The Modified Super-Ellipsoid Yield Criterion for Human Trabecular Bone

Abstract

Despite the importance of multiaxial failure of trabecular bone in many biomechanical applications, to date no complete multiaxial failure criterion for human trabecular bone has been developed. By using experimentally validated nonlinear high-resolution, micromechanical finite-element models as a surrogate for multiaxial loading experiments, we determined the three-dimensional normal strain yield surface and all combinations of the two-dimensional normal-shear strain yield envelope. High-resolution finite-element models of three human femoral neck trabecular bone specimens obtained through microcomputed tomography were used. In total, 889 multiaxial-loading cases were analyzed, requiring over 41,000 CPU hours on parallel supercomputers. Our results indicated that the multiaxial yield behavior of trabecular bone in strain space was homogeneous across the specimens and nearly isotropic. Analysis of stress-strain curves along each axis in the 3-D normal strain space indicated uncoupled yield behavior whereas substantial coupling was seen for normal-shear loading. A modified super-ellipsoid surface with only four parameters fit the normal strain yield data very well with an arithmetic error +/-SD less than -0.04 +/- 5.1%. Furthermore, the principal strains associated with normal-shear loading showed excellent agreement with the yield surface obtained for normal strain loading (arithmetic error +/- SD < 2.5 +/- 6.5%). We conclude that the four-parameter "Modified Super-Ellipsoid" yield surface presented here describes the multiaxial failure behavior of human femoral neck trabecular bone very well.

An experimental and computational investigation of the post-yield behaviour of trabecular bone during vertebral device subsidence

“…The non-linear behaviour of microstructural trabecular bone models have also been modelled by reducing the elastic modulus of the trabecular material to 5% when critical principal strain is computed at a material point (Bayraktar et al 2004a;2004b;Niebur et al 2000;Guillén et al 2011;Verhulp et al 2008;Harrison et al 2012). Niebur et al (2000) based such criteria on macroscopic testing of cortical bone by Reilly and Burstein (1975).…”

“…Van Rietbergen et al (1995) and Boyd et al (2002) considered confined compression loading that was limited to the linear elastic regime, hence offering no insight into multiaxial yield behaviour. Using the principal strain based criterion at a trabecular level, biaxial compression strain and normal-shear strain of μCT based trabecular specimens has been simulated (Bayraktar et al 2004a;Niebur et al 2002). Niebur et al (2002) Complex multiaxial loading of trabecular bone is encountered during bone fracture, screw pullout, press-fit device implantation (Cawley et al 2012 …”

“…A number of studies have simulated non-linear trabecular behaviour by reducing the elastic modulus of the trabecular material when the principal strain at a material point exceeds a predefined value (Bayraktar et al 2004a;2004b;Niebur et al 2000;Guillén et al 2011;Verhulp et al 2008). Verhulp et al (2008) simulated uniaxial compression using μCT based trabecular geometry with a perfectly-7 plastic VM plasticity formulation, while Harrison et al (2012) simulated material damage and fracture in the trabecular microarchitecture, also under uniaxial compression.…”

“…Verhulp et al (2008) simulated uniaxial compression using μCT based trabecular geometry with a perfectly-7 plastic VM plasticity formulation, while Harrison et al (2012) simulated material damage and fracture in the trabecular microarchitecture, also under uniaxial compression. Apart from the studies of Niebur et al (2002) and Bayraktar et al (2004a), in which biaxial (triaxial stress) and axial-shear testing was simulated using the principal strain based modulus reduction model, none of these μCT studies has investigated the role of microstructural architecture in the multiaxial yielding of trabecular bone at the apparent level by considering loading configurations other than uniaxial compression. Van Rietbergen et al (1995) and Boyd et al (2002) considered confined compression of microstructural trabecular specimens, which were restricted to the elastic regime with maximum apparent strains of 1%, again assuming linear elastic material behaviour.…”

“…Kelly and McGarry (2012) and the present study have demonstrated that continuum based plasticity formulations such as the DP and Mohr-Coulomb formulations that that have linear yield surfaces in the q-p plane are inappropriate for modelling trabecular bone, although they have been shown to capture the inelastic behaviour of cortical bone (Feerick and McGarry 2012;Mullins et al 2009). Plasticity formulations such as 22 the modified super-ellipsoid yield criterion (Bayraktar et al 2004a), the Tsai-Wu plasticity formulation (Fenech and Keaveny 1999;Keaveny et al 1999) and a cellular solid criterion (Fenech and Keaveny 1999) have also been proposed to describe the multiaxial yielding of trabecular bone. The Tsai-Wu plasticity formulation has been shown to reasonably predict the combined axial-shear strength (Fenech and Keaveny 1999) and to poorly predict the triaxial compressive stress In the present study the video imaging of the macroscale VVTs reveal regions of localised trabecular whitening immediately adjacent to the punch which remain almost constant with increased indentation.…”