Comparison of linear and nonlinear stepwise μFE displacement predictions to digital volume correlation measurements of trabecular bone biopsies

Abstract

Digital volume correlation (DVC) enables to evaluate the ability of μFE models in predicting experimental results on the mesoscale. In this study predicted displacement fields of three different linear and materially nonlinear μFE simulation methods were compared to DVC measured displacement fields at specific load steps in the elastic regime (StepEl) and after yield (StepUlt). Five human trabecular bone biopsies from a previous study were compressed in several displacement steps until failure. At every compression step, μCT images (resolution: 36 μm) were recorded. A global DVC algorithm was applied to compute the displacement fields at all loading steps. The unloaded 3D images were then used to generate homogeneous, isotropic, linear and materially nonlinear μFE models. Three different μFE simulation methods were used: linear (L), nonlinear (NL), and nonlinear stepwise (NLS). Regarding L and NL, the boundary conditions were derived from the interpolated displacement fields at StepEl and StepUlt, while for the NLS method nonlinear changes of the boundary conditions of the experiments were captured using the DVC displacement field of every available load step until StepEl and StepUlt. The predicted displacement fields of all μFE simulation methods were in good agreement with the DVC measured displacement fields (individual specimens: R2>0.83 at StepEl and R2>0.59 at StepUlt; pooled data: R2>0.97 at StepEl and R2>0.92 at StepUlt). At StepEl, all three simulation methods showed similar intercepts, slopes, and coefficients of determination while the nonlinear μFE models improved the prediction of the displacement fields slightly in all Cartesian directions at StepUlt (individual specimens: L: R2>0.59 and NL, NLS: R2>0.68; pooled data: L: R2>0.92 and NL, NLS: R2>0.94). Damaged/overstrained elements in L, NL, and NLS occurred at similar locations but the number of overstrained elements was overestimated when using the L simulation method. Considering the increased solving time of the nonlinear μFE models as well as the acceptable performance in displacement prediction of the linear μFE models, one can conclude that for similar use cases linear μFE models represent the best compromise between computational effort and accuracy of the displacement field predictions.