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5.15 Specific Reaction Rate

Specific reaction rate constitutive equations provide a relation for \(k\) as a function of solid matrix strain and the concentrations of solid-bound molecular species.

Constant Specific Reaction Rate

For this material model, \(k\) is constant.

Huiskes Remodeling

For this material, which is based on the bone remodeling framework of Weinans et al. 1, the specific reaction rate depends on the deviation of the specific strain energy from a threshold value,

\[ \begin{equation} k\left(\mathbf{F},\rho_{r}^{s}\right)=\frac{B}{\left(J-\varphi_{r}^{s}\right)M^{s}}\left(\frac{\Psi_{r}}{\rho_{r}^{s}}-\psi_{0}\right)\,,,\label{eq531} \end{equation} \]

where \(B\) is a constant, \(\Psi_{r}\) is the strain energy density of the solid, \(\rho_{r}^{s}\) is the referential mass density of the solid, \(\psi_{0}\) is the threshold value for the specific strain energy. In this relation, \(J=\det\mathbf{F}\) is evaluated from the solid deformation and \(\varphi_{r}^{s}\) is evaluated from (2.12-7).

In the subsequent study by Mullender et al. 2 it was proposed to let osteocytes serve as sensing cells, such that the remodeling rate may depend on cells in the neighborhood of the the point at which remodeling is calculated. In that case, the model of eq.\eqref{eq531} is extended to include the contribution from neighboring sensing cells,

\[ \begin{equation} k\left(\mathbf{F},\rho_{r}^{s}\right)=\frac{1}{\left(J-\varphi_{r}^{s}\right)M^{s}}\left(B\left(\frac{\Psi_{r}}{\rho_{r}^{s}}-\psi_{0}\right)+\sum_{i=1}^{N}e^{-d_{i}/D}B_{i}\left(\frac{\Psi_{ri}}{\rho_{ri}^{s}}-\psi_{0}\right)\right)\,.\label{eq:eq531b} \end{equation} \]

Here, \(N\) represents the number of elements in the neighborhood of the current material point element, which fall within a characteristic sensing distance \(D\), and \(d_{i}\) represents the distance between these neighboring points and the current material point.

  1. Weinans, H; Huiskes, R; Grootenboer, H J. "The behavior of adaptive bone-remodeling simulation models." J Biomech, vol. 25, pp. 1425-41 (1992). ↩

  2. Mullender, M G; Huiskes, R; Weinans, H. "A physiological approach to the simulation of bone remodeling as a self-organizational control process." J Biomech, vol. 27, pp. 1389-94 (1994). ↩