Fabric-based Tsai-Wu yield criteria for vertebral trabecular bone in stress and strain space

Osteoporosis related vertebral fractures are an increasing clinical problem in ageing societies. The prediction of vertebral fracture load from QCT-based anatomy-specific finite element simulations could be very useful in the management of patients with osteoporosis, especially with regard to a possible fracture prevention or treatment optimisation. A key property in finite element analysis is the yield surface for the trabecular bone material. This study is aimed at identifying continuum-level yield criteria for vertebral trabecular bone using micro-finite element models subjected to uni-axial, shear, and tri-axial loading. A fabric-dependent, orthotropic Tsai-Wu yield criterion is proposed in both stress and strain spaces. Nonlinear micro-finite element models of cubic vertebral trabecular bone samples with 5.62 mm edge length were generated from μCT-scans. Kinematic boundary conditions were imposed and the specimen was loaded force controlled beyond yield in 17 different load cases (six uni-axial, three shear and eight multi-axial). The proposed yield criteria were fitted to the resulting yield data. Yield strains on-axis were significantly lower (10% in tension and 6% in compression) than in the transverse directions. Average yield strains were 0.7% in tension, 1.1% in compression, 1.0% in shear and ranged from 0.6% to 1.1% under multi-axial loading. In axial direction, maximum yield stress was 2.6 MPa in tension and 4.7 MPa in compression. Lowest shear stress was found in the transverse plane with 1.3 MPa. Multi-axial yield stresses ranged between values for uni-axial tension and compression. Yield stresses depended significantly and substantially on both volume fraction and fabric. Yield strains depended also significantly on both bone volume fraction and fabric, but only weakly on the former. The standard error of the estimate and the concordance correlation coefficient of the yield surface were 5.47% and 0.93 in strain space and 13.58% and 0.96 in stress space. The results of this study are not only consistent with experimental data from the literature but also extend the current knowledge of yield to multi-axial load cases that can hardly be realised in a biomechanical experiment. The presented yield data and criteria will help improving the prediction of vertebral ultimate load using anatomy-specific finite element models.