Biomechanics and Modeling in Mechanobiology

, Volume 12, Issue 2, pp 201–213 | Cite as

An anisotropic elastic-viscoplastic damage model for bone tissue

  • J. J. SchwiedrzikEmail author
  • P. K. Zysset
Original Paper


A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain \({D(\kappa)}\) , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.


Bone Constitutive model Viscoplasticity Damage 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Institute of Lightweight Design and Structural BiomechanicsVienna University of TechnologyViennaAustria
  2. 2.Institute of Surgical Technology and BiomechanicsUniversity of BernBernSwitzerland

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