Representations of Polycrystalline Microstructure by n-Point Correlation Tensors

Pavel Etingof,Brent L Adams

doi:10.1155/tsm.21.17

Abstract

An important class of representations of polycrystalline microstructure consists of the n-point correlation tensors. In this paper the representation theory of groups is applied to a consideration of symmetries in the n-point correlation tensors. Three sources of symmetry are included in the development: indicial symmetry in the coefficients of tensors, symmetry associated with the crystal lattice, and statistical symmetries in the microstructure induced by processing. The central problem discussed here is the residence space, or the space of minimum dimension occupied by correlation tensors possessing such symmetries. In addition to the general case of correlation tensors possessing such symmetries, a model microstructure is also considered which embodies an assumption of no spatial coherence of lattice orientation between neighboring grains or crystallites. It is shown that the model microstructure generally results in residence spaces of lower dimension.

Highlights

Microstructure refers to any of the myriad of features observed by the experimentalist when probing the internal arrangement of components of solid materials
If neither M nor Mff is empty with respect to the crystal symmetry group F, the results for correlation tensors possessing indicial symmetry typical of elasticity tensors are identical with the results already presented for the case when F is trivial, except in the case M->2 when a no-correlation assumption is made
In the development described above, we have considered the n-point correlation tensors as a particular class of quantitative representation of polycrystalline microstructure

Summary

INTRODUCTION

Microstructure refers to any of the myriad of features observed by the experimentalist when probing the internal arrangement of components of solid materials. Deep physical insight into the associations between microstructural features and behavior is required to understand their importance. It is in this complicated setting that we shall consider a particular and specialized class of representations of polycrystalline microstructure that frequently appears in property bounding theories. We must seek to find a minimal representation of microstructure which correlates, to the desired resolution, with those properties and behaviors of interest It is in this context that consideration of symmetry becomes an important factor, since symmetry analysis helps identify the minimal required representation. In the sections which follow we first define the n-point correlation tensors in the context of materials science and the modern quantitative texture theory of microstructure. We elect to give results for a selected set of problems ranging from the most elementary to examples which have substantial technological importance

REPRESENTATION OF MICROSTRUCTURE BY n-POINT CORRELATION TENSORS

Model Microstructures Exhibiting Only Crystallite-Scale Auto-Coherence

REPRESENTATIONS OF GROUPS

REPRESENTATIONS OF CORRELATION TENSORS

Examples from Representations of Material Tensors of Order Two

SUMMARY