Volume effects on fatigue life of equine cortical bone

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Abstract

Materials, including bone, often fail due to loading in the presence of critical flaws. The relative amount, location, and interaction of these flaws within a stressed volume of material play a role in determining the failure properties of the structure. As materials are generally imperfect, larger volumes of material have higher probabilities of containing a flaw of critical size than do smaller volumes. Thus, larger volumes tend to fail at fewer cycles compared with smaller volumes when fatigue loaded to similar stress levels. A material is said to exhibit a volume effect if its failure properties are dependent on the specimen volume. Volume effects are well documented in brittle ceramics and composites and have been proposed for bone. We hypothesized that (1) smaller volumes of cortical bone have longer fatigue lives than similarly loaded larger volumes and (2) that compared with microstructural features, specimen volume was able to explain comparable amounts of variability in fatigue life. In this investigation, waisted rectangular specimens (n=18) with nominal cross-sections of 3×4 mm and gage lengths of 10.5, 21, or 42 mm, were isolated from the mid-diaphysis of the dorsal region of equine third metacarpal bones. These specimens were subjected to uniaxial load controlled fatigue tests, with an initial strain range of 4000 microstrain. The group having the smallest volume exhibited a trend of greater log fatigue life than the larger volume groups. Each volume group exhibited a significant positive correlation between the logarithm of fatigue life and the cumulative failure probability, indicating that the data follow the two-parameter Weibull distribution. Additionally, log fatigue life was negatively correlated with log volume, supporting the hypothesis that smaller stressed volumes of cortical bone possess longer fatigue lives than similarly tested larger stressed volumes.

Introduction

Cortical bone is analogous to a fiber-reinforced ceramic matrix composite with the osteons acting as fibers within an interstitial matrix (Currey, 1964; Hogan, 1992; Buckwalter et al., 1995; Martin et al., 1998; Bigley et al., 2006). Ceramic matrix composites often show fatigue life variation due to the number and distribution of defects (Weibull, 1951; Hertzberg, 1996; Wisnom, 1999; Cattell and Kibble, 2001; Rentzsch, 2003). Although osteonal bone possesses several advantageous properties associated with ceramic matrix composites, it also possesses defect populations. Microcracks, regions of poor collagen quality, and microstructural components acting as stress concentrators (e.g. osteons, Haversian canals, resorption cavities) have been associated with low toughness in cortical bone (Schaffler et al., 1995; Wang and Puram, 2004; O’Brien et al., 2005). These interactions of microcracks with the microstructural components suggest that a single value of material strength or toughness is not sufficient to characterize the failure behavior of bone tissue.

Brittle material failure can be caused by a single critical defect or by several small defects acting together to create a critical flaw (Wisnom, 1999). The Weibull “weakest link” approach enables failure characterization when defects can be assumed to be randomly distributed throughout a material (Weibull, 1951). Due to the statistical nature of the occurrence and size of such defects in ceramic materials, many exhibit a volume effect. A material is said to exhibit a volume effect if the probability of failure increases in proportion to the stressed volume of the specimen. Griffith (1920) reported that failure strength increased with decreased fiber diameter in his pioneering work on brittle solids. Such volume effects are fundamental to understanding failure probability (Wisnom, 1999; Rentzsch, 2003).

If failure is determined by the presence of a critical defect, and such defects occur randomly within a material, it follows that for a given stress, larger volumes of a material will have higher failure probabilities because they have a higher probability of possessing a critical defect (Hertzberg, 1996). Weibull's statistical theory can be used to characterize the variability in fatigue life associated with the size of a structure or test specimen (Wisnom, 1999; Cattell and Kibble, 2001).

The two-parameter Weibull fatigue life model can be used to describe the failure probability, P, of a volume, V, of a material subject to a cyclically applied uniform stress field:P=1-exp[V(NfN0)mf],where Nf is the number of load cycles, N0 is the characteristic fatigue life of a unit volume, and mf is the Weibull fatigue modulus (Weibull, 1951; Hertzberg, 1996; Wisnom, 1999; Cattell and Kibble, 2001). The fatigue modulus describes the shape of the distribution and is representative of the fatigue life variability among test specimens of the same volume tested under similar stress conditions. High values of mf correspond to reduced variability of fatigue life, while small values of mf are associated with increased variability in fatigue life (Rentzsch, 2003).

Taylor (1998) analyzed fatigue strength data from numerous sources and published a model for bone fatigue strength based on stressed volume. In this volume effects model, the fatigue strength of human bone was defined as the failure stress range in MPa, determined from a stress-life curve, for a fatigue life of 100,000 cycles for bone tested wet at 2 Hz in zero-to-tension corrected to 37 °C (Taylor, 1998; Taylor et al., 1999). The study demonstrated that prediction of bone fatigue behavior from small test specimens is not applicable to larger specimens unless the volume effect is considered (Taylor, 1998). Taylor and Kuiper later combined this stressed volume model with finite element modeling to predict clinical stress fracture probabilities in the human tibia (Taylor and Kuiper, 2001). The model was later refined to incorporate the effects of remodeling and damage (Taylor et al., 2004). Weibull theory has also been used to analyze equine and human fatigue data to investigate the implications of volume effects on bone adaptation (Yeh and Martin, 2003). This investigation revealed Weibull fatigue life moduli ranging between 0.5 and 1.5 for initial strain amplitudes between −5000 and +10,000 microstrain. Although these studies successfully used the Weibull distribution to analyze fatigue strength data, to our knowledge no a priori study has directly tested the applicability of the Weibull distribution to cortical bone strengths measured using specimens of different volumes.

We hypothesize that strength volume effects in cortical bone are manifested in fatigue life and can be characterized by Weibull theory. Consequently, for a given applied stress, smaller stressed volumes have a lower probability of failure and will thus possess a statistically longer fatigue life when compared with larger stressed volumes. Additionally, we hypothesized that the observed volume effect explains variability of the fatigue life of equine cortical bone not accounted for by microstructural variables.

Section snippets

Experimental preparation

Originally, 30 specimens were obtained from the third metacarpal (cannon) bones of 15 necropsied Thoroughbred racehorses. These racehorses consisted of 4 females, 3 males, and 8 castrated males with ages ranging between 2 and 7 years. The bone specimens were kept frozen at −20 °C except while machining or testing, when they were hydrated with saline at room temperature. One rectangular beam, nominally 140×15×6 mm, was cut with a bone saw (Hobart Corp, Troy, OH) from the dorsal region of the

Results

Twelve test specimens were excluded from the analysis because failure occurred outside of the waisted gage region. The current analysis consists of 18 test specimens from 14 racehorses (3 females, 3 males, 8 castrated males) between 2 and 6 years.

Discussion

Stress fractures that occur in military recruits, athletes, and Thoroughbred racehorses are clinical manifestations of bone biology under the influence of the mechanical processes of fatigue and fracture. The mechanistic principles causing these manifestations are not fully understood. However, it appears that variations in collagen, mineral, and other microstructural components of bone play significant roles in these failure processes (Schaffler et al., 1987; Schaffler et al., 1995; Currey,

Conflict of interest

None of these authors has conflicts of interest

Acknowledgments

This work was supported by the Doris Linn Chair of Bone Biology. The authors are grateful to Shane Curtiss, Justin Creel and Ron June for helpful discussions and technical assistance.

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Orthopaedic Research Laboratory, Research Building 1, Room 2000, UC Davis Medical Center, 4635 Second Ave., Sacramento, CA 95817

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