13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- 4. Discussion There is some scatter in the literature about the actual fracture toughness of enamel [13]. Table 1 shows the fracture toughness values obtained after applying the semi-empirical equations (1) to (4). These data were within the range of those reported by other authors. Hassan et al. (1981) [14] have reported values of human tooth enamel, using a Vickers indenter combined with a semi-empirical equation type 1 (Eq. (1)), in the range of 0.7 to 1.37 MPa·m1/2. Xu et al. (1998) [12] reported fracture toughness values of 0.84 MPa·m1/2 for labial human enamel also using a Vickers indenter with semi-empirical equation type 1 (Eq. (1)). Bajaj and Arola (2009) [15] reported fracture toughness values for human enamel that ranged from 1.79 MPa·m1/2 to 2.37 MPa·m1/2 obtained from R-curve analysis, and Baldassarri et al. (2008) [16] obtained values of 0.5 MPa·m1/2 and 1.3 MPa·m1/2 for transversal and midsagital enamel orientation, respectively, using a Vickers indenter on rat tooth. Rasmussen and Patchin (1984) [17] used SEM fractography and work-of-fracture techniques to investigate the fracture properties of human enamel and dentin as a function of the temperature of an aqueous environment. In this work, the specimens were notched in order to give controlled fracture in one of two preferred directions, either "perpendicular" or "parallel" to the rods for enamel. They reported values of work-of-fracture for human enamel that range from 0.13 J/m2 (fracture toughness of 1.09 MPa·m1/2) for parallel direction to 1.90-2.00 J/m2 (fracture toughness of 4.18-4.29 MPa·m1/2) for perpendicular orientation, at room temperature. The experimental load-displacement curves did not show, in our case, a constant load step during the loading branch (Fig. 5). In order to apply an energetic approach, an alternative methodology had to be developed. The proposed procedure is based on the contact stiffness variation due to cracking during the indentation process. It should be noted that besides the cracking process, could also have other phenomena that may affect the contact stiffness, as quasi-plastic deformation phenomena associated with the movement of the water and protein phase due to the indentation. However, given the conditions under which the indentation tests were made, with high frequencies of the loading and unloading cycles (45 Hz), and small penetration depths in each cycle (2 nm), the contribution of these phenomena to the variation of the contact stiffness can be considered negligible. It is well known that the area under the load-displacement curve is the work performed by the indenter during elastic-plastic deformation. However, if cracking occurs, part of the elastic energy stored will be released to create new crack surfaces. This will be reflected as a change in the contact stiffness and, consequently, the relationship between contact stiffness, S, and the squared root of the contact area, A, will not be linear as the contact theory predicts [10]: π A S 2 Er ⋅ = ⋅ (6) where Er is the reduced elastic modulus. For Berkovich indenters, the contact area is obtained from the following equation: K ⋅ + ⋅ + ⋅ + ⋅ + = 4 1 3 p 2 1 2 p 1 p 2 p A24.5h Ch Ch Ch (7)
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