Assessing the Feasibility of Cold Forming of Automotive Parts from Quenched and Partitioned Martensitic Stainless Steels

Sheet metal forming is among the most widely employed processes in manufacturing. Its economic feasibility relies on many factors, including the mechanical properties of the processed materials, energy efficiency of the metal‐forming processes, and quality of the manufactured parts. This work focuses on developing a computational approach to assess the mechanical behavior of quenched and partitioned (QP) martensitic stainless steels (MSSs) undergoing cold forming, aiming at the serial production of automotive parts. The Johnson–Cook (J–C) constitutive model and forming limit diagram (FLD) are derived and validated for each alloy using the experimental data from tensile and Nakajima tests. From the simulated FLD, a theoretical forming limit stress diagram (FLSD) is calculated. The latter is used as a fracture criterion for the cold‐forming process. Cold stamping of two automotive parts, B‐pillar and tunnel, is modeled using the J–C constitutive model and the FLSD. It is demonstrated that the tunnel can be successfully cold‐formed from all three studied steels. In contrast, extensive cracking is expected during the cold forming of the B‐pillar from all materials. It is envisaged that these computational models can be employed for the assessment of any comparable part manufacturing procedure from the QP MSSs.

of a matrix of martensite laths, and they can present thin films of retained austenite between laths. [5]Their mechanical and application-related properties are determined by microstructure, which, in turn, strongly depends on the applied heat treatment.Therefore, tuning the heat treatment enables engineers to achieve a wide range of properties in these materials.The QP process has been applied to reach higher ductility levels without losing strength in MSSs.The material is austenitized, and then quenched to a temperature between the start and finish temperatures of the martensitic transformation, retaining an amount of austenite within the newly formed martensite.During the next partitioning step, the samples are heated to a higher temperature to facilitate the diffusion of carbon from the supersaturated martensite into the austenite.This increase of carbon in the austenite stabilizes it and allows it to remain in the final microstructure at room temperature.The presence of metastable austenite enables the TRIP effect during plastic deformation.It improves the material's strain hardening ability, resulting in enhanced ductility.A few publications focused on QP processing of AISI grades 410 and 420.Regarding grade 410, Tobata et al. [6] investigate the effect of silicon content and QP parameters on microstructure and strength-ductility balance, and Tsuchiyama et al. [7] study the effect of carbon content.In terms of grade 420, Huang et al. [8] focus on the impact of the heat treatment on the microstructural parameters; Yuan et al. [9] analyze the stability of retained austenite during plastic deformation; Mola and De Cooman [10] investigate the microstructure-property relationship; and Sierra-Soraluce et al. [11] study the effect of chemistry on the microstructure, tensile deformation behavior, and microstructure evolution during plastic deformation of QP-treated MSSs.In these works, high volume fractions of retained austenite are achieved in the final microstructure: up to a 15% in 410 (low C, 0.12-0.13wt%) [6] and up to 57% in 420 (medium C, 0.47 wt%). [8]Such microstructures result in improved tensile behavior of MSS.In the 410 grade, while maintaining a similar level of strength (at %1200 MPa), the tensile ductility is increased by 5% compared to the standard quenched and tempered state. [7]n the case of 420 grade, the same behavior is observed with ultimate tensile strengths %1800 MPa. [8]lthough the formability of QP MSSs has not been studied, a first gross estimate can be drawn up, taking the ductility.It is well known that formability is closely related to a material's ductility.As mentioned above, the ductility of MSS improves after QP treatment.For other types of AHSS, the positive effect of QP treatment on formability has been demonstrated for low-carbon steel [12] and medium-carbon steel. [13]Independently, the potential of MSSs for cold-forming applications has already been proven with the cold forming of diverse automotive parts. [14]old forming has several advantages compared to hot forming.It is energy efficient and allows for high precision, high-quality surface finishes, and high-speed production.The main disadvantage of this process is the low level of geometry complexity achievable.The central objective of this work is to assess the feasibility of cold-forming automotive parts from QP MSSs.To achieve this objective, a computational approach, moderately based on experimental data, has been developed.The outcome of this approach is a set of numerical models that significantly reduce the number of required experiments and related material waste.In more detail, based on a limited collection of experimental data, the procedure involves a comparative selection of two distinct material constitutive models.The chosen model is then coupled with experimentally derived forming limit diagrams, which serve as a ductile fracture criterion.Through successive intermediate validations, these developments serve as computational tools for obtaining high-fidelity numerical models specifically designed for industrial cold-forming applications.These models were initially established as the primary goal of this study.

Johnson-Cook Model in Formability Studies
In this section, a comprehensive insight into the Johnson-Cook (J-C) constitutive model is given, including its mathematical formulation and the approach taken when using it for the development of formability simulations.

Application of J-C Model in Formability Studies
The use of the J-C constitutive model to reproduce the mechanical behavior of materials undergoing forming processes is fairly standardized.In particular, Wang et al. [15] employ the mentioned equation to simulate the stamping process of preheated boron steel samples.A good match between simulated results and experimental data is found, thus enabling numerical prediction before performing the physical process at an industrial level.On the other hand, in another work [16] the authors analyze the prediction capabilities of the model regarding the simulation of a steel material when subjected to the cold roll-beating, finding that the data of the true stress generated during the simulated procedure are consistent with those obtained with experimental methods.Likewise, in ref. [17] the authors find a good agreement of the numerical data with regard to the prediction of the behavior of copper and aluminum alloys when being processed through a cold wire drawing cycle, or in ref. [18], where the authors explore the simulation of the oscillating cold-forming process by using the J-C model coupled with different unloading models, finding that one of the combinations could successfully describe the experimentally observed deformation.It must be emphasized that while the J-C model is capable of simulating materials experiencing deformations under variable strain rate and temperature conditions, its application in scenarios where strain is the only variable does not diminish its simulation capabilities.In situations where only strain varies, as observed in previous studies [15][16][17][18] and the present article, the J-C model is capable of independently handling this variable.This is due to the model's formulation, which accounts for the inherent independence of strain, strain rate, and temperature by representing them as separate factors in the equation.
Nevertheless, when employing a widely applicable strength model such as the J-C model in this case, it is essential to consider its use in conjunction with another constitutive equation that incorporates the material's fracture behavior.The strength model establishes the material's limits for failure under the given simulated conditions.In particular, a similar approach has already been used in the previous work by Panich et al., [19] where a yield model is coupled with the forming limit diagram (FLD) and forming limit stress diagram (FLSD) fracture criteria to assess the formability of steel sheets under stamping processes, having as an outcome a successful prediction when comparing the data with those extracted from real experiments.
All of these works strengthen the initial hypothesis considered in this article, relying on the employment of the aforementioned constitutive equation coupled with either a strain-based or a stress-based criterion to properly assess the different martensitic stainless steels under study when used in industrial cold forming.

J-C Constitutive Relations
The J-C constitutive model has been widely employed in the reproduction of the plastic behavior of metals under conditions of large strains, high strain rates, and high temperatures.It is not considered to be a constitutive equation able to offer extremely accurate predictions for all the possible variable ranges, but among its advantages, it counts on fair robustness and its computational implementation is relatively simple.Due to this, it has been used over the last decades in a wide spectrum of applications: machining simulations, [20] hot stamping process, [21] and Charpy test simulations. [22]he J-C constitutive equation is one of the J 2 classical plasticity models, in which the yield stress σ y is assumed to be of the form where there are three main terms.The first part refers to the quasistatic hardening, in which the initial yield stress, A, together with the parameters representing the strain hardening, B, and the exponent n can be found, as well as the variable ε p , representing the equivalent plastic strain.The second one is related to the hardening provoked by the strain rate effects, where the strain rate parameter, C, and the dimensionless plastic strain rate, εÃ p ¼ εp εp 0 , including εp 0 as a reference plastic strain rate chosen in accordance to the lowest experimental values expected, are found.Finally, the third term, which accounts for the effects of temperature, includes the thermal softening exponent, m, and also the so-called "homologous temperature," as where Θ exp ≡ Θ represents the experimental temperature at which the material is being modeled, Θ melt is the melting temperature of the material, and Θ room is the ambient temperature for the minimum experimental reference.

Materials and Processing
Three alloys are selected for this study, and their chemical composition in weight percent is presented in Table 1.The first alloy (1) can be classified within the grade AISI 410, while the second alloy (2) falls in the grade AISI 420.The third alloy is a modified version of alloy 2, with an increased content in Mn and small additions of Ti and Nb.With the increased content of Mn, a stabilization of the austenitic phase is sought and the microalloying with Ti and Nb provides a grain refinement thanks to the formation of their nanocarbides.Slabs of casted material were hot rolled with a finishing temperature of 1000 °C.Then an annealing was performed for alloys 1 and 2 annealed at 740 °C for 24 h and air cooled, while alloy 3 was annealed at 700 °C for 24 h and furnace cooled.The final material had a form of sheets of 1.5 mm in thickness and 220-240 mm in width.Sheets of the selected alloys are QP treated using the parameters provided in Table 2.They were austenitized at 1100 °C for 15 min to dissolve the carbides present in the as-received material, with the exception of (Ti,Nb)C in alloy 3.This process also homogenized the distribution of alloying elements.The quenching temperatures for alloys 1 and 2 were chosen to yield the highest fraction of retained austenite in the final microstructure.For alloy 3, the quenching temperature was set to room temperature (30 °C) to investigate this specific processing route due to its industrial relevance, as the optimal quenching temperature was relatively close (62 °C).The quenching soaking time was sufficient to ensure temperature homogenization across the sheet.The partitioning parameters used in this study were determined in previous research to provide the most favorable partitioning for the obtained retained austenite. [11]

Mechanical Testing
From the QP-treated sheets, samples are machined for two different experimental tests: uniaxial tensile testing and Nakajima testing.For the tensile testing, standard dog bone samples (gauge length 50 mm and gauge width 12.5 mm) are machined with a 90°angle with the rolling direction.
Following the standard ASTM E8M, [23] tensile test are carried out in an universal testing machine at room temperature and strain rate of 10 À3 s À1 .Three tensile specimens are tested per alloy.A representative stress-strain curve of each alloy is displayed in Figure 2. From these curves, the yield strength (YS), the ultimate tensile strength (UTS), uniform elongation (ε u ), and elongation to fracture (ε f ) are obtained and are presented in Table 3.
In the case of Nakajima testing, five geometries (two coupons per geometry) are machined for every alloy, ranging from (quasi) uniaxial to equibiaxial stress states.After removing the oxide layer with sand blasting, a square grid is applied to each specimen.The samples are deformed until fracture using a hydraulic press with a hemispherical punch of 100 mm diameter.Lubrication is used to reduce the friction between the punch and the sheet samples.The load versus displacement curves of selected specimens are recorded.Using digital image correlation (DIC), the strain measurements are extracted from the deformed grid post-test.The forming limit curves (FLC) are then constructed for each alloy.The FLD is an extension of the FLC.These data are used to develop the finite element method (FEM) models described in the coming sections.

Microstructural Characterization
The samples are characterized using electron backscatter diffraction (EBSD), and the resulting maps are collected in Figure 3.The volume fraction of retained austenite increases from 9.7% in alloy 1 to 18.3% in alloy 2 to 18.7% in alloy 3.This trend is related to the increasing content of austenite-stabilizing alloying elements (C, Mn). [11]urther analysis is carried out for alloy 3 (undeformed, Geometry 0), estimating its fresh martensite fraction according to the work [24] and collected in Table 4.The prior austenite grains (PAG) are reconstructed using the MTEX package for MATLAB. [25]energy-dispersive X-ray spectroscopy (EDX) analysis in TEM was performed in different areas of alloy 3, and no presence of Ti nor Nb was found in solid solution in the matrix.Given the small fraction of Ti and Nb, and the superior fraction of C, it is known and established that all of them have precipitated in the form of carbides (consuming a low amount of C along), as it was proven in our recent work, where a thorough microstructural characterization of the studied alloys was performed. [11]Similar observations were also reported for a Nbmicroalloyed QP-treated carbon steel by Xia et al. [12] The effects of the modifications of alloy 3 are here observed, achieving a higher fraction of retained austenite and showing a general reduction of the grain size, which is related to the controlled prior austenite grain size (from PAG size 38.3 μm in alloy 1, to 20.4 μm in alloy 3) via nanocarbides precipitation, as recently reported in ref. [11].
Additionally, three geometries of alloy 3 are selected for their microstructural characterization (Geometries 1, 3, and 5).The same procedure applied to the undeformed material is used here again.The results are collected in Table 4, with the corresponding EBSD maps in Figure 4.The plastic strain (ε pl ) reported in Table 4 is calculated with the initial thickness of the sheet and the final one ε pl ¼ lnðt 0 =tÞ.Based on these results, it is clear to see that the volume fraction decreases with increasing plastic strain.The retained austenite is transformed into fresh martensite (Fresh M) which may be identified as such or nonindexed (NI), hence why both of these fractions increase as well.The remaining retained austenite is the more stable one, hence its grain size decreases, but most importantly its aspect ratio (a.r.).Similar microstructural evolution was also demonstrated by alloys 1 and 2.

Development of FEM Models
FEM models are employed in this work to accomplish two objectives: first, to replicate experimental tests and validate themselves by comparing the results with different material properties obtained from real data; second, to predict the outcome of forming processes, specifically the cold stamping of industrial parts.Nakajima experiments carried out with three different QP MSSs in the form of five different sample shapes are simulated, as shown in Figure 5, in order to validate the ductile fracture limits observed during the actual tests, being these expressed in the form of the equivalent plastic strain, ε p , and the FLD curves.The latter referred simulated models are created employing the computer-aided engineering (CAE) software ABAQUS, in which the experiment is replicated by modeling the die, holder, and the hemispheric punch parts as rigid 3D shells, being the first two fixed, holding the sample in place based on a penalty friction formulation with a coefficient of 0.02, while the punch moves vertically at a constant speed of 1.5 mm s À1 , as dictated by the ISO standards 12004-1 [26] and 12004-2. [27]All the parts count on axisymmetric displacement boundary conditions so that only a quarter of the whole setup can be simulated, saving important    computational effort.More in detail, each of the differently shaped quarter-sheet parts is modeled through deformable 3D shells, with meshes consisting of approximately 5500 linear-S4R elements, counting on 6 integration points equally distributed over the 1.5 mm thickness of the sheets.The simulations are solved by employing an explicit integration scheme assisted by mass scaling techniques, aiming at reducing the overall computational cost.
The simulation of the behavior of each steel is approached from the plasticity point of view, considering two initial perspectives.The first relied on the use of the stress-strain curves recorded for each alloy during tensile tests carried out beforehand.While the second, and eventually the one which is chosen, is based on the use of the J-C constitutive model, whose parameters are obtained also employing the data from the latter experiments, with a similar method as the one shown in ref. [28], and then validated employing the force versus displacement curves extracted from the Nakajima tests.Insights on this matter are given below in this article.
Thus, once the model is set, the quantities of interest (QoI) ε p and the FLD curves are recorded in the simulations and compared with those acquired during the tests, therefore aiming at validating them.Subsequently, the extraction of the forming limit stress diagram (FLSD) curves for each material is conducted through simulations of Nakajima tests.This approach proves advantageous as it circumvents the high cost associated with attempting to obtain these curves directly through experimental procedures.Recording the entire deformation path in the failure region is necessary to calculate these values, as indicated by the Levy-Mises relation where the tensor of incremental strain dε ij is proportional to the plastic slip dλ and the tensor of deviatoric stress s ij , which, in turn, depends on the nonincremental tensor of strain ε ij .Finally, after carrying out the extraction of this alternative fracture ductile criteria of the three steel materials through numerical methods, some relevant stamping simulations are performed, involving the cold stamping process of two industrial manufactured parts, namely, the B-pillar and the central transmission tunnel of generic automobiles, being both representative examples of the standard usage and application of the materials studied.

Results and Discussion
In this section, the results derived from the simulations developed aiming at replicating the Nakajima experiments computationally are shown and discussed, specifically with a focus on the different theoretical procedures applied and the outcome arising from them.

Approach to Plasticity Simulation: J-C versus Traditional Stress-Strain Curves
As mentioned previously, during the development of this work, two different approaches were initially considered when dealing with the method to simulate the plastic behavior of the material for the different Nakajima experiments that needed to be replicated numerically.Namely, these were the use of stress-strain curves extracted from the initial tensile tests and the utilization of the J-C model, which implies the additional finding of the corresponding material parameters to describe the characteristics of the martensitic steel properly.
More in detail, when the first of them was employed, the tensile stress-strain data points taken from the quasistatic tensile tests were introduced as the plastic model within the ABAQUS models.Once the Nakajima tests models were run, the simulated force versus displacement curves could be obtained and compared with those recorded during the actual tests.This way, it was observed that depending on the sample shape, and therefore, on the different stress states of uniaxial tension, plane strain or biaxial tension occurring during the tests, the fitting accuracy of the numerical data with respect to the actual data could notably vary, getting relative error percentages between the different curves in the range of 5-150%.When trying to apply scaling factors to the stress-strain data to try to reduce the higher errors, the smaller ones were increased, not being able to improve the overall results.
Thus, a different approach had to be taken into account if the Nakajima tests were to be replicated with high fidelity.In particular, ABAQUS offers a variety of constitutive material models, among which the J-C equation is included.Thus, this relation has been considered to deal appropriately with the simulation of metals subjected to high strain, as is the case here, although any similar one, as it could be the Zerilli-Armstrong model or Arrhenius-type equations could be also considered for the same purposes.As a result of its use, each element stress state of the simulated Nakajima samples can be calculated in each simulation time step with a significantly higher precision than in the more rudimentary case of employing a tabulated relation of points uniquely taken from uniaxial tensile tests, which do not represent either the actual complex state and varying triaxiality occurring during the tests.This leads to the achievement of a better match between the experimental and simulated forcedisplacement curves obtained.
In order to be used, first, different material parameters need to be determined, carrying out a similar procedure as the one presented in ref. [28].Unfortunately, as merely quasistatic tests at room temperature were available, only the A, B, and n parameters could be calculated, remaining thus C and m to be fixed, taking for them standard values found in the literature for the most similar martensitic steels regarding the chemical composition, assuming these to be the same for the three different materials (Table 5). [29]In any case, neither high strain rates nor high temperatures are expected to take place during the industrial processes aimed at being simulated, therefore no major importance is considered to lie upon these values.
Once fixed, all three alloys are simulated under the Nakajima test procedure, performing first the comparison between the force versus displacement data extracted from them, employing both the J-C and the FLD curves as strength and fracture constitutive methods, and the actual data from the experiments.
In Figure 6, the comparison of Nakajima force-displacement data between simulated and experimental tests is shown for the extreme cases of stress state, taking these place when samples 1 and 5 are simulated (see Figure 5), having then uniaxial tension and equibiaxial tension conditions, respectively.It can be observed how the errors in both displacement to fracture and force exerted during the tests are relatively limited and even in all cases.In particular, the global average errors found were 11.7% and 14.9%, respectively.It must be remarked that as the displacement to fracture has been assessed as very lowly dependent on the work absorbed by the samples during the simulations, only the maximum force, and not the whole loading curve, has been considered for the calculation of the latter error.
Analyzing these results, the error of 11.7% obtained when simulating the displacement to fracture for all samples can tentatively be considered a success.The cracking phenomenon is highly complex, and errors within similar ranges are typically encountered when attempting to simulate such intricate physical processes.Regarding the error of 14.9% obtained when simulating the maximum force exerted during the tests, it has been assessed that the significance of this factor for simulating a material's ductility is not as high as the importance of the displacement to fracture.Therefore, the consequences of this result are not considered to have a major impact on the simulations.This will be proved with the of the simulations of the next sections, where in spite of the fact of having observed higher maximum forces in the Nakajima numerical models, as shown in Figure 6, the simulated FLD curves obtained will be more conservative than those extracted from the real tests.
Bearing this analysis in mind, the previous results can be initially considered good enough to be able to assess the simulations as successful, provided that these will be further validated in the  next sections of the document on the basis of subsequent and more complex tests.

Plastic Strain Distribution and FLD Curves Comparison in the Nakajima Tested Samples
After the previously demonstrated initial validation of the constitutive model for simulating the strength and fracture behavior of the different alloys, a more comprehensive comparison is conducted.This comparison focuses on the key data extracted from the Nakajima experiments, with the aim of fully validating the computational models.Specifically, the two considered QoI are the plastic equivalent strain (ε p ) prior to the fracture point as well as the FLD curves, which are considered as the most critical criteria for predicting the material's ability to withstand ductile damage during forming processes.Therefore, its validation in the computational case attains considerable importance to employ these models as future useful tools.
More specifically, for the validation of the simulated ε p , the same procedure as the one used to extract the experimental data is employed in the simulations, taking the highest value registered in any of the sample's mesh elements after the fracture of the material has occurred.
Once checked all the comparative values for every test (e.g., Figure 7), it has been calculated that the global average error when considering the 15 cases for every alloy and sample shape is 32.7%.It might be regarded as too large taking into consideration that the attainment of a high-fidelity model is one of the main goals in this work.However, it must be taken into account that the computational model tends to increase excessively the value of plastic strain in those elements close to the fractured ones, distorting in an artificial manner the extracted values of ε p .This comparison raises concerns about the suitability of using this approach to validate the model due to the presence of this undesirable numerical side effect.
On the other hand, a second path is explored when comparing now the experimental and simulated FLD curves.Again, the standard criteria established in ISO 12004-2 [27] are replicated to find the latter.In this particular case, the complexity increases because in order to capture the major and minor principal strain values in each simulation it is necessary to take all the major and minor principal strain values from the deformed node elements, which are perpendicular to the fracture zone.Then, both series of data, experimental and simulated, are approximated by a seconddegree equation employing the least squares minimization technique.Finally, the relevant data pair of each simulation test is taken as the greatest and lowest values of both maximum and minimum strain approximated curves, which should ideally take place in the same element node.
By performing this procedure and comparing the experimental and the simulated data, the FLD error results can be found.The pairs of curves are shown in Figure 8.
On the whole, upon examining these results, they can be considered partially successful, as there is generally a good agreement between the pairs of data.There is, however, a notable discrepancy in the case of alloy 3, where a significant gap exists between the simulated and the experimental curves.This mismatch is likely due to the absence of simulated data in the minor strain range from 0.0 to 0.08, which results in a loss of information regarding the actual fracture behavior of the material in that zone.When the global average error considering the three alloys is calculated, the outcome results in 13.9%.This result has been notably affected by the large error found for alloy 3 (18.5%),as previously explained, in contrast to those achieved for alloys 1 and 2 (12.4% and 10.9%, respectively), which are in a more common and acceptable range, considering the complex features of the ductile fracture phenomena aimed at being simulated in this article.Also, it must be noted that the FLD curves found are, in theory, more conservative than the experimental ones.Nonetheless, it has been observed in posterior experimental testing that the experimental curves are, in some cases, too optimistic, finding that the fracture tends to occur before reaching the limits they predict.This would support the agreement of the simulated curves with the real behavior of the material.
In any case, the latter errors are considerably lower than the previous error found for the QoI ε p comparison.This is probably caused by the fact that not only a single point is taken to extract the given computational result but a set of them, having this a balancing effect on the numerical inaccuracies produced during the simulations.As an outcome, it can be considered that the simulated models are, on the whole, validated at the sight of the latter findings.

FLSD versus FLD: Improving Fracture Modeling
FLD ductile fracture criterion has been widely used for the simulation of forming processes in order to predict the onset of necking or fracture occurrence.Nonetheless, as pointed out by works such as refs.[30,31] or ref. [32], some disagreements between these limit curves and the actual fracture behavior of metallic materials have been identified.Thus, these findings provide support for considering the use of the FLSD criterion based on stress as a relatively more favorable alternative, as indicated by these research studies.This criterion has been proved to bring some advantages, as the invariance of the limit curves themselves with respect to the thickness of the alloy sheets in comparison to the FLD criterion, or its notable capability to accurately predict necking in forming processes, as shown in ref. [33] or ref. [34], and further investigated in ref. [35] or, also considering other kinds of extended stress-based FLC criteria, in ref. [36].
The FLSD criterion relies on the concept of setting pairs of minor and major stresses shaping a limiting curve, such that if the stress state of the material is under it, the necking or fracture of the material will not happen.
The procedure employed to extract these ductile fracture limiting curves is based on computational methods, due to the restriction and complexity of the experimental equipment needed to calculate these values, as explained previously.Alternatively, validated models can be used to find them by taking maximum and minimum in-plane stresses present on the elements at the very moment in which necking is taking place on them.Thus, once this technique is employed, the values for the FLSD curves for each alloy are found.These are shown in Figure 9.
It can be observed how the different QP MSSs share very similar values for their forming limits, finding to a low degree very few discrepancies when reaching high minor stresses.Also, it must be noted that the fluctuations on each of the curves are due to the simulated nature of the extracted data and the numerical variations obtained in each of the elements from which the maximum and minimum values are sampled.Each of the QP MSSs' FLSD curves, and thus, the simulated alloys' forming limits found, will be computationally tested when performing the numerical tests of sheet cold stamping for the manufacturing of usual car parts in the next section.This is regarded as a first testing bench for the proper prediction of the industrial procedure feasibility before any actual part gets physically into the assembly line.As observed in Figure 10, the complexity related to the shape of both parts is uneven because the pillar possesses a surface filled with numerous bends of a small radius and a much more intricate geometry in general in comparison to the tunnel part, subjecting the material to higher strains and stresses during the stamping, which can have critical consequences toward the success of the process.This makes the B-pillar a more challenging geometry for a high strength alloy, while the tunnel part represents, in relative terms, a moderately difficult one.For these simulations, all three alloys are employed, making use of the already validated data to set up the ABAQUS models (Figure 11).In particular, they consist of a lower rigid die, fixed on the lower part, acting as the base, and an upper rigid die, moving at a standard speed of 1.25 mm s À1 , whereas the martensitic alloy sheet with 1.25 mm thickness is positioned in between both at the start of the simulations, counting on approximately 38 000 linear-S3R elements.The rest of the setup features for these models are replicated from those already described in Section 3.4.No guide pins are considered in the simulation setup, as the misalignment of the sheet during the deformation process is negligible.In the same way, the appearance of wrinkling is practically avoided without any additional parts due to the action of the forces exerted by the dies themselves as the part gets pressed.
The results of the different simulations for both parts can be observed in Figure 12 and 13.
As shown in the latter, no significant differences with regard to the final results considering the different alloys can be reported, as the widespread fracturing occurring in the run of the B-pillar is not avoided for any of the three metal materials.Likewise, the outcome of the models replicating the stamping process of the tunnel part is quite similar, at least in what concerns the nononset of cracks in the sheets for each QP MSS case, having this already been predicted in view of the similarities observed in the found FLSD curves.Similarly, as expected due to the difference in complexity of the parts, there is a clear contrast between the forming processes of both car pieces, being far from successful in the case of the B-pillar, whereas the tunnel part does not show any fully damaged element, therefore according to these outcomes it could be foreseen that, under the same cold stamping conditions, only the production of one of the latter seems feasible.It is important to notice that the springback issues are relevant during metal forming of stainless steels. [37]is topic has been out of scope of our article because it is impossible to consider all the issues related with metal-forming process.However, it should be noted that there is a body of research in the current literature clearly showing the springback effect can be significantly reduced in martensitic steel sheets by electrically single-pulsed current. [38]

Conclusions
In this work, a fully computational approach has been developed on the basis of experimental data extracted from quasistatic tensile and Nakajima tests, employing them to feed the modeling of cold-forming processes applied on three different martensitic In particular, the J-C model has been coupled with FLSD curves to simulate the behavior during the plastic straining phase and later potential fracture of the material, extracting the main values of both methods from the abovementioned experimental sources.The resulting models have been validated to a satisfactory degree with complementary data from the same actual physical tests, being finally used to replicate the expected outcome as a result of performing a cold stamping process for the manufacturing of two widespread automotive metal parts.
The output of the latter simulations seems to fit reasonably well with the anticipated behavior according to the characteristics observed during the modeling process, predicting a low ductility, and therefore, poor capacity to withstand the forming process when trying to shape complex surfaces cluttered with small bending radii.These expectations are also well in line with the previous experimental ductility performance of the different alloys witnessed during the Nakajima tests campaign.Nonetheless, it will be necessary to validate these numerical stamping runs with physical tests to fully confirm the usefulness of the models.
Altogether, the procedure is relatively fast and not heavily dependent on experimental data, which is something considered positive because these data must be necessarily extracted from expensive physical tests.Despite this simplification in the procedure, the method employed has been proven to be reliable in comparison to others when examined in previous research work.
The modeling shows extensive cracking during the cold forming of the B-pillar for all the three studied alloys.In contrast, no single damage element was detected in the case of cold forming of the tunnel part for any of the three alloys.Therefore, producing the tunnel parts from the QP-treated MSSs seems feasible.
Finally, the proposed approach can be easily applied to predict the formability of other complex shape parts made of QP-treated MSSs.

Figure 2 .
Figure 2. Engineering stress-strain curves for the three alloys.

Figure 3 .
Figure 3. EBSD band contrast maps of the QP-treated alloys.In a green overlay, the retained austenite is highlighted: a) alloy 1, b) alloy 2, and c) alloy 3.

Figure 5 .
Figure 5. a) Nakajima test samples (1-5, from left to right) tested and b) computational layout of Nakajima test using ABAQUS software.

Figure 6 .
Figure 6.Nakajima force versus displacement data comparison between simulated and experimental tests for a,b) alloy 1, c,d) alloy 2, and e,f ) alloy 3.

Figure 7 .
Figure 7. Nakajima test comparison for ε p , showing a) experimental versus b) computational results, with maximum values of 0.29 and 0.343 marked with a white dot, respectively.

Figure 8 .
Figure 8.Comparison of experimental and simulated FLD curves for a) alloy 1, b) alloy 2, and c) alloy 3.

Figure 10 .
Figure 10.Actual final shape of the automotive parts whose stamping process is simulated: a) B-pillar and b) tunnel.[40]

Figure 11 .
Figure 11.Setup of the ABAQUS simulation models for both parts: a) B-pillar and b) tunnel.

Figure 13 .
Figure 13.Final results of transmission tunnel stamping simulations for a,b) alloy 1, c,d) alloy 2, and e,f ) alloy 3.

Table 1 .
Chemical composition of the studied alloys in wt%.

Table 2 .
QP treatment parameters of this study.

Table 3 .
Extracted data from tensile curves.

Table 4 .
Microstructural parameters of selected Nakajima samples of alloy 420ma.

Table 5 .
Values employed for the three different alloys in every parameter and constant considering the J-C model.