E. Inversion of x-ray diffraction data for macroscopic
stress estimate
The development of x-ray diffraction technique has allowed us
to explore the details of polycrystal deformation as summarized
before. These studies provided, for the first time, a means of
inferring the nature of stress distribution in a polycrystalline
(and a multi-phase) material. However, these studies have also
highlighted a need for a better understanding of stress-strain
distribution in a deforming material in order to obtain a macroscopic
stress from the x-ray diffraction of individual diffraction peaks.
The stress that we estimate from x-ray diffraction is calculated
from the elastic strain of a given group of crystals (with a specific
crystallographic orientation) in a polycrystalline sample. These
measurements therefore provide an estimate of a microscopic (i.e.,
grain-scale) stress. The relation between microscopic and macroscopic
stress is not straightforward. In the classic paper by (Singh,
1993a), which forms a basis for the stress estimate from radial
x-ray diffraction (x-ray diffraction for various directions),
an assumption of elastic accommodation is made. This leads to
the well-known upper and the lower bound (Voigt and Reuss average
respectively), but the experimental results by (Li et al., 2004b)
on MgO clearly indicated that the data exceed these bounds. This
is an indication of the breakdown of the assumption of elastic
accommodation.
We propose to follow two complementary strategies to develop
a better understanding of the relationship between microscopic
and macroscopic stress. First, we will use elastic plastic self
consistent modeling (e.g. Clausen, 1997; Turner and Tome, 1994)
to model the effect of plastic anisotropy on the stress distribution
between various grain orientation populations. By using the critical
resolved shear stresses (CRSS) of different slip systems as fitting
parameters we can reproduce the behavior of measured lattice reflections
and calculate a macroscopic stress for the polycrystal. Weidner,
Li and Burnley have already begun to explore this approach. One
short coming of this type of modeling is that the CRSS for the
slip systems in mantle minerals are not known, so at the moment,
we cannot say if CRSS required by the models are reasonable. Therefore,
it will be important to also investigate plastic properties of
single crystals using the Laue diffraction technique. White-beam
Laue diffraction allows one to measure deviatoric lattice strain
tensor of a single crystal under deformation with 2D detectors.
With the ability of photon energy scan to identify individual
diffraction spots, one obtains the complete lattice strain tensor,
which can be easily converted to stress tensor with known elastic
constants of the crystal. Wang and coworkers have tested the feasibility
of this technique at the bending magnet station at GSECARS. We
used a Si (111) monochromator to scan incident photon energies
up to 65 keV. We direct the white beam through the gap between
the two Si (111) crystals to produce Laue patterns on single crystals
in the DDIA and then move the monochromator into the beam the
scan energy. By manually scanning the monochromator, Laue spots
are identified with an energy accuracy of +/- 3 eV. Deformation
data have been collected on almandine single crystals at 3 GPa,
with more than 10% shortening. These data will be analyzed after
some software development (existing Laue software for stress measurement
assumes a "90°-geometry" - incident beam and scattered
beam are approximately perpendicular, whereas the DDIA setup has
a 0°-geometry). Dr. Wenge Yang at HPCAT, who has extensive
experience in microstress analysis, has been assisting us in this
development. Such studies will provide important data on plastic
anisotropy that can be incorporated in numerical or theoretical
models of polycrystal deformation. Second, we will conduct “benchmarking”
– tying high-pressure measurements to measurements made
at lower pressures in conventional machines. The “gold standard”
of such measurements is the gas machine, such as the Paterson
Apparatus, but that instrument has a maximum pressure capability
of approximately 0.5 GPa. The Griggs Apparatus with a molten salt
cell assembly allows truly hydrostatic confining pressure to be
applied to an encapsulated specimen at pressures up to ~ 3.5 GPa.
The axial stress applied to the specimen is measured externally,
however, and the friction on that piston has to be deconvolved
from the signal in order to obtain the stress on the specimen.
For large values of friction and small values of stress, that
is a nontrivial process. However, a procedure has been worked
out to achieve differential stress measurements as low as 5-10
MPa in favorable circumstances (Green and Borch, 1989; Green and
Borch, 1990). In addition, one can use the dislocation density
measured in a material after deformation to obtain an estimate
of the strength it had been subjected to. An advantage of this
approach is that one can investigate the orientation dependence
of local stress through orientation dependence of dislocation
density (KARATO and LEE, 1999). We will employ all of these techniques
on a standard material (polycrystalline olivine synthesized at
U. Minn.) to compare with similar experiments in the DDIA in which
the stress has been measured by x-rays. The Griggs-rig experiments
will be made in Green’s laboratory. At pressures of 2.0–3.5
GPa, direct comparison can be made between the Griggs Apparatus
and the DDIA.
