An alternative to periodic homogenization for dentin elastic stiffness
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Dentin, the main tissue of the tooth, is made of tubules surrounded by peri-tubular dentin (PTD), embedded in a matrix of inter-tubular dentin (ITD). The PTD and the ITD have different relative fractions of collagen and hydroxyapatite crystals. The ITD is typically less rigid than the PTD, which can be seen as a set of parallel hollow cylindrical reinforcements in the ITD matrix. In this paper, we extend Hashin and Rozen's homogenization scheme to a non-uniform distribution of hollow PTD cylinders, determined from image analysis. We relate the transverse isotropic elastic coeffcients of a Representative Elementary Volume (REV) of dentin to the elastic and topological properties of PTD and ITD. The model is calibrated against experimental data. Each sample tested is consistently characterized by Environmental Scanning Electron Microscopy (ESEM), nano-indentation and Resonant Ultrasound Spectroscopy (RUS), which ensures that macroscopic mechanical properties measured are correlated with microstructure observations. Despite the high variability of microstructure descriptors and mechanical properties, statistical analyses show that Hashin's bounds converge and that the proposed model can be used for back-calculating the microscopic Poisson's ratios of dentin constituents. Three-point bending tests conducted in the laboratory were simulated with the Finite Element Method (FEM). Elements were assigned transverse isotropic elastic parameters calculated by homogenization. The tubule orientation and the pdf of the ratio inner/outer tubule radius were determined in several zones of the beams before testing. The remainder of the micro-mechanical parameters were taken equal to those calibrated by RUS. The horizontal strains found experimentally by Digital Image Correlation (DIC) were compared to those found by FEM. The DIC and FEM horizontal strain fields showed a very good agreement in trend and order of magnitude, which verifies the calibration of the homogenization model. By contrast with previous studies of dentin, we fully calibrated a closed form mechanical model against experimental data and we explained the testing procedures. In elastic conditions, the proposed homogenization scheme gives a better account of microstructure variability than micro-macro dentin models with periodic microstructure.