Numerical Homogenization of Trabecular Bone Specimens using Composite Finite Elements

Numerical homogenization is a tool to determine effective macroscopic material properties for microstructured materials. This tool is tailored and applied to ensembles of young and elder human and of porcine and bovine vertebral bone specimens. On the microscale of the spongiosa a linearized Lam{\'{e}}--Navier type elasticity model is assumed and the computed macroscopic material properties are represented by a general elasticity tensor. The computation is based on a suitable set of microscopic simulations on the cubic specimens for macroscopic strain scenarios. The subsequent evaluation of the effective stresses is used to determine effective linear elasticity tensors. A Composite Finite Element discretization is taken into account to resolve the complicated domain. The classical strain--stress and a corresponding variational homogenization approach are compared. In case of an (artificial) periodic microstructure, a fundamental cell is easily identified and a macroscopic unit strain can be imposed using affine-periodic boundary conditions. In contrast, statistically periodic structures require the identification of statistically representative prototype cells. Unit macroscopic strains are then imposed only in an approximate sense using displacement boundary conditions. The impact of the resulting boundary artifacts on the solution are compensated for via restricting the evaluation of effective stress to a suitably selected smaller subset of the cubic specimen. Furthermore, an optimization approach is used to identify possible axes of orthotropy of the resulting linear elasticity tensor. Finally, the different specimens of human, porcine and bovine spongiosa are analyzed statistically.