On the Development of Life Prediction Methodologies for the Failure of Human Teeth

Human dentin is known to be susceptible to failure under cyclic loading. Surprisingly, there are few reports that quantify the effect of such loading, considering the fact that a typical tooth experiences a million or so loading cycles annually. In the present study, a systematic investigation is described of the effects of prolonged cyclic loading on human dentin in a simulated physiological environment. In vitro stress-life (S/N) data are discussed in the context of possible mechanisms of fatigue damage and failure. Stiffness loss data collected in situ during these tests are used to calculate crack-growth velocities and the fatigue thresholds, and are presented as plots of the crack-propagation rates (da/dN) as a function of the stress-intensity range (∆K). The S/N and da/dN-∆K data are discussed in light of a framework for a fracture mechanics-based methodology for the prediction of the fatigue life of human teeth. Introduction and Background Dentin lies between the exterior enamel and the soft pulp in the core of human teeth. It forms the bulk of the interior structure of the tooth and hence is critical to its structural integrity. Consequently, it is vital that the mechanical properties of dentin are known so that realistic predictions can be made for the effect of microstructural modifications due to caries, sclerosis, aging and dental restorative processes on tooth strength. While several studies have focused on evaluating these properties [e.g., 1-6], there is little consistency in the available data. Exposed root surfaces in teeth often exhibit non-carious notches in the dentin just below the enamelcementum junction; the etiology for such lesions is believed to involve a combination of erosion, abrasion and abfraction. These notches serve as very effective stress raisers and hence, can be sites for failure due to fracture. Although such fractures have not been studied extensively, it is generally believed that tooth failure is associated either with catastrophic events induced by very high occlusal stresses or, more plausibly, by cyclic fatigue-induced subcritical crack growth. In view of this, it is surprising that so few studies have investigated the effect of prolonged fatigue cycling on human dentin, particularly as such information is also important for the development of replacement materials for restorative dentistry. In engineering terms, fatigue is used to describe the response of a material to repeated application of stress (or strain). Obviously, the prediction of a time to failure under such loading is critical for engineering design and durability assurance. The classical approach to fatigue, often called the “stress-life” or “S/N” approach, has involved the characterization of the total life to failure in terms of a cyclic stress range. This approach involves the estimation of the number of such cycles required to induce complete failure of a nominally flaw-free “smoothbar” specimen at a given alternating (σa) and mean (σm) stress. The measured fatigue lifetime represents the number of the cycles needed to both initiate and propagate a (dominant) crack to failure. S/N curves for many materials (e.g., steels) exhibit a plateau at about 10-10 fatigue cycles, termed the fatigue limit, below which failure does not occur [7]. In the absence of a fatigue limit, a fatigue endurance strength is usually defined as the alternating stress needed to give a specific number of cycles to failure. Both terms are used for traditional fatigue design and life prediction, after adjusting for operational variables such as the presence of notches, the environment (e.g., temperature, pH, humidity), the history and spectrum of loading, etc. [7]. In reality, most structures, including human teeth, have an inherent population of flaws. In such cases, the crack initiation life may be non-existent, thus making lifetimes predicted from the S/N approach highly non-conservative. A more realistic approach here is to consider that the life is the cycles needed to propagate one such flaw to failure. To make such predictions, fracture mechanics (damage-tolerant) methodologies are generally used, where the number of cycles required for an incipient crack to grow subcritically to a critical size, defined by the limit load or fracture toughness [6], is computed from information relating the crack velocity to the mechanical driving force (e.g., the stress-intensity factor). The objective of the this study is to characterize the stress-life and crack-propagation fatigue behavior of human dentin in vitro, i.e., in Hank’s Balanced Salt Solution (HBSS), and to use this information as a preliminary basis to develop such lifetime prediction analyses for teeth. Materials and Experimental Procedures Recently extracted human molars, sterilized using gamma radiation, were used in this study. Sections, ~1.5-2.0 mm thick, were prepared from the central portion of the crown and the root vertically through the tooth. The typical microstructure of dentin is shown in Fig. 1. Human dentin is a hydrated composite composed of nanocrystalline apatite mineral (~45% by volume), type-I collagen fibrils (~30% by volume) and fluid, and other non-collageneous proteins (~25% by volume). The mineral is distributed in the form of fine crystallites (5 nm thick) in a scaffold created by the collagen fibrils (50-100 nm diameter). The distinctive feature of the “microstructure” of the dentin is a distribution of cylindrical tubules (~1-2 μm diameter) that run from the dentin-enamel junction to the pulp chamber. These tubules are surrounded by a collar of highly mineralized peritubular dentin (~1 μm thick) and are embedded within a matrix of mineralized collagen (intertubular dentin). The mineralized collagen fibrils form a planar felt-like structure oriented perpendicular to the tubules. There is evidence, although somewhat inconclusive, that the orientation of the tubules leads to anisotropic mechanical properties in human dentin [2,3]. The present study, however, is restricted to a single orientation with cracking nominally perpendicular to the long axis of the tubules, i.e., in the plane of the collagen fibrils. This orientation is believed to have the lowest fracture toughness [2,3]. Fig. 1: Micrograph illustrating the typical microstructure of human dentin. Beams of dentin (0.9 x 0.9 x 10.0 mm) were machined in an attempt to align the tubules to the long axis of the beam. In actuality, it is almost impossible to align the fracture plane precisely with the tubule axes a priori because, with the exception of the root, the tubules in 10 μm dentin do not run a straight course from the enamel to the pulp, but follow a complex, S-shaped curvature [8]. Consequently, the orientation of the crack plane was determined by examination of the fracture surfaces. Samples were obtained from these beams by wet polishing to a 600 grit finish; twenty-five such beams were used in the present study. Each beam included some root dentin and some coronal dentin such that the loading configuration shown in Fig. 2 could be achieved. In vitro first yield (σy) and maximum flexural (σF) strengths levels were measured in HBSS in bending to be σy ~ 75 MPa and σF ~ 160 MPa, respectively. It should be noted here that the “yielding” observed is the result of irrecoverable diffuse damage. Fig. 2: Schematic illustration of the cantilever beam geometry used for in vitro fatigue testing. In vitro S/N fatigue tests were conducted in ambient temperature HBSS with unnotched cantilever beams cycled on an ELF 3200 series acoustic testing machine. Tests were performed at a load ratio, R (minimum load/maximum load) of 0.1 at cyclic frequencies of 2 and 20 Hz. The beams were cycled to failure under displacement control, with the loads being monitored continuously. Stress-life curves were derived for both frequencies in terms of the stress amplitude, σa, (one half of the difference between the maximum and minimum nominal bending stresses). The minimum and maximum stress levels employed ranged between, respectively, ~5 and 135 MPa. Crack-propagation rates were estimated from the loss in stiffness of the test specimens during the S/N tests. Continuous in situ monitoring of the specimen stiffness, measured in terms of the bending load and load-line displacement, was used to yield an estimate of the specimen compliance, which was then related to a crack size using standard beam theory and fracture mechanics analyses. The sample compliance was calculated using the analysis for the additional remote-point displacement (rotation), θcrack, due to a crack in a cantilever bending beam [9]: θcrack = θtotal θno crack , (1) where θtotal is the total rotation and θno crack the rotation of an uncracked sample, respectively. These rotations can be calculated from standard beam theory, where θcrack is given by [9]: θcrack = 4σ S(a/b)/E' , (2) where σ is the maximum nominal bending stress (in the absence of a crack), a is the crack length, b is the beam width, E' is the appropriate Young’s modulus, and S(a/b) is given by [9]: S(a/b) = (a/b){5.93 19.69(a/b) + 37.14(a/b) 35.84(a/b) + 13.12(a/b)}/(1 (a/b)) . (3) 4 mm 2 mm Load