13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- Figure 4. Comparison of the experimental data (markers) and modeling results for applied compressive stress vs. elastic lattice strain for HAp crystals. 5. Discussion & Conclusion The agreement of the apparent modulus K results calculated using the HAp preferred orientation angles determined by SAXS and WAXS suggests that the SAXS patterns arising from gap scattering can be used to deduce the HAp orientation distribution, i.e. the gaps are almost parallel to HAp crystals inside the rod. Further validation can be demonstrated by examining in more detail the effect of crystal orientation on the apparent modulus. A 3D model of perfectly aligned crystals inside a rod is established (Fig. 5a) with the angle ϕdescribing the rotation of the alignment direction around the global z axis. When all HAp crystals are aligned along the global x-direction, ϕequals to 0°. By changing the perfect alignment direction (changing the transformation matrix in Eq. 5), the variation of aligned K obtained in the loading direction can be calculated (Fig. 5b). From Fig. 5b, the corresponding results using the real orientation angles found in the experiment (174° from SAXS and 14° from WAXS) are found to be _ aligned SAXS K = 133.19GPa and _ aligned WAXS K =128.25 GPa, i.e. closely similar values. Meanwhile, due to the high degree of alignment of HAp crystals in the enamel, the value of the overall apparent modulus _ HAp partial aligned K lies close to the value aligned K . The enamel displays strong microscopic elastic anisotropy. It is interesting to note that the stiffest orientation is, as expected, around 0° with respect to the loading direction. However, the most compliant orientation observed is not at 90°, i.e. perpendicular to the loading direction, but rather around 50° or 130°. In this study, the longitudinal apparent Young’s modulus of human enamel was measured during in situ elastic compression by the combination of synchrotron WAXS and SAXS for the first time. This provides access to the information on both the structural and mechanical aspects of the sample and allows us to make further progress compared to previous studies that only used WAXS [11,14]. A multi-scale Eshelby inclusion model was established to estimate the elastic material properties of the enamel in terms of its constituents, considering it as a two-level composite. Good agreement with the experimental data was obtained, indicating an improvement of the earlier proposed composite model [15,16] The model offers a powerful tool for the evaluation of the apparent modulus of enamel-like composites, and helps understand the relationship between the external load
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