The elastic anisotropy of bone

Katz, J. Lawrence<sup>1</sup>; Meunier, Alain<sup>2</sup>

Affiliations

¹Department of Biomedical Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, U.S.A.

²Laboratoire de Recherches Orthopediques, Faculte de Medecine Lariboisiere-St. Louis, 75010 Paris, France

Abstract

In modeling the anisotropic properties of hydroxyapatite (HAp), Katz found that two kinds of phenomenological relationships held among the elastic stiffness coefficients. Firstly, there are three linear combinations—(c₁₁ + c₂₂ + c₃₃), (c₄₄ + c₅₅ + c₆₆), (c₁₂ + c₁₃ + c₂₃)—which arise naturally when computing the isotropic averages of anisotropic crystal systems over all possible spatial orientations. Secondly, the degree of elastic anisotropy in such crystal systems is characterized by two specific factors: (a) the ratio of the linear compressibility along the unique axis to that perpendicular to it, (c₁₁ + c₁₂ − ₂c₂₃)(c₃₃ − c₁₃); and (b) the ratio of the two shear moduli, .

There have been a number of experiments in recent years which have used either mechanical methods or ultrasonic techniques to measure the anisotropic elastic properties of bovine and human cortical bone. Analyses of data from these experiments show that the above relationships also play a significant role in characterizing the elastic anisotropy in bone.

Links

DOI: 10.1016/0021-9290(87)90024-8

PubMed: 3323198

WoS: A1987L506100007

Cited Works (14)

Year	Entry
1980	Katz JL. Anisotropy of Young's modulus of bone. Nature. January 3, 1980;283(5742):106-107.
1975	Reilly DT, Burstein AH. The elastic and ultimate properties of compact bone tissue. J Biomech. 1975;8(6):393-405.
1969	Lang SB. Elastic coefficients of animal bone. Science. July 18, 1969;65(3890):287-288.
1976	Yoon HS, Katz JL. Ultrasonic wave propagation in human cortical bone, I: theoretical considerations for hexagonal symmetry. J Biomech. 1976;9(6):407-412.
1967	Evans FG, Bang S. Differences and relationships between the physical properties and the microscopic structure of human femoral, tibial and fibular cortical bone. Am J Anat. January 1967;120(1):79-88.
1984	Ashman RB, Cowin SC, Van Buskirk WC, Rice JC. A continuous wave technique for the measurement of the elastic properties of cortical bone. J Biomech. 1984;17(5):349-361.
1971	Katz JL, Ukraincik K. On the anisotropic elastic properties of hydroxyapatite. J Biomech. May 1971;4(3):221-227.
1971	Katz JL. Hard tissue as a composite material, I: bounds on the elastic behavior. J Biomech. October 1971;4(5):455-473.
1970	Lang SB. Ultrasonic method for measuring elastic coefficients of bone and results on fresh and dried bovine bones. IEEE Trans Biomed Eng. April 1970;17(2):101-105.
1976	Yoon HS, Katz JL. Ultrasonic wave propagation in human cortical bone, II: measurements of elastic properties and microhardness. J Biomech. 1976;9(7):459-464.
1981	Van Buskirk WC, Cowin SC, Ward RN. Ultrasonic measurement of orthotropic elastic constants of bovine femoral bone. J Biomech Eng. May 1981;103(2):67-72.
1979	Lakes RS, Katz JL. Viscoelastic properties of wet cortical bone, II: relaxation mechanisms. J Biomech. 1979;12(9):679-687.
1979	Lakes RS, Katz JL. Viscoelastic properties of wet cortical bone, III: a non-linear constitutive equation. J Biomech. 1979;12(9):689-698.
1969	Evans FG, Vincentelli R. Relation of collagen fiber orientation to some mechanical properties of human cortical bone. J Biomech. 1969;2(1):63-73.

Cited By (16)

Year	Entry
2010	Hamed E, Lee Y, Jasiuk I. Multiscale modeling of elastic properties of cortical bone. Acta Mech. August 2010;213(1):131-154.
1995	Turner CH, Chandran A, Pidaparti RMV. The anisotropy of osteonal bone and its ultrastructural implications. Bone. July 1995;17(1):85-89.
1997	Guo XE, Goldstein SA. Is trabecular bone tissue different from cortical bone tissue? Forma. 1997;12(3-4):185-196.
1989	Cowin SC, Mehrabadi MM. Identification of the elastic symmetry of bone and other materials. J Biomech. 1989;22(6-7):503-515.
1992	Gross TS, McLeod KJ, Rubin CT. Characterizing bone strain distributions in vivo using three triple rosette strain gages. J Biomech. September 1992;25(9):1081-1087.
1996	Pidaparti RMV, Chandran A, Takano Y, Turner CH. Bone mineral lies mainly outside collagen fibrils: predictions of a composite model for osternal bone. J Biomech. July 1996;29(7):909-916.
2000	Wirtz DC, Schiffers N, Pandorf T, Radermacher K, Weichert D, Forst R. Critical evaluation of known bone material properties to realize anisotropic fe-simulation of the proximal femur. J Biomech. October 2000;33(10):1325-1330.
2020	Alonso MG, Bertolino G, Yawny A. Mechanobiological based long bone growth model for the design of limb deformities correction devices. J Biomech. August 26, 2020;109:109905.
2020	Colabella L, Cisilino A, Fachinotti V, Capiel C, Kowalczyk P. Multiscale design of artificial bones with biomimetic elastic microstructures. J Mech Behav Biomed Mater. August 2020;108:103748.
2006	Peterlik H, Roschger P, Klaushofer K, Fratzl P. From brittle to ductile fracture of bone. Nat Mater. January 2006;5(1):52-55.
1995	Zioupos P, Currey JD, Mirza MS, Barton DC. Experimentally determined microcracking around a circular hole in a flat plate of bone: comparison with predicted stresses. Philos Trans R Soc Lond B Biol Sci. March 29, 1995;347(1322):383-396.
2016	Grigoriadis G. Heel Biomechanics Under Blast Conditions [PhD thesis]. Imperial College London; June 2016.
2011	Abdel-Wahab AA-GM. Experimental and Numerical Analysis of Deformation and Fracture of Cortical Bone Tissue [PhD thesis]. Loughborough University; August 2011.
2019	Persons AK. Finite Element Analysis of the Mechanisms of Impact Mitigation Inherent to the North American Bison (Bison Bison) Skull [Master's thesis]. Mississippi State University; December 2019.
1991	Pattin CAG. Cyclic Mechanical Property Degradation in Bone During Fatigue Loading [PhD thesis]. Stanford University; March 1991.
1992	Oxland TR. Burst Fractures of the Human Thoracolumbar Spine: A Biomechanical Investigation [PhD thesis]. New Haven, CT: Yale University; May 1992.