Stochastic simulation of functionally graded materials with random porosity and volume fraction

The stresses in traditional thermal barrier coatings (TBC's) can be unacceptably high due to sharp discontinuities in the thermal expansion coefficient between the thermally insulating material, often a ceramic, and the structural material, often a metal. Recently, functionally graded materials (FGM's), which are by definition non-homogeneous materials with a continuous variation in composition and/or microstructure in some spatial direction, have been developed in order to mitigate these high stresses at the material interface. This functional grading allows the material to achieve an optimal combination of properties such as strength, hardness, durability and wear/corrosion resistance, which distinguishes FGM's from conventional composites where uniform materials are joined along sharp boundaries. Computational efforts to successfully model and simulate FGM's are the subject of many studies, considering both analytical and finite element based techniques. Of fundamental importance to any such attempt is the accurate representation of the graded material properties. Despite the on-going efforts, very little work has been developed that treats the randomness inherent in their microstructure. This randomness is a natural consequence of the difficulty in manufacturing these materials to exact specifications. The major goal of this current research is to develop a numerical model that investigates the effects of microstructural randomness on the stress and temperature distribution in FGM's, based on stochastic simulation techniques. This will be achieved by assuming that porosity and metal volume fraction vary randomly around their specified through-thickness values and by assuming one-dimensional steady-state heat transfer. In order to perform stochastic simulations that reflect randomness in the FGM microstructure, the porosity and the metal/ceramic volume fraction are described by non-homogeneous stochastic fields, which are assumed homogeneous after normalization by the mean and variance. Previous work addressed the simulation of these fields assuming that they are homogeneous and that the material remains in the elastic range. In this work, based on the porosity and volume fraction samples generated by the simulation scheme, the thermal conductivity, thermal expansion coefficient, yield stress, and elastic modulus are calculated. A nonlinear analysis yields the resulting sample stress and temperature distributions, which are used to determine the variance of these quantities. Results show that the randomness in the metal/ceramic volume fraction leads to significant variability in the stress distribution but that it has little impact on the variability of the temperature distribution. Randomness in porosity has little impact on the results. Refs. 1 (author)