The anisotropic Hooke’s law for cancellous bone and wood

Yang, Guoyu<sup>1</sup>; Kabel, Jesper<sup>2</sup>; Van Rietbergen, Bert<sup>3</sup>; Odgaard, Anders<sup>2</sup>; Huiskes, Rik<sup>4</sup>; Cowin, Stephen C.<sup>1</sup>

Affiliations

¹The Center for Biomedical Engineering and The Department of Mechanical Engineering, The School of Engineering of The City College and The Graduate School of The City University of New York, New York, NY 10031, U.S.A.

²Orthopaedic Research Laboratory, Aarhus University Hospital (ÅKH), Nørrebrogade 44, 8000 Aarhus C, Denmark

³Institute for Biomedical Engineering, Swiss Federal Institute of Technology, Moussonstrasse 18, 8044 Zurich, Switzerland

⁴Biomechanics Section, Institute of Orthopaedics, University of Nijmegen, P.O. Box 9101, 6500 HB Nijmegen, The Netherlands

Abstract

A method of data analysis for a set of elastic constant measurements is applied to data bases for wood and cancellous bone. For these materials the identification of the type of elastic symmetry is complicated by the variable composition of the material. The data analysis method permits the identification of the type of elastic symmetry to be accomplished independent of the examination of the variable composition. This method of analysis may be applied to any set of elastic constant measurements, but is illustrated here by application to hardwoods and softwoods, and to an extraordinary data base of cancellous bone elastic constants. The solid volume fraction or bulk density is the compositional variable for the elastic constants of these natural materials. The final results are the solid volume fraction dependent orthotropic Hooke's law for cancellous bone and a bulk density dependent one for hardwoods and softwoods.

Links

DOI: 10.1023/A:1007575322693

PubMed: 11543211

WoS: 000081301300003

History

Received: 1999-01-19

Accepted: 1999-03-21

Cited Works (11)

Year	Entry
1990	Hodgskinson R, Currey JD. Effects of structural variation on Young's modulus of non-human cancellous bone. Proc Inst Mech Eng Part H-J Eng Med. 1990;204(1):43-52.
1999	Kabel J, van Rietbergen B, Odgaard A, Huiskes R. Constitutive relationships of fabric, density, and elastic properties in cancellous bone architecture. Bone. October 1999;25(4):481-486.
1997	Keaveny TM, Pinilla TP, Crawford RP, Kopperdahl DL, Lou A. Systematic and random errors in compression testing of trabecular bone [published correction appears in J Orthop Res. 1995;17(1):151]. J Orthop Res. 1997;15(1):101-110.
1997	Cowin SC, Yang G. Averaging anisotropic elastic constant data. J Elast. February 1997;46(2):151-180.
1988	Rice JC, Cowin SC, Bowman JA. On the dependence of the elasticity and strength of cancellous bone on apparent density. J Biomech. 1988;21(2):155-168.
1996	van Rietbergen B, Odgaard A, Kabel J, Huiskes R. Direct mechanics assessment of elastic symmetries and properties of trabecular bone architecture. J Biomech. December 1996;29(12):1653-1657.
1991	Hollister SJ, Fyhrie DP, Jepsen KJ, Goldstein SA. Application of homogenization theory to the study of trabecular bone mechanics. J Biomech. 1991;24(9):825-839.
1999	Kabel J, van Rietbergen B, Dalstra M, Odgaard A, Huiskes R. The role of an effective isotropic tissue modulus in the elastic properties of cancellous bone. J Biomech. 1999;32(7):673-680.
1998	Van Rietbergen B, Odgaard A, Kabel J, Huiskes R. Relationships between bone morphology and bone elastic properties can be accurately quantified using high‐resolution computer reconstructions. J Orthop Res. January 1998;16(1):23-28.
1997	Odgaard A. Three-dimensional methods for quantification of cancellous bone architecture. Bone. 1997;20(4):315-328.
1990	Hodgskinson R, Currey JD. The effect of variation in structure on the Young's modulus of cancellous bone: a comparison of human and non-human material. Proc Inst Mech Eng Part H-J Eng Med. 1990;204(2):115-121.

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2007	Cowin SC, Doty SB. Bone tissue. In: Tissue Mechanics. Boca Raton, FL: Springer; 2007:341-384.
2002	Currey JD. Bones: Structure and Mechanics. Princeton, NJ: Princeton University Press; 2002.
2004	Meyer F, Bourdet N, Deck C, Willinger R, Raul JS. Human neck finite element model development and validation against original experimental data. Stapp Car Crash J. 2004;48:177-206. SAE 2004-22-0008.
2006	Kimpara H, Nakahira Y, Iwamoto M, Miki K, Ichihara K, Kawano S-i, Taguchi T. Investigation of anteroposterior head-neck responses during severe frontal impacts using a brain-spinal cord complex FE model. Stapp Car Crash J. 2006;50:509-544. SAE 2006-22-0019.
2009	Mueller TL, Stauber M, Kohler T, Eckstein F, Müller R, van Lenthe GH. Non-invasive bone competence analysis by high-resolution pQCT: an in vitro reproducibility study on structural and mechanical properties at the human radius. Bone. February 2009;44(2):364-371.
2010	Lievers WB, Petryshyn AC, Poljsak AS, Waldman SD, Pilkey AK. Specimen diameter and "side artifacts" in cancellous bone evaluated using end-constrained elastic tension. Bone. August 2010;47(2):371-377.
2020	Silva-Henao JD, Rueda Esteban RJ, Marañon-Leon A, Casas-Rodríguez JP. Post-yield mechanical properties of bovine trabecular bone: relationships with bone volume fraction and strain rate. Eng Fract Mech. June 15, 2020;232:107053.
2003	Morgan EF, Bayraktar HH, Keaveny TM. Trabecular bone modulus–density relationships depend on anatomic site. J Biomech. July 2003;36(7):897-904.
2003	Zysset PK. A review of morphology–elasticity relationships in human trabecular bone: theories and experiments. J Biomech. October 2003;36(10):1469-1485.
2007	Lievers WB, Lee V, Arsenault SM, Waldman SD, Pilkey AK. Specimen size effect in the volumetric shrinkage of cancellous bone measured at two levels of dehydration. J Biomech. 2007;40(9):1903-1909.
2012	Hazrati Marangalou J, Ito K, van Rietbergen B. A new approach to determine the accuracy of morphology–elasticity relationships in continuum FE analyses of human proximal femur. J Biomech. November 15, 2012;45(15):2884-2892.
2001	Pistoia W, van Rietbergen B, Laib A, Rüegsegger P. High-resolution three-dimensional-pqct images can be an adequate basis for in-vivo μfe analysis of bone. J Biomech Eng. October 2001;123(2):176-183.
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.
2012	Moesen M, Cardoso L, Cowin SC. A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor. Mech Mater. November 2012;54:70-83.
2012	Hambli R, Bettamer A, Allaoui S. Finite element prediction of proximal femur fracture pattern based on orthotropic behaviour law coupled to quasi-brittle damage. Med Eng Phys. March 2012;34(2):202-210.
2007	Fratzl P, Weinkamer R. Nature’s hierarchical materials. Prog Mater Sci. November 2007;52(8):1263-1334.
2003	Bredbenner TL. Damage Modeling of Vertebral Trabecular Bone [PhD thesis]. Cleveland, OH: Case Western Reserve University; January 2003.
2010	Zhang XH. High Resolution Imaging Based Patient Specific Biomechanical Assessment of Bone Quality [PhD thesis]. New York, NY: Columbia University; 2010.
2012	Slyfield CR Jr. The Biomechanics of Bone Turnover [PhD thesis]. Ithaca, NY: Cornell University; January 2012.
2013	Souzanchi MF. The Effect of Microarchitecture and Fabric Anisotropy of Trabecular Bone on Its Mechanical Behavior [PhD thesis]. New York, NY: The City University of New York; 2013.
2016	Florencio FL. Multiscale Modelling of Trabecular Bone: From Micro to Macroscale [PhD thesis]. Edinburgh, Scotland: University of Edinburgh; 2016.
2005	Day JS. Bone Quality: The Mechanical Effects of Microarchitecture and Matrix Properties [PhD thesis]. Erasmus University Rotterdam; 2005.
2007	Voide R. Functional Phenotyping of Bone: A Hierarchical Assessment of Bone Failure Characteristics [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2007.
2010	Müller T. Bioimaging and Biomechanics of Bone Competence and Implant Stability [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2010.
2018	Enns-Bray WS. Identifying Individuals Predisposed to Hip Fracture [PhD thesis]. Swiss Federal Institute of Technology Zürich; 2018.
2009	Moesen M. Modeling of the Geometry and Mechanical Behavior of Bone Scaffolds [PhD thesis]. Leuven, Belgium: Katholieke Universiteit Leuven; June 2009.
2004	Wang X. Measurement and Analysis of Microdamage in Bone [PhD thesis]. University of Notre Dame; December 2004.
2010	Shi X. Effects of Architecture on Microdamage Susceptibility in Trabecular Bone [PhD thesis]. University of Notre Dame; April 2010.
2020	Karali A. Multi-Scale Evaluation of Bone Combining Indentation, in Situ XCT Mechanics and Digital Volume Correlation [PhD thesis]. Portsmouth, England: University of Portsmouth; 2020.
2009	Lievers WB. Effects of Geometric and Material Property Changes on the Apparent Elastic Properties of Cancellous Bone [PhD thesis]. Queen's University; April 2009.
2013	Morton JJ. An Investigation of Rat Vertebra Failure Behaviour Under Uniaxial Compression Through Time-Lapsed Micro-CT Imaging [Master's thesis]. Queen's University; 2013.
2023	Branni MG. Constitutive Models of Bone: The Human Femur [PhD thesis]. Queensland University of Technology; 2023.
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2008	Bevill GR. Micromechanical Modeling of Failure in Trabecular Bone [PhD thesis]. Berkeley, CA: Berkeley, University of California; 2008.
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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.
2010	Wald MJ. Mapping Trabecular Bone Fabric Tensor by in Vivo Magnetic Resonance Imaging [PhD thesis]. Philadelphia, PA: University of Pennsylvania; 2010.
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2022	Xiao P. Medical Image-Based AI Techniques in Prediction of Trabecular Bone Microarchitecture and Mechanical Properties [PhD thesis]. San Antionio, TX: University of Texas at San Antonio; December 2022.
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