Retardation of plastic instability via damage-enabled microstrain delocalization

As a first step toward the goal of statistically relevant characterization of ductile damage evolution up to failure, all possible damage mechanisms in the three DP microstructures (µ _FG, µ _CG, and µ _HM) and three strain paths (UAT, PST, and BAT) were extensively studied by exploiting the in-situ SEM capabilities of the miniaturized Marciniak setup. The five most relevant damage mechanisms are presented in Fig. 4. These five mechanisms, which are also the dominant mechanisms observed in the literature [23‐27, 29], were chosen as categories in the semi-automatic statistical damage identification algorithm (Fig. 3) as a starting point for the quantitative damage analysis, discussed next.

The analysis starts with the fine-grained (μ _FG) microstructure, for which the different damage mechanisms were quantified for the three loading states (UAT, PST, and BAT). The damage incident densities of the five different damage mechanisms (D _MC, D _MFI, D _FGB, D _FGI, and D _INC) are shown in Fig. 5 as function of the von Mises strain, with the vertical dashed lines denoting the strain level at the point of necking (i.e., global localization). Each data point was obtained by quantifying all damage incidents over five large-area (300 × 300 μm²) SEM images, i.e., a total area of 450,000 μm². This large amount of data allows for a very accurate determination of the averaged damage incident density. It should be noted, however, that the damage incident density inherently shows large variability due to the strong heterogeneity of the DP microstructure even in commercial grades, as can be observed by the wide error bands in Fig. 5. Perhaps this inherent variability may also explain why, to our knowledge, such an extensive quantification of the relevant damage mechanisms as a function of strain level and for different strain paths and microstructures has not been carried out before.

The first aspect to note from Fig. 5 is that D _FGB, D _FGI, and D _INC damage incidents are all clearly present, however, only to a limited extent; therefore, these mechanisms most probably do not play a critical role in controlling the necking and failure behavior. For this reason, the investigation will focus on the D _MFI and D _MC damage mechanisms, for which a number of interesting observations can be made as follows:

Interestingly, the first three observations are in agreement with those of Tasan et.al [46], where the total number of damage incidents was measured (only) at the point of necking and failure, for the commercial (parent) DP600 microstructure with the same chemical composition (note that no comparison with observation 4 could be made).

In order to understand these observations, a thorough experimental and numerical analysis, discussed below, was initiated, which led to the following hypothesis on a chain of events that links D _MC to D _MFI:

One can easily see that this hypothesis, in which D _MFI is caused by D _MC, can explain the peculiar similarity in BAT trend for D _MFI and D _MC (observation 4). It may also explain why most damage incidents initiate at low strain (observation 3), while at the same time the built up of stress in M explains the relatively high yield strength of DP steels. Moreover, the critical role of hydrostatic stress can explain why D _MC primarily occurs at BAT [D _MC is negligible for UAT (Fig. 5a) and small for PST (Fig. 5b)]. Furthermore, the coupling of D _MFI to D _MC can explain that D _MFI also increases from UAT to PST to BAT (observation 1). Lastly, the diffuse straining in combination with strain hardening may prevent the formation of percolation paths, and thus delaying global localization; such a necking retardation mechanism may explain the large necking strain at BAT (observation 2). Nevertheless, to test the validity of this D _MC-D _MFI hypothesis, additional numerical and experimental studies were conducted, which are presented next.

First, numerical simulations of the (measured) fine-grained microstructure loaded at UAT, PST, and BAT to 5 % strain are investigated. To this end, Fig. 6 shows the hydrostatic stress and plastic strain fields. Note that the deviatoric stress (or von Mises stress) and volumetric strain are not shown as they scale with the plastic strain and hydrostatic stress, respectively, in the isotropic elasto-plastic model used (“Methodology” section). Also no damage mechanisms were included in these simulations, as they would require the measurement of constitutive laws for damage initiation and growth; the fundamental challenges in obtaining such laws have been described in detail in [31]. Since these simulations do not include damage-induced strain relaxation and stress redistributions, care should be taken when comparing to experimental results. Nevertheless, the simulations do provide qualitative insight in the differences in stress and strain state for the different strain paths.

Figure 6a–c shows that the equivalent plastic strain is higher in the ferrite matrix than the martensite islands and shows strain bands between 45 and 60° to the main loading direction, in agreement with [30]. Regarding the plastic strain magnitude and distribution in the ferrite, it is observed that, from BAT to PST to UAT, the strain localizes increasingly into peaks. Based on this trend, a decrease in D _MFI from UAT to BAT would be expected; however, the opposite is observed in Fig. 5, which indicates that another mechanism for damage in ferrite becomes active at PST and especially BAT.

The plastic straining releases the deviatoric stress in the ferrite matrix and, through stress redistribution (bounded by stress equilibrium at the phase boundaries), also the hydrostatic stress. This is seen in Fig. 6d–f, which shows that the hydrostatic stress is (much) higher in the martensite islands. Naturally, the hydrostatic stress increases with the change of loading from UAT to PST to BAT. This increase in hydrostatic stress explains the observed increase in fracture of martensite (i.e., D _MC) from UAT to PST to BAT (Fig. 5).

The simulations thus support the first two steps of the D _MC–D _MFI hypothesis; however, because of the absence of damage mechanisms, the last two steps (regarding the coupling between D _MC and D _MFI) cannot be investigated. Hence, two additional experiments were performed to examine the connection between D _MC and D _MFI.

In the first additional experiment to study the evolution of individual damage incidents during the deformation, biaxial tension tests up to failure were performed in situ under SEM (SE-mode) observation using home-built miniaturized Marciniak setup, shown in Fig. 2a. The measured large-area (300 × 300 μm²) in-situ SEM movies were analyzed in detail with respect to martensite cracking incidents and further deformation around these D _MC sites. First of all, it was found that the areal density of D _MC incidents at the surface was significantly lower than in the bulk, which is attributed to the lower hydrostatic stress at the surface. Still, many D _MC incidents could be observed under biaxial loading, of which seven examples are given in Fig. 7. It was found that most D _MC incidents occurred in the smallest cross section of the irregularly shaped martensite islands, i.e., the thin martensite bridges. Moreover, it was observed that almost all D _MC incidents initiated at the early stages of deformation, see Fig. 7b, and that D _MC incidents were typically accompanied by one or more location of extreme plasticity in the surrounding ferrite, see Fig. 7c. This would be counted as D _MFI damage in the damage quantification methodology, giving direct evidence for the hypothesis that D _MC triggers D _MFI. Finally, it should be noted that around most D _MC–D _MFI locations the localized extreme plastic straining spreads out into the neighboring ferrite grains resulting in diffuse deformation zones that can cover the complete ferrite grain, see Fig. 7d, thus supporting the necking retardation mechanism of the hypothesis. This mechanism of ferrite damage (i.e., highly localized ferrite deformation) activating diffuse deformation zones in the adjacent ferrite grains was also observed in situ in the microstructural martensite bands observed in commercial DP600 sheet [30]. Combining Figs. 5 and 7, it can be concluded that the early-initiated martensite cracking incidents are well enough dispersed to postpone the formation of percolation paths, which explains the late global localization.

In the second additional experiment to investigate whether the coupling between D _MC and D _MFI damage initiation is also present in the specimen interior, high-resolution 3D depth profiling is performed on the cross section of a 16 % biaxially strained fine-grained specimen. To this end, a series of flat profiles are made approximately 300 nm apart. Note that the high requirements on surface roughness rule out the (Nital) surface etching, used before to distinguish between martensite and ferrite phases. Instead, precision polishing is used to reproducibly remove a ~300 nm surface layer, while SEM imaging in backscatter electron (BSE) imaging mode is used to identify the martensite and ferrite phases by the difference in channeling contrast (note that martensite shows much finer spatial variations in channeling contrast due to its much finer substructure compared to that of the relatively coarse ferrite sub-grains). This identification procedure was verified in detail using electron backscatter diffraction analysis (not shown). Note also that, due to the channeling contrast, D _MC and especially D _MFI damage locations appear differently.

Three typical examples of the detailed 3D shape of a D _MC damage location are shown in Fig. 8. A number of observations could be made from these and other depth profiles measured in the specimen interior.

In addition, all recorded high-resolution SEM-BSE images (with a total area of 38200 μm²) were processed with the above-mentioned damage quantification methodology, i.e., similar to Fig. 5. A total of 202 damage incidents were automatically found by the software and identified as D _MC, D _MFI, D _FGB, D _FGI, or D _INC. Again D _MFI and D _MC damage dominated showing a mutual ratio of ~1.7 in good agreement with the ratio found in Fig. 5c at 16 % strain, especially when considering the differences in image contrast mode used. Detailed investigation of the 3D connections revealed that the 202 damage counts in these stacked images could be traced back to 81 3D damage zones and approximately half of the D _MFI incidents originate from a martensite cracking event (D _MC), which may explain the increase in D _MFI from PST to BAT loading, observed in Fig. 5.

Finally, it is noted that, with this insight in the 3D character of coupled D _MC–D _MFI damage incidents, it cannot be excluded that the damage incidents at a ferrite grain boundary or inside the grain interior (D _FGB and D _FGI) are in fact caused by a martensite island above or below the surface of observation, and thus should have been counted as D _MFI. However, due to the relative unimportance of D _FGB compared to D _FGI, this would not alter the conclusions.

In all, it can be concluded that the D _MFI–D _MC hypothesis is supported by many different forms of experimental and numerical evidence. Especially, the mechanism that spreads out the deformation over a larger ferrite area (the diffuse deformation zones) is interesting, as it seems to be the cause for the delay of global localization. For this necking retardation mechanism to be effective, however, the damage incidents need to be well enough dispersed, such that the early burst of D _MC damage in BAT does not result in global localization by connection of D _MC damage localizations. Therefore, next, the influence of microstructure features (grain size and martensite volume percentage) is investigated.

The isolated influence of grain size is investigated by comparing the μ _FG and μ _CG microstructures: a reduction in grain size corresponds to an increase in D _MFI and D _MC densities and, especially, earlier damage initiation at low strains (Fig. 9a, b). These effects could be caused by the same grain size effect underlying the well-known Hall–Petch relation between the yield (and flow) strength and the grain size, which is explained by the obstruction of plastic slip at the grain and/or phase boundaries causing dislocation pile-up, thereby locally increasing the stress level at the boundaries. Indeed, the experimental global stress–strain curves in Fig. 1d show this increase in yield and flow strength. The D _MFI–D _MC hypothesis would predict that a faster rise of the stress level at the martensite–ferrite boundaries (due to a reduction in ferrite grain size) results in more and earlier D _MC damage and, due to the D _MC–D _MFI causality, in more D _MFI damage, thus explaining the observed differences between Fig. 9a, b. The evolutions of the simulated hydrostatic stress also show significantly higher stress concentrations in the martensite islands of the μ _FG microstructure, but this is a direct result of the higher ferrite yield strength used, see Table 1, which indirectly takes into account the Hall–Petch effect.

To investigate the isolated influence of martensite volume fraction, next, the μ _CG and μ _HM microstructures are compared: an increase in martensite volume fraction results in an increase in D _MC damage, whereas it does not seem to significantly impact D _MFI (Fig. 9b, c). The increase in D _MC is attributed to the stress increase due to the reduction of plastically deforming ferrite phase resulting in a compact network of the harder martensite phase. Indeed, a pronounced increase in stress level (at equal global strain) is seen in the simulated hydrostatic stress fields (Fig. 9e versus 9f). Interestingly, the increase in D _MC with increasing martensite volume fraction is not followed by an increase of D _MFI. This may be the result of the lower probability that a D _MC location is adjacent to an open ferrite area that is large enough (and thus the constraint by the surrounding martensite network low enough) to develop extreme localized plasticity, identified as D _MFI. As a direct consequence, the areal density of diffuse deformation zones, which are initiated from a D _MFI sites as shown in Fig. 7d, will also be lower. This is precisely what is also seen in the simulated fields of the plastic strain, which for higher martensite volume fraction shows large regions with low ferrite strain, see, e.g., the lower left corner of Fig. 9i. In other words, the compact martensite network in the μ _HM microstructure prevents the plastic straining around a D _MC location from spreading out to surrounding ferrite grains. Indeed, as a consequence of the fact that this spreading of plastic straining is hampered, Fig. 9i also reveals a number of local spots where the plastic strain peaks to a level far above the maximum strain found in Fig. 9g, h.

Let us next focus on the global localization behavior of these three microstructures. Comparing the necking behavior of μ _CG with μ _HM, a large reduction in global localization strain is observed, which can be related to the increase in martensite volume fraction. Global localization involves connection of the above-mentioned diffuse deformation zones into a global strain percolation path, which, for DP steel, will obviously run through the available ferrite grains. For μ _HM, less strain percolation paths form, and hence each percolation path must strain more to accommodate the same applied global strain, therefore earlier reaching the point of global localization. This reduction of the number of percolation paths is clearly seen in Fig. 9i, which only shows one pronounced percolation path (running from upper left to lower right corner).

Figure 9 also shows that necking takes place at higher equivalent strain for μ _FG compared to μ _CG. Because the martensite volume fraction is the same for μ _FG and μ _CG, another mechanism must be at play, which may be explained as follows. Global localization is controlled by the weakest percolation path and, for μ _CG compared to μ _FG, less diffuse deformation zones need to be connected to complete a percolation path over the full sample thickness or width. Therefore, taking into account the large spread of grain properties and geometries, the percolation paths in μ _CG will exhibit a larger variability. As a result, the strength of the critical (weakest) percolation path will be smaller in μ _CG, which explains its lower global localization strain. The same mechanism was found to control the necking behavior observed in tensile tests of aluminum strips with very few grains over the specimen width [48], for which in-situ DIC strain maps showed direct evidence that weaker localized percolation paths develop when the grain size is increased, triggering earlier global localization. For our case, this possible explanation would indeed be supported by the strain fields in Fig. 9g, h, which shows that the number of percolation paths is higher in the μ _FG microstructure.

Finally, when the case of μ _FG is directly compared to that of μ _HM, it is interesting to note that the damage evolution at small strains looks quite similar, see Fig. 9a, c. However, there is a major difference, which exhibits itself in the observation of a higher flow stress as well as a higher fracture strain, see Fig. 1d. Of course, the above-mentioned Hall–Petch effect could explain the increase in flow stress; however, there exists a well-known competition between high strength versus high elongation. Therefore, to explain the observed increase in fracture strain for μ _FG compared to μ _HM another mechanism is required. As was seen above, for μ _HM, the high hydrostatic stresses are a direct result from the limitation in the number of strain percolation paths, which also localizes the damage evolution causing earlier global localization and final fracture (Fig. 9c). For μ _FG, on the other hand, the damage is more dispersed due to its finer microstructure and more ferrite grains, which activates the necking retardation mechanism in which damage initiation triggers (many) diffuse deformations zones, as was seen in Fig. 7, thereby spreading out plastic straining and thus postponing global localization. Hence, for μ _FG, the high hydrostatic stress does not seem to be detrimental, but actually beneficial as it increases the global flow strength compared to µ_HM (shown in Fig. 1d for the global stress–strain curves under uniaxial tension). This would mean that the well-known competition between high strength versus high elongation can be overcome by inserting many barriers in the microstructure that increase the hydrostatic stresses. It is crucial, however, that these barriers break open easily enough (as is the case in μ _FG and not in μ _HM) such that plasticity spreads out subsequently to the surrounding matrix in order to prevent early necking.

The role of the damage mechanisms in the localization and fracture behavior is critical. Without damage mechanisms, there is no stress release by diverging localized plasticity to non-local (diffuse) plasticity, thus the stress keeps on building up, leading to early necking. Of course, stress release can only activate a necking retardation mechanism when damage sets in before strain percolation paths have formed. In turn, early damage formation requires high hydrostatic stress built up at early stages of deformation, which can be achieved by microstructural refinement due to the grain size effect, while it also strongly depends on the loading conditions. For instance, for BAT, much higher hydrostatic stresses build up compared to UAT and PST, see Fig. 6, which may explain the unusually high BAT necking strain (Fig. 5) compared to typical forming limit diagrams which show the highest necking strain for UAT.

Based on these insights, it is anticipated that the ideal microstructure combining high strength with high ductility can be achieved through microstructural refinement, e.g., by careful design of a nano-grained DP. The hard phase (e.g., martensite) should be tailored to surround the softer grains with an approximately uniform layer that is strong enough to drive up the stress, but with enough weak spots that can lead to damage relatively easily, resulting in a high dispersion of damage locations, each activating a diffuse deformation zone, and thereby effectively retarding global localization. This mechanism may be the underlying reason for the recent success of nano-grain dual-phase steels [9]. The diffuse deformation and resulting strain hardening in the ferrite grains adjacent to the voids may also explain earlier observations that for DP steels the classical mechanism of ductile failure through void initiation, growth, and coalescence only becomes relevant close to the moment of final failure, i.e., after global localization has set in [46].