Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach

A micromechanical multiscale model which estimates the elastic properties of cortical bone as a function of porosity and mineral content is presented. The steps of the model are divided into two main phases. In the first one, the elastic properties of the collagen fibril, collagen fiber and lamella are given. In the second phase, porosity is included in the lamella in the form of canaliculi, lacunae and Haversian canals, to provide the elastic properties of the osteonal tissue. Then, a symmetrization technique is used to estimate the transversely isotropic elasticity tensor of the osteon. Osteons are superimposed using a self-consistent scheme, and finally, the fluid filling the pores is included to estimate the elastic constants of the undrained cortical tissue. The main novelty of the model presented here is the possibility of varying the mineral content of bone, considering that mineralization begins from the inner levels, initially intrafibrillar and then interfibrillar. Correlations of the elastic properties of cortical bone obtained with this model on the one hand, and porosity and ash fraction on the other hand, are estimated.