Tail Departure of Log-Normal Grain Size Distributions in Synthetic Three-Dimensional Microstructures
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INTRODUCTION
COMPUTATIONAL modeling of materials has become a crucial and efficient field of materials science and engineering. Efforts are now under way to incorporate computational materials engineering into the development of materials and realize some savings in time and cost that have benefitted other areas of engineering such as aerospace.[1] Different strategies have been employed to generate three-dimensional (3-D) digital microstructures for use in the prediction of properties from microstructure.[2–8] Saylor et al.[2] and Brahme et al.[3] used experimental grain size, shape and orientation statistics in two dimensions to infer 3-D grain structures and to fit crystal orientations to the grains. Uchic et al.[4] reconstructed microstructures directly from serial section data. Groeber et al.[5,6] incorporated statistical nearest neighbor (NN) information into their generation procedure. Fan et al.[7] and StPierre et al.[8] used Voronoi tessellation to simulate polycrystalline structures. It is commonly accepted that many single-phase fully dense polycrystals are described by a log-normal grain size distribution (GSD).[9–11] These studies confirm a log-normal fit to a high confidence because of limited data and the use of histograms that emphasize the region around the mean. No mention is made, however, JOSEPH C. TUCKER, PhD Candidate, GREGORY S. ROHRER, Professor and Department Head, and ANTHONY D. ROLLETT, Professor, are with the Department of Materials Science and Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213-3890. Contact e-mail: [email protected]. edu LISA H. CHAN, Applications Engineer, is with EDAX Inc., 91 McKee Drive, Mahwah, NJ 07430. MICHAEL A. GROEBER, Engineer, is with the Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright Patterson AFB, OH 45433. Manuscript submitted April 4, 2011. Article published online August 30, 2011 2810—VOLUME 43A, AUGUST 2012
concerning the consequences of deviation from the lognormal description. Of particular concern is the upper tail because a ‘‘fat tail’’ may result in large grains being inserted into the microstructure if the probability density is finite at grain sizes of the order of the simulation domain. If, however, the actual upper tail of the grain size distribution is less strong than log-normal, then nonphysical large grains may be present in the digital microstructure and the model will not be representative. Thus, subsequent property calculations using, e.g., the finite-element method (FEM), will accordingly lose predictive power.[12,13] The fundamental microstructure feature is the grain, which is described by shape, size, and orientation. The term representative volume element (RVE) is applied frequently to digital microstructures.[14] What must be addressed carefully is what part of the experimentally observed input statistics that the volume element represents. If a digital microstructure is an RVE, then the distribution of the entities of the microstructure must match the real material such that an
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