Cosserat Anisotropic Models of Trabecular Bone from the Homogenization of the Trabecular Structure: 2D and 3D Frameworks

Abstract

Cosserat models of trabecular bone are constructed in 2D and 3D situations, based on micromechanical approaches to investigate microstructure-related scale effects on the macroscopic properties of bone. The effective mechanical properties of cancellous bones considered as cellular solids are obtained thanks to the discrete homogenization technique. The cell walls of the bone microstructure are modeled as Timoshenko thick beams. An anisotropic micropolar equivalent continuum model is constructed, the effective mechanical properties of which are identified. Closed form expressions of the equivalent properties are obtained versus the geometrical and mechanical microparameters, accounting for the effects of bending, axial, and transverse shear deformations; torsion is additionally considered for a 3D geometry. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the micropolar material constants. The stress distribution in a cracked bone sample is computed based on the effective micropolar model, highlighting the regularizing effect of the Cosserat continuum in comparison to a classical elasticity continuum model.