Biomechanics and Modeling in Mechanobiology

, Volume 11, Issue 6, pp 883–901 | Cite as

Analytical methods to determine the effective mesoscopic and macroscopic elastic properties of cortical bone

  • William J. Parnell
  • M. B. Vu
  • Q. Grimal
  • S. Naili
Original Paper


We compare theoretical predictions of the effective elastic moduli of cortical bone at both the meso- and macroscales. We consider the efficacy of three alternative approaches: the method of asymptotic homogenization, the Mori–Tanaka scheme and the Hashin–Rosen bounds. The methods concur for specific engineering moduli such as the axial Young’s modulus but can vary for others. In a past study, the effect of porosity alone on mesoscopic properties of cortical bone was considered, taking the matrix to be isotropic. Here, we consider the additional influence of the transverse isotropy of the matrix. We make the point that micromechanical approaches can be used in two alternative ways to predict either the macroscopic (size of cortical bone sample) or mesoscopic (in between micro- and macroscales) effective moduli, depending upon the choice of representative volume element size. It is widely accepted that the mesoscale behaviour is an important aspect of the mechanical behaviour of bone but models incorporating its effect have started to appear only relatively recently. Before this only macroscopic behaviour was addressed. Comparisons are drawn with experimental data and simulations from the literature for macroscale predictions with particularly good agreement in the case of dry bone. Finally, we show how predictions of the effective mesoscopic elastic moduli can be made which retain dependence on the well-known porosity gradient across the thickness of cortical bone.


Cortical bone Asymptotic homogenization and micromechanics Transverse isotropy Porosity Mesoscale Macroscale 


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • William J. Parnell
    • 1
  • M. B. Vu
    • 2
  • Q. Grimal
    • 3
  • S. Naili
    • 2
  1. 1.School of MathematicsUniversity of ManchesterManchesterUK
  2. 2.Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRSCréteil CedexFrance
  3. 3.Laboratoire d’Imagerie ParamétriqueUniversité Pierre et Marie Curie, LIP UMR 7623 CNRSParisFrance

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