The simulation results are presented in the following. First, to quantify the average elastic and plastic behaviour, the orientation dependent
Young’s modulus (
E) and yield stress (
) are given and compared to the corresponding results from the analytic calculations (
Section 4.1). Then the local stress–strain distribution of selected simulations is presented in
Section 4.2 to investigate in detail the differences at the micro-scale caused by the very different model assumptions.
4.1. Average Behaviour
Young’s modulus E resulting from the simulations is calculated as where is the average second Piola–Kirchhoff stress and the average Green–Lagrange strain along the loading direction at the first, purely elastic loading step.
Table 2 gives an overview of the obtained values for loading along ND, RD and TD.
Table 2a shows that the simulation results obtained from the individual sections differ by at most
and
from the analytic results and
Table 2b reveals even slightly smaller differences when using the combined texture (
and
). For both, analytic calculation and simulated results, the
Young’s modulus along ND calculated from the RD-section is approximately 10
higher than the value obtained from the TD-section. The differences between these sections are, hence, significantly higher than among all full-field simulation approaches.
Figure 5 displays the course of
Young’s modulus over the three mutually perpendicular sections corresponding to the measurements. As the symmetry of grain shape and crystallographic texture allows to average the values of loading directions with an angular difference of 90
around the sample normal, only values for half of the considered loading direction range (0
–180
) are shown. A cubic spline interpolation was performed to obtain values between the rotation angles for which a simulation was conducted. The analytic calculation has been performed at steps of 1
, making an interpolation unnecessary.
Figure 5a compares the results of the analytic calculation to both 2D simulations using the full set of orientations from the individual measurements (i.e., microstructure sets I a and I b). Additionally, the range observed among all five simulations (I a to I e) is given as a background color.
Figure 5b shows results from the analytic and numerical calculations from the combination of the full texture information and the cases of a random texture (models II b and II c only).
Among all simulation results obtained from the individual measurements (
Figure 5a) the relative difference computed as
is smaller than 2.0%, 3.0% and 4.0% for the RD-section, ND-section and the TD-section, respectively. Results obtained by the analytic calculation are very close to the simulation not taking the grain shape into account (microstructure variant I b). The largest deviations between the two simulation approaches in
Figure 5a can be seen for loading along ND (RD-section at 90
, TD-section at 0
), where the values obtained from the simulation including grain shape are lower by 4
and at 45
between ND and RD where the simulation including grain shape is higher by 4
. Overall, the influence of the grain morphology is rather small, a finding in agreement with a study by Jöchen et al. [
23].
There are virtually no differences observable between the results from the analytic calculations using the complete orientation information obtained from all three measurements and its sample consisting of 1000 representative orientations, see
Figure 5b. The same holds for the 3D models, where the use of globular (II b) and elongated (II c) grain shapes gives virtually the same results. Moreover, the differences between the simulations and the analytic calculations are smaller than 3
(less than 1.5%) for the whole orientation range (
Figure 5b).
The results from the tensile tests of the three samples from the flange material (
Figure 1) are given in
Figure 6;
Figure 6a shows the engineering stress–strain curves and
Figure 6b the extracted flow curves together with the 0.05% proof stress used to approximate the yield point. A clear influence of the loading direction can be seen: the sample oriented under 45
between RD and TD shows a significantly higher uniform elongation as well as a slightly higher strain hardening rate in comparison to the samples oriented in RD and TD, respectively. To determine the yield stress from the stress–strain curve, first the elastic portion of the strain is subtracted and the flow curves are plotted (
Figure 6b). Then, the stress at 0.05% plastic strain was defined as the proof stress/yield point
. The values determined in this manner amount to about 895
in RD, 890
in TD and 845
under 45
rotation between RD and TD.
A similar but automated procedure was employed to define the yield stress of each of the 384 crystal plasticity simulations. For the automatic determination, first a continuous representation has been created with a spline interpolation from the 25 stress–strain values per simulation. From this smooth stress–strain curve, the elastic part has been subtracted to evaluate the stress at 0.05% plastic strain. A comparison with results obtained by the method proposed by Christensen [
44] and the direct calculation of a plastic strain offset from the constitutive model (i.e., the plastic strain calculated from
) revealed only quantitative but no qualitative differences. It should be noted that adjusting the phenomenological constitutive parameters allows to reproduce the yield point or proof stress for any other method or threshold value as well.
Table 3 gives an overview of the obtained yield point values for loading along ND, RD and TD. In this table, the microstructure representations used to adjust the parameters are also indicated; those are the full orientation set for the
Taylor factor calculation and variant II c for the full-field simulations. An influence of both, orientation data and modeling approach, is observed:
The yield stress calculated for the individual sections with the analytic approach depends slightly on the data set, it differs by 30
(i.e., 3.4%) for the yield stress in TD direction
, see
Table 3a.
The various microstructure models used for the individual data (I a to I e) predict differences of up to 38
(
calculated from ND-section data), see
Table 3a.
The yield stress in RD, , predicted by all simulations is lower than the value obtained from the analytic expression.
Sampling 1000 orientations from the combined texture results in an increase of the predicted yield stress by 4
–12
when employing the analytic approach, see
Table 3b.
Employing the simpler models (II a: 3D microstructure without grain information and II b: 3D microstructure with globular grains) lowers
and
and increases
in comparison to model II c (3D microstructure with elongated grains) which has the most realistic grain geometry, see
Table 3b.
The course of
is presented in
Figure 7 in a similar fashion as for the
Young’s modulus in
Figure 5. For
, however, only results obtained from the combined texture data are presented as the inaccuracies resulting from the use of the individual measurements are already known. It can be seen that the two considered simulation approaches (II b and II c) form a narrow band (less than 15
deviation) of yield point values and cross at four rotation angles. Although both analytic results are also close to each other, a clear difference to the simulation results can be seen. More precisely, the simulations predict a rather constant yield point from TD to ND (RD-section) and a peak between ND and RD whereas the
Taylor factor calculation results in a decrease from TD to ND followed by a leveling-off increase between ND and RD. Qualitatively, the minimum at 45
between RD and TD is similarly predicted by the simulations and the analytic expression but the latter forecasts a higher value at RD. Comparison to the experimental results reveals a closer agreement for the crystal plasticity simulation at 45
between RD and TD and for the analytic expression at RD. When comparing the results of the simulations using a random texture, it can be seen that the grain morphology has only an effect when loading along ND, that is, perpendicular to the flat side of the elongated grains. More precisely,
of the microstructure with elongated grains is higher by approximately 40
in comparison to the microstructure having globular grains.
To investigate how the grain morphology influences the materials response at higher strain levels, the
Lankford coefficient
R, that is, the ratio between in-plane strain (perpendicular to the loading direction) and the out of plane strain (normal to the normal direction) is computed at a total strain of 10% in loading direction. Only 3D models using the combined and downsampled texture information are compared. The results are shown in
Figure 8. It can be seen that the incorporation of the grain shape results in a reduced
R value, while there is no significant difference whether spatially resolved globular grains or individual orientations per material point are used. It should be noted that the values of
R depend critically on the used method, that is, which strain level is selected and whether the strain increments or the total strain is used for the determination.
4.2. Micro-Mechanical Behaviour
The micro-mechanical behaviour presented in the following is based on the simulation results at step 20, that is, a strain of approximately 0.04%. This strain level corresponds to a stress just below the proof stress.
The spatial distribution of stress and strain in loading direction is shown exemplarily for the TD-section
Figure 9, that is, a model of type I a. In this figure, the local stress and strain in loading direction is shown at
,
,
to ND. The grain structure is clearly visible, where the elongated grains are most obvious in the strain map when loaded perpendicular to the long grain axis and in the stress map when loaded along the long axis. A similar pattern can be observed for the RD-section (not shown in this study). The clear patterning ranging over the whole microstructure is less pronounced for equiaxed grains, that is, the ND section and
Voronoi tessellated structures with globular grain morphology in two and three dimensions (I d, I e and II b). The pattern is totally missing for the random spatial distribution of crystallographic orientations (I b, I c and II a).
For a more quantitative inspection that also enables to systematically investigate the 3D microstructures, “heat maps” of the stress–strain response of each voxel of the employed microstructures are plotted. In
Figure 10, such maps are shown for the 2D microstructure models generated from all measured crystallographic orientations in the RD-section sample (model type I a and I b).
Figure 10a,c show the stress–strain response for loading along TD, that is, along the elongated grains, for the model including grain morphology and the model with random distribution, respectively. The corresponding plots for loading along ND, that is, perpendicular to the long axis of the grains are given in
Figure 10b,d. Independently of the microstructure model, a characteristic unimodal distribution results from the loading along TD while a bimodal distribution results from the loading along the ND. This bimodal distribution is approximately parallel to the strain axis and, hence, results in unimodal stress distributions (shown on the right side of the heat maps). In contrast, the shape of the strain distributions (shown on the top of the heat maps) depends on the microstructure model. Taking the grain shape into account (I a,
Figure 10a) results in a bimodal distribution while the minimum deteriorates to a plateau for the random orientation assignment (I b,
Figure 10d).
Figure 11 shows the heat maps for loading along ND from model variants I c (Random orientation assignment 3D), I d (2D
Voronoi tessellation), I e (3D
Voronoi tessellation) created from the RD-section and II c (i.e., using the combined texture information). Comparing
Figure 11a with
Figure 10d shows that for texture component modeling a difference in the stress–strain partitioning between the 2D and the 3D model (model I b and I c) is hard to ascertain. In contrast, using a 2D or a 3D model makes a difference for spatially resolved grains: The 2D variant of the model with 1000 grains (I d,
Figure 11b) shows significantly stronger clustering than the 3D counterpart (I e,
Figure 11c). The use of a realistic grain shape (II c,
Figure 11d) narrows the strain distribution in comparison to the use of globular grains (I d,
Figure 11c). The strain distribution is even more narrow when the measured microstructure is directly imported (I a,
Figure 10b).