Abstract

Using finite elements over Rodrigues space, methods are developed for the formation and inversion of pole figures. The methods take advantage of the properties of Rodrigues space, particularly the fact that geodesics corresponding to pole figure projection paths are straight lines. Both discrete and continuous pole figure data may be inverted to obtain orientation distribution functions (ODFs) in Rodrigues space, and we include sample applications for both types of data.

Highlights

  • Since the early work of Roe (1965) and Bunge (1969), the analysis of texture, or preferred crystallographic orientation, has remained an active research area

  • The texture is quantified by an orientation distribution function (ODF) which often is regarded as the fundamental entity of quantitative texture analysis and whose determination is of prime importance

  • While automated electron back-scatter diffraction (EBSD) methods offer a viable alternative to determine an ODF, the inversion of pole figure data from X-ray or neutron diffraction goniometers remains an important method of ODF determination

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Summary

INTRODUCTION

Since the early work of Roe (1965) and Bunge (1969), the analysis of texture, or preferred crystallographic orientation, has remained an active research area. There are further advantages, including: simple determination of the asymmetric domain (making for easy construction of the finite element mesh), well-behaved metric throughout the asymmetric domain, and straightforward enforcement of crystal symmetries Due to these appealing properties, Rodrigues parameter space is finding more widespread use in the representation and analysis of preferred orientation (see, for example, Frank, 1988; Becker and Panchanadeeswaran, 1989; Randle, 1990; Heinz and Neumann, 1991; Neumann, 1991; Morawiec, 1995; Morawiec and Field, 1996; Morawiec, 1997; Adams and Olson, 1998). Using piecewise interpolants circumvents the difficult task of determining global interpolants in the Rodrigues parameter space that satisfy the symmetry constraints a priori This representation of the ODF is based on a finite number of parameters (the number of degrees of freedom corresponds to the number of unconstrained nodal values in the mesh), so the first issue of indeterminacy is not a concern. Neutron diffraction methods were used to collect the data for the example applications, and we examine materials with both hexagonal and cubic crystal symmetry

THE USE OF FINITE ELEMENTS
ORIENTATIONS AND POLE FIGURES
PATH INTEGRALS IN RODRIGUES SPACE
Line Integral Parameterization in Rodrigues Space
Numerical Integration
METHODS
Example Application – Pole Figure Inversion by Least Squares on Discrete Data
Example Application – Pole Figure Inversion by Surface Integral Minimization
CONCLUSIONS

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