Computational Micromechanics Modeling of Polycrystalline Superalloys. . .
151
Table 3 Single crystal elastic constants of Inconel 718 at 400C
C 11 (GPa)
C 12 (GPa)
C 44 (GPa)
240
165
101
Table 4 Optimized crystal plasticity parameters for a wrought Inconel 718 at 400 ◦ C
Isotropic hardening
τ 0 (MPa)
τ s
h 0
q αβ
τ 0
0.71τ 0
−57.13τ 0
1
Kinematic hardening
c
d
mk
58.9τ 0
198.3
17.7
Cyclic softening
τ
cyc
s
h 1
h 2
0.076τ 0
0.07τ 0
2.33 10 −6 τ 0
The remaining parameters of the crystal plasticity model were obtained by means
of the inverse optimization procedure developed in [27, 28]. This inverse technique
is a minimum squares approach based on minimizing the differences between a
set of experimental stress-strain curves with respect to the response obtained by
means of computational homogenization using the Levenberg-Marquardt method.
The experimental data used in the optimization process are the stress-strain loops
obtained under uniaxial cyclic tension with strain control at three different strain
amplitudes, ε//ε min = 1, 1.75, and 2.25 (small, medium, and high) with respect
to the normalizing value ε min and with R ε = 0. The details of the fitting procedure
can be found in [15], and the resulting parameters are summarized in Table 4,
normalized by the initial critical resolved shear stress τ 0 .
6.2 Simulation of the Cyclic Behavior
The numerical simulation of the fine-grained Inconel 718 polycrystal alloy at 400 ◦ C
was carried out using a voxelized RVE (Fig. 10b) and the crystal plasticity model
presented above. The RVE contained 30 × 30 × 30 cubic elements, which ≈300
grains and ≈90 elements per grain. Voxelized RVEs were used (Fig. 10), and
the microstructures were synthetically generated using the Dream3D software to
fulfill the log-normal distribution obtained from microscope images of the material.
Similar to the case of the monotonic behavior, the macroscopic behavior was
relatively independent of the particular RVE realization [15], and a set of five
different grain realizations was used to obtain the macroscopic cyclic behavior.
The cyclic stress-strain curves under uniaxial tension with cyclic strain amplitudes min = 1.25, 1.5, and 2.75 with R ε = 0 and R ε = −1 were simulated
using the polycrystalline homogenization approach. The experimental results of the
two extremal cases, ε//ε min = 1.25 and 2.75, are plotted in Fig. 13 together with
the numerical predictions provided by computational homogenization. Figure 13
includes the cyclic stress-strain curves in the first cycle and in a cycle at ≈75% of
the fatigue life as well as the evolution of the stress amplitude, σ , and of the mean
151
Table 3 Single crystal elastic constants of Inconel 718 at 400C
C 11 (GPa)
C 12 (GPa)
C 44 (GPa)
240
165
101
Table 4 Optimized crystal plasticity parameters for a wrought Inconel 718 at 400 ◦ C
Isotropic hardening
τ 0 (MPa)
τ s
h 0
q αβ
τ 0
0.71τ 0
−57.13τ 0
1
Kinematic hardening
c
d
mk
58.9τ 0
198.3
17.7
Cyclic softening
τ
cyc
s
h 1
h 2
0.076τ 0
0.07τ 0
2.33 10 −6 τ 0
The remaining parameters of the crystal plasticity model were obtained by means
of the inverse optimization procedure developed in [27, 28]. This inverse technique
is a minimum squares approach based on minimizing the differences between a
set of experimental stress-strain curves with respect to the response obtained by
means of computational homogenization using the Levenberg-Marquardt method.
The experimental data used in the optimization process are the stress-strain loops
obtained under uniaxial cyclic tension with strain control at three different strain
amplitudes, ε//ε min = 1, 1.75, and 2.25 (small, medium, and high) with respect
to the normalizing value ε min and with R ε = 0. The details of the fitting procedure
can be found in [15], and the resulting parameters are summarized in Table 4,
normalized by the initial critical resolved shear stress τ 0 .
6.2 Simulation of the Cyclic Behavior
The numerical simulation of the fine-grained Inconel 718 polycrystal alloy at 400 ◦ C
was carried out using a voxelized RVE (Fig. 10b) and the crystal plasticity model
presented above. The RVE contained 30 × 30 × 30 cubic elements, which ≈300
grains and ≈90 elements per grain. Voxelized RVEs were used (Fig. 10), and
the microstructures were synthetically generated using the Dream3D software to
fulfill the log-normal distribution obtained from microscope images of the material.
Similar to the case of the monotonic behavior, the macroscopic behavior was
relatively independent of the particular RVE realization [15], and a set of five
different grain realizations was used to obtain the macroscopic cyclic behavior.
The cyclic stress-strain curves under uniaxial tension with cyclic strain amplitudes min = 1.25, 1.5, and 2.75 with R ε = 0 and R ε = −1 were simulated
using the polycrystalline homogenization approach. The experimental results of the
two extremal cases, ε//ε min = 1.25 and 2.75, are plotted in Fig. 13 together with
the numerical predictions provided by computational homogenization. Figure 13
includes the cyclic stress-strain curves in the first cycle and in a cycle at ≈75% of
the fatigue life as well as the evolution of the stress amplitude, σ , and of the mean