13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- oriented total A A ρ= (2) where total A is the total area below the curve, and oriented A refers to the area above the constant background level umoriented A . Thus, the value of ρ ranges from 0 to 1, with 0 ρ= indicating no predominant orientation within the plane of the section, while 1 ρ= indicates a perfect alignment of all crystals [25,26]. Figure 2. Representative WAXS and SAXS patterns of enamel 3. Model formulation Human enamel has a hierarchical two-level composite structure, where the first level is represented by the keyhole-like rod and the second level by the bunch of HAp crystals within one rod [27]. Fig. 3a shows the keyhole-like microstructure of enamel, modified from Habelitz [28], and demonstrates how the HAp crystals are distributed within the rod in 3D. Fig. 3b-d provide schematic illustration of the geometric model derived from the enamel structure, where the first level regards the whole enamel sample as composed of aligned rods within a collagen matrix phase (Fig. 3b), and the second level considers the rod as a composite in detail, consisting of partially aligned HAp crystals and a collagen matrix. Both levels are non-dilute systems with a number of inhomogeneous inclusions. For simplicity, both rod and HAp crystals are assumed to be of needle shape (Fig. 3b-d). 3.1. First level model: multiple aligned rod inclusions within enamel The purpose of the first-level model is to establish the relationship between the externally measured stress Aσ and the stress in the rod inclusions inclusion rod σ σ = . According to the Eshelby model derivation [29], the stress in the inclusion can be expressed as [ ] { } { } 1 1 1 1 1 1 1 1 1 1 1 (1 ) ( ) ( ) ( ) inclusion T A matrix rod matrix matrix rod matrix T f C C S f S I C C C T C σ σ − − − = − − − − − − − or, simplified 1 1 inclusion A K σ σ = (3) where 1f is the volume fraction of rod inclusion, 1S the Eshelby tensor for a cylinder corresponding to the rod shape, 1 matrix C and rod C are the stiffness of collagen matrix and rod, respectively, and T is the tensor transformation (rotation) matrix that depends on the Euler angles giving the orientation of the rods with respect to the fixed laboratory system. In the present model this was fixed, as it was
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