ELASTIC STRAIN AND STRESS BY RIETVELD REFINEMENT
Davor Balzar and Hassel Ledbetter
Materials Science and Engineering Laboratory, National Institute of Standards and Technology, Boulder, CO 80303, U. S. A.
Keywords: diffraction; strain; stress;
Rietveld refinement.
Diffraction was used in strain/stress determination for more than 70 years. The strain is generally anisotropic and the diffraction method relies on measuring interplanar d spacings along different directions. Usually, only a single d spacing is measured, which may not be possible in cases of strong texture. If the complete diffraction pattern is collected, all available diffraction lines can be used. Then, the analysis of diffraction-line shifts can be incorporated in the full-diffraction-powder-pattern-fitting approaches, such as the Rietveld refinement [1], which is being developed into a procedure to obtain a variety of information about a specimen. Recently, texture determination by Rietveld refinement from relatively few specimen orientations was introduced by simultaneous refinement of texture harmonic coefficients [2] along with other refinable parameters in the Rietveld program GSAS [3].
An approach to obtain all components of the elastic-strain tensor with a high precision by the simultaneous Rietveld refinement of several tens of diffraction patterns was recently proposed [4]. The interplanar spacing d is in Rietveld refinement averaged over all the specimen orientations, defined by angles y and f. For a sufficiently large number of collected patterns evenly distributed over the specimen orientation angles,
During the Rietveld refinement, strain-related parameters are adjusted for each diffraction pattern simultaneously with other refinable parameters. There are two possibilities:
(i) Lattice parameters are refined and the unstressed interplanar spacing d0 is known:
Hydrostatic component of strain is obtained from <d>, and the refined strain parameters yield directly a deviatoric strain tensor eNij.
(ii) Lattice parameters are held constant at the previously determined value for the unstressed specimen:
Tr (e'ij) = 0.
The refined strain parameters yield the complete strain tensor
eij.
To obtain the components of strain tensor eij, we simultaneously solve the overdetermined system of equations:
As an example, we show results obtained at the High Intensity Powder Diffractometer (HIPD) at the Manuel Lujan Jr. Neutron Scattering Center (MLNSC), Los Alamos National Laboratory. The material considered is an extruded Al-6061 alloy reinforced with SiC (hexagonal) whiskers (27.9 volume %). The measurements were collected at 18 different specimen positions (each defined by a set of y and f angles) and only the measurements from high-angle detector banks were used in the Rietveld refinement to obtain a desired resolution. This gave a system of 72 equations that was solved by least squares to obtain the complete strain tensor. The refined strain components eyf are also plotted in the figure as a function of the tilt angle y for the Al matrix for both cases described: (i) when the lattice parameter is refined; (ii) when the lattice parameter is fixed to the value of the unstressed specimen.
The specimen's cylindrical symmetry is evident and linear fits were drawn through the points according to:
Therefore, the difference of ordinate intercepts yields
directly eH, whereas slopes must stay equal.
From the same refinements, the texture harmonic coefficients
are determined, which allow calculations of the
orientation-distribution function (ODF) and consequent weighting
of monocrystal elastic constants and estimations of accurate
elastic stresses. An anisotropic strain component may be refined
for cubic and hexagonal crystal structures along with the
isotropic. The method is also applicable to x-ray or neutron
constant-wavelength sources but is especially convenient with
polychromatic sources, such as TOF neutrons or synchrotron
energy-dispersive diffraction.