114
D. W. Eastman et al.
strength of each cross-sectional layer, which contains grains loaded in parallel, was
defined to be equal to the strength of the grain with the lowest Schmid factor in
that layer. Then, the yield strength of the sample was defined using a weakest link
approximation along the length of the gage. The strength of the layers loaded serially
was taken to be the same as the weakest layer.
In developing this model, an interesting trend with regards to Schmid factor
distributions in a randomly textured FCC alloy was observed. When considering
<110>{111} FCC slip systems, randomly oriented grains generate a high propensity
for high Schmid factors; 50% of grains have a maximum resolved Schmid factor
of 0.45 or higher [60]. By contrast, the probability of finding a grain with a low
maximum resolved Schmid factor is much smaller. The chance of having a sample
with all layers dominated by grains with low maximum resolved Schmid factors
(high strength) is in fact very rare and decreases with increasing sample size.
Calculations were conducted for thousands of synthetic samples, from as small
as one grain through the thickness to up to 30 grains through the thickness. The
results showed that while it is geometrically possible to arrange the grains in small
samples such that they have a higher-than-average resolved Schmid factor, and thus
a lower-than-average overall strength, the converse is much less likely [60]. This
outcome helps explain the increased scatter that is observed in the experimentally
measured yield strength with decreasing sample size, as well as why the strength of
oligocrystals (with finite numbers of grains) can be weaker, but not stronger, than the
bulk strength of the polycrystalline alloy. Although this simple numerical analysis
contains many assumptions and ignores the effects of load shedding, free surfaces,
and grain size and shape, the qualitative influence of how the finite sampling of grain
orientations results in decreasing strength and increasing scatter with decreasing
sample size is demonstrated.
Physics-based CPFEM offers a more quantitative approach for modeling the
strength of oligocrystals, and the experimentally measured size dependence of
strength can be used to benchmark and validate such models. The use of oligocrystals allows for measurements that are much more sensitive to microstructural
variations than bulk samples. One such example involves the work of Bagri et al.,
who employed statistically equivalent representative volume elements (SERVEs)
to determine convergence of properties as a function of the size of a sample
microstructural volume [109]. This approach is typically used to find a measurement
of the proper size RVE needed for convergence of a property and can also be used
to predict the effect of sample size on strength. Starting from a large SERVE that
produces a bulk value for yield strength, and then decreasing the SERVE size,
leads to an increased scatter in strength that is also seen experimentally. However,
theses simulations predict that the sample strength will both increase and decrease at
smaller sample sizes, while the experiments only documented a decrease in strength
at smaller sample sizes. There is a difference in the SERVE geometry as compared
to the tested sample geometry, which may account for the difference between
simulation and experiment. The SERVE geometry is a cube, whereas the sample
geometry has a rectangular shape in order to comply with ASTM tensile testing
standards. The influence of microstructure is different for cubes and rectangles, with
D. W. Eastman et al.
strength of each cross-sectional layer, which contains grains loaded in parallel, was
defined to be equal to the strength of the grain with the lowest Schmid factor in
that layer. Then, the yield strength of the sample was defined using a weakest link
approximation along the length of the gage. The strength of the layers loaded serially
was taken to be the same as the weakest layer.
In developing this model, an interesting trend with regards to Schmid factor
distributions in a randomly textured FCC alloy was observed. When considering
<110>{111} FCC slip systems, randomly oriented grains generate a high propensity
for high Schmid factors; 50% of grains have a maximum resolved Schmid factor
of 0.45 or higher [60]. By contrast, the probability of finding a grain with a low
maximum resolved Schmid factor is much smaller. The chance of having a sample
with all layers dominated by grains with low maximum resolved Schmid factors
(high strength) is in fact very rare and decreases with increasing sample size.
Calculations were conducted for thousands of synthetic samples, from as small
as one grain through the thickness to up to 30 grains through the thickness. The
results showed that while it is geometrically possible to arrange the grains in small
samples such that they have a higher-than-average resolved Schmid factor, and thus
a lower-than-average overall strength, the converse is much less likely [60]. This
outcome helps explain the increased scatter that is observed in the experimentally
measured yield strength with decreasing sample size, as well as why the strength of
oligocrystals (with finite numbers of grains) can be weaker, but not stronger, than the
bulk strength of the polycrystalline alloy. Although this simple numerical analysis
contains many assumptions and ignores the effects of load shedding, free surfaces,
and grain size and shape, the qualitative influence of how the finite sampling of grain
orientations results in decreasing strength and increasing scatter with decreasing
sample size is demonstrated.
Physics-based CPFEM offers a more quantitative approach for modeling the
strength of oligocrystals, and the experimentally measured size dependence of
strength can be used to benchmark and validate such models. The use of oligocrystals allows for measurements that are much more sensitive to microstructural
variations than bulk samples. One such example involves the work of Bagri et al.,
who employed statistically equivalent representative volume elements (SERVEs)
to determine convergence of properties as a function of the size of a sample
microstructural volume [109]. This approach is typically used to find a measurement
of the proper size RVE needed for convergence of a property and can also be used
to predict the effect of sample size on strength. Starting from a large SERVE that
produces a bulk value for yield strength, and then decreasing the SERVE size,
leads to an increased scatter in strength that is also seen experimentally. However,
theses simulations predict that the sample strength will both increase and decrease at
smaller sample sizes, while the experiments only documented a decrease in strength
at smaller sample sizes. There is a difference in the SERVE geometry as compared
to the tested sample geometry, which may account for the difference between
simulation and experiment. The SERVE geometry is a cube, whereas the sample
geometry has a rectangular shape in order to comply with ASTM tensile testing
standards. The influence of microstructure is different for cubes and rectangles, with