Formulation of an effective growth response of trabecular bone based on micromechanical analyses at the trabecular level

摘要

We construct in the present paper the effective growth response of trabecular bone, based on micromechanical analyses at the scale of a representative volume element consisting of individual trabeculae defining the representative unit cell (RUC). From a fundamental perspective, we adopt the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the surface remodeling of the individual trabeculae, the averaging of which leads to a net growth at the mesoscopic scale of the RUC. The effective model for the growing unit cell relies on an average kinematics which has been formulated in terms of the effective growth velocity gradient and effective elastic rate of deformation tensor, both functions of the rate of the effective density and stress applied over the RUC. The evaluation of the relation between the average rate of growth tensor and elastic growth rate versus the external stress applied to the RUC is obtained from a split of the average rate of growth tensor into its spherical and deviatoric parts, each of which being related to the corresponding applied stress quantities. The effective growth model is written for three different loading conditions as a polynomial relation between the components of the rate of growth tensor and similar components of the stress tensor under planar conditions, with a nonlinear function of the effective density as a weighting factor. The developed methodology is quite general and is applicable to any choice of the RUC morphology representative of the internal bone architecture.