A RUC is a small material region that can structurally represent the entire material by repeating itself infinitely. Here the smallest such a model is a single prism rod. However, imposing loading conditions would be difficult along non-planar surfaces of key-holed geometry. Thus we have chosen a brick or cuboid geometry as the unit cell as shown in Fig. 4. Essentially this cell contains two equivalent prism rods in the model. A great care was taken to construct 3D mesh of the brick RUC model. First since there is no material variation through thickness, one layer element is only needed along the z-axis. On the xy plane, since the periodic boundary condition must be imposed on the outer boundaries, nodes on the opposing sides must match exactly. Furthermore some of the interior nodes must follow the physical interfaces of prism rods. Other interior nodes must be located so that the symmetry and conjugate elements in two prism rods are identical. The constructed mesh shown in Fig. 4 contains 240 8noded brick elements and 546 nodes. Here the dimensions are chosen as 7×7×7Pm although the dimensional scales of RUC model do not enter in determining the effective properties. 2.2. Local Properties within Prism Rod Locally, the enamel prisms are composed of inorganic mineralhydroxyapatite and organic matrixproteins with specific orientations. The structure of the hydroxyapatiteprotein matrix is similar to that of a fiber reinforced composite, which is axisymmetric in the long direction as shown in Fig. 3. It is suitable to treat the local properties as transversely isotropic. The local properties of the elements are modeled as transversely isotropic (with 5 material constants) and variable orientations of the HAp crystal composite with the elements. The transversely isotropic material contains 5 constants as EL, ET, ȞLT, ȞTT and ȝLT. Although some values were reported for the elastic moduli Fig. 3. Schematics of HAp fiber orientations within prism rod shown in two separate perspectives. Fig. 4. The cross-sectional illustration of repeated prism rods highlighting finite element mesh for unit cell. Actual 3D mesh for cuboid geometry is also shown. x y Symmetry plane 173
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