Shear strength behavior of human trabecular bone

Abstract

The shear strength of human trabecular bone may influence overall bone strength under fall loading conditions and failure at bone-implant interfaces. Here, we sought to compare shear and compressive yield strengths of human trabecular bone and elucidate the underlying failure mechanisms. We analyzed 54 specimens (5-mm cubes), all aligned with the main trabecular orientation and spanning four anatomic sites, 44 different cadavers, and a wide range of bone volume fraction (0.06-0.38). Micro-CT-based non-linear finite element analysis was used to assess the compressive and shear strengths and the spatial distribution of yielded tissue; the tissue-level constitutive model allowed for kinematic non-linearity and yielding with strength asymmetry. We found that the computed values of both the shear and compressive strengths depended on bone volume fraction via power law relations having an exponent of 1.7 (R(2)=0.95 shear; R(2)=0.97 compression). The ratio of shear to compressive strengths (mean±SD, 0.44±0.16) did not depend on bone volume fraction (p=0.24) but did depend on microarchitecture, most notably the intra-trabecular standard deviation in trabecular spacing (R(2)=0.23, p<0.005). For shear, the main tissue-level failure mode was tensile yield of the obliquely oriented trabeculae. By contrast, for compression, specimens having low bone volume fraction failed primarily by large-deformation-related tensile yield of horizontal trabeculae and those having high bone volume failed primarily by compressive yield of vertical trabeculae. We conclude that human trabecular bone is generally much weaker in shear than compression at the apparent level, reflecting different failure mechanisms at the tissue level.

Figure 1

Validation of predictions of compressive yield strength, for the 22 specimens in this study that had both micro-CT scans and experimental measures of compressive yield strength. The same tissue-level effective modulus of 18.0 GPa was used in all models. Orthogonal regression was used since there were measurement errors in both experimental and finite element strength measures. Values in brackets represent the 95% confidence interval for the slope.

Figure 2

(a) A 5mm cube trabecular bone specimen with its Z-axis oriented along the main material orientation (the X-axis was selected randomly in the transverse plane) (b) An engineering XZ shear strain of 1.5% was applied for shear strength analysis, the dashed lines denoting the prescribed displacement boundary condition on four faces, the other two faces remaining unconstrained (c) A compressive normal strain of 1.0% was applied for compressive strength analysis, the dashed lines denoting the prescribed displacement boundary condition on the top face, the bottom face minimally fixed on rollers and the sides faces unconstrained.

Figure 3

Variation of compressive (solid line), shear (dashed line), and torsion (light line) apparent-level 0.2% offset yield strengths with bone volume fraction.

Figure 4

Variation of the ratio of shear to compressive yield strengths with the bone volume fraction. This ratio (mean ± SD = 0.44 ± 0.16 for 54 specimens) displayed additional scatter below a bone volume fraction of about 0.20. This ratio for the traditional von Mises criterion is 0.58.

Figure 5

Variation of the proportion of yielded tissue (expressed as a percentage of total tissue in the bone specimen) at the apparent-level 0.2% offset yield point for shear and compression loading.

Figure 6

Variation of the ratio of tissue yielded in tension to tissue yielded in compression at the apparent-level 0.2% offset yield point, for compression (left) and shear (right) loading. Note the 10-fold difference in the vertical scales between plots.

Figure 7

Distribution of tissue-level yielding for a thin slice for two specimens, of low (BV/TV = 0.07) and high (BV/TV = 0.26) bone volume fraction. Red regions denote tissue-level yield in tension and blue regions denote tissue-level yield in compression. The Z-axis denotes the main trabecular orientation.

Figure 8

Literature comparison (Garnier et al. 1999; Ford and Keaveny 1996; Rincón-Kohli and Zysset 2009; Garrison et al. 2011) of torsional yield strength vs. bone volume fraction, for 15 specimens from this study. The error bars show one standard deviation around the mean value of bone volume fraction and yield strength. For studies that did not report apparent density, the bone volume fraction was calculated assuming a tissue density of 2.0 g/cm³. The data from (Bevill et al. 2009a) is unpublished (n=6 vertebral body specimens).