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TECHNICAL PAPERS

# Methodology for Determining the Variance of the Taylor Factor: Application in $Fe-3%Si$

[+] Author and Article Information
Craig P. Przybyla1

Dept. of Mat. Sci. and Eng., Georgia Institute of Technology, 771 Ferst Drive, Atlanta, Georgia 30332-0245cpriz@gatech.edu

Dept. of Mech. Eng., Brigham Young University, 435 CTB, Provo, UT 84602

Michael P. Miles

Manufacturing and Eng. Technology, Brigham Young University, 265 CTB, Provo, UT 84602

1

This research was conducted while the author was a grad. Research Assistant at Brigham Young University.

J. Eng. Mater. Technol 129(1), 82-93 (Sep 27, 2006) (12 pages) doi:10.1115/1.2400268 History: Received September 13, 2005; Revised September 27, 2006

## Abstract

A method is proposed to determine the variance of an arbitrary material property based on the statistics of the texture of polycrystalline materials for a specified volume. This method is applied to determine the variance of the Taylor factor (i.e., measure of plastic deformation in crystal plasticity) and is compared to a random sampling method. The results from the random sampling method correlated well with the statistical variance relationship when the magnitude of the variance was greater than that of the numerical errors observed in the statistical calculation. An empirical relation was also shown to model the results, and the constants for this relationship were determined for pseudo-three-dimensional $Fe-3%Si$. Implementation of the statistical variance relationship in true three-dimensional microstructures is not limited by material opacity, since it depends only on the two-point pair correlation functions. The connection between the variance of the R-value and variance of the Taylor factor is considered. Although only a weak connection was found, it was observed that relatively small variations in the Taylor factor yield large variances in the R-value.

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## Figures

Figure 1

Definition of material space. The ith sample is shown cut out of the parent material. The various mathematical constructs used in the derivation of the statistical variance relationship are indicated.

Figure 2

The local state space as defined by the complete set all physically distinctive orientations for cubic sample symmetry known as the fundamental zone or SO(3)∕G. This space is three-dimensional and parametrized by the Euler angles φ1, Φ, and φ2. An orientation g in the tessellated space, SO(3)∕G, is shown within the tessera hj.

Figure 3

The volume of overlap of two equal rectangular regions (each defined by Ω) whose centers are separated by r⃗

Figure 4

Coordinate system used to define the R-value in terms of a tensile sample where F is the applied force and w and t are the width and thickness of the tensile sample, respectively

Figure 5

Six principal directions defined by the hexagonal scanning grid that was used for the OIM measurements. (The negative directions are shown but not labeled.)

Figure 6

OIM scan of the surface of Fe-3%Si with grain orientation indicated by different false shades of gray. The window, Ω, placed in the scan defines the area in which the two-point statistics are sampled. Each material point in Ω (that was sampled in OIM to determine lattice orientation) is sampled by r⃗. The tail of the sampled vectors, r⃗, is allowed to fall outside Ω but the head is not as shown by the arbitrarily placed vector r⃗ in the figure. (The dimensions of the OIM scan must at least twice that of Ω so the statistics sampled by r⃗ are equally weighted for all lengths of r⃗.)

Figure 7

The tensile samples cut from Fe-3%Si with the circle grid etched onto the surface. The base and height of the rectangles defined by Ω1 and Ω1 were determined by taking the average of the measurements taken at each indicated corner.

Figure 8

The effective Taylor factors are compared for 0<λ<1 as calculated according to Eq. 33. For this calculation the fundamental zone of orientations (SO(3)∕G) was tessellated using 512, 1728, and 4096 tesserae. The complete continuous (nondiscretized) dataset obtained from the OIM scan was also included for comparison.

Figure 9

Standard deviation of the effective Taylor factor over Ω=2.2×2.2mm

Figure 10

Standard deviation of the effective Taylor factor over Ω=4.7752×9.5504mm. The terms λl=0.00 and λl=0.93 define the range of possible contractile strain ratios within the variation of M¯(λ).

Figure 11

Standard deviation of the effective Taylor factor over Ω=9.5504×9.5504mm. The terms λl=0.02 and λl=0.87 define the range of possible contractile strain ratios within the variation of M¯(λ).

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