Chapter 5 Micromechanical modeling of plasticity and fracture of DP steels
5.2 Plastic properties of DP steels
5.2.1 Effect of martensite volume fraction on strength and ductility of DP
The first problem to address is the effect of the martensite volume fraction. The distribution and properties of martensite are kept the same independent of Vm in the unit cell calculation. Indeed, according to the experiment design in chapter 2 and 3, starting from the same spheroidized microstructure, the DP steels have the same spatial distribution of martensite phase (mainly along the ferrite grain boundaries).
Furthermore, the hardness of martensite is roughly the same at a fixed annealing temperature (Figure 2.25). Therefore, the simplified unit cell model can be considered as representative to the real microstructure in this sense, but, of course, the periodic distribution remains a strong assumption that will be discussed.
5.2.1.1 Simulation of the tensile curves and parameters identification
The flow behavior of QT-700 with various martensite volume fractions can be properly captured using the Voce law, as shown in Figure 5.3. All the curves are shown up to necking. For the case of low Vm values (such as 15% and 19%), a sharp elasto-plastic transition is shown in the simulated curves (Figure 5.3a and b). For the case of 37% of martensite, such transition is progressive, and the tensile curve shows high initial hardening rate (Figure 5.3d). However, except for the case of 15%
martensite, necking as defined by the attainment of the Considere criterion occurs too early in the simulation curves, compared to the experiemental curves.
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Figure 5.3. Comparison of experimental and simulated tensile curves for QT-700 with 15% (a), 19% (b), 28% (c) and 37% (d) of martensite. All the curves are shown up to
necking.
The ferrite parameters are identified by fitting with experimental data, as shown in Figure 5.4. As Vm is increased, the mean free path of ferrite, which can be approximated as equal to the ferrite grain size, is decreased (see chapter 2).
Therefore, the yield strength is increased accordingly (Figure 5.4a). The dislocation storage in the vicinity of the grain boundary will contribute both to forest hardening giving an isotropic hardening contribution [147] and to the building up of back stresses (giving a kinematic hardening contribution) [77, 164, 165]. As the grain size is refined, more grain boundaries are generated, and then the isotropic hardening is more efficient and the kinematic hardening becomes significant. These can explain higher value with finer grain sizes. Additionally, an increased value can be explained by the enhanced dynamic recovery with more grain boundaries [166].
Therefore, the identified parameters from the experimental results are physically sound.
Chapter 5 Micromechanical modeling
mean free path of ferrite (μm)
a
mean free path of ferrite (μm)
b
mean free path of ferrite (μm)
c
Figure 5.4. Identified ferrite parameters: evolution of (a) yield strength, (b)and (c)
with mean free path of ferrite. These parameters are identified by fitting with the experimental data.
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5.2.1.2 Predictions of tensile strength and uniform elongation
As shown in Figure 5.3, the experimental tensile curves can be well simulated by the model, which provides the basis for the properties predictions. As shown in Figure 5.5a, the tensile strength as a function of martensite volume fraction is accurately described by the model. Both experimental and predicted tensile strengths have an approximately linear relationship with martensite volume fraction.
The predicted uniform elongation decreases with increasing Vm (Figure 5.5b), which also fits the experimental trend (Figure 3.3). However, necking starts earlier in the simulation (Figure 5.3) and the model underestimates the uniform elongation. Except for the cases of 15% and 19% martensite, the predicted uniform elongation is about 2% (in absolute terms) less than the experimental one. This reflects an underestimation of the work-hardening rate at a given flow stress.
0.16 0.20 0.24 0.28 0.32 0.36
650
0.16 0.20 0.24 0.28 0.32 0.36
0.09
Figure 5.5. Predictions of tensile strength (a) and uniform elongation (b).
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5.2.1.3 Improved predictions with stage-IV hardening of ferrite
This section will firstly analyze the origin of the deficiency of the cell calculation regarding the proper prediction of the hardening rate at large strain, and then propose to improve the prediction with a modified behavior law of ferrite (equation 5.5).
Strain concentration: the necessity of improving the hardening law at large strain
Since martensite is much stronger than ferrite, strain is higher in the ferrite matrix with a heterogeneous distribution of plastic strain. As shown in Figure 5.6, the equivalent plastic strain (PEEQ) is strongly concentrated in the ferrite at the ferrite/martensite interface, and there is a tendency for strain localization. The plastic strain of the elements at the interface can be up to 5.
In stage III, the work-hardening rate of metals keeps decreasing during deformation, but this trend changes at large strain after entering stage IV and a constant work-hardening rate is reached [147]. This kind of linear hardening in the large strain regime has been observed in several engineering alloys [167] and the transition from stage-III to stage-IV occurs in polycrystals at strain typically between 0.15 and 0.6.
Farrell et al. [167] found for a large variety of metallic alloys that the post-necking flow response is well represented by a linear hardening (with necking starting at strain between 0.15 and 0.4). Simar [168] found that the prediction of fracture strain, which depends on the behavior law at large strain, requires accounting for stage-IV hardening. Lecarme [169] also found that a linear hardening regime at large strain can influence the damage evolution.
Referring to the simulated tensile curves in Figure 5.3, the uniform elongation is underestimated by the model because the work-hardening rate is decreasing relatively fast. It is proposed that taking into account for stage-IV hardening of ferrite (equation 5.5) by a moderate constant hardening rate can improve the prediction of uniform elongation, counteracting also partly the tendency for strain localization in the ferrite in the unit cell. The hardening rate will be fitted with the experimental results of QT-700-37% and kept constant for all the martensite volume fractions.
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Figure 5.6. Equivalent plastic strain (PEEQ) distribution within the unit cell (QT-700-37%). The macroscopic strain is 0.3.
Tensile curves and work-hardening
Figure 5.7 shows the comparison of the simulated tensile curves with and without stage-IV hardening of ferrite. The stage-IV hardening rate (θIV) is fitted to be 100MPa (about μ/800, slightly higher than the values reported in [147]). The macroscopic response of the unit cell is changed, especially when Vm is large. With the modified law for ferrite, the initial deformation is not influenced and the tensile strength is not affected obviously. But the onset of necking is postponed due to the increased work-hardening rate at large strain. Figure 5.8 shows the effect of the modified law on the work-hardening of RVE at large strain. The work-hardening rate is the same until large strain, and the Voce law of ferrite results in a faster drop of the work-hardening rate. Although 100MPa is a small value comparing with the initial hardening rate, the modified behavior of the ferrite can lead to a higher work-hardening of the composite at large strain, which postpones the onset of necking and improves the prediction of uniform elongation.
For pure metals, the stage-IV hardening rate is very small [147]. As to engineering metals, values between 40 and 110 MPa were identified for aluminum alloys, and the heavily overaged materials were found to reach stage-IV even before the onset of
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necking [168]. In this sense, 100 MPa in this work is a physically acceptable value.
However, this value is actually empirical. It also accounts for other microstructural factors, such as morphological effects not properly taken into account and the inaccuracy of martensite behavior law.
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Figure 5.7. Comparison of the tensile curves with and without considering stage-IV hardening of ferrite for QT-700-15% (a), 19% (b), 28% (c) and 37% (d).
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Figure 5.8. Effect of stage-IV hardening of ferrite on work-hardening of the unit cell.
Effect on properties prediction
Figure 5.9 shows the experimental and predicted tensile strength and uniform elongation. The tensile strength is only slightly influenced by the stage-IV hardening of ferrite. But the uniform elongation can be better predicted after using the modified behavior law of ferrite.
The decrease in uniform elongation with increasing Vm has been reported [5, 52] and theoretically predicted by Delince [77], but the underlying physics must be discussed.
If the martensite remains elastic, an increase in Vm can increase significantly the back stress [50], which tends to enhance work-hardening and to improve the uniform elongation. However, the co-deformation of martensite with ferrite is promoted when Vm is large [170]. This argument is supported by the calculated results shown in Figure 5.10 that the plastic strain in the martensite is larger for a higher martensite volume fraction. According to the tensile behavior of martensite, the work-hardening rate decreases dramatically with strain and the flow stress tends to become saturated at large strain [40]. Therefore, the reduced plastic incompatibility between martensite and ferrite counteracts the effect of increasing Vm, resulting in a rapid decrease of work-hardening rate (see chapter 3).
Chapter 5 Micromechanical modeling
Figure 5.9. Prediction of the tensile strength (a) and uniform elongation (b) with and without accounting for the stage-IV hardening of ferrite.
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
Figure 5.10. Evolutions of the average equivalent plastic strain in the martensite with the macroscopic strain as predicted by the unit cell calculations (with stage-IV hardening taken into account).