13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- the rods. Fig. 4a shows the experimental results of the applied stress vs. HAp lattice strain, indicating the expected linear relationship. The ratio of the stress and the average HAp lattice strain gives the apparent modulus [31]. The preferential orientation of HAp crystals obtained by WAXS pattern analysis is shown to be roughly perpendicular to the arc, while that obtained by SAXS pattern is shown to be roughly along the short axis of the elliptical SAXS pattern (Fig. 2). The detailed values from experiment and model are listed in Table 1. 4.2. Evaluation and testing of multi-scale Eshelby model In the model, the material properties and other parameters were derived from the literature, and refined by fitting with the experimental data. The average mineral concentration (HAp volume fraction) has been reported to be ~95% in each level of enamel, so that the overall volume fraction of HAp could be close to 85%, which is consistent with [6]. In general, Young’s modulus of 1GPa for collagen is found in the literature, without taking into account the viscoelasticity or viscoplasticity [19,20]. Polycrystalline HAp is considered to be transversely isotropic with five independent elastic constants [32]. To describe the shape of the rod and of the HAp crystallites for each level, the Eshelby tensor for the cylinder was used. It was assumed that the lengths of ellisptical axes a1 and a2 within the transverse cross-section were the same, a1/a2=1). The apparent modulus K was calculated based on the different preferred orientation angles of the HAp crystals obtained by both SAXS and WAXS. All the parameters refined to obtain the best fit are listed in Table 1. A comparison of the experimental results and the model evaluation is plotted in Fig. 4. Good agreement is observed. Table 1 Experimental results from SAXS/WAXS, and the refined structural parameters of the enamel used in the two-level Eshelby model Parameter Enamel Orientation (from SAXS) 174° Orientation (from WAXS) 14° Degree of alignment 0.6 K_exp. 124.3 GPa 1 2 f f = 96% 1 matix C Em=1GPa, vm=0.3 2 matix C Em=1GPa, vm=0.3 HAp C Exx=140GPa, Gxy= Gxz=39.6GPa vxy= vxz =0.3, vyz=0.2 Eyy= Ezz=114.3GPa 1 2 cylinder cylinder S S= 1 2 / 1 a a = K_model (from SAXS) 122.9 GPa K_model (from WAXS) 120.4 Gpa
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