Nonlinear Behavior of Trabecular Bone at Small Strains

Study of the behavior of trabecular bone at strains below 0.40 percent is of clinical and biomechanical importance. The goal of this work was to characterize, with respect to anatomic site, loading mode, and apparent density, the subtle concave downward stress–strain nonlinearity that has been observed recently for trabecular bone at these strains. Using protocols designed to minimize end-artifacts, 155 cylindrical cores from human vertebrae, proximal tibiae, proximal femora, and bovine proximal tibiae were mechanically tested to yield at 0.50 percent strain per second in tension or compression. The nonlinearity was quantified by the reduction in tangent modulus at 0.20 percent and 0.40 percent strain as compared to the initial modulus. For the pooled data, the $mean±SD$ percentage reduction in tangent modulus at 0.20 percent strain was $9.07±3.24$ percent in compression and $13.8±4.79$ percent in tension. At 0.40 percent strain, these values were $23.5±5.71$ and $35.7±7.10$ percent, respectively. The magnitude of the nonlinearity depended on both anatomic site $p<0.001$ and loading mode $p<0.001,$ and in tension was positively correlated with density. Calculated values of elastic modulus and yield properties depended on the strain range chosen to define modulus via a linear curve fit $p<0.005.$ Mean percent differences in 0.20 percent offset yield strains were as large as 10.65 percent for some human sites. These results establish that trabecular bone exhibits nonlinearity at low strains, and that this behavior can confound intersite comparisons of mechanical properties. A nonlinear characterization of the small strain behavior of trabecular bone was introduced to characterize the initial stress–strain behavior more thoroughly.