A multiscale approach to modeling formability of dual ...

Modelling and Simulation in Materials Science and Engineering Modelling Simul. Mater. Sci. Eng. 24 (2016) 025011 (30pp)

doi:10.1088/0965-0393/24/2/025011

A multiscale approach to modeling formability of dual-phase steels A Srivastava1,5, A F Bower1, L G Hector Jr2, J E Carsley2, L Zhang3 and F Abu-Farha4 1

School of Engineering, Brown University, Providence, RI 02912, USA General Motors Research & Development, Warren, MI 48090, USA 3 Department of Mechanical Engineering, Tongji University, Shanghai, China 4 Department of Automotive Engineering, Clemson University, Greenville, S.C. 29607, USA 2

E-mail: [email protected] and [email protected] Received 16 August 2015, revised 11 December 2015 Accepted for publication 28 December 2015 Published 19 January 2016 Abstract

A multiscale modeling approach is used to predict how the formability of dual-phase (DP) steels depend on the properties of their constituent phases and microstructure. First, the flow behavior of the steels is predicted using microstructure-based finite element simulations of their 3D representative volume elements, wherein the two phases (ferrite and martensite) are discretely modeled using crystal plasticity constitutive models. These results are then used to calibrate homogenized constitutive models which are then used in large-scale finite element simulations to compute the forming limit diagrams (FLDs). The multiscale approach is validated by predicting the FLDs of two commercial DP steels and comparing the predictions with experimental measurements. Subsequently, the approach is used to compute flow behavior and FLDs of a series of ‘virtual’ DP steels, constructed by varying the microstructural parameters in the commercial DP steels. The results of these computations suggest that combining the ferrite from one of the two commercial steels with the martensite of the other and optimizing the phase volume fractions can yield ‘virtual’ steels with substantially improved properties. These include a material with an FLD0 (plane strain) that exceeds those of the commercial steels by 75% without a degradation in strength; and a material with a flow strength (0.2% offset) that exceeds those of the commercial steels by ~30% without degradation of formability.

5 Present address: Department of Materials Science and Engineering, Texas A&M University, College Station, TX 77843, USA.

0965-0393/16/025011+30$33.00 © 2016 IOP Publishing Ltd Printed in the UK

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Modelling Simul. Mater. Sci. Eng. 24 (2016) 025011

Keywords: advanced high strength steel, microstructure design, formability, crystal plasticity, constitutive modeling, finite element modeling, digital image correlation (DIC) (Some figures may appear in colour only in the online journal) 1. Introduction The microstructures of Advanced High Strength Steels (AHSS), such as dual-phase, transformation-induced plasticity, and twinning induced plasticity are multiphase with various combinations of ferrite, bainite, austenite, and martensite. These phase combinations, together with alloying constituents such as Si, Mn, Nb, Al, and the sophisticated thermomechanical processing treatments to which these materials are subject result in properties that are favorable for a variety of technological applications. For example, their strength and ductility (e.g. yield and tensile strengths in excess of 300 MPa and 600 MPa, respectively [1]) relative to conventional steel grades [1, 2] make them especially attractive for automotive applications where vehicle weight reduction and crashworthiness are of paramount importance [3]. Of the various AHSS, dual-phase steels, introduced in the mid-1970s [4], have experienced the fastest growth in the automotive industry [5]. These are produced on continuous annealing lines that allow intercritical heating into the ferrite-austenite field followed by rapid cooling to cause austenite to transform to martensite [4, 6, 7]. In the fully heat-treated condition, the microstructure of dual-phase steels primarily consists of hard martensite islands dispersed in a soft ferrite matrix, but sometimes the microstructure can also include traces of other phases such as retained austenite [8]. Hence, the deformation behavior of dual-phase steels, especially those with increased martensite content and tensile strength of order 1GPa, is quite complex. These advanced high strength dual-phase steels exhibit varying strain-hardening capabilities in the low, intermediate and high strain regions [7, 9, 10]. Nevertheless, dual-phase steels provide adequate compromises on strength and formability that are attractive for automotive applications [4, 5]. The strength and formability of dual-phase steels are determined partly by the properties of the phases themselves, and partly by microstructural features, such as the volume fraction of the phases, their sizes and morphologies. Numerous experimental efforts have aimed to correlate the strength and volume fraction of the constituent phases with overall strength and formability of dual-phase steels [4, 7, 11–18]. In general, these experiments suggest that the strength of dual-phase steels increases with an increase in the volume fraction and/or strength of the martensite phase; however, an increase in dual-phase steel strength tends to have a deleterious effect on formability, typically characterized by a forming limit diagram (FLD) [19, 20]. The FLD is measured by deforming a sheet metal under in-plane biaxial straining. The values of the critical in-plane strains at which the sheet fails (by through-thickness necking or fracture) are then recorded, as a function of the ratio of two strains [21]. For many dual-phase steel grades, diffuse necking generally precedes fracture, and fracture occurs shortly after strain localization. Under these conditions, formability is controlled by the resistance of the sheet to necking, which depends on both the geometry of the sheet (in particular, on variations in its thickness), as well as the flow behavior of the material. In principle, the properties of the constituent phases and the microstructural features of dualphase steels can be tuned and optimized to achieve a particular performance matrix. But in the absence of any knowledge of optimum properties of the constituent phases and the microstructural features, conventional material design that relies on extensive trial-and-error experimentation may be quite time consuming. Hence, the ability to predict the formability of dual-phase steels using microstructure-based multiscale simulations with limited experimental inputs is 2

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intriguing and offers the potential to reduce material development time. In fact, such a capability will address the three primary elements of the ‘materials by design’ approach [22, 23], namely: structure, properties and performance. However, computer-aided materials design or integrated computational materials engineering is particularly challenging because of uncertainties in experiments, models and model parameters [24–26]. With these ideas and challenges in mind, we address the following three questions in this paper: (i) If the properties of the constituent phases and microstructural features of dual-phase steels are known, is it possible to predict their FLDs by means of multiscale simulations with only minimal experimental inputs? (ii) How do changes in the microstructural features, such as volume fractions of the phases influence the FLD of the existing dual-phase steels? (iii) Is it possible to computationally design a new dual-phase steel microstructure using information from existing (or parent) steels that has better (predicted) formability (relative to the parent steels) without degrading strength, or which has improved strength without a corresponding degradation in formability? To answer the first question, we have selected two commercial low-carbon dual-phase sheet steels, DP980 and DF140T (the ‘DF’ designation stands for ‘DiForm’), as the parent steels. DP980 and DF140T are particularly well suited for our purposes as the properties of their constituent phases, ferrite and martensite, were determined experimentally via micropillar compression tests in Chen et al [10]. Both steels exhibit a tensile strength of the order of 1GPa; however, they have significantly different microstructures [10]. We predict the FLDs of both steels using multiscale finite element simulations and validate the predictions by comparing them with experimental measurements. Having fully validated the multiscale simulation approach, we create ‘virtual’ dual-phase steels by changing the volume fraction of the constituent phases in the two parent steels. We then compute the macroscopic flow response and FLDs of all the ‘virtual’ steels. This allows us to analyze the influence of microstructure on the formability of the parent steels. Following this, we then create new ‘virtual’ dual-phase steels by taking advantage of the best ferrite phase from one of the parent steels and the best martensite phase from the other. The macroscopic flow response and FLD of the optimized ‘virtual’ steels are then computed and compared with those of the parent steels. Our results suggest two possible scenarios for the optimized steels: a dual-phase steel microstructure with an FLD0 (plane strain) that exceeds those of the parent steels by 75% without a degradation in strength; and a dual-phase steel with a yield strength (0.2% offset) that exceeds those of the parent steels by approximately 30% without significant changes to the FLDs of either of the parent steels. The remainder of this paper is organized as follows. In section 2, we outline our multiscale approach to predict FLDs for the dual-phase steels. In section 3, we describe the experimental procedures and results, viz.: a brief review of microstructural characterization, micropillar compression tests, and macroscopic tensile tests from [10]; surface roughness measurements to estimate initial thickness variations in the parent sheet steels; and measurement of FLDs of the two steels. The multiscale simulation procedure and its validation are described in section 4. The effect of the microstructure in controlling the formability of the parent steels, as suggested by the computed macroscopic flow response and FLDs of the ‘virtual’ steels, is discussed in section 5. In section 6, we present the optimized ‘virtual’ dual-phase steels and compare their macroscopic flow response and FLDs with the parent steels. The conclusions of this work are summarized in section 7. 2. Multiscale approach The multiscale approach for predicting FLDs for dual-phase steels, together with the experiments used to validate the predictions, is illustrated in detail in figure 1. The procedure involves the following steps: 3

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Figure 1. Schematic of the multiscale approach to predict forming limit diagram of dual-phase steel and its validation.

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1. The dual-phase steel microstructure is characterized. Specifically, the volume fractions of the constituent phases are measured, and the shape, size and orientation of the ferrite grains and microstructural features of the martensite particles are characterized qualitatively. 2. To determine the mechanical behavior of the individual phases of the steels, microscale compression specimens are extracted from each phase using focused ion-beam milling. Each pillar is then deformed under compression using a flat-punch nanoindenter. These experiments provide flow (stress-strain) curves for single crystal specimens of ferrite with several known crystallographic orientations, and for martensite specimens consisting of single or multiple blocks. 3. The respective crystal plasticity constitutive models for ferrite and martensite phases are calibrated, by fitting the predictions of finite element simulations of micropillar compression to the measurements in step 2. 4. A 3D representative volume element (RVE) of the steel microstructure is then generated, and its uniaxial stress-strain response under tension is computed. The prediction at this point is validated by comparing the result with macroscale uniaxial tensile test data (steps I-II). 5. The results obtained in step 4 are then used to calibrate a homogenized constitutive model that captures the various stages of strain-hardening in the dual-phase steel. 6. The surface roughness of the as-received sheet steel is measured using a 3D digital image correlation technique (DIC). The roughness measurements are used to estimate the initial thickness variations in the as-received sheet steel as required for the FLD predictions. 7. The FLD for the dual-phase steel of interest is then predicted using the MarciniakKuczynski (M-K) analysis [27]. In this approach, finite element simulations are used to predict the critical strains required to initiate necking (modeled using the calibrated constitutive model from step 5) due to initial thickness variations in the sheet steel (characterized in step 6) that is subjected to proportional biaxial straining. The predicted FLD at this point is validated by comparing it with experimentally determined FLD, as indicated in steps III-IV. Steps 1 through 4 are detailed in [10]. In this previous study, micropillar compression specimens were extracted from both the ferrite and martensite phases of the DP980 and DF140T (parent) steels, and their flow (stress-strain) curves under compression were measured. These measurements provided flow curves for single crystal specimens of ferrite with known initial orientations. For martensite, the measurements provided flow stresses for micropillars containing single martensite blocks, as well as multiple martensite blocks. These measurements were used to calibrate the crystal plasticity constitutive models of both the ferrite and martensite phases, which then allowed the macroscopic behavior of 3D RVEs of the two steel microstructures to be computed. The computed flow behavior under uniaxial tension of the computational RVEs of the parent steels was shown to be in very good agreement with their measured (macroscopic) uniaxial tensile behavior. In the present study, we first focus on steps 5 through 7 of the approach illustrated in figure 1. The results of microstructure-based crystal plasticity finite element simulations of the parent steels are used to calibrate a homogenized constitutive model suitable for use in large scale finite element simulations to predict the FLD (a similar two scale simulation approach has been previously used to analyze single phase polycrystalline aggregates, see e.g. [28], as well as two phase ferrite-pearlite steels, see e.g. [29]). The homogenized constitutive model accurately captures the varying strain-hardening capabilities of the parent steels. The surface roughness of the as-received sheet steels was measured using 3D DIC. The FLD for the parent steels was then predicted using a computational approach to the M-K analysis. The predicted FLDs 5

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Table 1. Nominal chemical compositions (in wt.%) of the DP980 and DF140T steels.

Dual-phase steel

C

Mn

Si

DP980 DF140T

0.09 0.15

2.15 1.45

0.60 0.30

for both the dual-phase steels were validated using FLDs determined experimentally according to the Marciniak [27] and Nakazima [30] tests. In the experiments, the DIC technique is used to accurately measure the strains at the onset of necking during the FLD measurements. The predicted FLDs are shown to be in good agreement with those from the experiments. After validating the multiscale simulation approach for the parent steels (step II and IV), we then explore the influence of their microstructures on their formability. This involves the following steps: creation of 3D RVEs of ‘virtual’ dual-phase steel microstructures by changing the volume fraction of the constituent phases in the two parent steels and computing their tensile flow behavior (step 4); calibration of homogenized constitutive models (step 5); and FLD predictions (step 7). Based on the lessons learned from this exercise we then create optimized ‘virtual’ dual-phase steels and compute their macroscopic flow response and FLD by repeating steps 4, 5 and 7, and compare the predictions with the tensile flow responses and FLDs of the parent steels. 3. Experiments In this section, we describe the parent dual-phase steels, DP980 and DF140T, the experiments that provided the required inputs for the multiscale simulations, and the experiments used to validate the simulation results. The experiments in figure 1, viz: (i) microstructural characterization (step 1); micropillar compression tests (step 2); and (ii) surface roughness measurements (step 6), provide the necessary data to generate computational 3D RVEs of the parent steels, calibrate the crystal plasticity constitutive models for the individual phases present in the parent steels, and estimate the initial thickness variation in the parent steels, respectively. As shown in figure 1, uniaxial tensile tests (step I) and experimental measurements of the FLDs (step III) are required to validate the predictions of the multiscale simulations. The results of microstructural characterization, micropillar compression tests as well as uniaxial tensile tests for the parent steels are reported in [10]. For the sake of completeness, the relevant results from [10] are briefly summarized in section 3.1. The procedure used to measure the surface roughness of the parent steels to estimate the initial thickness variations is detailed in section 3.2 while the experimental measurement of FLDs for the parent steels are detailed in section 3.3. 3.1. Material description and characterization

The parent DP980 and DF140T dual-phase steels are commercial, low-carbon steels, each with a 1.4 mm nominal thickness. Their chemical compositions are detailed in table 1. Both steels were produced on a commercial water-quenched continuous annealing line. They have significantly different microstructures, as shown in figure 2(a), but very similar macroscopic tensile flow responses, figure 2(b). The microstructures of both steels consist of martensite islands/particles dispersed in a matrix of ferrite grains (figure 2(a)). The martensite particles have a hierarchical microstructure consisting of laths, blocks and packets (shown schematically in figure 1, further details on the hierarchical microstructure of martensite can be 6

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Figure 2. (a) Secondary electron SEM images showing the microstructures of the parent DP980 (left) and DF140T (right) sheet steels which are composed of ferrite (F) and martensite (M) phases [10]. (b) Uniaxial tensile flow (nominal stress-strain) curves for the parent steels with loading axis aligned parallel to the rolling direction [10].

found in [10, 31]). The martensite volume fraction in DP980 is ~60%, whereas the martensite volume fraction in DF140T is ~40%. The distribution of martensite particles in both steels is fairly homogeneous. The variation in the area fractions of the martensite phase measured from multiple secondary electron SEM images was found to be within 2–3% [10]. As shown in figure 2(b), the ultimate tensile strength (UTS) of both steels is ~1.0GPa, and final fracture of the tensile specimens with loading axis aligned parallel to the rolling direction occurs beyond the UTS at a plastic strain of ~14% and ~15.5% for DF140T and DP980, respectively. In [10], tensile specimens with loading axis aligned at 45° and 90° to the rolling direction were also tested in uniaxial tension and in-plane anisotropy was found to be insignificant in both the steels (at least up to the UTS). In addition, EBSD analysis reported in [10] confirmed that the parent sheet steels do not have pronounced texture. The flow behavior of the ferrite and martensite phases in the parent steels was also characterized in [10]. Cylindrical specimens with diameters ranging from 0.3 μm–2 μm and a height/ diameter ratio of the order of 1.5 were extracted from the individual phases using focused ionbeam milling. These specimens were then deformed under compression using a flat punch nanoindenter. A full description of these experiments can also be found in [9, 31, 32]. The data critical to the present work has been reproduced from [10] in figure 3, which shows the measured flow curves of micropillars extracted from the ferrite and martensite phases of both parent steels. These microscale experiments provided compressive stress-strain curves for single crystal specimens of ferrite with known initial orientations, figures 3(a) and (b), and for specimens consisting a single martensite block, as well as multiple martensite blocks, figures 3(c) and (d). These stress-strain curves were then used to determine the parameters of the crystal plasticity constitutive models for the two phases present in the parent steels. The predictions of the calibrated crystal plasticity constitutive models (detailed in section 4.1) are also compared with the experimental measurements in figure 3. 3.2. Measuring and characterizing sheet steel surface roughness

The forming limits of the DP980 and DF140T steels are controlled by the onset of necking instability that precedes crack nucleation, and fracture occurs shortly after strain localization, 7

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Figure 3. Comparison of the measured (Exp) flow curves of micropillars extracted from the individual phases with the predictions of the calibrated crystal plasticity constitutive models of the same (Fit) [10] for the parent dual-phase steels. (a) Ferrite from DP980; (b) Ferrite from DF140T; (c) Martensite from DP980; (d) Martensite from DF140T. The initial orientations of the ferrite micropillars extracted from various ferrite grains are given in the brackets in the legends of (a) and (b).

as shown in detail in the section to follow. The onset of necking instability is in general sensitive to intrinsic geometric and material defects, such as variations in sheet thickness and/or microstructural inhomogeneity (see for example [33–41]). The two parent dual-phase steels considered here have a very homogeneous microstructure as detailed in the previous section, so necking is very likely triggered by initial variations in the sheet thickness. Also, in the computational approach to the M-K analysis, an initial thickness imperfection in the sheet steel is introduced to predict the onset of necking under biaxial straining of the sheet. In general, the initial thickness imperfection in this approach is a fitting parameter. Hence, it is not truly a predictive approach. In order to predict FLDs for materials that neck prior to fracture using a computational approach to the M-K analysis, it is therefore necessary to quantify the variations in the initial sheet thickness. Extreme thickness variations in the sheet steel that are more likely to trigger strain localization can be estimated by measuring the waviness of the surface roughness, and assuming that these surface undulations (with wavelengths comparable to the sheet thickness) on either 8

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side of the sheet are completely out of phase. In this case, the trough of the waviness of the surface roughness on one side of the sheet coincides with the trough of the waviness of the surface roughness on the other side leading to a maximum reduction in the sheet thickness. To this end, we have used a 3D DIC technique to determine the waviness of the initial surface roughness of the parent sheet steels. Prior to measurement, each specimen was cleaned and lightly coated with white spray paint followed by the application of black spray paint droplets. Each droplet was no more that about 33% of the area of the chosen pixel subset for DIC post-processing. The DIC system was calibrated with the VIC Snap software from Correlated Solutions, Inc. (CSI) and several static images were recorded. The statistics of the intrinsic surface roughness of the two sheet steels were extracted from the 3D DIC measurements as follows. The coordinates (both in-plane and out of plane) of equispaced points, separated by a distance of 0.5 mm (comparable to the nominal half thickness, 0.7 mm, of the parent sheet steels) along a straight line were extracted using the Vic3D-7 software from CSI. The surface roughness along this line was then defined as δz = x3 − x3, where x3 is the out of plane coordinate of the point and x3 is the mean out of plane coordinate of the line. The surface roughness δz(x ) was measured along several lines for a given dualphase sheet steel specimen, and measurements were repeated for several sheet steel specimens taken from both parent steels. Typical surface profiles obtained in this way are shown in figure 4(a). The surface roughness profiles of the parent steels at the chosen length scale of 0.5 mm were found to be statistically identical; hence, the results in figure 4(a) do not distinguish between them. Next, for a given surface roughness profile shown in figure 4(a), we estimate the height variation, ∆z, between two points separated by a distance, δx. The value of ∆z is defined as ∆ z (δx ) = z(x + δx ) − z(x ) (1)

In equation (1), the value of δx is varied in integer multiples of 0.5 mm. Since the onset of localization depends on the extreme thickness variations, we confine our attention to the maximum height variations, ∆zmax = max{∆z(δx )}, for a given δx. The variation of ∆zmax with δx for a few representative cases are shown in figure 4(b). For all cases shown in figure 4(b), the value of ∆zmax increases with δx up to δx ≈ 2 − 3mm and thereafter either decreases or saturates to an approximately constant value. The descending cumulative distribution function (CDF) of ∆zmax for a given δx obtained from hundreds of surface roughness line scans from several steel blanks of both parent steels is shown in figure 4(c). Here, the CDF approaches a distribution that is insensitive to δx once δx exceeds 1.5 − 2 mm. Figure 4(d) shows the variation of the maximum, mean and minimum values of ∆zmax for a given δx and is obtained from the CDF in figure 4(c). The maximum value of ∆zmax is approximately 0.03 mm and occurs for δx ≈ 2 − 3 mm. The statistics of ∆zmax and corresponding δx values can now be used to characterize the initial thickness variation in the sheet steel specimens (as detailed in section 4.3) and to carry out FLD prediction using a computational approach to the M-K analysis. 3.3. Forming limit diagram measurement

The formability of both parent steels was assessed by measuring strains at the onset of throughthickness necking in both the Marciniak [27] and Nakazima [30] tests which are common tests to assess formability of sheet metals. Specimens with varying geometries were machined from the sheets by water jet cutting along the rolling and transverse directions to determine the forming limits for several strain paths. The specimen geometries required to cover the strain paths corresponding to both the left and right hand sides of the FLD are detailed in [42, 43]. 9

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Figure 4. (a) The variation of the surface roughness, δz (x ), measured along the lines drawn on the surfaces of sheet steel specimens taken from the parent dual-phase steels. Roughness data from a few representative line scans (different colors represent different line scans) are shown. (b) The maximum height variation, ∆zmax, between two points separated by a distance, δx for a few representative surface roughness profiles shown in (a). (c) The descending cumulative distribution function, CDF, of ∆zmax for δx ranging from 0.5 mm to 3.0 mm obtained from hundreds of surface roughness profiles. (d) The maximum, mean and minimum values of ∆zmax for a given δx obtained from their cumulative distribution function shown in (c).

The FLD specimens were stretched in an Interlaken servo-controlled hydraulic press with a 101.6 mm diameter flat-topped punch having a 10 mm profile radius (Marciniak), and a 101.6 mm diameter hemispherical punch (Nakazima). Teflon sheet (0.05 mm) was utilized as the lubricant between the punch and the carrier blank for the Marciniak tests and between the punch and the test specimen for the Nakazima tests. The carrier blank contains a 33 mm diameter hole that isolates the test specimen from the punch. The blank holding force and punch speed were 600kN and 0.5 mm s−1, respectively. In the Marciniak tests, strain localization and fracture occur at or very close to the pole of the specimen within the hole of the carrier blank. However, strain localization and fracture in the Nakazima tests can occur anywhere between the pole of the specimen and the unsupported region where there is no contact with the punch. Test results from specimens that fractured at locations other than these were discarded. 10

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The 3D DIC system described previously was used to measure strains that developed on one surface of each test specimen. A sequence of images were captured during testing to obtain the deformation history using a flexible capture setting of 2 frames per second (fps) in the first 10 s, 5 fps for the next 20 s and finally at 20 fps up to the fracture event. Post processing with the Vic3D-7 software included a pixel subset of 29 and step size of 7 as recommended by CSI for the speckle pattern used (thin layer of white spray paint with a random pattern of black paint droplets). The first image before the onset of any deformation was used as the reference image to calculate the accumulated strain field. A representative contour plot of true major strain just before fracture near the center of a DP980 Nakazima specimen is shown in figure 5(a). The highly localized major strain (dark red contours in figure 5(a)) near the center of the specimen confirms that diffuse necking acts as the precursor to final fracture and serves to limit ductility. The DIC strain measurements were used to determine the critical strains at the onset of necking. There are several different procedures for constructing a forming limit diagram from DIC measurements: see Wang et al [42] for a review. Here, we have adopted the ‘DICgrid’ method proposed by Zhang et al [44] which is based on the work of Hecker [45]. This approach resolves shortcomings of traditional ‘circular grid’ analysis, such as fracture within the grids and subjective measurement errors. The forming limit data points obtained from raw data with the ‘DIC-grid’ method for the parent dual-phase steels are shown in figure 5(b) (which combines the results of both Marciniak and Nakazima tests). The two steels have similar forming limits, but the forming limit curve (i.e. the curve that separates forming with no necking and forming with necking) of the DP980 steel appears to be slightly higher than that of the DF140T steel in both positive and negative minor strain regions. This is consistent with the slightly higher ductility of the DP980 steel than that of DF140T steel in uniaxial tension as shown in figure 2(b). For both sheet steels, the orientation of the FLD specimens, i.e. parallel or transverse to the rolling direction, does not seem to have a significant effect on formability which is in accord with the absence of in-plane anisotropy at least up to the UTS (see section 3.1). Note also the increase in scatter of the forming limit strains when moving from plane strain into the biaxial tension regime (large positive minor strains) for both dual-phase steels. Examination of the DIC data showed that in these tests the material was subjected to nonproportional strain paths during the forming tests. Nevertheless, the data has been included in the figure for completeness. 4. Multiscale finite element simulations In this section, we describe the multiscale finite element simulation procedure used to model the formability of dual-phase sheet steels. The computations involve three stages (see figure 1). First, microstructure-based crystal plasticity finite element simulations are used to predict macroscopic flow behavior of a computational RVE (step 4). Secondly, the predicted macroscopic flow behavior is used to calibrate a homogenized constitutive model (step 5). Finally, this calibrated homogenized constitutive model is used in finite element simulations to predict the critical strains at the onset of necking under biaxial straining (step 7). We used this procedure to predict the FLDs of the parent DP980 and DF140T steels. Here, the crystal plasticity constitutive models for individual phases required to carry out RVE simulations are calibrated using the results of the micropillar compression tests (step 3 in figure 1). The predictions of the microstructure level simulations are validated by comparing the results with the experimentally measured uniaxial flow behavior (step II). These steps have been discussed in detail in [10]. For the sake of completeness, the microstructure level simulation 11

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Figure 5. (a) Contour plot of true major strain extracted from DIC post-processing

analysis prior to final fracture near the center of a representative FLD specimen of DP980. The result is for a Nakazima specimen deformed with a ratio of minor strain to major strain of 0.78. Red/dark red contours denote necking. (b) Forming limit data pairs of major and minor strain of the parent DP980 and DF140T sheet steels, measured according to both Nakazima (N) and Marciniak (M) tests. The FLD specimens were taken along directions both parallel (RD) and transverse (TD) to the rolling direction.

procedure and relevant results for the parent steels are summarized briefly in section 4.1. In section 4.2, we describe the macroscopic constitutive model used in the FLD simulations, as well as the procedure for calibrating the macroscopic constitutive model using the results of the RVE simulations. Validation of the predicted FLDs of both parent steels with experiments is detailed in section 4.3. 4.1. Microstructure-based finite element simulations

Both the martensite and ferrite phases in the parent steels are idealized using crystal plasticity constitutive models. For the ferrite phase, we use a constitutive model that accounts for departures from Schmid’s law commonly observed in bcc crystals (see [46]). The shearing rate is therefore characterized by a constitutive relation that accounts for non-Schmid effects as formulated in [46] and detailed in [10], in which the plastic flow is assumed to take place by shearing on {1 0 0} 1 1 1 slip systems, with the shearing strain rate ⎛ τ (α) ⎞1/ m (α ) ⋅ ⋅ γ = γ0⎜⎜ (α) ⎟⎟ sign(τ (α)) (2) ⎝ g* ⎠

where τ (α) = m*(α) ⋅ σ ⋅ s*(α) is the resolved shear stress on slip system (α) characterized by unit vectors s*(α) and m*(α) in the deformed crystal, and g (α) is the modified slip system * (α ) , 0), where g(α) represents strength. Note that g (α) is calculated from g (α) = max(g(α) − gNS * * (α ) classical strain-hardening while gNS is a correction to account for non-Schmid behavior. This latter quantity is given by (α ) *(α)σ s*(α) + a2 m*(α)σ(m*(α) × s*(α)) + a3p*(α)σ(p*(α) × s*(α)) g(3) NS = a1p

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where p(α) is a unit vector normal to the {1 1 0} plane in the zone of s(α) that makes an angle of −60 with the reference plane m(α) in the undeformed configuration. The vectors marked with asterisks in equation (3) are in the deformed configuration. All of the relevant vectors in the reference undeformed configuration are given in [47]. The classical strain-hardening is modeled as N

(α )

(β )

˙ = ∑ hαβ γ˙ g(4) β=1

where, following [48], the self-hardening moduli are given as

⎧ ⎫ (α ) ⎞ ⎛ 2 (h 0 − hs )γ h(5) ⎟ + hs⎬G (γ (β ); β ≠ α) αα = ⎨(h 0 − hs )sech ⎜ ⎝ g0 − gs ⎠ ⎩ ⎭ ⎪

⎪

⎪

⎪

⎛ (β ) ⎞ G (γ (β ); β ≠ α) = 1 + ∑ f0 tanh⎜ γ γ ⎟ (6) 0⎠ ⎝ β≠α h(7) β≠α αβ = qhαα ⋅

In equations (2)–(7), γ0, m, a1, a2, a3, h 0, hs, g0, gs, f0, γ0 and q, are constitutive parameters. Martensite is also treated using a crystal plasticity constitutive model. The hierarchical microstructure of martensite is modeled by treating each martensite block as a single crystal in which the laths are represented by an evolving dislocation density (as in [49]). The plastic flow in a block is assumed to take place by shearing on {1 1 0} 1 1 1 and {1 1 2} 1 1 1 slip systems and the shearing rate is characterized using equation (2), but non-Schmid effects are not considered, and hence a1 = a2 = a3 = 0. The strain-hardening in martensite is modeled as N

(α ) g(8) = Gb ∑ aαβ ρ (β ) β=1

where aαβ = m(α)ζ(β ) are constants with ζ(β ) being the line direction of the forest dislocations intersecting the (α)th slip plane. The evolution of dislocation density, ρ (α), on slip system (α), following [50] is given by

⎡ ⎤ N dρ (α) k b (α ) ⎥ (α ) ⎢ 1 (β ) = γ ρ − ρ ˙ (9) ∑ ⎢⎣ bka β ⎥⎦ dt b

where b (Burgers vector) and G (shear modulus) are known constants, and γ˙0, m, ka, kb, together with the initial dislocation density (value of ρ (α) at time t = 0) are five constitutive parameters. The crystal plasticity constitutive models for both the ferrite and martensite phases are implemented as user-material subroutines in the commercial finite element code ABAQUS Standard v6.10 [51]. The constitutive parameters for the crystal plasticity constitutive models of both phases were fit to the micropillar compression data shown in figure 3. The details of the crystal plasticity constitutive model calibration and the resulting constitutive parameters for the parent DP980 and DF140T steels from [10] are reproduced in the Appendix for the sake of completeness. The predictions of the calibrated constitutive models for both phases in the parent steels are compared with the experimental data in figure 3. The calibrated crystal plasticity constitutive models for the ferrite and martensite phases are then used to predict the overall flow behavior of the dual-phase steel. For this a full 3D 13

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computational RVE of the dual-phase microstructure is generated, as described in detail in [10]. The procedure allows the phase volume fractions, orientation distributions, and morph ology to be controlled. In this procedure, first, a 3D ferrite grain structure is generated using a simulated annealing process [52] with brick meshing. Random crystallographic orientations are assigned to the ferrite grains, which generates a random initial texture. Simulations account for the evolution of texture during deformation. Secondly, prior austenite grains (to-be-martensite) are nucleated randomly on the ferrite grain boundaries and then grown at a fast rate along ferrite grain boundaries and a slow rate along other directions to generate a microstructure similar to the experimental micrographs shown in figure 2(a). The growth of prior austenite grains is terminated when the volume fraction reaches the prescribed target value (the volume fraction of martensite in the steel of interest). Note that the prior austenite grain nucleation and growth processes are based on brick meshing; that is, brick elements representing ferrite grains will be converted to represent prior austenite grains. A martensitic microstructure is then generated within each of the prior austenite grains, following the procedure described in detail in [31]. A representative computational RVE of the dual-phase microstructure is shown in figure 6(a). A typical computational RVE contains 125 000 C3D8 brick elements from the ABAQUS/ Standard element library [51]. Depending on the volume fraction of the phases in the RVE, the number of ferrite grains varies from approximately 1500 to 3000 and that of martensite blocks varies from 1000 to 2500. Within each ferrite grain and martensite block the properties are assumed to be homogeneous and the RVE is initially stress free. In the simulations, fully periodic boundary conditions are imposed on all six faces of the RVE and uniaxial tensile deformation is simulated by subjecting the RVE to a prescribed constant nominal strain rate parallel to the x1 axis, while maintaining zero resultant forces on planes perpendicular to the x2 and x3 axes. This allows us to compute the stress and strain distribution in the microstructure as well as to predict uniaxial tensile stress-strain curves for the dual-phase steel microstructures. The predictions for both parent steels are compared with experimental measurements in figure 6(b) and (c). The results of the RVE calculations for the two parent steels, as detailed in [10], suggest that a typical tensile stress-strain curve for dual-phase steels has three characteristic stages. At low stress, the response is elastic. The dual-phase microstructure first reaches yield in the ferrite. Following initial yield, we observe a period of steep strain-hardening. In this regime, the martensite remains elastic while ferrite contributes 100% of the plastic strain. This stage continues until martensite reaches yield. At this point, there is a significant reduction in the strain-hardening rate, and both ferrite and martensite continue to deform plastically. 4.2. Homogenized constitutive model for forming simulations

In theory, it is possible to model the FLD of dual-phase steels using microstructure-based crystal plasticity finite element simulations. However, the mesh density required to discretely model micron-size microstructural features, such as ferrite grains and martensite blocks in a 3D sheet specimen whose dimensions are of the order of tens of millimeters (recall the nominal sheet thickness of the two dual-phase steels alone is 1.4 mm and the other two in-plane dimensions of any sheet specimen are at least ten times that of the sheet thickness) would make finite element simulations prohibitively time consuming. Hence to model the FLD, we use a standard rate dependent, isotropic plasticity model. To provide a link to microstructure in this model, the constitutive parameters are chosen to fit the predicted uniaxial stressstrain curve of the microstructure-based simulations described in the preceding section. This homogenized constitutive model can also be used to simulate component level forming. The viscoplastic constitutive equation relates plastic strain rate, ε˙ijp, to stress by 14

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Figure 6. (a) A representative volume element (RVE) of dual-phase steel. The

different colors of element groups in ferrite mark different grains whereas the different colors of element groups in martensite mark martensite blocks. Next, comparison of uniaxial tensile nominal stress-strain curves of (b) DP980 and (c) DF140T parent dualphase steels obtained from uniaxial tensile tests, RVE simulations and finite element simulations of uniaxial tension using homogenized constitutive model (Fit). Note that necking cannot be predicted by the RVE calculations which (by construction) impose a uniform macroscopic strain on the RVE; hence, results are shown for nominal strains up to 0.1 (or 10%), which roughly corresponds to the onset of necking in the macroscopic tensile tests (figure 2(b)).

⎛ σ ⎞1/ m 3 Sij ˙ijp = ε˙0⎜⎜ ep ⎟⎟ ε(10) ⎝ g(εe ) ⎠ 2 σe

where ε˙0 is a characteristic strain rate, m is the strain-rate sensitivity parameter, Sij = σij − σkkδij /3 is the deviatoric part of the Cauchy stress σij, and σe = 3SijSij /2 . Strain-hardening is represented by the function g(εep ), where εep is effective plastic strain. At this point it is worth noting that the forming predictions are very sensitive to the choice of the strain-hardening function. Existing hardening functions, for example Swift or Voce, cannot predict the various stages of the flow behavior of dual-phase steels [53, 54]. Hence, to capture the various stages of the flow behavior of dual-phase steels we use a five parameter function, g(εep ), which is given by g(11) = σ0 + k1(1 − exp(−k2εep)) + hs(εep)n 15

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where σ0, k1, k2, hs and n are the constitutive parameters. The strain-hardening function in equation (11) can be viewed as a further generalization of the Voce or Ludwik type isotropic hardening functions. The equation (11) reduces to Voce hardening function for hs = 0 whereas with k1 = 0 it reduces to Ludwik hardening function. This effectively captures the three stages of the macroscopic flow behavior of dual-phase steels: (i) low initial yielding due to onset of plastic flow in the soft ferrite (first term in equation (11)); (ii) steep strain-hardening until the onset of plastic flow in the hard martensite (second term in equation (11)); and (iii) strainhardening when both phases are undergoing plastic flow (third term in equation (11)). This constitutive model is implemented as a user-material subroutine (UMAT) in the commercial finite element code ABAQUS standard version 6.10 [51]. The homogenized constitutive model described above has seven constitutive parameters that need to be determined: two in equation (10) and five in equation (11). We first focus on the strain-rate sensitivity parameter, m, which governs the constitutive response of the mat erial to the change in strain-rate induced by strain localization [55]. The strain-rate sensitivity parameters for rate dependent crystal plasticity constitutive models, equation (2), of the individual phases in the two steels are approximately in the range 0.01 to 0.02 [10]. In addition, the overall strain-rate sensitivity parameter of the DP980 steel estimated from macroscopic tensile tests in the strain-rate region 10−3 to 101 s−1 was found to be approximately 0.01 in [56]. Hence, for both parent steels, the value of m is taken to be 0.01. We next focus on the parameter σ0 in equation (11) which corresponds to the onset of plastic flow in the ferrite phase. The flow strength of the ferrite phase of both the steels is found to be ~430 MPa as shown in section 5. Hence, the value of σ0 is fixed to 430 MPa. An iterative optimization procedure was used to determine the other five constitutive parameters that best fit the uniaxial stress-strain data and the strain-hardening rate obtained from microstructurebased crystal plasticity finite element simulation of the dual-phase steel microstructure. The iterative optimization scheme was implemented as a MATLAB function. The MATLAB function carries out the finite element simulation of uniaxial tensile test using ABAQUS; calculates the mean squared average of the difference between the predicted and target uniaxial stress-strain data and the strain-hardening rate; and minimizes the error by adjusting the five constitutive parameters following the Nelder-Mead simplex algorithm. The target data are the results of microstructure-based crystal plasticity finite element simulation of the dual-phase steel microstructure. In all the cases the mean squared error between the predictions and the target is minimized down to the order of 10° or less. The final values of all seven constitutive parameters for both parent steels are listed in table 2. The table also shows parameters for several ‘virtual’ dual-phase steels. For all the steels (parent as well as ‘virtual’) the value of Young’s modulus is taken to be 210GPa and that of Poisson’s ratio is taken to be 0.3. The ‘virtual’ dual-phase steels will be discussed in more detail in sections 5 and 6. Briefly, the ‘virtual’ steels VDP980M50 and VDP980M70 are created by varying the volume fraction of the phases present in DP980 steel, similarly VDF140TM30 and VDF140TM50 are created by varying the volume fraction of the phases present in DF140T steel. These ‘virtual’ steels allow us to explore the influence of microstructure on the FLDs of the parent steels. The other two ‘virtual’ steels, VDPM40 and VDPM60, have the ferrite taken from DP980 and martensite taken from DF140T. These two ‘virtual’ steels are created in an attempt to increase the formability while keeping the strength same (or vice versa) as compared to the two parent steels. Figures 6(b) and (c) compare the uniaxial tensile stress-strain curves of the two parent steels, DP980 and DF140T, obtained from single element, quasi-static finite deformation finite element simulations using the calibrated homogenized constitutive model detailed above, with the predictions of microstructure-based crystal plasticity finite simulations detailed in 16

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Table 2. Constitutive parameters for the homogenized constitutive model (equations

(10) and (11)) obtained from the microstructure-based crystal plasticity finite element simulations of the parent DP980 and DF140T dual-phase steels, and other ‘virtual’ dual-phase steels. The martensite content in the representative volume element of the steels in percentage volume fraction, fmart, is also given.

Dual-Phase Steels

ε˙0

m

σ0 (MPa) k1 (MPa) k2

hs (MPa) n

DP980 ( fmart ≈ 60%) DF140T ( fmart ≈ 40%) VDP980M50 ( fmart ≈ 50%) VDP980M70 ( fmart ≈ 70%) VDF140TM30 ( fmart ≈ 30%) VDF140TM50 ( fmart ≈ 50%) VDPM40 ( fmart ≈ 40%) VDPM60 ( fmart ≈ 60%)

0.21 0.10 0.12 0.15 0.16 0.14 0.18 0.10

0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01

430 430 430 430 430 430 430 430

1222.6 1149.2 1440.8 1003.8 1110.6 1297.7 2452.8 2065.4

309.4 367.9 273.4 311.4 323.6 503.3 305.9 418.3

634.8 464.8 605.4 634.2 403.5 419.9 416.4 537.6

0.44 0.45 0.52 0.33 0.54 0.45 0.54 0.37

section 4.1 and good agreement is noted. Since equations (10) and (11) only captures the isotropic hardening in the material, it will not predict flow behavior accurately under cyclic or highly non-proportional loading. For the FLD simulations reported here, loading remains proportional throughout the deformation history. 4.3. Computing forming limit diagrams

A computational approach to the M-K [27] analysis was used to predict the FLDs for the parent steels. In this approach, it is assumed that formability is limited by the onset of necking and necking acts as the precursor to fracture. The failure locus can then be predicted by calculating the critical strains at the onset of necking in a specimen of sheet steel that is subjected to a prescribed biaxial straining. We consider a sheet specimen with thickness 2h, width (dimension along x1) 2W and length (dimension along x2) 2L as shown schematically in figure 7(a). Cartesian tensor notation is used and the origin of the coordinate system is taken to be at the center of the sheet specimen. Symmetry along all three xi axes is assumed so that only the region 0 ⩽ x1 ⩽ W , 0 ⩽ x2 ⩽ L and 0 ⩽ x3 ⩽ h of the sheet specimen needs to be analyzed computationally as marked with the dashed line in figure 7(a). The symmetry boundary conditions are u1(0, x2, x3) = 0, T2(0, x2, x3) = 0, T3(0, x2, x3) = 0 u2(x1, 0, x3) = 0, T1(x1, 0, x3) = 0, T3(x1, 0, x3) = 0 (12) u3(x1, x2, 0) = 0, T1(x1, x2, 0) = 0, T2(x1, x2, 0) = 0

where ui are the displacements and Ti are the tractions. The sheet specimen is deformed under in-plane biaxial straining by prescribing velocities v1 on the surface (W , x2, x3) and v2 on the surface (x1, L , x3). The two velocities are kept at a fixed ratio, vL ρ(13) = 1 v2W

where v2 > v1, so that x2 is the direction of major applied strain and x1 is the direction of minor applied strain. In the M-K analysis, necking is assumed to be triggered by a small variation in thickness of the sheet. Here we assume that the thickness of the sheet, h, varies along x2 in a ‘defect’ band of width 2λ and is normal to x2 (direction of applied major strain rate) while outside the 17

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Figure 7. (a) Schematic of a sheet blank of length 2L, width 2W and thickness 2h that varies along the x2 axis in the ‘defect’ band of width 2λ as per equation (14). The nominal thickness of the sheet is 2h 0. The sheet blank is subjected to biaxial straining by applying the velocities, v1 and v2. (b) Evolution of strain (along the x2, direction of the Band prescribed nominal major strain rate) in the ‘defect’ band, ε22 , with the evolution of Matrix , for two linear strain paths, ρ = 0 and 0.2 (see strain (along x2) in the matrix, ε22 equation (13)), defined as the ratio of applied nominal strain rate along x1 to applied nominal strain rate along x2 for the parent DP980 and DF140T steels. The onset of strain localization/necking is marked with a cross.

band the sheet thickness corresponds to the nominal thickness, h 0, of the parent steels. In the ‘defect’ band the thickness variation is idealized as

⎧ ⎛ x −λ ⎞ x2 ⎪ h 0 − ξ sin⎜ 2 π⎟ 10 (15) d ε22 Matrix 18

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where 1 ε22 Band = Band ∑ ε22(k )Vel(k ) (16) V el ∈ Band 1 ε22 Matrix = Matrix ∑ ε22(k )Vel(k ) (17) V el ∈ Matrix

The sum in equation (16) is taken over elements within the ‘defect’ band for ε22 Band, and over elements in the matrix for ε22 Matrix in equation (17). Note that Vel(k ) is the volume of the (k ) kth element; and ε22 is the strain along x2 in the kth element. As an example, the evolution Band of ε22 as a function of ε22 Matrix for two representative simulations of the parent DP980 and DF140T (characterized by the calibrated homogenized constitutive model, see section 4.2 and table 2) steels are shown in figure 7(b). The ‘defect’ band in the sheets in these simulations is characterized by ξ /h 0 ≈ 0.036 and λ /h 0 ≈ 1.43. The simulations were carried out for two linear strain paths, viz. ρ = 0 and ρ = 0.2, for both parent steels. Figure 7(b) shows that ε22 Band → ∞ as ε22 Matrix approaches a critical value, indicating a necking instability induced by the ‘defect’ band. The onset of necking as defined in equation (15) is marked with a cross in figure 7(b). The value of ε22 Matrix at the onset of necking will now be referred as the ‘major strain’ and the corresponding value of ε11 Matrix (analogous to ε22 Matrix in equation (17)) will be referred to as the ‘minor strain’. As shown in figure 7(b), the value of the major strain for the DP980 steel matrix for both values of ρ is slightly greater than that of the DF140T steel. The ranges of ξ and λ in equation (14) for the parent sheet steels can be estimated from the measurements of the surface roughness of the sheets described in section 3.2. Recall that figure 4(d), shows the variation of maximum, mean and minimum values of ∆zmax with δx. We interpret ∆zmax as ξ, the amplitude of the ‘defect’, and δx as λ, the half width of the ‘defect’ band. The measurements show that ∆zmax increases with δx for δx < 2.5 mm, but is roughly constant for δx > 2.5 mm. For δx ≈ 2.5 mm, the value of ∆zmax is ~0.03 mm. Based on this data, the upper bound value of ξ can be fixed at 0.03 mm which gives ξ /h 0 ≈ 0.04. Next, we analyze the influence of the half width λ on the major strain predictions. The variation in major strain with the variation in λ for a linear strain path ρ = 0, for DP980 steel for two values of the amplitude ξ is shown in figure 8(a). For a fixed value of ξ, the value of major strain decreases with increasing λ up to λ ≈ 2.5 mm; beyond this the critical major strain is approximately insensitive to an increase in λ. Based on this observation, the upper bound value of λ can be fixed at 2.5 mm which gives λ /h 0 ≈ 3.6. The influence of ξ on the major strain for a linear strain path ρ = 0, for DP980 steel for two values of λ is shown in figure 8(b). Here, the value of the major strain decreases almost monotonically with an increase in the value of ξ for a fixed value of λ. The effect of ξ and λ on the major strain for DF140T steel is also similar to DP980 steel. A qualitatively similar effect of the defect geometry on the onset of necking in ferritic high strength low alloy steels was experimentally demonstrated by McCarron et al [34]. To predict the FLDs of the parent sheet steels, finite element simulations of in-plane biaxial straining of sheet steel specimens characterized by varying amplitude and width of the ‘defect’ band (equation (14)) were carried out. The ‘defect’ band amplitude was varied in the range, 0.02 < ξ /h 0 < 0.04, and for each amplitude, ‘defect’ band width was varied in the range 1.4 < λ /h 0 < 3.6. All of the sheet specimens were subjected to linear strain paths in the range −0.4 < ρ < 1. These simulations used the calibrated homogenized constitutive model for the respective dual-phase steel as detailed in section 4.2. In total, 512 finite element simulations were 19

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Figure 8. The variation of major strain in DP980 steel for a linear straining path, ρ = 0, with the (a) variation in the half width, λ, of the ‘defect’ band for two values of the amplitude, ξ, of the ‘defect’, and (b) variation in the value of ξ for two values of λ.

carried out to predict the FLDs (with error bars showing the effects of the statistical nature of thickness variation) of both steels. The predicted FLDs for the parent steels together with the experimentally measured FLDs (detailed in section 3.3) are shown in figure 9(a). The lines in the figure were fit to the mean values of the major and minor strains for a given strain path. Note that the vertical error bars are the variations in the major strains and the horizontal error bars are the variations in the minor strains due to variations in amplitude and width of the ‘defect’ band. The predicted FLDs for the parent DP980 and DF140T steels are in good agreement for minor strains in the −0.05 < ε11 < 0.2 range, figure 9(b). Predictions over-estimate the values of major strains for ε11 < −0.05. Also the difference between the predicted major strains and the experimentally measured major strains increases with decreasing value of ε11 . The possible reason for this discrepancy is that the ‘defect’ band (shown schematically in figure 7(a)) in all simulations was assumed to lie normal to the prescribed major strain rate direction hence constraining the orientation of strain localization. Although it is well known that for linear strain paths ρ < 0, the strain localization occurs at an angle inclined to the prescribed major strain rate direction [57]. The discrepancies between the predicted FLDs and the experimental measurements for ε11 > 0.2 for DF140T and ε11 > 0.3 for DP980 are due to the development of non-linear/non-proportional straining paths during the forming experiments as also noted in section 3.3. These discrepancies notwithstanding, we take the agreement between the predictions and experiments for −0.05 < ε11 < 0.2 (which encompasses most of the straining paths encountered during the room temperature forming of automotive sheet steels) as a demonstration that the multiscale simulations carried out in the current work successfully predict the FLDs of the parent dual-phase steels. In particular, it is interesting to note that the predicted forming limit curve (i.e. the curve that separates forming with no necking and forming with necking) of the DP980 steel is also slightly higher than that of the DF140T steel, in agreement with the experimental measurements. The effect of the material parameters characterizing the strain-hardening behavior (equation (11)) in the homogenized constitutive model on the FLD predictions was also analyzed. Attention was confined to four parameters, viz., k1, k2, hs and n: these characterize the steep strain-hardening response of the material until the onset of plastic flow in the martensite phase and the strain-hardening behavior of the material when both the phases are undergoing plastic 20

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Figure 9. (a) Comparison of predicted (lines) and experimentally measured (symbols)

forming limit diagrams of the parent dual-phase sheet steels, DP980 and DF140T. The zoomed in forming limit diagram for minor strains in range −0.05 to 0.2 is shown in (b). The error bars denote the variations in the critical strain predictions from a set of simulations with varying amplitude and width of the ‘defect’ band (equation (14)). The lines are a fit through the average values of these simulations.

flow. The effect of these material parameters on FLD predictions are as follows. Increasing the value of k1 decreases the values of the limit strains whereas the values of the limit strains are almost insensitive to changes in the values of k2. On the other hand, the values of the limit strains increase almost linearly with an increase in the values of hs or n, with the effect of n being significantly greater than that of hs. 5. The influence of microstructure on formability We next use the multiscale simulations described and validated in the preceding section to investigate the influence of the volume fractions of the phases on controlling the formability of the parent dual-phase steels. To this end, we create computational RVEs of dual-phase steel microstructures by varying the volume fraction of constituent phases by ±10% in the parent dual-phase steels, DP980 and DF140T. The volume fraction of martensite in DP980 steel is approximately 60%; hence, we consider two ‘virtual’ DP980 dual-phase microstructures designated VDP980M50 and VDP980M70 with martensite volume fractions of 50% and 70%, respectively. Similarly the volume fraction of martensite in DF140T steel is approximately 40%. We therefore constructed two ‘virtual’ DF140T dual-phase microstructures designated VDF140TM30 and VDF140TM50 with martensite volume fractions of 30% and 50%, respectively. Microstructure-based crystal plasticity finite element simulations (as detailed in section 4.1) were then conducted to predict the macroscopic flow behavior of these ‘virtual’ steels under uniaxial tension. The constituent phases in the ‘virtual’ steels VDP980M50 and VDP980M70 were idealized using the microscale calibrated crystal plasticity constitutive models of ferrite and martensite of the parent DP980 (figures 3(a) and (c)). Similarly, the microscale calibrated crystal plasticity constitutive models of ferrite and martensite of the parent DF140T (figures 3(b) and (d)) were used in the microstructure-based simulations of the ‘virtual’ steels VDF140TM30 and VDF140TM50. Note that we assume that the flow 21

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behavior of the individual phases in the ‘virtual’ steels is the same as that of their parent steels. The results of the microstructure-based simulations were then used to calibrate the homogenized constitutive model (as detailed in section 4.2) for the four ‘virtual’ steels. The values of the material parameters of the homogenized constitutive model for these four ‘virtual’ steels are listed in table 2. Using the calibrated homogenized constitutive model and a fixed ‘defect’ band characterized by ξ /h 0 ≈ 0.036 and λ/h 0 ≈ 1.43, the FLDs of the four ‘virtual’ sheet steels were also computed. This enabled us to analyze the influence of phase volume fraction on the macroscopic flow behavior and formability of the two parent steels. Figure 10 shows how a ±10% variation in the constituent phase volume fraction influences the macroscopic flow behavior of the parent dual-phase steels. As shown in figure 10(a), the flow behavior of DP980 is less sensitive to a ±10% variation in the volume fraction of the constituent phases as compared to DF140T (figure 10(b)). Also, the difference in the flow behavior due to a ±10% variation in the volume fraction of the constituent phases in DP980 steel decreases with progressive tensile deformation. For example, the flow stress of ‘virtual’ DP980 with 70% martensite content (VDP980M70) is roughly 14% greater than that of the ‘virtual’ DP980 with 50% martensite content (VDP980M50) at about 2% strain, whereas this difference decreases to roughly 7% at about 10% strain. Alternatively, the flow behavior of DF140T is not only more sensitive to a ±10% variation in the volume fraction of the constituent phases, but the difference in the flow behavior almost remains constant with progressive tensile deformation. For example, the flow stress of ‘virtual’ DF140T with 50% martensite content (VDF140TM50) is roughly 30% greater than that of the ‘virtual’ DF140T with 30% martensite content ((VDF140TM30) at about 2% strain, and the percentage difference remains roughly constant up to 10% strain. Similarly, the effect of a ±10% variation of the volume fraction of the constituent phases on the FLD of the parent sheet steels is shown in figure 11. Unlike the flow behavior shown in figure 10, the FLD of DP980 is more sensitive to a ±10% variation in the volume fraction of the constituent phases compared to DF140T. As shown in figure 11(a), the forming limit strains of DP980 increase with decreasing martensite content. This increase is most significant in the plane strain regime, ρ = 0, which is commonly referred to as FLD0. Here, the major strain increases by roughly a factor of two for a corresponding 20% decrease in martensite content. This increase in the major strain decreases as we move along the minor strain axis either towards the right (positive minor strain) or towards the left (negative minor strain). With 50% and 40% martensite content, the formability of DF140T shown in figure 11(b) is roughly the same and increases slightly with an additional 10% decrease in the martensite content. This result, together with the flow curves shown in figure 10, suggests that the formability of DP980 can be slightly increased by decreasing its martensite content without significantly degrading its strength. Alternatively, the flow strength of DF140T can be slightly increased, without significantly degrading its formability, by increasing its martensite content. Note however, that our simulations do not account for the possibility of fracture occurring in the material before the onset of necking. Dual-phase steels with higher martensite content may be more susceptible to fracture. To probe the micromechanical origin of the contrasting effects that emerge from changes to the phase volume fractions in DP980 and DF140T, we next examine the flow behavior of their constituent phases in more detail. Figure 3 shows the results of micropillar compression experiments on ferrite and martensite pillars extracted from the two parent steels. It is difficult to compare the two steels directly from this data since variations in crystallographic orientations in the ferrite micropillars and variations in number of blocks in the martensite micropillars can cause large changes in flow stresses that mask systematic differences between the 22

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Figure 10. The influence of the volume fraction of martensite ( fmart ) on the macroscopic

flow behavior of the (a) DP980 and (b) DF140T parent dual-phase steels. Both the predictions of microstructure-based crystal plasticity finite element simulations of representative volume elements (RVE) and finite element simulations of uniaxial tension using the homogenized constitutive model (designated as Fit), are shown.

Figure 11. The influence of the volume fraction of martensite ( fmart ) on the formability

of (a) DP980 and (b) DF140T parent dual-phase steels.

two steels. To remove these effects, we created ‘virtual’ single-phase microstructures of both ferrite and martensite phases of both parent steels. The single-phase microstructure of ferrite consisted of over 3000 grains (with random orientation) and the single-phase microstructure of martensite consisted of over 3000 blocks. Microstructure-based crystal plasticity finite element simulations of uniaxial tension tests of these ‘virtual’ single-phase microstructures were then carried out using the microscale calibrated crystal plasticity constitutive models of the ferrite and martensite of the parent steels. These calculations homogenized the orientation effect of ferrite grains and martensite blocks and allowed us to compare their isotropic flow behavior. The computed isotropic flow curves of the ferrite phase present in the two steels are compared in figure 12(a) and the flow curves of the martensite phase of these two steels are compared in figure 12(b). As shown in figure 12(a), the initial yielding in the ferrite of both parent 23

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steels occurs at ~430 MPa. This is followed by steep hardening for a brief period during which time the ferrite grains with soft orientations are undergoing plastic deformation while the martensite grains with hard orientations are still elastic. After all grains have yielded, the DP980 ferrite exhibits strain-hardening whereas the ferrite of DF140T does not exhibit strainhardening. Alternatively, the DF140T martensite yield strength is significantly greater than that of DP980, as shown in figure 12(b). The initial yielding of the DP980 martensite occurs at ~800 MPa whereas that of the DF140T martensite occurs at ~1500 MPa. Neither the DP980 nor the DF140T martensite exhibits strain-hardening. The differences in the flow behavior of the constituent phases of the two steels are consistent with differences in their nominal chemical compositions (see table 1). Based on the nominal chemical and phase composition of the parent steels, and assuming all carbon partitions to martensite, the carbon content of martensite in DF140T should be approximately twice that of carbon content of martensite in DP980. The strength of martensite is known to increase with its carbon content [4, 16] and hence it explains the difference in the strength level of martensite of the parent steels. Similarly, the higher silicon as well as higher manganese contents in DP980 can explain the enhanced strain-hardenability of ferrite in DP980. Both silicon and manganese are known to be ferrite solid solution strengtheners [4, 16, 58]. However, further chemical analysis is needed to confirm the partitioning of these alloying elements among the two constituent phases. The observation that changing the phase volume fraction has a different effect on the flow strength and formability of the parent steels can now be explained by differences in the flow behavior of their constituent phases. In the DP980, the initial yield strength of martensite is only twice that of ferrite, whereas in the DF140T the initial yield strength of martensite is roughly 3.5 times that of ferrite. Hence, a small change in the martensite content causes a large change in the flow stress of DF140T, but only a small change in DP980. The increased strainhardenability of ferrite in DP980 also tends to compensate for any reduction in flow stress that occurs when the martensite volume fraction is reduced. Similarly, the changes in the FLDs are explained by differences in the strain-hardening behavior of the ferrite in the two steels. In general, following Considère’s criterion, increasing strain-hardenability and reducing flow stress tends to delay the onset of necking. Reducing the volume fraction of martensite in DP980 reduces its flow stress, and since the ferrite in DP980 hardens significantly, its overall rate of hardening is also increased due to increased ferrite content. As a result, reducing the volume fraction of martensite improves the formability of DP980. In contrast, the ferrite in DF140T exhibits little strain-hardening at strains exceeding 0.01 (or 1%) so changing the volume fraction of martensite has negligible or only a small effect on the formability of DF140T. 6. Optimization of dual-phase microstructure for Improved Formability The observations in the preceding section suggest a possible approach to developing new dual-phase steel microstructures, with improved properties, by exploiting the strain-hardenability of ferrite in DP980 together with the enhanced strength of martensite in DF140T. We focus first on identifying a dual-phase steel with flow strength comparable to that of DP980, but with improved formability compared to the parent dual-phase steels. Reducing the volume fraction of martensite in DP980, while increasing its strength will result in greater strain-hardenability, and hence will improve the resistance of the sheet steel to necking. To this end, we modeled a ‘virtual’ steel microstructure, designated VDPM40, which contains a 40% volume fraction of martensite, with flow behavior corresponding to that of martensite extracted from DF140T. The remaining 60% volume fraction consists of ferrite with the properties of the DP980 ferrite. Secondly, we suggest a dual-phase 24

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Figure 12. Comparison of computed overall uniaxial tensile flow curves of (a) ferrite phase and (b) martensite phase of the parent DP980 and DF140T dual-phase steels. These curves were generated by deforming the corresponding polycrystalline 3D RVEs in tension.

microstructure with enhanced strength, but with formability comparable to both parent steels. For this purpose, we modeled a ‘virtual’ steel microstructure, designated VDPM60, with 60% martensite having properties corresponding to martensite in DF140T, and 40% ferrite with properties corresponding to ferrite in DP980. The macroscopic flow behavior of the two ‘virtual’ dual-phase steels, VDPM40 and VDPM60, were then computed using microstructure-based crystal plasticity finite element simulations (as detailed in section 4.1). The results of the microstructure-based simulations were then used to calibrate the homogenized constitutive model (as detailed in section 4.2) for FLD simulations. The values of the material parameters of the homogenized constitutive model for these two ‘virtual’ steels are listed in table 2. The computed macroscopic flow behavior of the two ‘virtual’ dual-phase steels, VDPM40 and VDPM60, are compared with the computed macroscopic flow behavior of the parent DP980 and DF140T steels in figure 13(a). Similarly, the computed FLDs of the two ‘virtual’ dual-phase steels are shown together with the computed FLDs of the parent steels in figure 13(b). In these FLD computations, the ‘defect’ band is characterized by ξ /h 0 ≈ 0.036 and λ/h 0 ≈ 1.43, for all sheet steels (‘virtual’ or existing). As shown in figure 13(a), the 0.2% offset flow strength of VDPM40 is comparable to that of the parent DP980 and DF140T steels; however, it has significantly better strain-hardenability compared to the two parent steels. The enhanced strain-hardenability of VDPM40 leads to far better formability by significantly delaying the onset of necking compared to the parent steels as shown in figure 13(b). Alternatively, the formability of VDPM60 is comparable to that of the parent steels (figure 13(b)). However, VDPM60 exhibits significantly better macroscopic flow behavior relative to the parent dual-phase steels (figure 13(a)). The two proposed ‘virtual’ dual-phase steel microstructures can, in principle, be created by selecting the chemical composition and heat treatment conditions appropriately. As described in section 5, the required strength level of martensite can be achieved by increasing the nominal carbon content of the steel, as is the case in DF140T steel. The required strain-hardenability of the ferrite can be achieved through the addition of the alloying elements that lead to solid solution strengthening, such as silicon and manganese, as is the case in the ferrite of DP980 steel. Also, the parent dual-phase steels were produced on continuous 25

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Figure 13. Comparison of computed (a) macroscopic flow behavior and (b) FLDs of the two ‘virtual’ dual-phase steel microstructures (VDPM60 and VDPM40) with parent DP980 and DF140T dual-phase steels.

annealing lines that allow intercritical heating into the ferrite-austenite field followed by rapid cooling to cause austenite to martensite transformation [4, 6, 7]. The intercritical heat treatment temperature controls the volume fraction of austenite and hence to an extent the volume fraction of martensite formed following rapid cooling. During rapid cooling martensite begins to form at a critical temperature, defined as the martensite start temperature. The martensite start temperature is the second factor that controls the amount of martensite formed at room temperature [59] and is a function of the carbon and alloy content of the steel [6]. Hence, by optimizing these heat treatment conditions required phase compositions can also be achieved. 7. Concluding remarks In this paper, we have described a multiscale approach for predicting the forming limit diagrams of dual-phase steels. The flow behavior of the dual-phase microstructures were predicted with microstructure-based crystal plasticity finite element simulations of 3D representative volume elements. The results of the microstructure-based simulations were then used to determine parameters of a rate dependent, isotropic constitutive model that describes the various stages of strain-hardening that are characteristic of dual-phase steels. The forming limit diagrams of the sheet steels were predicted following a computational approach to the Marciniak-Kuczynski analysis. The method was used to predict the forming limit diagrams for two commercial dualphase steels, DP980 and DF140T ‘parent’ materials, for which the properties of the ferrite and martensite phases were previously measured in [10]. The initial thickness variation in the parent sheet steels required for forming limit diagram predictions were determined using 3D DIC. The predicted forming limit diagrams for these two sheet steels were shown to be in good agreement with the forming limit diagrams measured by means of Nakazima and Marciniak tests. The multiscale simulations were then used to determine the role of individual phases in controlling the flow stress, strain-hardenability, and forming limit diagrams of the parent dual-phase steels. This was accomplished by computing the changes in flow strength and forming limit diagrams resulting from changing the volume fractions of the phases in the parent steels. The insights obtained from these analyses were used to identify changes to the microstructure of the two parent steels that would either improve the formability without reducing the overall flow strength of the steel, or which would improve the flow strength without degrading the formability. 26

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The principal conclusions of our study are as follows: 1. A multiscale approach was developed to predict the forming limit diagram of dual-phase advanced high strength steels and explore the influence of the properties of each phase and the overall microstructure of dual-phase steels on their forming limit diagrams. The predictions via this approach for the two parent dual-phase steels (DP980 and DF140T) were found to be in good agreement with experimental measurements. 2. Although both simulations and experimental measurements show that the parent steels have similar overall flow strength and formability, the properties of the phases in these steels differ significantly. Results detailed in [10] revealed that the ferrite phases in the two steels have similar flow strength; however, the martensite in DF140T has a flow strength that is roughly twice that of DP980. A re-examination of the data from [10] revealed that the ferrite in DP980 shows significant strain-hardenability, while ferrite in DF140T has an approximately constant flow stress beyond a plastic strain of 1%. This difference is consistent with the differences in the nominal chemical compositions of the DP980 and DF140T steels. 3. The differences in flow behavior of the phases in DP980 and DF140T result in differences in the influence of the volume fractions of phases on their macroscopic flow behavior and formability. For instance, the macroscopic flow behavior of DF140T was found to be more sensitive to a ±10% variation in the volume fraction of the phases compared to DP980. The flow strength of the DF140T was found to decrease significantly with decreasing martensite content. The formability of DP980, as inferred from its forming limit diagram, was found to be more sensitive to a ±10% variation in the volume fraction of the phases compared to DF140T. The formability of DP980 was found to improve with decreasing martensite content. 4. Simulations suggest that the sensitivity of the flow behavior and formability to changes in phase volume fractions can be exploited to create new dual-phase steels with 0.2% offset flow strength comparable to the parent DP980 and DF140T steels, but with significantly improved formability. Specifically, a dual-phase steel microstructure with 40% volume fraction of martensite and flow strength comparable to the martensite phase in DF140T, and 60% ferrite with flow strength and strain-hardenability comparable to the ferrite phase in DP980 is predicted to have a flow strength comparable to the parent dual-phase steels but with significantly improved formability. This ‘virtual’ dual-phase steel microstructure can, in principle, be created by appropriately selecting the chemical composition and heat treatment conditions as detailed in section 6. 5. Similar to Conclusion 4, the sensitivity of the flow behavior and formability to changes in phase volume fractions can be exploited to create new dual-phase steels with formability that is comparable to the parent DP980 and DF140T materials, but with significantly enhanced flow strength. For this purpose, a dual-phase steel microstructure with a 60% volume fraction of martensite with flow strength comparable to the martensite phase in DF140T, and a 40% ferrite with flow strength and strain-hardenability comparable to the ferrite phase in DP980 is proposed. Acknowledgments The DP980 and DF140T sheet steels were kindly provided by Dr Shrikant Bhat of ArcelorMittal. The authors are very grateful for the many fruitful discussions with Drs Shrikant Bhat and Sriram Sadagopan of ArcelorMittal and Dr Anil K Sachdev of General Motors. The work reported in this paper was supported by General Motors through the GM/Brown Collaborative Research Laboratory on Computational Materials Research. 27

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Appendix Table A1. The crystal plasticity constitutive model parameters determined by 3D finite element simulations of micropillar compression tests on the ferritic and martensitic phases of DF140T and DP980 dual-phase steels from [10].

Two parent dual-phase steels

DF140T

Material parameters for ferrite Reference strain rate γ˙0 (s−1) Stress exponent m Initial hardening rate h 0 (MPa) Saturation hardening rate hs (MPa) Initial flow stress g0 (MPa) Saturation flow stress gs (GPa) Characteristic shear strain γ0 Latent hardening coefficient f0 Latent hardening coefficient q Non-Schmid effect parameter a1 Non-Schmid effect parameter a2 Non-Schmid effect parameter a3 Material parameters for martensite Reference strain rate γ˙0 (s−1) Stress exponent m Shear modulus G (GPa) Burgers vector b (nm) Dislocation nucleation rate ka Dislocation annihilation rate kb Initial dislocation density ρ0 (m−2)

DP980

0.002 41.0 97.1 5.36 253.8 3.22 65.1 99.9 1.4 0.6 0.01 0.2

0.009 67.8 556.3 188.9 238.7 0.61 55.2 0.08 1.0 0.6 0.01 0.2

1.6 × 10−3 63.7 75 0.3 86.4 0.0935

1.7 × 10−3 71.2 75 0.3 73.0 0.14

8.2 × 1013

3.0 × 1013

Note: For both steels the elastic constants for both the phases were assumed a priori and were not fitted to the micropillar compression test data. For the ferrite phase of the two steels the elastic constants were assumed to have cubic symmetry and were specified by C11 = 310GPa, C12 = 145GPa and C44 = 105GPa whereas for the martensite phase of both the steels, Young’s modulus was taken to be 195 GPa and Poisson’s ratio was taken to be 0.3.

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