M-SERVE and P-SERVE
87
0.1μ, where μ is the mean value property. Reasonable agreement is seen with
experimental results. With this criterion, convergence is achieved for SERVEs for
which the upper and lower bound values of material properties are within 10% of
the mean value. Room temperature simulation results for the largest SERVE with
a size of ∼ 150 μm are used for the identification of the P-SERVE. The results in
Fig. 18 indicate that convergence is achieved for a SERVE size of 100 μm. This
implies that the P-SERVE size is ≈ 100 μm, above which material properties are
convergent. This is also corroborated with experimental results. In comparison, the
M-SERVE size in Sect. 3.3 is found to be 150 μm. The smaller P-SERVE compared
to the M-SERVE indicates that the characteristic features of twins implemented in
determining the M-SERVE are more stringent than those necessary for determining
material properties like strength and hardness. This exercise implicitly establishes
a relation between the M-SERVE and P-SERVE. Finally, while the M-SERVE and
P-SERVE are relatively small compared to the scale of the grain structure for these
properties, other properties such as fatigue [57] may require larger volumes of the
material to be analyzed.
3.5 Summary of the Polycrystalline Scale Analysis
This section establishes microstructure-based statistically equivalent RVEs or MSERVEs and property-based statistically equivalent RVEs or P-SERVEs from
statistically equivalent virtual polycrystalline microstructures of polycrystalline
Ni-based superalloys containing annealing twins. Results from the KolmogorovSmirnov (KS) convergence test show that the minimum size of the M-SERVE
that captures the experimentally obtained statistics is ≈ 150 μm. On the other
hand, a P-SERVE size of ≈ 100 μm is adequate to reproduce the macroscopic
experimental material response. The smaller P-SERVE size compared to the MSERVE size implies that the characteristic features of twins needed for determining
the M-SERVE are more stringent than those necessary for determining material
properties like strength and hardness. This M-SERVE-P-SERVE study implicitly
establishes a dependency of a property on the microstructure, i.e., microstructure
property relations. In conclusion, the methodologies developed in this paper provide
a solid foundation for micromechanical analysis leading to the evaluation of multiscale properties of complex polycrystalline materials.
In summary, this chapter develops a cogent framework for statistically equivalent
virtual microstructures and subsequently M-SERVEs and P-SERVEs of nickelbased superalloy microstructures at multiple scales. The framework couples experimental methods, image extraction, statistical analysis and finite element modeling
for establishing a robust methodology that can be applied to a wide variety of
heterogeneous materials. This framework lays the foundation for the development of
parametrically homogenized constitutive models as functions of the microstructural
morphology and crystallography.
87
0.1μ, where μ is the mean value property. Reasonable agreement is seen with
experimental results. With this criterion, convergence is achieved for SERVEs for
which the upper and lower bound values of material properties are within 10% of
the mean value. Room temperature simulation results for the largest SERVE with
a size of ∼ 150 μm are used for the identification of the P-SERVE. The results in
Fig. 18 indicate that convergence is achieved for a SERVE size of 100 μm. This
implies that the P-SERVE size is ≈ 100 μm, above which material properties are
convergent. This is also corroborated with experimental results. In comparison, the
M-SERVE size in Sect. 3.3 is found to be 150 μm. The smaller P-SERVE compared
to the M-SERVE indicates that the characteristic features of twins implemented in
determining the M-SERVE are more stringent than those necessary for determining
material properties like strength and hardness. This exercise implicitly establishes
a relation between the M-SERVE and P-SERVE. Finally, while the M-SERVE and
P-SERVE are relatively small compared to the scale of the grain structure for these
properties, other properties such as fatigue [57] may require larger volumes of the
material to be analyzed.
3.5 Summary of the Polycrystalline Scale Analysis
This section establishes microstructure-based statistically equivalent RVEs or MSERVEs and property-based statistically equivalent RVEs or P-SERVEs from
statistically equivalent virtual polycrystalline microstructures of polycrystalline
Ni-based superalloys containing annealing twins. Results from the KolmogorovSmirnov (KS) convergence test show that the minimum size of the M-SERVE
that captures the experimentally obtained statistics is ≈ 150 μm. On the other
hand, a P-SERVE size of ≈ 100 μm is adequate to reproduce the macroscopic
experimental material response. The smaller P-SERVE size compared to the MSERVE size implies that the characteristic features of twins needed for determining
the M-SERVE are more stringent than those necessary for determining material
properties like strength and hardness. This M-SERVE-P-SERVE study implicitly
establishes a dependency of a property on the microstructure, i.e., microstructure
property relations. In conclusion, the methodologies developed in this paper provide
a solid foundation for micromechanical analysis leading to the evaluation of multiscale properties of complex polycrystalline materials.
In summary, this chapter develops a cogent framework for statistically equivalent
virtual microstructures and subsequently M-SERVEs and P-SERVEs of nickelbased superalloy microstructures at multiple scales. The framework couples experimental methods, image extraction, statistical analysis and finite element modeling
for establishing a robust methodology that can be applied to a wide variety of
heterogeneous materials. This framework lays the foundation for the development of
parametrically homogenized constitutive models as functions of the microstructural
morphology and crystallography.