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On the in vitro Fatigue Behavior of Human Dentin: Effect of Mean Stress
1 Materials Sciences Division, Lawrence Berkeley National Laboratory, and Department of Materials Science and Engineering, Hearst Mining Building, University of California, Berkeley, CA 94720; and Correspondence: * corresponding author, RORitchie{at}lbl.gov
Human dentin is susceptible to failure under repetitive cyclic-fatigue loading. This investigation seeks to address the paucity of data that reliably quantify this phenomenon. Specifically, the effect of alternating vs. mean stresses, characterized by the stress- or load-ratio R (ratio of minimum-to-maximum stress), was investigated for three R values (–1, 0.1, and 0.5). Dentin was observed to be prone to fatigue failure under cyclic stresses, with susceptibility varying, depending upon the stress level. The "stress-life" (S/N) data obtained are discussed in the context of constant-life diagrams for fatigue failure. The results provide the first fatigue data for human dentin under tension-compression loading and serve to map out safe and unsafe regimes for failure over a wide range of in vitro fatigue lives (< 103 to > 106 cycles).
Key Words: dentin • fatigue • S/N behavior • stress-ratio • fractography • constant-life diagram
Dentin, the most abundant mineralized tissue in teeth, is a hydrated composite of collagen fibrils and nanocrystalline apatite (Kinney et al., 2003). Fluid-filled cylindrical tubules course from the dentin-enamel and cementum-enamel junctions to the pulp; they are surrounded by a thin, highly mineralized peritubular dentin cuff. The mineralized collagen fibrils are arranged orthogonally to the tubules (Jones and Boyde, 1984), forming a planar, felt-like structure called the intertubular dentin matrix. Since dentin forms the bulk of the human tooth and is critical to its structural integrity, an understanding of its mechanical properties is important for life prediction, particularly because of the effect of microstructural modifications from caries, sclerosis, aging, and restorative processes. While several studies have focused on evaluating the properties of dentin (e.g., Craig and Peyton, 1958; Lehman, 1967; Rasmussen et al., 1976; Goodis et al., 1993; Sano et al., 1994; Tonami and Takahashi, 1997; Arola et al., 2002; Kinney et al., 2003; Nalla et al., 2003a), there are only limited data on the role of fatigue loading in affecting the mechanical behavior of dentin (Tonami and Takahashi, 1997; Arola et al., 2002; Nalla et al., 2003a). Exposed root surfaces commonly exhibit non-caries lesions in the form of notches in the dentin just below the enamel-cementum junction. The etiology for such lesions is believed to involve a combination of erosion, abrasion, and abfraction (Levitch et al., 1994). Such notches serve as effective stress-raisers and sites of fracture-induced failure. While cusp fractures sometimes occur in posterior teeth, anterior teeth are more susceptible to fracture at the gingiva, severing the crown. Although such fractures have not been studied extensively, it is believed that tooth failure is often associated with either catastrophic events induced by high occlusal stresses or subcritical crack-growth induced by cyclic (fatigue) loading (Wiskott et al., 1995).
To date, only limited studies have been performed on fatigue in dentin. Tonami and Takahashi (1997) used a "staircase" method to determine values of 46.9–51.0 MPa for the 105-cycle endurance strength of bovine dentin, the lower value being obtained for dentin from older animals. Although this study suggested an effect of aging, it was restricted to a narrow range of purely tensile stress-ratios (R = 0.15–0.25), a low number of cycles to failure (105-cycle fatigue strengths may not be "conservative", since a typical human tooth experiences ~ 106 cycles annually), and tests in water rather than simulated body fluid, raising concerns of demineralization. [The stress (or load) ratio, R, defined as the ratio of the minimum-to-maximum stress (or load), is a measure of the magnitude of the mean vs. alternating stresses (or loads); see Fig. 1a
where
Materials Human molars, recently extracted according to protocols approved by the University of California, San Francisco, Committee on Human Research, and by the Lawrence Livermore National Laboratory, were used in this study. We obtained samples by wet polishing (600-grit finish) sections (~ 1.5–2.0 mm thick) cut from the central portion of the crown and the root vertically through the tooth. Seventeen such beams (0.9–1.5 x 0.9–1.5 mm cross-section, 10.0–11.5 mm long) were used, with the long axes of the tubules aligned along the length of the beam. This "perpendicular" orientation (relative to the orientation of the crack plane) has the lowest toughness (e.g., Rasmussen et al., 1976; Nalla et al., 2003b). In actuality, it is impossible to align the fracture plane precisely with the tubule axes a priori because, with the exception of the root, the tubules do not run a straight course from the enamel to the pulp. Rather, from the cervical margin through the crown, the tubules have a complex S-shaped curvature (Ten Cate, 1994). Hence, the precise orientation of the crack plane was determined afterward from examination of the fracture surfaces. In vitro first yield (  y) and maximum flexural tensile (ts) strengths were measured in Hanks’ Balanced Salt Solution (HBSS) in bending to be y ~ 75 MPa and ts ~ 160 MPa (Fig. 1d ); these values were used for selection of stress levels for fatigue testing and in the constant-life approach. All specimens were kept hydrated throughout preparation and testing.
Stress-life (S/N) Fatigue Testing
Fractography
Stress-life Fatigue Behavior Stress-life (S/N) data are shown in Fig. 2a  in the form of number of fatigue cycles to failure, Nf, as a function of the stress-amplitude, a. Fatigue data spanning greater than three decades of lifetimes (~ 103 to > 106 cycles) are shown (each data point represents the reported lifetime from a single specimen). S/N data for various stress-ratios (R = –1, 0.1, and 0.5) at a constant frequency (10 Hz) and various frequencies (2, 10, and 20 Hz; 2 and 20 Hz data from Nalla et al., 2003a) at constant stress-ratio (R = 0.1), are shown in Figs. 2a and 2b, respectively.
For R = –1, the stress-amplitude at zero mean-stress, ao, can be mathematically related to the number of stress reversals, 2Nf, to failure (twice the number of cycles) by the Basquin (1910) formulation:
where
This formulation yields more reasonable estimates for lifetimes at R = 0.1 (R2 = 0.8), as compared with R = 0.5 (R2 = 0.6). Overall, Eq. 3
Fractographic Observations
In common engineering parlance, fatigue refers to the response of a material to repeated application of stress or strain. For most structural components, the prediction of number of cycles to failure under such repetitive loading is often a crucial step for design and durability assurance. The traditional S/N approach used herein serves to determine the cycles required to induce failure of nominally flaw-free samples loaded at specific alternating- and mean-stresses (as in Fig. 1b ); the consequent fatigue lifetime therefore represents the cycles to both initiate and propagate a crack to failure.
Consistent with previous studies (Nalla et al., 2003a), dentin was seen to display "metal-like" stress-life behavior, with an apparent fatigue limit for lives in excess of 106 cycles (Fig. 2a
The effect of mean-stress on the fatigue lifetime is often represented in terms of constant-life diagrams (Fig. 1c
where
In Fig. 4
With the Gerber and Goodman lines, respectively, as upper and lower bounds (Fig. 4d ), these diagrams can provide a reasonable definition of safe and unsafe regions for in vitro fatigue failure of dentin. For example, if the loading involves a maximum stress of 50 MPa and a minimum stress of –50 MPa (a = 50 MPa, m = 0 MPa), failure would probably occur in under one million cycles. Conversely, a maximum stress of 25 MPa and a minimum stress of –15 MPa (a = 20 MPa, m = 5 MPa) are unlikely to induce failure in a million cycles. Since human teeth experience ~ 5 x 105 to 106 loading cycles per year with typical masticatory stress levels on the order of 20 MPa (Anderson, 1956), the current results are comforting, since the S/N approach does not predict failures in dentin under these conditions (except perhaps with very high mean stresses), since stresses lie below the relevant endurance strengths. In summary, S/N data are presented for in vitro fatigue of human dentin over a wide range of alternating stresses (involving tension and compression loading), tensile mean-stresses, and stress-ratios from –1 to 0.5. The results are quantified in terms of both (i) a mean-stress modified Basquin equation to describe the relationship between applied mean/alternating stresses and lifetime, and (ii) constant-life diagrams based on the Gerber, Goodman, and Söderberg relationships. It was found that human dentin displays "metal-like" S/N behavior with an apparent fatigue limit. To account for the effects of tensile mean-stresses, the lifetime data were found to be bounded by the Gerber parabola at lower lives and by the Goodman line at longer lives. By using these relationships, we have been able to span the entire range of possible load/unload conditions with relatively few S/N measurements. These results suggest that an undamaged tooth is designed not to fail from normal mastication. However, it should be noted that our measurements have been made with undamaged specimens in a controlled environment and a single orientation. Therefore, it is possible that these fatigue measurements may not provide a conservative estimate for tooth lifetimes. Further studies designed to include such potentially deleterious effects are warranted. Nevertheless, these results are of significance for the development of a micro-mechanistic understanding of how cyclic-fatigue loading can affect the behavior of human dentin, with consequent implications for lifetime prediction.
This work was supported by Grant #P01DE09859 from National Institutes of Health/National Institute of Dental and Craniofacial Research and by Grant #DE-AC03-76SF00098 from the US Department of Energy, Office of Science-Basic Energy Sciences, Division of Materials Sciences and Engineering (for ROR). The authors thank Grace Nonomura for specimen preparation. Received for publication March 27, 2003. Revision received July 28, 2003. Accepted for publication December 15, 2003.
Journal of Dental Research, Vol. 83, No. 3,
211-215 (2004)
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ABSTRACT
INTRODUCTION
m) of: 
K is the stress-intensity range (difference in maximum and minimum stress intensities) and C and m are experimentally determined scaling constants. Taken as a whole, these studies represent only a limited understanding of how dentin responds to cyclic vs. mean stresses. To address this specific issue, the present work aims to document the role of the stress-ratio (i.e., mean vs. alternating stress; Fig. 1a
0), 