Evaluating the macroscopic yield behaviour of trabecular bone using a nonlinear homogenisation approach

Computational homogenisation approaches using high resolution images and finite element (FE) modelling have been extensively employed to evaluate the anisotropic elastic properties of trabecular bone. The aim of this study was to extend its application to characterise the macroscopic yield behaviour of trabecular bone. Twenty trabecular bone samples were scanned using a micro-computed tomography device, converted to voxelised FE meshes and subjected to 160 load cases each (to define a homogenised multiaxial yield surface which represents several possible strain combinations). Simulations were carried out using a parallel code developed in-house. The nonlinear algorithms included both geometrical and material nonlinearities. The study found that for tension-tension and compression-compression regimes in normal strain space, the yield strains have an isotropic behaviour. However, in the tension-compression quadrants, pure shear and combined normal-shear planes, the macroscopic strain norms at yield have a relatively large variation. Also, our treatment of clockwise and counter-clockwise shears as separate loading cases showed that the differences in these two directions cannot be ignored. A quadric yield surface, used to evaluate the goodness of fit, showed that an isotropic criterion adequately represents yield in strain space though errors with orthotropic and anisotropic criteria are slightly smaller. Consequently, although the isotropic yield surface presents itself as the most suitable assumption, it may not work well for all load cases. This work provides a comprehensive assessment of material symmetries of trabecular bone at the macroscale and describes in detail its macroscopic yield and its underlying microscopic mechanics.