Systematic error in mechanical measures of damage during four-point bending fatigue of cortical bone

Abstract

Accumulation of fatigue microdamage in cortical bone specimens is commonly measured by a modulus or stiffness degradation after normalizing tissue heterogeneity by the initial modulus or stiffness of each specimen measured during a preloading step. In the first experiment, the initial specimen modulus defined using linear elastic beam theory (LEBT) was shown to be nonlinearly dependent on the preload level, which subsequently caused systematic error in the amount and rate of damage accumulation measured by the LEBT modulus degradation. Therefore, the secant modulus is recommended for measurements of the initial specimen modulus during preloading. In the second experiment, different measures of mechanical degradation were directly compared and shown to result in widely varying estimates of damage accumulation during fatigue. After loading to 400,000 cycles, the normalized LEBT modulus decreased by 26% and the creep strain ratio decreased by 58%, but the normalized secant modulus experienced no degradation and histology revealed no significant differences in microcrack density. The LEBT modulus was shown to include the combined effect of both elastic (recovered) and creep (accumulated) strain. Therefore, at minimum, both the secant modulus and creep should be measured throughout a test to most accurately indicate damage accumulation and account for different damage mechanisms. Histology revealed indentation of tissue adjacent to roller supports, with significant sub-surface damage beneath large indentations, accounting for 22% of the creep strain on average. The indentation of roller supports resulted in inflated measures of the LEBT modulus degradation and creep. The results of this study suggest that investigations of fatigue microdamage in cortical bone should avoid the use of four-point bending unless no other option is possible.

Figure 1

The dependence of the initial specimen modulus measured using linear elastic beam theory (LEBT) on the magnitude of the applied preload for specimens loaded from 40 to 300 N and 300 to 40 N, increasing or decreasing, respectively, the load by 20 N every 30 cycles. Error bars show one standard deviation. Data was fit by nonlinear least squares regression using a power-law (R² = 0.78 and 0.80, respectively). Shaded regions show the range of initial specimen moduli, E_o, for preload levels of 100 and 200 N employed for the measurements in Figure 3.

Figure 2

Hysteresis loops for cyclic four-point bending fatigue of a representative specimen at the initial and final loading cycle showing various mechanical measures of damage in terms of (a) stiffness or modulus, and (b) deflection or strain for elastic, e, (recovered) and “plastic,” p, (accumulated) deformation at the initial, o, and n-th loading cycle. Note that the accumulated deformation during fatigue is hereafter, and more accurately, termed creep. Also, note that in the second experiment conversion of a beam deflection into strain was not necessary since all measures of mechanical degradation were reported by a normalized stiffness or strain ratio (Eqs. 3-5), which was equivalent to the corresponding deflection ratio since the dimensions of all specimens were identical.

Figure 2

Hysteresis loops for cyclic four-point bending fatigue of a representative specimen at the initial and final loading cycle showing various mechanical measures of damage in terms of (a) stiffness or modulus, and (b) deflection or strain for elastic, e, (recovered) and “plastic,” p, (accumulated) deformation at the initial, o, and n-th loading cycle. Note that the accumulated deformation during fatigue is hereafter, and more accurately, termed creep. Also, note that in the second experiment conversion of a beam deflection into strain was not necessary since all measures of mechanical degradation were reported by a normalized stiffness or strain ratio (Eqs. 3-5), which was equivalent to the corresponding deflection ratio since the dimensions of all specimens were identical.

Figure 3

Modulus degradation, E/E_o, measured using LEBT during load-controlled four-point bending fatigue of specimens normalized to an initial maximum strain of 6000 μstrain using an initial specimen modulus determined at a preload of either 100 N or 200 N (Fig. 1). Error bars show one standard deviation.

Figure 4

Mechanical degradation based on measurements of the LEBT modulus (or total strain), secant modulus (or elastic strain), and creep (or accumulated strain) during four-point bending fatigue. Error bars show one standard deviation. Note that the normalized LEBT and secant moduli were equivalent to strain ratios enabling a direct comparison to the creep strain ratio.

Figure 5

Mechanical degradation based on measurements of the secant, loading, and unloading stiffness during four-point bending fatigue. Error bars show one standard deviation.

Figure 6

Optical micrographs using green epifluorescence of calcein stained longitudinal beam sections showing (a) minimal and (b) significant indentation adjacent to the roller supports. Note the prominence of vasculature within the tissue underlying the larger indentation.