A software for diffraction stress factor calculations for textured materials

Thomas Gnäupel-Herold1,2,a)

1

Materials Science and Engineering, University of Maryland, Building 090, Room 2135, College Park, Maryland 20742

2

Center for Neutron Research, NIST, 100 Bureau Dr stop 6102, Gaithersburg, Maryland 20899-6102

(Received 8 February 2012; accepted 29 February 2012)

A software for the calculation of diffraction elastic constants (DEC) for materials both with and without preferred orientation was developed. All grain-interaction models that can use the crystallite orientation distribution function (ODF) are incorporated, including Kröner, Hill, inverse Kröner, and

Reuss. The functions of the software include: reading the ODF in common textual formats, pole figure

calculation, calculation of DEC for different (hkl,w,ψ), calculation of anisotropic bulk constants from

the ODF, calculation of macro-stress from lattice strain and vice versa, as well as mixture ratios of

(hkl) of overlapped reflections in textured materials. © International Centre for Diffraction Data

[doi:10.1017/S0885715612000267]

Key words: diffraction, elastic constants, texture, strain, stress

I. INTRODUCTION

Diffraction-based stress analysis depends critically on the

use of the correct diffraction elastic constants (DEC). A survey

through the literature makes it abundantly clear that in the vast

majority of cases in which lattice strain needed conversion to

stress-preferred grain orientation (texture) is – justifiably or

not – disregarded, and isotropic diffraction elastic constants

were used. The main reasons for this apparent oversimplification are the added need to quantify the degree of preferred

grain orientation through pole figure measurements, and the

use of pole figures to determine the orientation distribution

function (ODF). Even with the ODF, further calculations are

hampered by the lack of freely available software tools to perform calculations of anisotropic diffraction elastic constants.

The IsoDEC (Gnäupel-Herold, 2011) software program introduced here was developed to address this need.

II. FEATURES AND FUNCTIONALITY

The main purpose of IsoDEC is the calculation of the

orientation and (hkl) dependent stress factors using the ODF

(in the textual form as provided by the output of freely available texture packages), the single crystal elastic constants and

the bulk elastic constants as input. The latter can be calculated

from the ODF. The software can also be used to calculate

stress from measured lattice strains using the thus determined

stress factors or, in reverse, calculate lattice strains from a

given stress tensor. Some of the implemented models for

DEC calculations also allow including the grain shape,

expressed as a general ellipsoid, into the calculation of the DEC.

Evidently the ODF is crucial. It needs to be supplied in

textual form, i.e., using the output from popLA (Kallend

et al., 1991) or MTEX (Hielscher and Schaeben, 2008). The

ODF is expected in a 5° × 5° × 5° equidistant spacing for the

three Euler angles w1Φw2 (Bunge notation). This tabular

form is internally expanded to the full interval [0...2π;

0...π; 0...2π] and made quasi-continuous by means of trilinear interpolation. This allows the computation of pole

figures (in popLA format) and orientation fibers (i.e., ODF

values for a series of Euler angle combinations) through

Euler space. Orientation fibers are a key element of DEC calculations that use the ODF. The models for DEC calculations

are the Kröner model (Kröner, 1958; Bollenrath et al., 1967;

Behnken and Hauk, 1986), the Reuss model (Reuss, 1929;

Möller & Martin, 1939), the Hill model (Voigt, 1928;

Reuss, 1929; Hill, 1952), the inverse Kröner model (Kröner,

1958; Gnäupel-Herold, 2012), and the modified Voigt

model (Murray and Noyan, 1999). The two Kröner-type

models allow the user to devise complex elastic interactions:

first, one can approximate multi-phase composites (e.g. elastically hard particles in a soft matrix such as SiC in Al) by using

appropriate set of single crystal constants and bulk constants.

Second, the effects of elongated or otherwise non-spherical

grains can be included by changing the grain-shape parameters in IsoDEC.

Input and output in IsoDEC can be done through the

spreadsheet that allows the import and export of text files with

its content as well as basic copy and paste exchanges

with common spreadsheet programs such as EXCEL. Large

datasets consisting of d-spacings, their uncertainties, unstressed

d-spacings with uncertainties, and measurement directions (wψ)

can be treated this way. Stress tensor components and the

unstressed d-spacing can be fixed or free parameters in the

stress fit, depending on boundary/equilibrium conditions.

III. OPERABILITY

IsoDEC offers two visual interfaces: one for isotropic calculations and one for calculations that include preferred orientation. The interface for isotropic calculations is shown in

Figure 1. The fewest possible steps for calculating the isotropic constants s1(hkl) and 1⁄2s2(hkl) are loading a set of material

constants from the drop-down list, entering the indices h, k,

and l, and starting the calculation. Elastic constants that are

not found in the database included can be entered manually,

but must include the lattice parameters and, after entering

the single crystal elastic constants, a calculation of the isotropic bulk constants (upper left).

For anisotropic calculations, the amount of information

that has to be provided increases as shown in Figure 2.

a)Author to whom correspondence should be addressed. Electronic mail:

tg-h@nist.gov

114 Powder Diffraction 27 (2), June 2012 0885-7156/2012/27(2)/114/3/$18.00 © 2012 JCPDS-ICDD 114
Generally the ODF has to be provided either in popLA format

or in MTEX format (“Load ODF”). There is also the possibility of not using an ODF, and entering user values for the

bulk constants, e.g. using a rotated single crystal constants tensor to simulate the effects of individual texture components on

the stress factors. Of course, the anisotropic bulk constants can

be calculated from the ODF, the average grain shape (as ellipsoid axis ratios) can be entered, and the model (Hill, Reuss,

Kröner, inverse Kröner and Voigt/modified Voigt) can be chosen. The stress factors are calculated for the fields in which h,

k, l, phi and psi are given. This step has to precede any stress or

strain calculations that require the stress factors. A stress calculation needs the additional input of d-spacings, the d-uncertainties, the reference or unstressed d-spacing and their

uncertainties, and a sorting qualifier (column 1 in the

worksheet) that allows to group those data together that are

to be used for the calculation of the same stress tensor (i.e.,

same location on the sample or similar). The reverse calculation d-spacing/strain from stress naturally requires stress

values entered in their appropriate columns. The output of

this calculation is in the column “d_calc”, which means that

if strain is desired one has to perform the explicit calculation

(d − d0)/d0 oneself.

IV. PLATFORM AND AVAILABILITY

IsoDEC was written in Turbo Delphi for the Windows

operating system. It can also be used under MacOS X+ and

x86 linux if Windows emulator software is present. Both

Darwine (MacOS X and Wine) have been tested successfully.

Figure 2. Interface for anisotropic calculations.

Figure 1. Screenshot of IsoDEC for isotropic calculations.

115 Powder Diffr., Vol. 27, No. 2, June 2012 A software for diffraction stress factor calculations for textured materials 115
The installation package contains a help file, a database of

single crystal elastic constants and several ODFs in textual

form that are readable by IsoDEC for use in further calculations

of DEC, pole figures, anisotropic bulk constants, and ODF

intensities for given specimen directions. The program home

page and the download link can be found at http://www.ncnr.

nist.gov/programs/crystallography/software/isodec/.

DISCLAIMER

Certain commercial firms and trade names are identified

in this report in order to specify aspects of the experimental

procedure adequately. Such identification is not intended to

imply recommendation or endorsement by the National

Institute of Standards and Technology, nor is it intended to

imply that the materials or equipment identified are necessarily the best available for the purpose.

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