M-SERVE and P-SERVE
57
temperatures θ ≤ 650 ◦ C and higher stresses, creep is governed by different types of
dislocation-based shearing processes, while at higher temperatures θ ≥ 800 ◦ C, the
creep deformation is controlled by Orowan looping and cross-slip mechanisms [9].
Deformation behavior under various loading and temperature conditions has been
studied both for single crystal and polycrystalline Ni-based superalloys [10, 11].
Phenomenological crystal plasticity models, based on the power law or thermally
activated models of plasticity, have been implemented to model creep and deformation response of single crystal and polycrystalline Ni-based superalloys in [12–16].
These are generally single-scale models and lack information on the underlying
microstructural characteristics at the intragranular or subgrain scale, which affect
the single crystal and polycrystalline behavior. Three scales are relevant, when
modeling polycrystalline behavior of Ni-based superalloys using crystal plasticity
models. They are:
1. Intragranular, subgrain scale, characterized by the size of γ precipitates and their
spacing designated as the γ -matrix channel-width;
2. Grain-scale of single crystals characterized by the grain-boundary distance;
3. Scale corresponding to representative volume elements of polycrystalline aggregates.
It is computationally intractable to simulate the behavior of polycrystalline
microstructures with explicit representation of the subgrain-scale microstructure.
To represent the effects of lower-scale morphology and deformation mechanisms
on higher-scale response models, it is necessary to develop crystal plasticity models
that hierarchically incorporate microstructural information from the lower scales.
In [17], hardening parameters are expressed as functions of the average size of
precipitates. Hierarchical approaches for Ni-based superalloys have been proposed
in [18], where artificial neural network algorithms are used to develop grain size
and volume fraction-dependent dislocation density-based crystal plasticity models
for creep and fatigue. Ghosh et. al. have homogenized subgrain scale response to
develop hierarchical grain-scale crystal plasticity models for Ni-based superalloys
in [19–23]. Parametric forms of subgrain-scale morphological characteristics are
incorporated in grain-level constitutive relations in these models.
An important aspect of hierarchical modeling is the establishment of the
“representative volume element” or RVE [24] for conducting direct numerical
simulations (DNS) of the micromechanical problem. The RVE is defined as a
microstructural domain that optimally represents the morphological characteristics
and effective response of the entire microstructure. However, it is difficult to define
an RVE in the strictest sense for microstructures with nonuniformly dispersed
heterogeneities as shown in Fig. 1, due to the lack of uniformity or periodicity
[25]. To facilitate a computational domain for nonuniform microstructures, the
statistically equivalent RVE or SERVE has been introduced in [26–28]. It is
designated as an optimal microstructural domain, for which statistical distribution
functions of morphological parameters, as well as material properties converge to
those for the entire microstructure. The associated exterior statistics-based boundary
conditions are discussed in another chapter of this book.
57
temperatures θ ≤ 650 ◦ C and higher stresses, creep is governed by different types of
dislocation-based shearing processes, while at higher temperatures θ ≥ 800 ◦ C, the
creep deformation is controlled by Orowan looping and cross-slip mechanisms [9].
Deformation behavior under various loading and temperature conditions has been
studied both for single crystal and polycrystalline Ni-based superalloys [10, 11].
Phenomenological crystal plasticity models, based on the power law or thermally
activated models of plasticity, have been implemented to model creep and deformation response of single crystal and polycrystalline Ni-based superalloys in [12–16].
These are generally single-scale models and lack information on the underlying
microstructural characteristics at the intragranular or subgrain scale, which affect
the single crystal and polycrystalline behavior. Three scales are relevant, when
modeling polycrystalline behavior of Ni-based superalloys using crystal plasticity
models. They are:
1. Intragranular, subgrain scale, characterized by the size of γ precipitates and their
spacing designated as the γ -matrix channel-width;
2. Grain-scale of single crystals characterized by the grain-boundary distance;
3. Scale corresponding to representative volume elements of polycrystalline aggregates.
It is computationally intractable to simulate the behavior of polycrystalline
microstructures with explicit representation of the subgrain-scale microstructure.
To represent the effects of lower-scale morphology and deformation mechanisms
on higher-scale response models, it is necessary to develop crystal plasticity models
that hierarchically incorporate microstructural information from the lower scales.
In [17], hardening parameters are expressed as functions of the average size of
precipitates. Hierarchical approaches for Ni-based superalloys have been proposed
in [18], where artificial neural network algorithms are used to develop grain size
and volume fraction-dependent dislocation density-based crystal plasticity models
for creep and fatigue. Ghosh et. al. have homogenized subgrain scale response to
develop hierarchical grain-scale crystal plasticity models for Ni-based superalloys
in [19–23]. Parametric forms of subgrain-scale morphological characteristics are
incorporated in grain-level constitutive relations in these models.
An important aspect of hierarchical modeling is the establishment of the
“representative volume element” or RVE [24] for conducting direct numerical
simulations (DNS) of the micromechanical problem. The RVE is defined as a
microstructural domain that optimally represents the morphological characteristics
and effective response of the entire microstructure. However, it is difficult to define
an RVE in the strictest sense for microstructures with nonuniformly dispersed
heterogeneities as shown in Fig. 1, due to the lack of uniformity or periodicity
[25]. To facilitate a computational domain for nonuniform microstructures, the
statistically equivalent RVE or SERVE has been introduced in [26–28]. It is
designated as an optimal microstructural domain, for which statistical distribution
functions of morphological parameters, as well as material properties converge to
those for the entire microstructure. The associated exterior statistics-based boundary
conditions are discussed in another chapter of this book.