A multiscale homogenization procedure combining the fabric tensor with a natural neighbour meshless method

Abstract

Bone is a material that can be classified as a hierarchical structure, where the different structural levels can be identified from the microscale to macroscale. Multiscale models enable to model the material using homogenization techniques. In this work a innovative homogenization technique for trabecular bone tissue is proposed, which uses the fabric tensor concept and a bone phenomenological material law, linking the apparent density with the trabecular bone mechanical proprieties. The proposed methodology efficiently homogenize the trabecular bone highly heterogeneous medium, allowing to define its homogenized microscale mechanical properties and to reduce the analysis computational cost (when compared with classical homogenization techniques). The homogenization technique is combined with the natural neighbour radial point interpolation method (NNRPIM). The NNRPIM uses the natural neighbour mathematical concept to enforce the nodal connectivity and build the background integration mesh, required to numerically integrate the integro-differential elasticity equations. Furthermore, the NNRPIM uses the radial point interpolators technique to construct its interpolating shape functions. In order to verify the efficiency of the technique several examples are solved using a confined square patch of trabecular bone under compression. In the end, the results obtained with a classic homogenization technique and the proposed methodology are compared.