Application of Homogenization Theory to Study the Mechanics of Cortical Bone

Homogenization theory is utilized to study the effect on the axial stiffness of secondary osteons in cortical bone due to the presence of micro porous features (e.g., lacunae, canaliculi clusters, and Haversian canals). Specifically, 2 geometric characteristics were used to describe these features within the secondary osteons: volume fraction (% porosity) and shape (circular- or elliptical-shaped). Such information was determined for each individual porous feature from an image segmentation methodology developed earlier by Hage and Hamade. For each feature, aspect ratio vectors (or arrays of ratios for each individual porous feature) were used to classify each pore inhomogeneity as cylindrical, elliptical or irregular shape. Two prominent homogenization theories were used: the Mori-Tanaka (MT) and the generalized self-consistent method (GSCM). Using the results of image segmentation, it was possible to calculate the respective Eshelby tensors of each porous feature. To calculate the isotropic stiffness tensors for matrix (C_m) and pores (C_p) the Young’s modulus and Poisson’s ratio for the matrix (E_m, ν_m) were assigned as obtained from literature and as those of blood (E_p=10MPa, ν_p= 0.3), respectively. The effective elastic stiffness tensors (C*) for the secondary osteons were obtained from which axial Young’s modulus was obtained as function of volume fraction (% porosity) of each pore type and their individual shapes. The normalized axial Young’s modulus was found to 1) significantly decrease with increasing volume fraction (%) of porosity and 2) for the same % porosity, to slightly decrease (increase) with increasing ratio of circular-shaped to elliptical-shaped (elliptical-shaped to circular-shaped) porous features. These findings were validated using experimental micro-indentation study performed on secondary osteons.