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mathematical Modeling and Design

In order to calculate the temperature distribution and the thermal-stress distribution in the FGM, effective material properties (such as the thermal conductivity A, the coefficient of thermal expansion a, and the elastic constants—including Young's modulus E and Poisson's ratio v) are required for intermediate compositions of the FGM. There are two keys to estimate the effective material properties of the intermediate compositions: heuristic approaches and micromechanical approaches.

Tohru Hirano and Kenji Wakashima Inverse Design Procedure for FGMs For the design of functionally gradient materials (FGMs), necessary material properties, such as thermal-expansioncoefficient and Young's modulus in the specific region, are optimized by controlling the distribution profiles of composition and microstructures, as well as micropores in the materials. For this purpose, our research team employs the inverse design procedure in which both the basic material combination and the optimum profile of the composition and microstructures are determined with respect to the objective structural shape and the thermomechanical boundary conditions.1 Figure 1 shows the inverse design proce-

Heuristic Approach

dure for FGM, in which the final structure to be developed, as well as the boundary conditions, are specified first. After the fabrication method and an allowable material combination are selected from the FGM database, the estimation rules for the material properties of the intermediate compositions are determined based upon the micro-structure. Then, the temperature distribution and the thermal-stress distribution are calculated with the assumed profiles of the distribution functions for the constituents. Other possible combinations and different profiles are also investigated until the optimum is obtained.

Distribution Functions Assuming that the constituents of FGM comprise phase A (ceramic), phase B (metal), and micropores, a dimensionless (normalized) parameter is introduced, related to the fractional volumes for the constituents, expressed as VA, VB, Vp, respectively, as follows: VB' = VB/(VA + VB)

(1)

The distribution function is defined for VB' with dimensionless thickness z, where 1 is the full thickness of the film.

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