Monte Carlo Type Simulations of Mineralized Collagen Fibril Based on Two Scale Asymptotic Homogenization

A multiscale model for mineralized collagen fibril (MCF) is proposed by taking into account the uncertainties associated with the geometrical properties of the mineral phase and its distribution in the organic matrix. The asymptotic homogenization approach along with periodic boundary conditions has been used to derive the effective elastic moduli of bone's nanostructure at two hierarchical length scales, namely: microfibril (MF) and MCF. The uncertainties associated with the mineral plates have been directly included in the finite element mesh by randomly varying their sizes and structural arrangements. A total of 100 realizations for the MCF model with random distribution have been generated using an in-house MATLAB code, and Monte Carlo type of simulations have been performed under tension load to obtain the statistical equivalent modulus. The deformation response has been studied in both small (⁠$≤$10%) and large (⁠$≥$10%) strain regimes. The stress transformation mechanism has also been explored in MF which showed stress relaxation in the organic phase upon different stages of mineralization. The elastic moduli for MF under small and large strains have been obtained as 1.88 and 6.102 GPa, respectively, and have been used as an input for the upper scale homogenization procedure. Finally, the characteristic longitudinal moduli of the MCF in the small and large strain regimes are obtained as 4.08 $±$ 0.062 and 12.93 $±$ 0.148 GPa, respectively. All the results are in good agreement to those obtained from previous experiments and molecular dynamics (MD) simulations in the literature with a significant reduction in the computational cost.