Appl. Phys. Lett., Vol. 76, No. 6, 7 February 2000
Ustundag, Clausen, and Bourke
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FIG. 3. Finite element model predictions of average composite and unre-
duced spinel elastic strains as a function of unreduced spinel volume frac-
tion. Two cases are compared: pure elastic deformation and creep in all
regions.
FIG. 2. Change in the lattice constants ͑‘‘strain’’͒ during reduction. The
‘‘strain’’ in the reduced composite was calculated from the measured
‘‘strain’’ in Ni and ␣-Al2O3 using the rule of mixtures ͑20 vol. % Ni in an
␣-Al2O3 matrix was assumed͒. The insert shows a schematic of the speci-
men cross section and the stress state predicted to result from the volume
shrinkage in the reduced layer.
metric finite element model was created and calculations per-
formed using the ABAQUS™ code. In the model, both the
unreduced and reduced regions were treated as single phase,
isotropic materials with experimentally determined elastic
constants ͓spinel, Young’s modulus, Eϭ240 GPa, Poisson’s
ratio, ϭ0.3; reduced region, Eϭ150 GPa, ϭ0.2 ͑Ref. 4͔͒.
The strain evolution for 5.1% volume shrinkage is shown in
Fig. 3 for two cases: one assumes elastic behavior ͑no stress
relaxation͒, and the other assumes power-law creep with
the sample. Moreover, one would need the instantaneous
density at each time ͑to convert from weight fraction to vol-
ume fraction͒. Although the assumption of a uniform reac-
tion front is validated,2–4 the prediction of instantaneous den-
sity is difficult, especially in view of the low observed
shrinkage and generation of porosity in ␣-Al2O3. A further
complication in kinetics analysis is the cylindrical geometry
of the specimen. This invalidates the use of the ‘‘traditional’’
kinetics models that assume planar reaction fronts.7 All of
these issues are discussed in a future publication.8
Turning instead to the lattice parameters, we observed
significant changes in the spinel lattice parameter, a, during
the reduction. These changes were converted to ‘‘strains’’
with respect to the value of a ͑ϳ8.14 Å͒ at the reduction
temperature, but prior to reduction ͑Fig. 2͒. Since there is no
Ni and ␣-Al2O3 at the beginning of the reaction, the lattice
constants of these phases obtained at the end of the reduction
͑before cooling͒ were taken as ‘‘strain-free’’ references. This
was justified by the fact that the reaction proceeded to
completion in this sample. This way, lattice parameter
changes due to, for instance, dissolution of other elements in
either Ni or ␣-Al2O3 were also accounted for.
There are at least two possible explanations for the
changes in the lattice constants during reduction: ͑i͒ elastic
strain due to the volume shrinkage, and ͑ii͒ structural
changes in the unreduced spinel. Shrinkage of the exterior is
expected to introduce a compressive in-plane strain in the
unreduced spinel ͑Fig. 2, insert͒. In the absence of relaxation,
the strain in unreduced spinel would increase in compression
as the reaction proceeds. In the reduced region, a balancing
strain of radial compression and tangential ͑or hoop͒ tension
would be expected perpendicular to sample axis ͑Fig. 2, in-
sert͒. Note that, since the whole specimen is sampled by the
neutron beam, the measurement integrates the component of
strain along the scattering vector and yields an average of
radial and tangential strain components ͑the details are ex-
plained elsewhere9͒.
typical ␣-Al2O3 behavior. The constitutive law10 used in all
n
˙
regions of the sample was steady-state creep rate, ⑀ϭA ,
where Aϭ3.33ϫ10Ϫ10 sϪ1 ͑MPa͒Ϫn, is equivalent ͑von
Mises͒ stress in MPa and nϭ1.69.11 The material parameters
were determined by Robertson et al.11 using an ␣-Al2O3
with average grain size of 1–2 m under uniaxial tension
and compression at 1250 °C. The justification for our use of
these parameters is that grain size, temperature, and stress
level in Ref. 11 were similar to those in this study. More-
over, there are no experimental data on the creep behavior of
either spinel or the reduced composite. The creep calculation
should therefore be viewed as representative and qualitative.
In comparing Figs. 2 and 3, the most obvious observa-
tion is that the strain response of the sample does not com-
pletely agree with either the elastic or the creep calculations.
At this stage, it is not possible to rule out stress relaxation via
creep because the lack of elastic strain in the composite can
be attributed to this mechanism. Since the elastic and creep
calculations should, in principle, bracket the measured re-
sponse, we explored alternative mechanisms that might ex-
plain the changes in lattice parameters in Fig. 2.
One possible mechanism that can modify the spinel lat-
tice constant is the cationic disorder. NiAl2O4 can transform
from a normal spinel, ͑Ni2ϩ͓͒Al23ϩ͔O4, to an inverse spinel,
͑Al͓͒NiAl͔O4 and vice versa with a small energy change.12 In
this process, the larger Ni2ϩ ions which occupy the tetrahe-
dral (8a) sites in normal spinel move to the octahedral
(16d) sites in inverse spinel, while half of the smaller Al3ϩ
ions move in the opposite direction.12 The extent of this
move is measured by the disorder or inversion parameter, I
For insight into the elastic strain evolution, an axisym-
͑the fraction of tetrahedral sites occupied by the Al ions, I
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Ustundag
