On the anisotropic elastic properties of hydroxyapatite

Katz, J. L.<sup>1</sup>; Ukraincik, K.<sup>1</sup>

Affiliations

¹Laboratory for Crystalographic Research, Department of Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, N.Y. 12181, U.S.A.

Abstract

Experimental measurements of the isotropic elastic moduli on polycrystalline specimens of hydroxyapatite and fluorapatite are compared with elastic constants measured directly from single crystals of fluorapatite in order to derive a set of pseudo single crystal elastic constants for hydroxyapatite. The stiffness coefficients thus derived are: c₁₁ = 13.7, c₁₂ = 4.25, c₁₃ = 5.49, c₃₃ = 17.2, and c₄₄ = 3.96 (all in 10ⁱⁱ dyn/cm²). Anisotropic and isotropic elastic properties are then computed and compared with similar properties derived from experimental observations of the anisotropic behavior of bone.

Links

DOI: 10.1016/0021-9290(71)90007-8

PubMed: 5119417

WoS: A1971J295000007

History

Received: 1970-08-24

Cited Works (10)

Year	Entry
1969	Lang SB. Elastic coefficients of animal bone. Science. July 18, 1969;65(3890):287-288.
1965	Sedlin ED. A rheologic model for cortical bone: a study of physical properties of human femoral samples. Acta Orthop Scand. 1965;(suppl 83):1-77.
1967	Bonfield W, Li CH. Anisotropy of nonelastic flow in bone. J Appl Phys. May 1967;38(6):2450-2455.
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.
1961	Dempster WT, Coleman RF. Tensile strength of bone along and across the grain. J Appl Physiol. March 1961;16(2):355-360.
1970	Piekarski K. Fracture of bone. J Appl Phys. January 1970;41(1):215-223.
1964	Currey JD. Three analogies to explain the mechanical properties of bone. Biorheology. 1964;2(1):1-10.
1966	Bonfield W, Li CH. Deformation and fracture of bone. J Appl Phys. February 1966;37(2):869-875.
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.

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2004	Hellmich C, Ulm F-J, Dormieux L. Can the diverse elastic properties of trabecular and cortical bone be attributed to only a few tissue-independent phase properties and their interactions? arguments from a multiscale approach. Biomech Model Mechanobiol. June 2004;2(4):219-238.
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.
1994	Hasegawa K, Turner CH, Burr DB. Contribution of collagen and mineral to the elastic anisotropy of bone. Calcif Tiss Int. November 1994;55(5):381-386.
2018	Grandfield K, Vuong V, Schwarcz HP. Ultrastructure of bone: hierarchical features from nanometer to micrometer scale revealed in focused ion beam sections in the TEM. Calcif Tiss Int. December 2018;103(6):606-616.
2008	Budyn E, Hoc T, Jonvaux J. Fracture strength assessment and aging signs detection in human cortical bone using an X-FEM multiple scale approach. Comput Mech. September 2008;42(4):579-591.
2020	Maghsoudi-Ganjeh M, Wang X, Zeng X. Nanomechanics and ultrastructure of bone: a review. Comput Model Eng Sci. 2020;125(1):1-32.
2007	Budyn É, Hoc T. Multiple scale modeling for cortical bone fracture in tension using X-FEM. Eur J Comput Mech. 2007;16(2):213-236.
2004	Hellmich C, Barthélémy J-F, Dormieux L. Mineral–collagen interactions in elasticity of bone ultrastructure: a continuum micromechanics approach. Eur J Mech – A/Solids. September–October 2004;23(5):783-810.
2009	Budyn É, Hoc T. Analysis of micro fracture in human Haversian cortical bone under transverse tension using extended physical imaging. Int J Num Meth Eng. 2009;82(8):940-965.
1974	Lakes RS, Katz JL. Interrelationships among the viscoelastic functions for anisotropic solids: application to calcified tissues and related systems. J Biomech. May 1974;7(3):259-270.
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.
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.
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.
1986	Fischer RA, Arms SW, Pope MH, Seligson D. Analysis of the effect of using two different strain rates on the acoustic emission in bone. J Biomech. 1986;19(2):119-127.
1987	Katz JL, Meunier A. The elastic anisotropy of bone. J Biomech. 1987;20(11-12):1063-1070.
1993	Mammone JF, Hudson SM. Micromechanics of bone strength and fracture. J Biomech. April–May 1993;26(4-5):439-446.
1993	Crolet JM, Aoubiza B, Meunier A. Compact bone: numerical simulation of mechanical characteristics. J Biomech. June 1993;26(6):677-687.
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.
1996	Aoubiza B, Crolet JM, Meunier A. On the mechanical characterization of compact bone structure using the homogenization theory. J Biomech. December 1996;29(12):1539-1547.
2002	Habelitz S, Marshall GW Jr, Balooch M, Marshall SJ. Nanoindentation and storage of teeth. J Biomech. July 2002;35(7):995-998.
2008	Siegmund T, Allen MR, Burr DB. Failure of mineralized collagen fibrils: modeling the role of collagen cross-linking. J Biomech. 2008;41(7):1427-1435.
2011	Zamiri A, De S. Mechanical properties of hydroxyapatite single crystals from nanoindentation data. J Mech Behav Biomed Mater. February 2011;4(2):146-152.
2011	Luo Q, Nakade R, Dong X, Rong Q, Wang X. Effect of mineral–collagen interfacial behavior on the microdamage progression in bone using a probabilistic cohesive finite element model. J Mech Behav Biomed Mater. October 2011;4(7):943-952.
2007	Fritsch A, Hellmich C. "Universal" microstructural patterns in bone: micromechanics-based prediction of anisotropic material behavior. J Theo Biol. February 21, 2007;244(4):597-620.
2009	Fritsch A, Hellmich C, Dormieux L. Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: experimentally supported micromechanical explanation of bone strength. J Theo Biol. September 21, 2009;260(2):230-252.
1998	Rho J-Y, Kuhn-Spearing L, Zioupos P. Mechanical properties and the hierarchical structure of bone. Med Eng Phys. 1998;20(2):92-102.
2012	Hamed E. Multiscale Modeling of Elastic Moduli and Strength of Bone [PhD thesis]. University of Illinois at Urbana-Champaign; 2012.
2015	Oftadeh R. Hierarchical Analysis and Multiscale Modelling of Cellular Structures: From Meta Materials to Bone Structure [PhD thesis]. Northeastern University; December 2015.
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.
2006	Yue W. Micromechanical Modeling of Hydroxyapatite Whisker Reinforced Polymer Composites and Cortical Bone Tissue [PhD thesis]. University of Notre Dame; July 2006.
2010	Deuerling JM. Effects of Ultrastructural Organization on the Mechanical Properties and Anisotropy of Human Cortical Bone [PhD thesis]. University of Notre Dame; December 2010.
1975	Lakes RS. Viscoelastic and Dielectric Relaxation in Cortical Bone [PhD thesis]. Troy, NY: Rensselaer Polytechnic Institute; July 1975.
2012	Evdokimenko E. Investigation Into Mechanical Properties of Bone and Its Main Constituents [PhD thesis]. San Diego (UCSD), University of California; 2012.
1994	Jepsen KJ. Characterization of the Hierarchical Composite Properties of Cortical Bone: A Transgenic Approach [PhD thesis]. University of Michigan; 1994.
2005	Oyen ML. Ultrastructural Characterization of Time-Dependent, Inhomogeneous Materials and Tissues [PhD thesis]. Minneapolis, MN: University of Minnesota; June 2005.
2012	Barkaoui A. Modélisation multiéchelle du comportement mécano-biologique de l’os humain: De l’ultrastructure au remodelage osseux [PhD thesis]. University of Orleans; 2012.
2021	Atthapreyangkul A. Multi-Scale Finite Element Modelling of Human Cortical Bone: An Analysis of the Mechanical Properties and Microdamage Onset of Cortical Bone [PhD thesis]. University of New South Wales; April 2021.
2009	Nakade R. Influence of Interfacial Properties and Inhomogeneity on Formation of Microdamage in Bone [Master's thesis]. San Antionio, TX: University of Texas at San Antonio; December 2009.
2017	Ahsan AS. Effect of Intrafibrillar Mineralization on the Mechanical Properties of Osteogenesis Imperfecta Bone Using a Cohesive Finite Element Approach [Master's thesis]. San Antionio, TX: University of Texas at San Antonio; May 2017.
2020	Maghsoudi-Ganjeh M. Computational Investigation of Ultrastructural Behavior of Bone and Bone-Inspired Materials [PhD thesis]. San Antionio, TX: University of Texas at San Antonio; May 2020.