Continuum Mechanics and Thermodynamics

, Volume 30, Issue 3, pp 529–551 | Cite as

Identification of a constitutive law for trabecular bone samples under remodeling in the framework of irreversible thermodynamics

  • Zineeddine Louna
  • Ibrahim Goda
  • Jean-François Ganghoffer
Original Article


We construct in the present paper constitutive models for bone remodeling based on micromechanical analyses at the scale of a representative unit cell (RUC) including a porous trabecular microstructure. The time evolution of the microstructure is simulated as a surface remodeling process by relating the surface growth remodeling velocity to a surface driving force incorporating a (surface) Eshelby tensor. Adopting the framework of irreversible thermodynamics, a 2D constitutive model based on the setting up of the free energy density and a dissipation potential is identified from FE simulations performed over a unit cell representative of the trabecular architecture obtained from real bone microstructures. The static and evolutive effective properties of bone at the scale of the RUC are obtained by combining a methodology for the evaluation of the average kinematic and static variables over a prototype unit cell and numerical simulations with controlled imposed first gradient rates. The formulated effective growth constitutive law at the scale of the homogenized set of trabeculae within the RUC is of viscoplastic type and relates the average growth strain rate to the homogenized stress tensor. The postulated model includes a power law function of an effective stress chosen to depend on the first and second stress invariants. The model coefficients are calibrated from a set of virtual testing performed over the RUC subjected to a sequence of loadings. Numerical simulations show that overall bone growth does not show any growth kinematic hardening. The obtained results quantify the strength and importance of different types of external loads (uniaxial tension, simple shear, and biaxial loading) on the overall remodeling process and the development of elastic deformations within the RUC.


Bone remodeling Surface growth Effective growth model Irreversible thermodynamics Micromechanics FE simulations 

List of symbols


Tensor of effective elastic moduli

\(\overline{\mathbf{D}} \)

Average rate of deformation tensor

\(\overline{\mathbf{D}} _\mathrm{e} \)

Average elastic rate of deformation tensor

\(\overline{\mathbf{D}} _\mathrm{g} \)

Average rate of growth tensor

\(\mathbf{E}\left( x \right) \)

Average kinematic first gradient tensors


Effective homogenized moduli in x, y, respectively

\({\varvec{\varepsilon }}\)

Classical infinitesimal strain tensor

\(\overline{{{\varvec{\upvarepsilon }} }_\mathrm{e} } ,\overline{{{\varvec{\upvarepsilon }}}_\mathrm{g}}\)

A average elastic and growth tensors (small strain)

\(\mathbf{f}_{\mathrm{S}} \)

Surface force field


Surface deformation gradient


Surface accommodation mapping


Surface growth mapping

\(\mathbf{g}_\mathrm{a} \)

Acceleration due to gravity

\(\mathbf{I}_\mathrm{S} \)

Surface identity tensor


Jacobean of the total deformation gradient

\(\tilde{J}_{\mathrm{a}}, \tilde{J}_{\mathrm{g}}\)

Determinants of the accommodation and growth tensors

\({J}_{1}, {J}_{2}\),

First and second invariants of the applied stress


Curvature tensor of the surface


Constant measuring the rate of surface adaptation

\(\overline{\mathbf{L}} _\mathrm{g} \)

Average growth velocity gradient


Unit normal vector

\(\overline{{\dot{p}}} _\mathrm{g} \)

Effective growth strain rate (second invariant of \(\overline{\mathbf{D}} _\mathrm{g} )\)


Isotropic hardening


Surface nominal stress

\(\mathbf{u}=\left( {u_x ,u_y } \right) \)

Displacement vector in 2D


Strain energy over the unit cell

\(V_{{\mathrm{RUC}}} =\left| \Omega \right| \)

Volume of the RUC


Total velocity field

\({\tilde{\mathbf{V}}}_{\mathrm{g}} \)

Surface growth velocity field


In-plane effective Poisson’s ratios


Surface energy density


Growth position

\(\varphi ^{*}\)

Dissipation potential

\(\rho _{\mathrm{S}} \)

Surface density of nutrients

\({r\rho }^{\mathrm{eff}}\)

Effective density

\(\dot{\rho }^{\mathrm{eff}}\)

Rate of effective density

\(\sigma ^{\mathrm{eq}}\)

Equivalent stress

\({{\varvec{\upsigma }} }^{\mathrm{ext}}\)

Applied stress tensor

\(\sigma ^{\mathrm{g}}\)

Growth threshold corresponding to the minimal effective stress

\(\tilde{{\varvec{\Sigma }}}_{a}\)

Surface Eshelby stress

\(\Gamma ^{{\mathrm{S}}}\)

Rate of mass surface growth

\(\Omega _{\mathrm{g}} \)

The scalar growth potential

\(\alpha ,\beta ,K,N\)

Four material parameters of the constitutive growth model

\(\psi _\mathrm{e} \)

Elastic part of the free energy density

\(\psi _\mathrm{g} \)

Growth part of the free energy density


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Zineeddine Louna
    • 1
  • Ibrahim Goda
    • 2
    • 3
  • Jean-François Ganghoffer
    • 4
  1. 1.LMFTA, Faculté de physiqueUSTHBBab Ezzouar, AlgiersAlgeria
  2. 2.LPMTUniversité de Haute-AlsaceMulhouse CedexFrance
  3. 3.Department of Industrial Engineering, Faculty of EngineeringFayoum UniversityFayoumEgypt
  4. 4.LEM3 - Université de Lorraine and CNRS. 7MetzFrance

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