A cross-linking model for estimating Young's modulus of artificial bone tissue grown on carbon nanotube scaffold

Carbon nanotubes (CNTs) provide a suitable environment for growth and proliferation of bone cells. The elastic properties exhibited by CNTs can enhance mechanical characteristics of bone mineral phase, hydroxyapatite (HAp), precipitated on such a scaffold. In this article, a simplified model for estimating the axial Young's modulus of a representative volume element (RVE) of CNT-HAp composite is presented. The model is based on the idea of HAp formation on functionalized sites on CNTs as cross-links between HAp matrix and CNT. Modeling results show that the reinforcement role contributed by CNT in the RVE causes a significant increase in the Young's modulus of the composite material which is a direct consequence of transferring stresses from the HAp matrix to the CNT through the cross-links. Similar conclusions may be suggested regarding the improvement of overall mechanical properties of the material. The prediction made by the model lies reasonably well within the limits proposed by conventional Rule-of-Mixtures, and sliding below Voigt's model. The Young's modulus predicted by the model lies adjacent to the Hashin-Shtrikman upper bound as a function of the RVE length (or equivalently CNT aspect ratio). The model simulation indicates that an increase in the CNT aspect ratio and/or number of cross-links in the RVE, results in the prediction to move closer to the estimation made by Voigt as the assumption of perfect bonding between composite phases is approached.