13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- assumed that the rods were all aligned with the loading direction. Figure 3. Schematic diagrams of hieratical enamel structure and multi-scale Eshelby model. To determine the stress within the inclusion, knowledge of the rod stiffness matrix is required. The rod is regarded as a composite consisting of the same volume fraction of collagen and mineral crystals as the whole enamel. The stiffness of the rod rod C can thus be determined and is given as: ( ) ( ) { } ( ) 1 1 1 1 2 2 _2 2 2 2 2 2 2 rod matrix matrix matrix matrix matrix HAp HAp C C f C C S f S I C C C C − − − − ⎧ ⎫ ⎡ ⎤ = − − − − + − ⎨ ⎬ ⎣ ⎦ ⎩ ⎭ (4) 3.2. Second level model: HAp inclusion within one rod The purpose of the second-level model is to establish the relationship between the external stress and the average strain in the HAp crystals in a rod: HAp ε ( 1 HAp inclusion K σ ε = ). Here the stress within the rod inclusion 1 inclusion σ found from the first-level model serves as the external stress in the second-level model. The overall stiffness of the rod rod C is required to link the average strain in the HAp crystals with the external stress obtained in the first-level model. A single HAp crystal can be regarded as a single inhomogeneity with needle-like shape, and thus the corresponding Eshelby tensor 2S should again be that for the cylinder. The alignment of HAp crystals within the rod is also partial, so that the real value of the apparent stiffness K is given by a combination of the values for two separate cases, namely, that of fully random distribution, and perfect alignment. 3.2.1 Multiple perfectly aligned HAp crystals Supposing the alignment direction of HAp crystals is described by the transformation matrix T, the measured crystal strain corresponds to the mean strain value for all the crystals within the considered gauge volume. The relationship between the average HAp strain and the external load (rod stress from the first-level model) can be established as follows:
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