Biomechanics and Modeling in Mechanobiology

, Volume 9, Issue 5, pp 597–611 | Cite as

A nonlocal constitutive model for trabecular bone softening in compression

  • Mathieu Charlebois
  • Milan Jirásek
  • Philippe K. Zysset
Original Paper


Using the three-dimensional morphological data provided by computed tomography, finite element (FE) models can be generated and used to compute the stiffness and strength of whole bones. Three-dimensional constitutive laws capturing the main features of bone mechanical behavior can be developed and implemented into FE software to enable simulations on complex bone structures. For this purpose, a constitutive law is proposed, which captures the compressive behavior of trabecular bone as a porous material with accumulation of irreversible strain and loss of stiffness beyond its yield point and softening beyond its ultimate point. To account for these features, a constitutive law based on damage coupled with hardening anisotropic elastoplasticity is formulated using density and fabric-based tensors. To prevent mesh dependence of the solution, a nonlocal averaging technique is adopted. The law has been implemented into a FE software and some simple simulations are first presented to illustrate its behavior. Finally, examples dealing with compression of vertebral bodies clearly show the impact of softening on the localization of the inelastic process.


Trabecular bone Constitutive law Softening Nonlocal Damage Plasticity Anisotropy Fabric 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Mathieu Charlebois
    • 1
  • Milan Jirásek
    • 1
  • Philippe K. Zysset
    • 2
  1. 1.Department of Mechanics, Faculty of Civil EngineeringCzech Technical University in PraguePragueCzech Republic
  2. 2.Institute of Lightweight Design and Structural BiomechanicsVienna University of TechnologyViennaAustria

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