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Modeling of anisotropic remodeling of trabecular bone coupled to fracture

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Abstract

As a living tissue, bone is subjected to internal evolutions of its trabecular architecture under normal everyday mechanical loadings leading to damage. The repeating bone remodeling cycle aims at repairing the damaged zones in order to maintain bone structural integrity; this activity of sensing the peak stress at locations where damage or microcracks have occurred, removing old bone and apposing new bone is achieved thanks to a complicated machinery at the cellular level involving specialized cells (osteocytes, osteoclasts, and osteoblasts). This work aims at developing an integrated remodeling-to-fracture model to simulate the bone remodeling process, considering trabecular bone anisotropy. The effective anisotropic continuum mechanical properties of the trabecular bone are derived from an initially discrete planar hexagonal structure representative of femur bone microstructure, relying on the asymptotic homogenization technique. This leads to scaling laws of the effective elastic properties of bone versus effective density at an intermediate mesoscopic scale. An evolution law for the local bone apparent density is formulated in the framework of the thermodynamics of irreversible processes, whereby the driving force for density evolutions is identified as the local strain energy density weighted by the locally accumulated microdamage. We adopt a classical nonlinear damage model for high cycle fatigue under purely elastic strains, where the assumed homogeneous damage is related to the number of cycles bone experiences. Based on this model, we simulate bone remodeling for the chosen initial microstructure, showing the influence of the external mechanical stimuli on the evolution of the density of bone and the incidence of this evolution on trabecular bone effective mechanical properties.

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Abbreviations

B :

proportionality constant measuring the rate of adaptation process;

\(B_{R}\) :

Set of beams within the reference unit cell;

C :

Elastic damage stiffness tensor;

\(\mathbf{C}^{0}\) :

Virgin (undamaged) elasticity stiffness tensor;

D :

The fatigue damage variable;

\(E_x ,E_y \) :

Homogenized Young’s moduli in x and y directions, respectively;

\(E_s \) :

Young’s modulus of trabecular material;

\(\mathbf{F}^{b}\) :

Resultant of forces at the nodes of a beam b;

F \(_{e}\) :

Accommodation tensor;

F \(_{g}\) :

Irreversible growth tensor;

\(G_{xy}\) :

Homogenized shear modulus;

\(G_{s}\) :

Shear modulus of trabeculae material;

g:

Jacobian associated with the function transformation from Cartesian to curvilinear coordinates;

\(\hbox {g}_{\mathrm{a}}\) :

Gravitational acceleration;

I :

Identity tensor;

J :

Jacobian of the total deformation gradient;

\(J_{\mathrm{e}}\) :

Jacobian of the accommodation tensor;

\(J_{g }\) :

Jacobian of the growth tensor;

k :

Timoshenko shear correction factor;

\(\mathbf{M}^{n}\) :

Bending moment at node n;

\({N/N_ f} \) :

Proportion of fatigue life;

N :

Daily loading cycle number;

\(N_ f \) :

Number of cycles to failure;

O, E :

Origin and end node of a beam, respectively;

P :

Traction vector;

R :

Position vector;

\(\mathbf{S}^{\mathrm {i}}\) :

Stress vector;

\(\mathbf{u}\left( {\hbox {u}_{\mathrm{x}} ,\hbox {u}_{\mathrm{y}} } \right) \) :

Displacement vector;

v :

Test (vector) function;

v \(^{\mathrm{n}}\) :

Virtual velocity field of node n;

\(\nu _{xy} ,\nu _{yx} \) :

Homogenized in-plane Poisson’s ratio;

\(\mathbf{w}^{n}\) :

Virtual rotational velocity of node n;

\(\alpha ,\beta \) :

Material fatigue parameters of bone;

\(\delta ^{ib}\) :

Shift factor for nodes belonging to a neighboring cell;

\(\phi ^{O\left( b \right) }, \phi ^{E\left( b \right) }\) :

The two discrete nodal rotations

\(\psi \) :

Strain energy density;

\({[}{{\varvec{\upvarepsilon }} }{]}\) :

Elastic strain tensor;

\(\rho _a \) :

Apparent density of trabecular bone;

\(\rho _s \) :

Density of trabeculae material;

\(\Gamma ^{s}\) :

Growth rate;

\({\varvec{\Sigma }}(\hbox {u})\) :

Actual volumetric Eshelby stress tensor;

\({{\varvec{\Sigma }} }_0 \) :

Threshold value for the volumetric Eshelby stress stimulus;

\({{\varvec{\upsigma }} }\left( \mathbf{u} \right) \) :

Stress tensor;

\(\lambda ^{i}\) :

Curvilinear Lagrangian coordinates suitable for a general parametrization of any material point

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Authors and Affiliations

  1. LPMT, Université de Haute-Alsace, 11 rue Alfred Werner, 68093, Mulhouse Cedex, France

    Ibrahim Goda

  2. Department of Industrial Engineering, Faculty of Engineering, Fayoum University, Fayoum, 63514, Egypt

    Ibrahim Goda

  3. LEM3, Université de Lorraine-CNRS, 7, rue Félix Savart, 57073, Metz, France

    Jean-François Ganghoffer

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  1. Ibrahim Goda
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Correspondence to Ibrahim Goda.

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Goda, I., Ganghoffer, JF. Modeling of anisotropic remodeling of trabecular bone coupled to fracture. Arch Appl Mech 88, 2101–2121 (2018). https://doi.org/10.1007/s00419-018-1438-y

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