Response classification of simple polycrystalline microstructures

A new method for approximate solution of mechanics problems is presented that uses a classifier to identify regions in a random heterogeneous material where stress is likely to be highly concentrated under a prescribed set of boundary conditions. The example problem studied is an aggregate of hexagonal grains, each modeled as orthotropic and linear elastic, and subject to uniaxial extension. It is shown that the Sobol’ decomposition can be used to determine which surrounding grains mechanical properties play the largest role in determining the average effective stress in any particular grain. It is also shown that the constituent functions of the Sobol’ decomposition determine a unique material pattern that corresponds to maximum stress concentration. A reduced order representation of the microstructure is developed that is in essence a projection of the microstructure description onto the material pattern. Finally, a classifier is developed that operates on this reduced order representation to predict the level of stress concentration. This classifier is shown to be over 90% accurate, and, when implemented in a moving window algorithm, to provide very good predictions of the subregions in a large microstructure where large stress concentration is likely. 2007 Elsevier B.V. All rights reserved.