Ultrasonic wave propagation in human cortical bone, I: theoretical considerations for hexagonal symmetry

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Yoon, Hyo Sub¹; Katz, J. Lawrence¹

Ultrasonic wave propagation in human cortical bone, I: theoretical considerations for hexagonal symmetry

J Biomech. 1976;9(6):407-412

Affiliations

¹Laboratory for Crystallographic Biophysics, Department of Physics, Rensselaer Polytechnic Institute, Troy, N.Y. 12181, U.S.A.
Abstract

From the theory of wave propagation in an anisotropic elastic medium are derived the basic equations relating the five independent second-order elastic stiffness constants (fourth-rank tensor) to the ultrasonic wave speeds in a hexagonal medium, with special emphasis on determining the microtextural symmetry of human cortical bone. In addition, the three pure mode directions of high symmetry in a hexagonal medium are explicitly shown. Finally, expressions relating the ‘technical moduli’ such as Young's modulus, shear modulus and bulk modulus to the elastic compliances are presented for the most general case (triclinic symmetry) and then are specialized for the hexagonal system.
Links

DOI: 10.1016/0021-9290(76)90118-4

PubMed: 932054

WoS: A1976BV06800006
History

Received: 1975-07-25

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Cited By (32)

Year	Entry
1981	Katz JL. Composite material models for cortical bone. In: Cowin SC, ed. Mechanical Properties of Bone. The Joint ASME-ASCE Applied Mechnics, Fluids Engineering and Bioengineering Conference; June 22-24, 1981; Boulder, CO. New York, NY: American Society of Mechical Engineers; 1981:171-184.
2001	Guo XE. Mechanical properties of cortical and cancellous bone tissue. In: Cowin SC, ed. Bone Mechanics Handbook. 2nd ed. Boca Raton, FL: CRC Press; 2001:10-1–10-23.
1998	Martin RB, Burr DB, Sharkey NA. Skeletal Tissue Mechanics. New York, NY: Springer-Verlag; 1998.
2012	Parnell WJ, Vu MB, Grimal Q, Naili S. Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone. Biomech Model Mechanobiol. July 2012;11(6):883-901.
1995	Turner CH, Chandran A, Pidaparti RMV. The anisotropy of osteonal bone and its ultrastructural implications. Bone. July 1995;17(1):85-89.
1984	Katz JL, Yoon HS, Lipson S, Maharidge R, Meunier A, Christel P. The effects of remodeling on the elastic properties of bone. Calcif Tiss Int. 1984;36(suppl 1):S31-S36.
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.
1979	Lakes RS, Katz JL, Sternstein SS. Viscoelastic properties of wet cortical bone, I: torsional and biaxial studies. J Biomech. 1979;12(9):657-678.
1980	Gottesman T, Hashin Z. Analysis of viscoelastic behaviour of bones on the basis of microstructure. J Biomech. 1980;13(2):89-96.
1984	Lipson SF, Katz JL. The relationship between elastic properties and microstructure of bovine cortical bone. J Biomech. 1984;17(4):231-240.
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.
1987	Katz JL, Meunier A. The elastic anisotropy of bone. J Biomech. 1987;20(11-12):1063-1070.
2004	Dong XN, Guo XE. The dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. J Biomech. August 2004;37(8):1281-1287.
2007	Shahar R, Zaslansky P, Barak M, Friesem AA, Currey JD, Weiner S. Anisotropic Poisson's ratio and compression modulus of cortical bone determined by speckle interferometry. J Biomech. 2007;40(2):252-264.
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.
1999	Fondrk MT, Bahniuk EH, Davy DT. Inelastic strain accumulation in cortical bone during rapid transient tensile loading. J Biomech Eng. December 1999;121(6):616-621.
2009	Espinoza Orías AA, Deuerling JM, Landrigan MD, Renaud JE, Roeder RK. Anatomic variation in the elastic anisotropy of cortical bone tissue in the human femur. J Mech Behav Biomed Mater. July 2009;2(3):255-263.
2012	Russell NA, Pelletier MH, Bruce WJ, Walsh WR. The effect of gamma irradiation on the anisotropy of bovine cortical bone. Med Eng Phys. October 2012;34(8):1117-1122.
1980	Katz JL. Anisotropy of Young's modulus of bone. Nature. January 3, 1980;283(5742):106-107.
1996	Rho J-Y. An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone. Ultrasonics. December 1996;34(8):777-783.
1989	Fondrk MT. An Experimental and Analytical Investigation Into the Nonlinear Constitutive Equations of Cortical Bone [PhD thesis]. Cleveland, OH: Case Western Reserve University; May 1989.
2002	Dong X. Micromechanics of Osteonal Cortical Bone [PhD thesis]. Columbia University; 2002.
2005	Yoon YJ. Estimates of Bone Elastic Constants At Sequential Hierarchical Levels [PhD thesis]. New York, NY: The City University of New York; 2005.
2011	Donaldson FE. On Incorporating Bone Microstructure in Macro-Finite-Element Models [PhD thesis]. Edinburgh, UK: University of Edinburgh; March 2011.
1988	Lotz JC. Hip Fracture Risk Predictions by X-Ray Computed Tomography [PhD thesis]. Cambridge, MA: Massachusetts Institute of Technology; August 1988.
2005	Espinoza Orias AA. The Relationship Between the Mechanical Anisotropy of Human Cortical Bone Tissue and Its Microstructure [PhD thesis]. University of Notre Dame; April 2005.
2014	Kim S. Contact Stress Analysis of the Native Radial Head and Radial Head Implants [PhD thesis]. University of Pittsburgh; 2014.
2001	Kim DG. Micromechanical Analysis of Bone At a Bone-Implant Interface [PhD thesis]. Troy, NY: Rensselaer Polytechnic Institute; April 2001.
2001	Lin W. Development of Quantitative Ultrasound to Determine the Physical Properties of Bone [PhD thesis]. Stony Brook, NY: State University of New York at Stony Brook; May 2001.
1982	Ashman RB. Ultrasonic Determination of the Elastic Properties of Cortical Bone: Techniques and Limitations [PhD thesis]. Tulane University; 1982.
2018	Khor F. Computational Modeling of Hard Tissue Response and Fracture in the Lower Cervical Spine Under Compression Including Age Effects [Master's thesis]. University of Waterloo; 2018.
2013	Faingold A. Bone Elementary Units: Mechanical Properties and Structure [PhD thesis]. Weizmann Institute of Science; February 2013.