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Although previous investigations provide a good understanding of the relationship between microstructure and fracture resistance of enamel, the role of rod decussation is still elusive. It is well known that the crossing of rod bundles generates the parazone and diazone in the inner enamel (Bajaj and Arola, 2009; Chai, 2014); however, crack growth in the parazone or diazone of enamel has been essentially unexplored. How does the diazone contribute to fracture resistance of enamel? What is the role of the parazone of enamel? How does crack interact with the two distinct zones in enamel? These questions call for a good understanding of the effect of rod decussation on fracture of enamel. The primary goal of this study is to unveil the individual roles of the parazone and diazone on crack growth in enamel. The structural features of rod decussation are experimentally explored, and motivated by the experimental observations, calculations are carried out for propagation of a crack initiating in the outer enamel into the inner enamel. The effects of structural features of rod decussation including decussation angle of rods are revealed, and the mechanisms of crack interacting with the parazone and diazone are identified.

The boundary value problem is solved using finite element method based on the principle of virtual work. The model is meshed using 4-node bilinear quadrilateral element and a total of approximately 66,000 elements are involved. The cohesive zone models for enamel rods and rod sheaths are implemented within the framework of extended finite element method (XFEM) (Wang and Ural, 2018), so that the crack path is the natural outcomes of imposed loading.

To identify the toughening mechanisms responsible for the dependence of fracture resistance on microstructural features of inner enamel, the crack paths in specimen P and specimen D corresponding to three values of decussation angle are calculated. As shown in Fig. 16, the crack is oriented along the long axis of enamel rods in the parazone when crack extends into the inner enamel, giving rise to inclined crack path, which indicates that crack deflection is the prominent toughening mechanism. It is important to note that the crack path, which is aligned with the long axis of enamel rods, is in good agreement with experimental observations (Bajaj and Arola, 2009; Bechtle et al., 2010), substantiating that the numerical analyses can capture the major features of fracture in the inner enamel. We recognize that the complex 3D features of crack growth in enamel including crack branching and crack tortuosity can be described by 3D micromechanical models (Pro and Barthelat, 2014). We emphasize that although the 2D numerical model is adopted in the present study, the salient features of crack deflection in the inner enamel are captured, which is the important focus of this study. Due to the arrangement of enamel rods in the parazone, the crack deflection angle increases with increasing decussation angle. Interestingly, crack growth in the microstructure of enamel with intermediate decussation angle ($\theta = 20^{\circ}$) can activate multiple crack deflections, leading to the amplified toughness of enamel (shown in Fig. 15). According to the study by Koester et al., 2008, large crack deflection angle can provide high degree of toughening, and hence the large crack deflection angle in the case of $\theta = 25^{\circ}$ enables slightly higher toughness compared with the case of $\theta = 10^{\circ}$. Fig. 17 shows crack paths in the specimen D. It can be seen that for $\theta = 10^{\circ}$, the crack deflects towards the right half of the specimen, while deflection of crack path towards the left half of the specimen takes place in the case of large decussation angle ($\theta = 25^{\circ}$). It is interesting to find that crack growth in the microstructure of enamel with intermediate decussation angle ($\theta = 20^{\circ}$) is along the axis of symmetry of the specimen, and crack deflection does not emerge. This suggests that the intermediate decussation angle of enamel rods suppresses crack deflection mechanism, thereby leading to low toughness of enamel. Compared with specimen P, the crack propagating in specimen D undergoes local deflection, giving rise to low degree of crack deflection and thereby leading to relatively low toughness. 2b1af7f3a8