0
TECHNICAL PAPERS

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

References

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(4), pp. 361–65.
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(4), pp. 353–60.
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(4), pp. 424–34.
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(2), pp. 209–14.
Keyak,  J. H., 2001, “Improved Prediction of Proximal Femoral Fracture Load Using Nonlinear Finite Element Models,” Med. Eng. Phys., 23, pp. 165–73.
Fenech,  C., and Keaveny,  T. M., 1999, “A Cellular Solid Criterion for Predicting the Axial-Shear Failure Properties of Trabecular Bone.,” J. Biomech. Eng., 121, pp. 414–22.
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.
Gibson, L. J., and Ashby, M. F., 1988, Cellular Solids: Structures & Properties, Oxford, Pergamon Press.
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(9), pp. 635–63.
Gibson,  L. J., 1985, “The Mechanical Behavior of Cancellous Bone,” J. Biomech., 18(5), pp. 317–28.
Cowin,  S. C., 1986, “Fabric Dependence of an Anisotropic Strength Criterion,” Mech. Mater., 5, pp. 251–60.
Brown,  T. D., and Ferguson,  A. B., 1980, “Mechanical Property Distributions in the Cancellous Bone of the Human Proximal Femur,” Acta Orthop. Scand., 51(3), pp. 429–37.
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. 1137–46.
Morgan,  E. F., and Keaveny,  T. M., 2001, “Dependence of Yield Strain of Human Trabecular Bone on Anatomic Site,” J. Biomech., 34(5), pp. 569–77.
Ford,  C. M., and Keaveny,  T. M., 1996, “The Dependence of Shear Failure Properties of Bovine Tibial Trabecular Bone on Apparent Density and Trabecular Orientation,” J. Biomech., 29, pp. 1309–17.
Stone,  J. L., Beaupre,  G. S., and Hayes,  W. C., 1983, “Multiaxial Strength Characteristics of Trabecular Bone,” J. Biomech., 16(9), pp. 743–52.
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(12), pp. 1575–83.
Keaveny,  T. M., Pinilla,  T. P., Crawford,  R. P., Kopperdahl,  D. L., and Lou,  A., 1997, “Systematic and Random Errors in Compression Testing of Trabecular Bone,” J. Orthop. Res., 15, pp. 101–10.
Turner,  C. H., Cowin,  S. C., Rho,  J. Y., Ashman,  R. B., and Rice,  J. C., 1990, “The Fabric Dependence of the Orthotropic Elastic Constants of Cancellous Bone,” J. Biomech., 23(6), pp. 549–61.
Van Rietbergen,  B., Odgaard,  A., Kabel,  J., and Huiskes,  R., 1998, “Relationships between Bone Morphology and Bone Elastic Properties Can be Accurately Quantified Using High-Resolution Computer Reconstructions,” J. Orthop. Res., 16(1), pp. 23–28.
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., 121(6), pp. 629–35.
Guldberg,  R. E., Hollister,  S. J., and Charras,  G. T., 1998, “The Accuracy of Digital Image-Based Finite Element Models,” J. Biomech., 120 (4), pp. 289–95.
Van Rietbergen,  B., Weinans,  H., Huiskes,  R., and Odgaard,  A., 1995, “A New Method to Determine Trabecular Bone Elastic Properties and Loading Using Micromechanical Finite Element Models,” J. Biomech., 28(1), pp. 69–81.
Rohl,  L., Larsen,  E., Linde,  F., Odgaard,  A., and Jorgensen,  J., 1991, “Tensile and Compressive Properties of Cancellous Bone,” J. Biomech., 24(12), pp. 1143–49.
Kopperdahl,  D. L., and Keaveny,  T. M., 1998, “Yield Strain Behavior of Trabecular Bone,” J. Biomech., 31(7), pp. 601–08.
Keller,  T. S., 1994, “Predicting the Compressive Mechanical Behavior of Bone,” J. Biomech., 27(9), pp. 1159–68.
Carter,  D. R., and Hayes,  W. C., 1977, “The Compressive Behavior of Bone as a Two-Phase Porous Structure,” J. Bone Jt. Surg., 59-A, pp. 954–62.
Rice,  J. C., Cowin,  S. C., and Bowman,  J. A., 1988, “On the Dependence of the Elasticity and Strength of Cancellous Bone on Apparent Density,” J. Biomech., 21(2), pp. 155–68.
Cowin,  S. C., 1985, “The Relationship between the Elasticity Tensor and the Fabric Tensor,” Mech. Mater., 4, pp. 137–47.
Turner,  C. H., and Cowin,  S. C., 1987, “Dependence of Elastic Constants of an Anisotropic Porous Material Upon Porosity and Fabric,” Mech. Mater., 22(9), pp. 3178–84.
Turner,  C. H., and Cowin,  S. C., 1988, “Errors Introduced by Off-Axis Measurements of the Elastic Properties of Bone,” J. Biomech., 110, pp. 213–14.
Odgaard,  A., Kabel,  J., Van Rietbergen,  B., Dalstra,  M., and Huiskes,  R., 1997, “Fabric and Elastic Principal Directions of Cancellous Bone Are Closely Related,” J. Biomech., 30(5), pp. 487–95.
Wolff, J., 1892, Das Gesetz Der Transformation Der Knochen., Berlin: Hirschwald.
Wolff, J., 1986, The Law of Bone Remodelling. Berlin; New York: Springer-Verlag. xii. 126.
Fyhrie,  D. P., and Vashishth,  D., 2000, “Bone Stiffness Predicts Strength Similarly for Human Vertebral Cancellous Bone in Compression and for Cortical Bone in Tension,” Bone (N.Y.), 26(2), pp. 169–73.
Vahey,  J. W., Lewis,  J. L., and Vanderby,  R. J., 1987, “Elastic Moduli, Yield Stress, and Ultimate Stress of Cancellous Bone in the Canine Proximal Femur,” J. Biomech., 20(1), pp. 29–33.
Oden,  Z. M., Selvitelli,  D., and Bouxsein,  M., 1999, “Effect of Local Density Increases on the Failure Load of the Proximal Femur,” J. Orthop. Res., 17(5), pp. 661–67.
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. 582–85.
Turner,  C. H., 1989, “Yield Behavior of Bovine Cancellous Bone,” J. Biomech., 111(3), pp. 256–60.
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(9), pp. 665–78.
Triantafillou,  T. C., and Gibson,  L. J., 1990, “Multiaxial Failure Criteria for Brittle Foams,” Int. J. Mech. Sci., 32(6), pp. 479–96.
Gibson,  L. J., and Ashby,  M. F., 1982, “The Mechanics of Three-Dimensional Cellular Materials,” Proceedings of the Royal Society (London), 382, pp. 43–59.
Benaissa,  R., Uhthoff,  H. K., and Mercier,  P., 1989, “Repair of Trabecular Fatigue Fractures. Cadaver Studies of the Upper Femur,” Acta Orthop. Scand., 60(5), pp. 585–89.
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(12), pp. 1653–57.
Arramon, Y. P., Yeh, O. C., Morgan, E. F., and Keaveny, T. M., 1999, “Axial-Shear Failure Behavior of Human Femoral Trabecular Bone,” International Mechanical Engineering Conference and Exposition, Vol. BED-43. Nashville TN: ASME.

Figures

Grahic Jump Location
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.
Grahic Jump Location
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).
Grahic Jump Location
(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).
Grahic Jump Location
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).
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.
Grahic Jump Location
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.

Tables

Errata

Discussions

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