An integrated experimental-numerical study of martensite/ferrite interface damage initiation in dual-phase steels

Martensite

Despite a variety of attempts to optimize the mechanical properties, the full potential of DP steels remains limited by the well-known strength/ductility trade-off [1].In particular, the underlying microscale mechanisms that cause failure and hence control ductility, are not yet fully understood.Extensive experimental observations have shown that M/F interface damage is relevant to failure of many DP steels, especially those with low (10%) to moderate (50%) martensite volume fractions (e.g.[2][3][4]).Commonly, M/F interface damage is expected to originate from the large M/F phase contrast since martensite is regarded as a hard constituent compared to ferrite (e.g.[5][6][7][8]).This understanding, however, is contradicted by recent experimental observations (e.g.DP600 [9], DP800 [10,11] and DP1000 [12]), showing interface damage accompanied by apparent martensite plasticity.
Recent crystal plasticity simulations [39] indicated that the substructure boundary sliding might also trigger and dominate M/F interface damage initiation upon the occurrence of apparent martensite plasticity.This sliding-triggered interface damage mechanism has been supported by experimental observations [40].In this work, an integrated experimental-numerical study is conducted to examine this hypothesis [39] and gain further insights for the interface damage initiation.The interface damage initiation is predicted numerically by applying the recently developed multi-scale framework [41]   Experimental specimens were prepared from a heat-treated DP600 (Fe-0.092C-1.68Mn-0.57Cr-0.24Siwt.%).The as-received steel sheet was first annealed for 10 minutes at 1100 • C and then intercritically annealed for 30 minutes at 750 • C, followed by water quenching to room temperature.The resulting DP steel microstructure revealed a martensite volume fraction ∼ 60% and martensite island sizes 50−100 μm.After the heat treatment, specimens with dimensions 30 mm × 10 mm × 1 mm were cut, followed by mechanical grinding and polishing with colloidal silica for 30 minutes.
Hemming tests were carried out on the prepared specimens.A microstructure area, Fig. 1(a), with an overall stress state close to uniaxial tension [42], was tracked.One region containing a large martensite island with a clearly visible substructure, Fig. 1(b), is focused.The crystallographic orientations of the individual grains, Fig. 1(c), were mapped with Electron BackScatter Diffraction (EBSD) employing spherical indexing [43].A misorientation threshold of 2.5 • was chosen to identify individual martensite sub-blocks.Based on the martensite crystallography data and considering the Kurdjumov-Sachs (KS) orientation relationship, prior austenite grains (PAG) as well as martensite packets and (sub-)blocks were reconstructed using the PAG reconstruction toolbox in MTEX [44], see [38,45] for more details.The resulting PAG crystallographic orientation map and KS packets (or variant groups, according to the variant list in [14]), together with the in-plane projections of the habit plane trace, are shown in Figs.1(d) and (e), respectively.As expected, the clearly visible lath boundaries are approximately parallel to the corresponding habit plane traces.A habit plane sliding factor is defined as  hp = cos  hp sin  hp , with  hp = arccos( ⃗  hp ⋅ ⃗   ), the angle between the habit plane normal ⃗  hp and the local loading direction ⃗   , indicating how favourable the sliding is within each packet (see also [37,38]).The resulting habit plane sliding factor map is shown in Fig. 1(f).
Numerical simulation of the region of interest (Fig. 1(b)) was conducted using a multi-scale framework [41].Two scales are considered: mesoscale representing the DP steel structure consisting martensite islands in a ferrite matrix, and microscale resolving the martensite substructure consisting of laths and RA films.The mesostructure is assumed to be invariable in the out-of-plane direction ( axis).The inplane mesostructure model is constructed from Fig. 1(b) and shown in Fig. 2(a).To mitigate potential artificial boundary effects [46], a ferrite buffer layer with a width of 5 μm has been added accordingly.It has been verified that the simulation results are insensitive to the choice of the buffer layer width.Within each martensite packet, a RA volume fraction of 10% 1 is assumed (see also [41]), i.e. ∼ 5% over the whole DP steel.The crystallographic orientations of the BCC laths and RA films are assigned according to Figs. 1(c) and (d), respectively.For the ferrite matrix, the crystallographic orientation effects are neglected, and an isotropic elasto-(visco-)plasticity model is used [41].For the martensite islands, a reduced lath martensite model (Fig. 2(b)) is adopted.The martensite packet is described by a lamella model, which represents a stack of laths and RA films [28].The FCC RA films are described by a reduced crystal plasticity model [47], where only the three slip systems parallel to the habit plane are incorporated, while the plastic deformation in other directions is modelled by von Mises (visco-)plasticity.The relatively hard martensite laths are represented using isotropic elasto-(visco-)plasticity. For the damage initiation prediction at the M/F interfaces (with normal ⃗  M/F ), a microphysics-based interface damage indicator model (Fig. 2(c)) [41] is employed.This model relies on the

Table 1
Isotropic plasticity model parameters of the ferrite matrix and martensite lath.The model parameters of the RA film are the same as used in [41] and thus not listed here.hypothesis [39], that the substructure boundary sliding dominates the interface damage initiation by inducing large plastic strain concentrations in the near-interface ferrite around the RA film tips, acting as damage 'hot spot' locations (yellow dots in Fig. 2(c)).The effective interface damage indicator is a function of the effective sliding representing the overall martensite shear deformation along the habit plane, and depends on the relative orientation of the interface with respect to the habit plane.
The parameters of the ferrite model have been identified by fitting the experimental stress-strain data in [48] and listed in Table 1.The parameters of the reduced lath martensite model used in [41] were based on a martensite carbon content ∼ 0.4 wt.% C. In the tested grade, the martensite carbon content is estimated to be ∼ 0.13 wt.% C.This entails a substantial decrease of the martensite hardness [49], whereas the austenite hardness is known to be much less sensitive to the carbon content [50].Therefore, the material parameters of the martensite laths have been adjusted accordingly in Table 1.The parameters of the M/F interface damage indicator model are taken from [41].
The region of interest including the ferrite buffer layer, is discretized using 3D linear finite elements with one element in the out-of-plane direction and an average in-plane element size ∼ 1 μm.The resulting discretization is shown in Fig. 2(d).Periodic boundary conditions are enforced in all three spatial directions.In-plane uniaxial tension along the positive  axis is applied.For the out-of-plane direction, different conditions including overall plane stress, overall plane strain and (point-wise) plane strain conditions, have been examined, revealing only minor influences on the simulation results.Only the results for the plane strain condition will therefore be presented.
The (accumulated) equivalent plastic strain maps of the region of interest predicted by the numerical simulation, are reported in Fig. 3(a).Highly heterogeneous plastic deformations are found in both ferrite and martensite phases, with comparable local strain levels.To rational-ize these observations, the effective sliding of each martensite packet is computed in Fig. 3(b) (greyscale colorbar).The locations in the martensite islands that present a large apparent plasticity (Fig. 3(a)) correspond with the packets that have a large sliding factor (Fig. 1(f)) revealing a pronounced sliding activity (Fig. 3(b)).This shows that the substructure boundary sliding is indeed the favourable deformation mode for the martensite islands.To predict the M/F interface damage initiation, the effective interface damage indicators are evaluated and plotted at the corresponding interface locations in Fig. 3(b) (cyan-to-magenta colorbar).Multiple interface damage initiation sites can be observed near the martensite packets with a pronounced sliding activity.Note, however, that a pronounced sliding activity does not necessarily imply a high interface damage indicator level, since the latter depends strongly on the relative orientation of the interface with respect to the habit plane (see also [41]).
Next, the in-situ SEM micrographs, Fig. 3(c), are considered.Comparing Figs.3(b) and (c) demonstrates an adequate qualitative agreement between the martensite island sliding activities predicted by the numerical simulation and those identified in the experiments.Extensive slip traces can be identified along the habit planes (Fig. 1(e)) of the martensite packets with a high sliding factor (Fig. 1(f)), experimentally demonstrating the sliding occurrence (see also [37]).Pronounced interface damage activities are visible in Fig. 3(c).Three mesoscopic M/F interfacial zones (boxes in Fig. 3(b)) with a high damage indicator level, are investigated in more detail.Tracking the corresponding regions in Fig. 3(c) together with the magnifications in Fig. 3(d), reveals that the laths slide along the substructure boundaries (black arrows) and gradually penetrate into the near-interface ferrite matrix, inducing strain localization (yellow arrows).This confirms that the interface damage initiation is indeed triggered by the substructure boundary sliding (see also [40]).As the loading proceeds, more sliding-triggered interface damage initiates, potentially leading to eventual interface separation (e.g. the green box).Furthermore, the red and green regions tend to initiate the interface damage earlier than the blue region.All these experimental observations are well predicted by the numerical simulation (Fig. 3(b)).Without incorporating the sliding mechanism, i.e. the RA films, the numerical simulation cannot reproduce the experimental observations of martensite plasticity and interface damage, see Supplementary material 1 for more details.Finally, notice that the interface damage can also develop due to severe localized ferrite plasticity alone, which, however, is only observed in very few regions (Fig. 3(c)).
The sliding activity in two martensite regions (orange triangles) is however underestimated by the numerical simulation, despite their high sliding favourability (Fig. 1(f)).As suggested in [37], this may originate from the modelling assumption of an invariable (extruded) out-of-plane mesostructure in combination with the out-of-plane periodic boundary conditions.In reality, the subsurface mesostructure is unlikely to be constant through the out-of-plane direction.Moreover, Fig. 3(c) shows that the substructure boundary sliding occurrence is highly discrete and carried by a few substructure boundaries only [31,32,51].This may be attributed to the presence of nano-precipitates or irregular lath morphology at some substructure boundaries [34], potentially obstructing the sliding activation.The substructure boundary sliding discreteness has not been included in the model.This could be incorporated in the future through a discrete slip plane model framework [52].
To enable a more quantitative comparison, the experimentally observed M/F interface damage initiation is further quantified based on the SEM micrographs.To this end, it is assumed that a higher image in-tensity contrast (i.e.greyscale value contrast) represents a higher strain localization [53] and thus a higher probability of damage initiation.At each sampled interface pixel, a circular interfacial zone with a radius  IZ is defined and all greyscale values  belonging to the included ferrite region are collected into a set { F }. Two subsets, {  The M/F interface damage indicator contours predicted by the numerical simulation and quantified from the experiments are compared in Fig. 4(a).Despite a minor mismatch between the numerical and experimental deformed configurations, the damage indicator levels at most interface locations agree adequately.Since only the slidingtriggered interface damage initiation has been assumed in the model, this good agreement implies that the M/F interface damage initiation is indeed dominated by the substructure boundary sliding.
To gain additional insights on the correlation between the M/F interface damage initiation and the deformation of the bulk phases, the predicted interface damage indicators at all interface locations are plotted in Fig. 4(b) against the near-interface ferrite and martensite equivalent strains.Clearly, the M/F interface damage initiation seems to be accompanied by a low M/F strain partitioning, i.e. along the diagonal where the martensite and ferrite strains are comparable (red ellipse).This is in sharp contrast to most literature where the M/F interface damage occurrence has been presumed to be due to a high M/F strain partitioning considering the hard nature of martensite (e.g.[5][6][7][8]).Furthermore, it has been recently demonstrated that martensite can behave in a soft and ductile manner (see e.g.martensitic steels [31][32][33] and DP steels [34][35][36][37]).The intrinsic correlation found here between the M/F interface damage initiation and the low M/F strain partitioning (also supported by recent experimental observations [9][10][11][12]) emphasizes that the physical origin behind martensite ductility, i.e. substructure boundary sliding, induces a highly localized plasticity, and often damage, in the near-interface ferrite.This new understanding can be instrumental for fine tuning the microstructure to further optimize the well-known strength/ductility trade-off of DP steels.
To summarize, an integrated experimental-numerical study has been carried out to investigate M/F interface damage initiation, which is an importance failure mechanism in many DP steels that exhibit apparent martensite plasticity.The main findings include: (i) the substructure boundary sliding, which acts as the favourable deformation mode for the martensite islands, triggers and dominates M/F interface damage initiation accompanied by apparent martensite plasticity, as originally hypothesized in [39]; (ii) as a consequence, the M/F interface damage initiation correlates with a low M/F strain partitioning, rather than with pronounced M/F strain partitioning as commonly presumed in literature; (iii) the M/F interface damage indicator model developed in [41] based on the sliding-triggered interface damage initiation mechanism, enables efficient and quantitative predictions of the M/F interface damage initiation sites in DP steels.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Importance of the substructure boundary sliding mechanism
This section investigates the influence of disregarding the substructure boundary sliding mechanism in the modelling approach employed in the article.
In Figure S1, the equivalent plastic strain maps for the DP steel region of interest predicted by the numerical simulations with and without the substructure boundary sliding mechanism, i.e. the RA films, and the corresponding SEM micrograph, are compared.As expected, the simulation without the sliding mechanism (see Figure S1(b)) predicts much less apparent martensite plasticity with, instead, more ferrite plasticity, compared to the simulation with the sliding mechanism included (see Figure S1(a)).Furthermore, comparing Figures S1(b) and (c) shows that with the sliding mechanism disregarded, the simulation cannot reproduce the martensite plasticity that is clearly visible in the experiments.These results evidence the importance of incorporating the substructure boundary sliding mechanism (even with a low RA volume fraction) into the modelling for the adequate prediction of the martensite island behaviour (see also [1,2]), which is crucial to determine the DP steel microstructure response as well as the M/F interface damage initiation.

M/F interface damage initiation quantification based on the SEM micrographs
This section details the quantification method of the experimentally observed M/F interface damage initiation employed in the article.

1
A SEM micrograph of the DP steel region of interest after deformation is shown in Figure S2(a).Several ferrite regions in the proximity of the M/F interfaces exhibit high image intensity contrast, indicating high deformation localization and thus potential damage occurrence at the M/F interfaces (see also [3]).For example, based on visual inspection, the red region is expected to exhibit damage while the blue region is not, Figure S2(b).Accordingly, the image intensity contrast may be considered as a suitable measure for the damage initiation sites observed in experiments [4].Let ⃗ x denote the location vector of an arbitrary pixel in an image, with the corresponding image intensity (i.e.greyscale value in case of SEM micrograph), indicated by p.By tracking the M/F interfaces, a set of interface pixels can be sampled (indicated by the yellow dots in Figure S2(a)).The locations of these pixels are collected in {⃗ x M/F }.For the i-th location ⃗ x i M/F , a near-interface ferrite region {⃗ x i F } is selected according to the following conditions (indicated by the shadowed regions in Figure S2(b)): where R IZ is the radius parameter of the defined interfacial zone and ⃗ , with the interface tangent vector ⃗ T i M/F that can be approximately determined using two M/F interface locations next to To quantify the greyscale value contrast of {p i F }, two subsets of {p i F } are extracted: where p H and p L represent the thresholds to classify, respectively, the highest and lowest η% greyscale values, controlled by a percentile parameter η.With {p H,i F } and {p L,i F }, the greyscale value contrast of {p i F } is defined as the absolute difference between their mean values: with ( •) denoting the mean operator.After standard min-max normalization to the range 0−1.0,C i IZ is used to quantify the experimentally observed M/F interface damage initiation sites.
In principle, C i IZ measures the maximum contrast over the greyscale values belonging to {p i F } collected from a finite ferrite region around ⃗ x i M/F .The radius parameter R IZ affects the resolution of the near-interface ferrite deformation localization and the influence of the far-interface ferrite region.The percentile parameter η together with the mean operator affects noise suppression and contrast smoothing.As a result, a trade-off always exists for choosing suitable values of R IZ and η.
The particular choice of the parameters R IZ and η may influence the M/F interface damage initiation quantification from the experimental micrographs, and therefore their influence is examined in Figure S3.It is observed that different parameter choices lead to similar M/F interface damage indicator patterns, demonstrating the robustness of the introduced quantification method.

Fig. 1 .
Fig. 1.(a) SEM micrograph of the DP steel microstructure area before deformation, showing several martensite islands (martensite/austenite laminates, M/A) embedded in a ferrite matrix; (b) magnification of the region of interest (blue box); (c) EBSD crystallographic orientation map of the martensite and ferrite grains.Black lines indicate the M/F interfaces, as determined from (b); (d) the PAG crystallographic orientation map; (e) the Kurdjumov-Sachs (KS) packets separated by the black lines.The in-plane projections of the habit plane traces within individual packets are indicated by the pink lines; (f) the so-called habit plane sliding factor map, computed with respect to the horizontal loading direction ⃗   .(For interpretation of the colours in the figure(s), the reader is referred to the web version of this article.)

Fig. 2 .
Fig. 2. (a-c) The multi-scale framework adopted to predict the M/F interface damage initiation sites in the DP steel region of interest shown in Fig. 1(b).The mesoscopic M/F interfaces and KS packet boundaries are indicated by the yellow and white lines, respectively.The ferrite buffer layer around the DP steel mesoscale model is indicated by the blue shadow.The underlying martensite substructure which consists of a stack of martensite laths (ML) and RA films, is not spatially resolved at the mesoscale, but accounted for through the microphysics-based constitutive relations: (b) the reduced lath martensite model for the interior of martensite island and (c) the interface damage indicator model for the M/F interfacial zone.The sliding-triggered damage hot spot locations in the M/F interface microstructure are indicated by the yellow dots; (d) the adopted in-plane finite element discretization.

Fig. 3 .
Fig. 3. Comparison between the numerical and experimental results for the DP steel region of interest at different levels of uniaxial tensile strain ⟨⟩  : (a) the equivalent plastic strain maps; (b) the martensite island sliding maps (greyscale colorbar)) and the M/F interface damage indicators (cyan-to-magenta colorbar); (c) the in-situ SEM micrographs; (d) magnifications of three selected mesoscopic M/F interfacial zones (boxes), revealing the interface damage initiation (yellow arrows) triggered by the substructure boundary sliding (black arrows).Due to convergence limitations, the numerical simulation did not reach the final deformation stage (⟨⟩  = 34%) as imposed in the experimental test.The two martensite regions with a sliding activity that is underestimated by the numerical simulation are indicated by the orange triangles in (b) and (c).
H F } and { L F }, representing the highest and lowest % greyscale values over the whole { F }, are extracted.The contrast is then quantified using the absolute difference between the mean values of { H F } and { L F } as  IZ = | p H F − p L F |.After standard min-max normalization to the range 0−1.0,  IZ serves to quantify the M/F interface damage initiation in the experiments.The two parameters  IZ and  are chosen as  IZ = 2 μm and  = 25.It has been verified that different parameter choices lead to similar quantitative results for the interface damage initiation.More details can be found in Supplementary material 2.

Fig. 4 .
Fig. 4. (a) Comparison between the M/F interface damage indicator contours for the considered region of interest obtained by the numerical simulation (SIM) and the experimental analysis (EXP) at a uniaxial tensile strain of ⟨⟩  = 7.4%.The martensite islands and ferrite matrix are not displayed for clarity; (b) the predicted M/F interface damage indicator versus the near-interface ferrite and martensite equivalent strains.For reference, the black dashed line represents  eq M/A =  eq F .The region with a high damage indicator level is indicated by the red ellipse.

Figure S1 :
Figure S1: Comparison between the numerical and experimental results for the DP steel region of interest at a uniaxial tensile strain of ⟨ϵ⟩xx = 7.4%: the equivalent plastic strain maps predicted (a) with and (b) without the substructure boundary sliding mechanism; (c) the SEM micrograph.

Figure
Figure S2: (a) SEM micrograph of the DP steel region of interest at a uniaxial tensile strain of ⟨ϵ⟩xx = 7.4%.The sampled M/F interface pixels are indicated by the yellow dots; (b) magnifications of two selected M/F interfacial zones, with the shadow regions identified as ferrite.The M/F interface and its local normal are indicated by the dashed line and arrow, respectively; (c) the corresponding PDF of the greyscale values for the two ferrite regions shown in (b).

)
All greyscale values belonging to the determined ferrite region {⃗ x i F } are collected into a set {p i F }.The probability density functions (PDF) of {p A F } and {p B F } (see shadowed regions in Figure S2(b)) are shown in Figure S2(c) as examples.Here, the greyscale values have been scaled to the range 0−1.0.A broad distribution of {p A F } can be observed, while the distribution of {p B F } is rather narrow, implying high and low greyscale value contrasts, respectively, consistent with the visual observations from Figure S2(b).
on the experimental mesoscale morphology and crystallography.The predicted damage initiation sites are compared against those observed in the experiments.