trabecular bone apparent yield strain,” J. Biomech., In Press.
Guo, X. E., 2001, “Mechanical properties of cortical bone and cancellous tissue,” Bone Mechanics Handbook, S. C. Cowin, ed., CRC Press, Boca Raton, pp. 10.11–10.23.
Bayraktar,  H. H., Morgan,  E. F., Niebur,  G. L., Morris,  G., Wong,  E. K., and Keaveny,  T. M., 2004, “Comparison of the elastic and yield properties of human femoral trabecular and cortical bone tissue,” J. Biomech., 37, pp. 27–35.
Beck,  J. D., Canfield,  B. L., Haddock,  S. M., Chen,  T. J. H., Kothari,  M., and Keaveny,  T. M., 1997, “Three-dimensional imaging of trabecular bone using the computer numerically controlled milling technique,” Bone (N.Y.), 21, pp. 281–287.
Rüegsegger,  P., Koller,  B., and Müller,  R., 1996, “A microtomographic system for the nondestructive evaluation of bone architecture,” Calcif. Tissue Int., 58, pp. 24–29.
Niebur,  G. L., Feldstein,  M. J., Yuen,  J. C., Chen,  T. J., and Keaveny,  T. M., 2000, “High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone,” J. Biomech., 33, pp. 1575–1583.
Niebur,  G. L., Yuen,  J. C., Hsia,  A. C., and Keaveny,  T. M., 1999, “Convergence behavior of high-resolution finite element models of trabecular bone,” J. Biomech. Eng., 121, pp. 629–635.
Van Rietbergen,  B., Odgaard,  A., Kabel,  J., and Huiskes,  R., 1996, “Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture,” J. Biomech., 29, pp. 1653–1657.
Barr,  A. H., 1981, “Superquadratics and angle-preserving transformations,” IEEE Comput. Graphics Appl., 1, pp. 11–23.
Hildebrand,  T., Laib,  A., Müller,  R., Dequeker,  J., and Rüegsegger,  P., 1999, “Direct three-dimensional morphometric analysis of human cancellous bone: microstructural data from spine, femur, iliac crest, and calcaneus,” J. Bone Miner. Res., 14, pp. 1167–1174.
Müller,  R., Gerber,  S. C., and Hayes,  W. C., 1998, “Micro-compression: a novel technique for the nondestructive assessment of local bone failure,” Technol. Health Care, 6, pp. 433–444.
Stölken,  J. S., and Kinney,  J. H., 2003, “On the importance of geometric nonlinearity in finite-element simulations of trabecular bone failure,” Bone (N.Y.), 33, pp. 494–504.
Stone,  J. L., Beaupre,  G. S., and Hayes,  W. C., 1983, “Multiaxial strength characteristics of trabecular bone,” J. Biomech., 16, pp. 743–752.
Turner,  C. H., 1989, “Yield behavior of bovine cancellous bone,” J. Biomech. Eng., 111, pp. 256–260.
Chang,  W. C. W., Christensen,  T. M., Pinilla,  T. P., and Keaveny,  T. M., 1999, “Isotropy of uniaxial yield strains for bovine trabecular bone,” J. Orthop. Res., 17, pp. 582–585.
Simo, J. C., and Hughes, T. J. R., 1998, “Computational Inelasticity,” Springer-Verlag, New York.
Adams,  M., 2002, “Evaluation of three unstructured multigrid methods on 3D finite element problems in solid mechanics,” Int. J. Numer. Methods Eng., 55, pp. 519–534.


Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In