Characterization and Prediction of Plane Strain Bendability in Advanced High-Strength Steels

: The rapid development of new classes of automotive steels such as the 3rd generation of advanced high-strength steels has created the need for the efﬁcient characterization of their mechanical properties in loading scenarios other than uniaxial tension. The VDA 238-100 tight radius bend test has gained widespread acceptance in recent years for characterizing performance in plane strain bending, but there is uncertainty surrounding the use of the bend angle and its interrelation with the test parameters. The objective of the present study is to investigate the intertwined effects of the sheet thickness, bend radius, and tensile properties upon the bendability of seven advanced high-strength steels in different thicknesses for a total of 83 conditions. Practical correlations are developed to predict the bend angle and plane strain fracture strain as functions of the bending conditions and tensile mechanical properties. An extensive dataset comprising 26 additional advanced high-strength steel test cases was compiled from the literature to evaluate the proposed correlation for the plane strain fracture strain.


Introduction
The development of advanced-high strength steels (AHSS) has enabled the downgauging of automotive body structures without compromising manufacturability and safety requirements [1,2].The need for structural lightweighting has further accelerated in recent years with the introduction of electric vehicles to compensate for the significant weight addition of the battery pack [3,4].This trend of using increasingly higher strength grades of AHSS for lightweighting has tightened product design windows such that applicationspecific performance metrics are needed to fully exploit the formability of these steels with increasingly complex microstructures, such as the 3rd generation of AHSS (GEN3 AHSS).
A persistent misconception in material selection has been the assumption of a direct relationship between global and local formability.For example, the uniform elongation and forming limit curves (FLCs) are metrics related to global formability, defined by the onset of instability and acute neck formation.These metrics do not necessarily correlate with local formability, which is synonymous with fracture resistance in loading scenarios when a neck does not form, such as in tight radius bending, stretch flanging, and hole extrusion operations.A correlation between global and local formability may hold for select material classes of AHSS, but it is not a general rule, as clearly seen in third-generation steels.Shen et al. [5] conducted a comprehensive study on dual-phase DP1000 and medium-Mn steel (MMnS) to demonstrate the distinguishable behavior of global and local formability.It was reported that the transformation-induced plasticity (TRIP) effect significantly improved the global formability of the MMnS.However, the local formability was lower than the DP1000 steel despite the tensile fracture elongation of MMnS being significantly higher than the DP1000.Therefore, it is imperative to perform local formability tests such as the tight radius v-bend test in parallel with conventional global formability tests to obtain a proper evaluation of material performance.
The VDA238-100 tight radius v-bend test [6] is ideal for characterizing local formability due to its simplicity, and how it provides loading in plane strain, which is the limiting stress state for both the FLC and fracture characterization [7][8][9][10].Plane strain bending can be obtained by bending a rectangular blank (minimum width of about 20 times the sheet thickness) between two rollers and a sharp punch of a 0.2 mm or 0.4 mm tip radius.The sharp bend radius induces severe through-thickness stress and strain gradients where the inner (concave) layer undergoes plane strain compression, while the outer (convex) surface experiences plane strain tension.Necking is inhibited due to the through-thickness strain gradients with fracture initiated at, or just below, the convex surface in tension under proportional plane stress loading [10][11][12].The severity of the plane strain bend can be expressed by a ratio between the sheet thickness and the punch radius, t/R p .
The VDA 238-100 specification recommends that the fracture be identified based upon a specified reduction of the peak punch force (30 N or 60 N depending on the material type and strength) to coincide with the formation of surface cracking.The fracture bend angle is computed from geometric relations and used as the performance metric.To retrieve the fracture strains in the v-bend test for the calibration of stress-state dependent phenomenological fracture models, Roth and Mohr [13] and Cheong et al. [11] inverted the VDA 238-100 bend setup to enable the DIC strain measurement on the convex surface where the fracture is initiated.Despite this advancement, two major challenges remain in using the VDA bend test for reliable fracture characterization.The first concern is related to the bend severity and ductility of the sheet metal since the specimen may perform a full bend without material rupture, which is referred to as folding over.To mitigate this limitation, the VDA 238-100 specification suggests pre-straining the flat strips prior to bending.However, Noder et al. [14] demonstrated that this procedure constitutes nonproportional loading of uniaxial tension followed by plane strain tension.The non-linear strain path produced by the uniaxial pre-straining affected the fracture strains and bend angles for DP1180, AA5182, and ZEK100 steel.For thin and/or ductile materials that fold over without fracture in the v-bend test, the dihedral punch test of Grolleau et al. [15] can be employed.
The second concern in the VDA bend test is its use of a load-based criterion for fracture detection.It was found in the studies by Larour et al. [16], Cheong et al. [17], and Noder et al. [18] that a reduction in the punch force does not necessarily coincide with material cracking.The punch force may drop even in the absence of material rupture.Troive [19] and Noder et al. [18] explained this phenomenon, referred to as a false positive, through a transition from an ideal three-point to a four-point bending scenario wherein the vertical force component naturally reduces.Troive [19] proposed fracture identification using the bending moment instead of the punch force, and Noder et al. [18] defined a bending stress metric that accounts for the geometry, punch force, and thinning of the cross-section.
While extensive research has been performed to critically assess the reliability of fracture detection methods in the VDA 238-100 bend test, relatively limited attention has been paid to the rationale behind using the bend angle to characterize bendability, given that it is not a unique property.The experimental study of Cheong et al. [17] demonstrated that the fracture strain is inherent to the material and should be used instead of the bend angle by conducting v-bend experiments on multiple grades of AHSS (DP600, DP980, and HS1500) with three different bend severities, adopting 0.2 mm, 0.4 mm, and 1 mm punch radii.Cai et al. [20] also reported the utility of the fracture strain as an independent performance metric by investigating press-hardened steels (PHS) with different surface conditions and thicknesses.Kim and Hance [21] conducted a regression analysis of tensile properties and the bend angle to relate the bend angle to the sheet thickness, uniform elongation, and local fracture strain of the minimum cross-section in uniaxial tension.
While these studies have advanced the understanding of the v-bend test, it remains unclear how transferable the results are across the range of AHSS.A comprehensive experimental study was undertaken for seven AHSS encompassing various strength levels from 780-1500 MPa, supplied in three to four different sheet thicknesses (1.0 to 1.6 mm).Punches with four different radii were employed to explore a wide range of bend severities, resulting in a total of 83 unique test conditions.The intention is to experimentally decouple the intertwined effect of bend parameters such as sheet thickness and punch radius on the bend performance.The interrelations and universality of the bending parameters are critically accessed and correlated with the tensile mechanical properties.Building upon the study of Kim and Hance [21], prediction models were developed to estimate the bend angle from the tensile properties and plane strain bending fracture strain.The validity of the fracture strain correlation was then evaluated using a separate set of AHSS data compiled from the literature.

Experimental Setup
Figure 1 provides an overview of the inverted v-bend test apparatus developed by Cheong et al. [11].The punch is stationary to have a fixed focal distance for DIC strain measurement with the bending performed by two rollers travelling downward.The rollers have a 30 mm nominal diameter and are chamfered to provide a wider viewing space for the DIC cameras.The roller surfaces were lubricated with the PTFE (Teflon) spray to reduce friction, and the rollers were on bearings for smooth rotation.properties and the bend angle to relate the bend angle to the sheet thickness, uniform elongation, and local fracture strain of the minimum cross-section in uniaxial tension.While these studies have advanced the understanding of the v-bend test, it remains unclear how transferable the results are across the range of AHSS.A comprehensive experimental study was undertaken for seven AHSS encompassing various strength levels from 780-1500 MPa, supplied in three to four different sheet thicknesses (1.0 to 1.6 mm).Punches with four different radii were employed to explore a wide range of bend severities, resulting in a total of 83 unique test conditions.The intention is to experimentally decouple the intertwined effect of bend parameters such as sheet thickness and punch radius on the bend performance.The interrelations and universality of the bending parameters are critically accessed and correlated with the tensile mechanical properties.Building upon the study of Kim and Hance [21], prediction models were developed to estimate the bend angle from the tensile properties and plane strain bending fracture strain.The validity of the fracture strain correlation was then evaluated using a separate set of AHSS data compiled from the literature.

Experimental Setup
Figure 1 provides an overview of the inverted v-bend test apparatus developed by Cheong et al. [11].The punch is stationary to have a fixed focal distance for DIC strain measurement with the bending performed by two rollers travelling downward.The rollers have a 30 mm nominal diameter and are chamfered to provide a wider viewing space for the DIC cameras.The roller surfaces were lubricated with the PTFE (Teflon) spray to reduce friction, and the rollers were on bearings for smooth rotation.All specimens were prepared as 60 mm × 60 mm squares and tested in the transverse direction (TD), which is generally recognized as the limiting fracture direction of AHSS.The transverse direction tends to have the highest strength and lowest ductility.In the ISO 12004-2 (2008) [22] standard for formability characterization, the TD is recommended for steel.Testing of the TD in the v-bend test corresponds to the punch being parallel to the rolling direction, so the major strain is in the direction of the TD, as shown in Figure 2b.All specimens were prepared as 60 mm × 60 mm squares and tested in the transverse direction (TD), which is generally recognized as the limiting fracture direction of AHSS.The transverse direction tends to have the highest strength and lowest ductility.In the ISO 12004-2 (2008) [22] standard for formability characterization, the TD is recommended for steel.Testing of the TD in the v-bend test corresponds to the punch being parallel to the rolling direction, so the major strain is in the direction of the TD, as shown in Figure 2b.The VDA 238-100 test reports the load response and the punch displacement as primary measurements.For AHSS, the onset of failure is determined by a 60 N reduction from the peak load.The bend angle, α, is estimated using the roller radius, Rr; sheet thickness, t; stroke, S; roller gap, L; and punch radius, Rp.Note that since 2017, the VDA 238-100 specification uses the same bend angle formula as ISO 7438 (2005) [23].Larour et al. [16] proposed an equivalent version of this bend angle formula in a simpler form, as Four different punch radii were considered to change the bend severity along with considering different gauges for each material grade.The two VDA-standardized punches with tip radii of 0.2 and 0.4 mm were considered along with two additional punches with 1.0 and 2.0 mm radii.The VDA roller gap does not consider the punch radius, Rp, which will lead to insufficient clearance to perform a complete bend for punches larger than 0.2 mm.Therefore, the roller gap formula was modified as  (mm) = 2 + 0.5 + 2  − 0.2 . ( A constant virtual strain gauge length (VSGL) of 0.50 mm was employed in the DIC strain analysis consistent with the convergence study performed by Cheong et al. [17].The VSGL is defined as As illustrated in Figure 3, surface strain profiles were extracted using five equallyspaced line slices across the specimen width, and averaged to obtain the representative strain measurement.The VDA 238-100 test reports the load response and the punch displacement as primary measurements.For AHSS, the onset of failure is determined by a 60 N reduction from the peak load.The bend angle, α, is estimated using the roller radius, R r ; sheet thickness, t; stroke, S; roller gap, L; and punch radius, R p .Note that since 2017, the VDA 238-100 specification uses the same bend angle formula as ISO 7438 (2005) [23].Larour et al. [16] proposed an equivalent version of this bend angle formula in a simpler form, as Four different punch radii were considered to change the bend severity along with considering different gauges for each material grade.The two VDA-standardized punches with tip radii of 0.2 and 0.4 mm were considered along with two additional punches with 1.0 and 2.0 mm radii.The VDA roller gap does not consider the punch radius, R p , which will lead to insufficient clearance to perform a complete bend for punches larger than 0.2 mm.Therefore, the roller gap formula was modified as A constant virtual strain gauge length (VSGL) of 0.50 mm was employed in the DIC strain analysis consistent with the convergence study performed by Cheong et al. [17].The VSGL is defined as As illustrated in Figure 3, surface strain profiles were extracted using five equallyspaced line slices across the specimen width, and averaged to obtain the representative strain measurement.For this study, the VDA load-based fracture definition was employed as a primary method to determine the failure, since the tested AHSS steels had 780 MPa and higher tensile strength.A clear identification of a fracture coinciding with the peak load was observed for most cases; see Figure 4 for an example in MS1500 steel.During the study, the influence of the punch radius upon the global load response was also observed.The larger punches flatten the load curves, and depending upon the material of interest, it can significantly alter the impact of the 60 N load offset of the VDA load-based analysis.This phenomenon is further discussed in Section 4.2.To overcome this phenomenon, a local strain-rate method consistent with the linear best fit (LBF) methodology of Volk and Hora [24] was considered for supplementary analysis in select test cases for verification purposes.As demonstrated in Figure 5, the fracture strain can be identified by a rapid increase in surface strain.Linear fits for the stable and unstable strain rate regimes are established with their intersection corresponding to the onset of surface cracking for failure strain determination.For this study, the VDA load-based fracture definition was employed as a primary method to determine the failure, since the tested AHSS steels had 780 MPa and higher tensile strength.A clear identification of a fracture coinciding with the peak load was observed for most cases; see Figure 4 for an example in MS1500 steel.For this study, the VDA load-based fracture definition was employed as a primary method to determine the failure, since the tested AHSS steels had 780 MPa and higher tensile strength.A clear identification of a fracture coinciding with the peak load was observed for most cases; see Figure 4 for an example in MS1500 steel.During the study, the influence of the punch radius upon the global load response was also observed.The larger punches flatten the load curves, and depending upon the material of interest, it can significantly alter the impact of the 60 N load offset of the VDA load-based analysis.This phenomenon is further discussed in Section 4.2.To overcome this phenomenon, a local strain-rate method consistent with the linear best fit (LBF) methodology of Volk and Hora [24] was considered for supplementary analysis in select test cases for verification purposes.As demonstrated in Figure 5, the fracture strain can be identified by a rapid increase in surface strain.Linear fits for the stable and unstable strain rate regimes are established with their intersection corresponding to the onset of surface cracking for failure strain determination.During the study, the influence of the punch radius upon the global load response was also observed.The larger punches flatten the load curves, and depending upon the material of interest, it can significantly alter the impact of the 60 N load offset of the VDA load-based analysis.This phenomenon is further discussed in Section 4.2.To overcome this phenomenon, a local strain-rate method consistent with the linear best fit (LBF) methodology of Volk and Hora [24] was considered for supplementary analysis in select test cases for verification purposes.As demonstrated in Figure 5, the fracture strain can be identified by a rapid increase in surface strain.Linear fits for the stable and unstable strain rate regimes are established with their intersection corresponding to the onset of surface cracking for failure strain determination.

Material Selection
This study examined seven grades of commercial AHSS ranging from 780 MPa to 1500 MPa in 3 to 4 gauges ranging from 1.0 to 1.6 mm, as shown in Table 1.The 980-GEN3 and the 1180-GEN3 were included in the investigation representing newer-generation steels with superior formability.All the steels were provided by the American Iron and Steel Institute (AISI) material sample bank, where the steels were donated by its member companies.
Table 1.Classes and sheet thicknesses of the provided advanced high-strength steels in the study.

Material
Sheet Thickness [mm] The tensile properties were characterized using JIS No. 5 tensile specimens machined in the transverse direction (TD) to be consistent with the bend test.The JIS tensile geometry is shown in Figure 6, and the engineering stress-strain curves are in Figure 7 with the mechanical properties summarized in Table 2.The tensile properties were measured using 50 mm vertical extensometers, and the local surface fracture strain was extracted using DIC.The r-value was determined over a range of axial true strain from 0.01 to 80% of the uniform elongation.

Material Selection
This study examined seven grades of commercial AHSS ranging from 780 MPa to 1500 MPa in 3 to 4 gauges ranging from 1.0 to 1.6 mm, as shown in Table 1.The 980-GEN3 and the 1180-GEN3 were included in the investigation representing newer-generation steels with superior formability.All the steels were provided by the American Iron and Steel Institute (AISI) material sample bank, where the steels were donated by its member companies.The tensile properties were characterized using JIS No. 5 tensile specimens machined in the transverse direction (TD) to be consistent with the bend test.The JIS tensile geometry is shown in Figure 6, and the engineering stress-strain curves are in Figure 7 with the mechanical properties summarized in Table 2.The tensile properties were measured using 50 mm vertical extensometers, and the local surface fracture strain was extracted using DIC.The r-value was determined over a range of axial true strain from 0.01 to 80% of the uniform elongation.All materials provided for this study met the nominal ultimate tensile strength (UTS).A noticeable variation in mechanical properties and hardening behavior with sheet thickness was seen for the material groups of DP780, MP980, and 1180-GEN3 steels.These observations will be considered further in Section 4 when interpreting the VDA test results.

VDA 238-100 Bend Test: Results and Analysis
It is important to first discuss how the punch tip lift-off and fold-over can affect the interpretation of the VDA test data.Punch tip lift-off, shown in Figure 8a, invalidates the common assumption that the inner radius of the specimen is the same as the punch radius.Tests with punch tip lift-off are still suitable for fracture characterization using surface strain measurement to determine the plane strain fracture limit.It is noted that lift-off will increase the effective bend severity and affect the bend angle.Specimen fold-over occurs when the bend severity is not sufficient to induce fracture, leading to the specimen wrapping around the punch without fracture.The vertical load will still drop due to the mechanics of the test, which may be incorrectly identified as the onset of fracture according to the VDA load threshold, as discussed by Noder et al. [18], and shown schematically in Figure 8b.Punch tip lift-off may also occur during fold-over due to the large bend angles involved.
Fortunately, lift-off and fold-over can be readily detected using DIC.As shown in Figure 9a, the cross-section thins during severe bending, which can be tracked by the out-of-plane displacement since the punch is stationary.The reversal of thinning indicates that the specimen has lifted off from the punch surface.Punch tip lift-off can also create an additional force against the punch, resulting in a load increase as depicted in Figure 9a.The onset of folding over can be identified by tracking the surface strain evolution and the strain rate.As shown in Figure 9b, the stagnation of the surface strain development with a decreasing strain rate indicates folding over.A slow and gradual load drop after the stagnation is a false positive in this case.Consequently, tests in which lift-off or fold-over occurred have been omitted from further analyses with the cases summarized in Table 3. Punch lift-off was most observed in tests with punch radii larger than 0.4 mm, whereas the folding-over was more frequent for the higher ductility materials and for the thinner gauges due to reduced bend severity.Fortunately, lift-off and fold-over can be readily detected using DIC.As sh Figure 9a, the cross-section thins during severe bending, which can be tracked by t of-plane displacement since the punch is stationary.The reversal of thinning indica the specimen has lifted off from the punch surface.Punch tip lift-off can also cr additional force against the punch, resulting in a load increase as depicted in Fig The onset of folding over can be identified by tracking the surface strain evolution strain rate.As shown in Figure 9b, the stagnation of the surface strain developme a decreasing strain rate indicates folding over.A slow and gradual load drop a stagnation is a false positive in this case.Consequently, tests in which lift-off or fo occurred have been omitted from further analyses with the cases summarized in T Punch lift-off was most observed in tests with punch radii larger than 0.4 mm, w the folding-over was more frequent for the higher ductility materials and for the gauges due to reduced bend severity.Fortunately, lift-off and fold-over can be readily detected using DIC.As shown in Figure 9a, the cross-section thins during severe bending, which can be tracked by the outof-plane displacement since the punch is stationary.The reversal of thinning indicates that the specimen has lifted off from the punch surface.Punch tip lift-off can also create an additional force against the punch, resulting in a load increase as depicted in Figure 9a.The onset of folding over can be identified by tracking the surface strain evolution and the strain rate.As shown in Figure 9b, the stagnation of the surface strain development with a decreasing strain rate indicates folding over.A slow and gradual load drop after the stagnation is a false positive in this case.Consequently, tests in which lift-off or fold-over occurred have been omitted from further analyses with the cases summarized in Table 3. Punch lift-off was most observed in tests with punch radii larger than 0.4 mm, whereas the folding-over was more frequent for the higher ductility materials and for the thinner gauges due to reduced bend severity.

Influence of Bend Severity on Bending Performance
The MS1500 steel had the largest number of successful test conditions at 14, with the results shown in Figure 10.In line with the observations of Cheong et al. [17], the plane strain fracture limit was insensitive to the bend severity so long as the mechanical properties were consistent for the different sheet gauges.In contrast to the fracture strain, the bend angle at the VDA load threshold was found to decrease with increasing sheet thickness, indicating its dependence on the bend severity, as seen in Figure 11.In addition, the results of the MP980 material group are also shown in Figure 12, where the same trends were observed for MS1500.Similar behavior was observed for all other AHSS considered, except for the DP780 material group for which discrepancies were reported in tensile properties across the different gauges (1.0 mm, 1.2 mm, and 1.4 mm), as illustrated in Figure 13.Specifically, a 25% difference in uniform elongation, an 18% difference in total elongation, and an 80% difference in the local DIC fracture strain were observed.Note that these differences are the percentage differences defined as where x is the variable of interest.
where x is the variable of interest.
The large variation in the DP780 is attributed to its microstructure, and the difference in tensile properties with sheet thickness implies that the microstructures were not as consistent as other materials in the study provided by the steel producers.Therefore, the DP780 was excluded from the comparative analysis between the gauges, but still considered in the analysis based on other relevant parameters, such as material strength and the bend radius.

Bend Radius and Time-Dependent Analysis
During the investigation using the VDA 238-100 load-based criterion, a subtle decreasing trend in the fracture strain was observed with larger punch radii.An example of this decreasing trend is shown in Figure 14a for PHS1500 and is attributed to the punch radius effect on the load response, as seen in Figure 14b.Due to the wider roller gap associated with the larger punch, the load curve flattens, leading to a premature detection of fracture where the peak load point is shifted away from the rapid drop of the load.The large variation in the DP780 is attributed to its microstructure, and the difference in tensile properties with sheet thickness implies that the microstructures were not as consistent as other materials in the study provided by the steel producers.Therefore, the DP780 was excluded from the comparative analysis between the gauges, but still considered in the analysis based on other relevant parameters, such as material strength and the bend radius.

Bend Radius and Time-Dependent Analysis
During the investigation using the VDA 238-100 load-based criterion, a subtle decreasing trend in the fracture strain was observed with larger punch radii.An example of this decreasing trend is shown in Figure 14a for PHS1500 and is attributed to the punch radius effect on the load response, as seen in Figure 14b.Due to the wider roller gap associated with the larger punch, the load curve flattens, leading to a premature detection of fracture where the peak load point is shifted away from the rapid drop of the load.To this end, the local strain-rate dependent fracture detection method-or LBF method-was considered, since it is independent of the load history.The test cases for local analysis were selected based on a percentage difference in the fracture strain between the smallest and the largest bend severities used.The average percentage differences for each material are listed in Table 4.
Table 4. Average percentage differences in the fracture strain of each material in different gauges and punch radii.The VDA load drop criterion was used to determine the fracture strain in all of the tests.To this end, the local strain-rate dependent fracture detection method-or LBF method -was considered, since it is independent of the load history.The test cases for local analysis were selected based on a percentage difference in the fracture strain between the smallest and the largest bend severities used.The average percentage differences for each material are listed in Table 4. Table 4. Average percentage differences in the fracture strain of each material in different gauges and punch radii.The VDA load drop criterion was used to determine the fracture strain in all of the tests.The significant variation in the DP780 material group was anticipated from the inconsistent tensile mechanical properties, as illustrated in Figure 13.Three other grades with a percentage difference higher than 10%, namely 980-GEN3, 1180-GEN3, and PHS1500, were re-evaluated using the LBF method.The fracture strains obtained using the LBF method restored the observation that the plane strain bending fracture limit is insensitive to the bend severity (see Figure 15 for the PHS1500) if the material properties are consistent.The updated percentage differences of the fracture strains for 980-GEN3, 1180-GEN3, and PHS1500 were reduced to 1.4%, 2.0%, and 1.1%, respectively.The LBF data are tabulated in Table A2.These results highlight the need to incorporate DIC strain measurement into the VDA test to supplement or replace its dependence upon a load threshold for fracture detection.The test results of 1180-GEN3 and PHS1300 showed similar limit strains from both VDA and time-dependent methods, with average differences of 3.3% and 4.8%, respectively.The 980-GEN3 showed an 18% difference between the VDA and the time-dependent limit strains.In this case, the LBF fracture strain appears to be in better agreement with the formation of a visible surface crack, while the VDA threshold was conservative, as demonstrated in Figure 16.The test results of 1180-GEN3 and PHS1300 showed similar limit strains from both VDA and time-dependent methods, with average differences of 3.3% and 4.8%, respectively.The 980-GEN3 showed an 18% difference between the VDA and the time-dependent limit strains.In this case, the LBF fracture strain appears to be in better agreement with the formation of a visible surface crack, while the VDA threshold was conservative, as demonstrated in Figure 16.

Materials
The test results of 1180-GEN3 and PHS1300 showed similar limit strains from both VDA and time-dependent methods, with average differences of 3.3% and 4.8%, respectively.The 980-GEN3 showed an 18% difference between the VDA and the time-dependent limit strains.In this case, the LBF fracture strain appears to be in better agreement with the formation of a visible surface crack, while the VDA threshold was conservative, as demonstrated in Figure 16.

Evaluation of Bend Performance with Tensile Properties
A relationship between plane strain formability and conventional tensile properties was investigated for practical application during grade development or when bend data is not available.As shown in Figure 17, there appears to be a first-order linear relationship between the bend angle and both the uniform and total elongations.However, there was no direct correlation between the elongation metrics and the plane strain fracture strain, which governs material bendability.

Evaluation of Bend Performance with Tensile Properties
A relationship between plane strain formability and conventional tensile properties was investigated for practical application during grade development or when bend data is not available.As shown in Figure 17, there appears to be a first-order linear relationship between the bend angle and both the uniform and total elongations.However, there was no direct correlation between the elongation metrics and the plane strain fracture strain, which governs material bendability.
Nevertheless, it is worthwhile to perform a multi-variable correlation of the bend angle with respect to the mechanical properties available from the tensile test.The bend angle correlation proposed by Kim and Hance [21] has been modified to use the bend severity, t/R p , instead of sheet thickness, and incorporate the tensile fracture strain obtained from stereoscopic DIC.The proposed bend angle correlation is: where ε u = ln[1 + (UE/100)] is the tensile strain calculated by the uniform elongation (UE), and ε DIC, fracture is the major surface strain at fracture measured using DIC.The coefficients, C 1 -C 5 , of the prediction model and the corresponding results are summarized in Figure 18 and Table 5 along with the 95% confidence interval (CI) and the prediction interval (PI).
The critical variables to construct the 95% confidence and prediction interval bands are also listed in Table 5.The t-statistic is the coefficient-to-standard error ratio describing the distribution of the dataset, and the p-value indicates the probability of each variable, which relates to the significance level of the variable in the correlation.The confidence and the prediction range can be calculated as where x predicted is a predicted value, t critical is the critical value of the t-statistic for the 95% confidence interval, and SE is the standard error of the interval.Nevertheless, it is worthwhile to perform a multi-variable correlation of the bend angle with respect to the mechanical properties available from the tensile test.The bend angle correlation proposed by Kim and Hance [21] has been modified to use the bend severity, t/Rp, instead of sheet thickness, and incorporate the tensile fracture strain obtained from stereoscopic DIC.The proposed bend angle correlation is: where εu = ln[1 + (UE/100)] is the tensile strain calculated by the uniform elongation (UE), and εDIC, fracture is the major surface strain at fracture measured using DIC.The coefficients, C1-C5, of the prediction model and the corresponding results are summarized in Figure 18 and Table 5 along with the 95% confidence interval (CI) and the prediction interval (PI).
The critical variables to construct the 95% confidence and prediction interval bands are also listed in Table 5.The t-statistic is the coefficient-to-standard error ratio describing the distribution of the dataset, and the p-value indicates the probability of each variable, which relates to the significance level of the variable in the correlation.The confidence and the prediction range can be calculated as where xpredicted is a predicted value, tcritical is the critical value of the t-statistic for the 95% confidence interval, and SE is the standard error of the interval.The UTS and the uniform elongation are clearly significant based on their p-values, but it is interesting to note that the DIC fracture strain and the bend severity were not statistically significant with their p-values of 0.83 and 0.20, respectively.It is noted that Kim and Hance [21] performed area reduction measurements of the tensile tests instead of the DIC surface strain.The local DIC surface strain was selected in the present study because it is readily available, with DIC becoming commonly used, while area measurements require additional post-mortem measurements.The area reduction approach is attractive as it provides the major strain of the cross-section, but its value also varies with the cross-section geometry of the tensile specimen.A large cross-sectional aspect ratio promotes shear band localization, while an aspect ratio lower than four tends to promote a triaxial necking mode.Consequently, the local DIC surface strain, while also imperfect, was used for its simplicity and availability.
Instead of using the tensile fracture strain, it is instructive to re-evaluate the bend angle correlation of Equation ( 5) by replacing the uniaxial fracture strain with the plane strain fracture strain at the VDA threshold, ε f,VDA , resulting in The multiple linear regression models for Equation (7) using the entire dataset from this study are shown in Figure 19 and Table 6.An excellent correlation with R-sq of 0.98 was obtained with all variables significant to 99% confidence.
However, the plane strain fracture limit is often unknown, since DIC is currently not a requirement in the VDA bend test.Tensile data and the VDA bend angle are now generally considered to be baseline characterization tests for AHSS, with the data typically available from the material producer.Therefore, it is informative to re-arrange Equation (7) to solve for the plane strain fracture strain as where the corresponding coefficients for Equation (8) are shown in Table 7 with an R-sq of 0.92.However, the plane strain fracture limit is often unknown, since DIC is currently not a requirement in the VDA bend test.Tensile data and the VDA bend angle are now generally considered to be baseline characterization tests for AHSS, with the data typically available from the material producer.Therefore, it is informative to re-arrange Equation (7) to solve for the plane strain fracture strain as where the corresponding coefficients for Equation (8) are shown in Table 7with an R-sq of 0.92.The influence of ultimate tensile strength (UTS) on the fracture strain prediction model was reported to be insignificant and should be excluded from the correlation in Equation (8).It is important to carefully interpret this outcome because the UTS indeed has a strong linear correlation with the bend angle parameter as shown in Table 6.One could substitute the linear relation for the bend angle of Equation (7) into Equation (8) to see that an explicit dependence upon the UTS would emerge.From a physical perspective, there should be no intercept in Equation ( 8) with the fracture strain being zero when its predictors are zero, such as in the case of the bend angle.The regression analysis confirms that the intercept should be removed with its p-value of 0.28.This sequence of adjustments will yield a final correlation as with an R-sq of 0.92 based on the coefficients introduced in Table 8, and all predictors achieved a level of significance of 99% based upon the p-values that are less than 0.01.Materials and CAE engineers can use Equation ( 9) to readily estimate the plane strain fracture limit for use in forming and crash simulations of AHSS when DIC data is not available from the v-bend test.Note that Equation (9) provides major strain at the fracture in plane strain, and that the CAE analyst would need to assume a constitutive model to convert it to a work-conjugate equivalent strain.The first term of the correlation in Equation ( 9) represents the combined impact of the fracture bend angle and the bend severity.The bend severity, t/R p , serves to adjust the measured bend angle as the fracture strain should be independent of bend severity.The second term related to uniform elongation provides insight into the hardening behavior, as it is equivalent to the n-value for a power law hardening model.From this perspective, the negative dependence upon the hardening rate aligns with empirical observations that local formability generally improves for lower n-values of the same grade.However, it is important to note the strong positive correlation of the uniform elongation (n-value) with the bend angle observed in Equation (7).A higher n-value contributes to reaching higher levels of deformation, but there is a trade-off in terms of the fracture strain.Since necking is suppressed due to the mechanics of tight radius bending, it is postulated that the higher n-value allows for more diffuse strain partitioning over the bend to achieve a higher bend angle for a lower value of strain.The higher n-value will also produce higher stress gradients across the bend and through-thickness that may promote fracture in AHSS, which typically possess multiple phases of different strengths such as martensite and ferrite in DP steels.In the limit of a material with no hardening (n = 0), there would still be a through-thickness strain gradient due to plastic bending, but no stress gradient, which may serve to inhibit damage development when necking is suppressed.Future studies are required to investigate the bend performance of AHSS that can account for the microstructure and explain the observed relationships in the phenomenological models of Equations ( 7)- (9).
To independently evaluate the fracture strain correlation in Equation ( 9), a separate dataset comprising 26 different AHSS test conditions ranging from 590 MPa to 1500 MPa was compiled from the literature [11,14,17,18,[25][26][27]. It is important to note that this diverse range of AHSS includes 13 steels with strengths lower than 780 MPa, which was the lowest strength level used in the calibration of Equation (9).AHSS of the same grade provided by different suppliers are identified with M1, M2, or M3 designations.Ductibor ® 500 and Usibor ® 1500 are press-hardened steels referred to as D500 and U1500, respectively, with a brief indication of the thermal process.The Ductibor ® 500 and Usibor ® 1500 were subjected to different quench conditions, resulting in significant variations in strength and ductility.Moreover, it is important to note that the fracture strains of 590R and Ductibor ® 500 were identified using the stress metric method to accommodate their exceptional ductility beyond the capability of the VDA's load-based analysis [18].A summary of the material conditions is in Table 9, and the detailed dataset can be found in Table A3 in Appendix A.

Material
Thickness Figure 20 demonstrates that the regression model exhibits a slight underprediction of the experimental fracture strains with a trendline slope of 0.95.Nonetheless, the R-sq value with respect to the ideal prediction (y = x) is 0.86, indicating a strong correlation.It is emphasized that analysts using the correlation should carefully consider the 95% confidence and prediction intervals in interpreting the predicted mean fracture strains due to the inherent variability associated with fracture.The majority of the data is captured within the 95% confidence interval for the mean fracture strain.The 95% prediction interval provides the expected range in which the fracture strain may occur in a test of the AHSS.The prediction interval does not correspond to the mean, but instead corresponds to the value of a new measurement.9)) using a new set of materials in Table A3.The R-sq value is calculated based on the ideal prediction case of y = x.

Conclusions
A comprehensive experimental study was conducted to investigate the plane strain bending performance of seven grades of advanced high-strength steels under test conditions that span a range of tensile strengths from 780 MPa to 1500 Mpa, and bend severities, y = 0.95x  9)) using a new set of materials in Table A3.The R-sq value is calculated based on the ideal prediction case of y = x.

Conclusions
A comprehensive experimental study was conducted to investigate the plane strain bending performance of seven grades of advanced high-strength steels under test conditions that span a range of tensile strengths from 780 MPa to 1500 Mpa, and bend severities, t/R p , from 0.5 to 8. The main conclusions are summarized as follows:

•
The plane strain bending fracture limit can be taken as an inherent material property.The fracture limit obtained for a grade of AHSS appears transferable between gauges if the tensile mechanical properties are comparable, implying that the microstructures are sufficiently similar.The fracture bend angle is not a unique metric, and is influenced by bend severity.The bend angle is instructive for material ranking in conditions with similar bend severities.

•
The applicability of the VDA load-based analysis with a threshold of 60 N is dependent upon the imposed bend severity.Larger punches produce a flatter profile in the load curve, and can produce a mild but false dependency of the fracture strain with bend severity.

•
In addition to providing the fracture strain, the use of DIC strain measurement enables the detection of punch lift-off and false positives for fractures associated with fold-over.

•
A correlation was developed to predict the bend angle as a function of bend severity and tensile properties when the bend test data is not available.The local DIC fracture strain from tensile tests and bend severity were not significant predictors, whereas the uniform elongation and UTS were the primary predictors.This result demonstrates that tensile properties do not correspond directly with local formability, but can be estimated to first-order accuracy.The correlation for bend angle was markedly more accurate when the plane strain fracture limit was employed instead of the tensile fracture strain.

•
A robust correlation was developed to predict the plane strain fracture limit as a function of the bend angle, uniform elongation, and bend severity when VDA test data with DIC is not readily available.

•
The correlation developed in this study was evaluated using an independent set of data compiled from the literature consisting of 26 different test cases of AHSS with tensile strength levels ranging from 590 MPa to 1500 MPa.The model showed a strong correlation with an R-sq value of 0.86.It is recommended that analysts use the correlation to predict the mean fracture strain along with the corresponding 95% confidence and prediction intervals due to the inherent variability associated with fractures.

Figure 1 .
Figure 1.An overview of the test apparatus-the inverted version of VDA 238-100 bend tester with the optical 3D DIC system.

Figure 1 .
Figure 1.An overview of the test apparatus-the inverted version of VDA 238-100 bend tester with the optical 3D DIC system.

Figure 2 .
Figure 2. (a) An overview of the VDA 238-100 bend test specimen geometry, and (b) an illustration of a transverse direction (TD) test.

Figure 2 .
Figure 2. (a) An overview of the VDA 238-100 bend test specimen geometry, and (b) an illustration of a transverse direction (TD) test.

Figure 3 .
Figure 3. (a) An example of DIC strain contour plots of major and minor in-plane principal strains during a v-bend test for an MP980 steel sample with 1.5 mm thickness, and (b) extracted surface strain profiles at the VDA load threshold.

Figure 4 .
Figure 4.An example of the surface failure of MS1500 (t = 1.55 mm) at the VDA load threshold.

Figure 3 .
Figure 3. (a) An example of DIC strain contour plots of major and minor in-plane principal strains during a v-bend test for an MP980 steel sample with 1.5 mm thickness, and (b) extracted surface strain profiles at the VDA load threshold.

Figure 3 .
Figure 3. (a) An example of DIC strain contour plots of major and minor in-plane principal strains during a v-bend test for an MP980 steel sample with 1.5 mm thickness, and (b) extracted surface strain profiles at the VDA load threshold.

Figure 4 .
Figure 4.An example of the surface failure of MS1500 (t = 1.55 mm) at the VDA load threshold.

Figure 4 .
Figure 4.An example of the surface failure of MS1500 (t = 1.55 mm) at the VDA load threshold.

Figure 5 .
Figure 5.An example of the time-dependent fracture identification method of 980-GEN3, t = 1.4 mm.

Figure 5 .
Figure 5.An example of the time-dependent fracture identification method of 980-GEN3, t = 1.4 mm.

Figure 6 .
Figure 6.(a) An overview of the JIS tensile specimen geometry, and (b) the layout of extensometers for DIC measurements.

Figure 7 .
Figure 7. Representative engineering stress-strain curves of the AHSS obtained using a 50 mm extensometer.

Figure 6 . 24 Figure 6 .
Figure 6.(a) An overview of the JIS tensile specimen geometry, and (b) the layout of extensometers for DIC measurements.

Figure 7 .
Figure 7. Representative engineering stress-strain curves of the AHSS obtained using a 50 mm extensometer.

Figure 7 .
Figure 7. Representative engineering stress-strain curves of the AHSS obtained using a 50 mm extensometer.

Figure 8 .
Figure 8.An illustration of the introduced test cases: (a) the specimen surface is lifted off f punch tip, and (b) the specimen is folded over and wrapped around the punch.

Figure 8 .
Figure 8.An illustration of the introduced test cases: (a) the specimen surface is lifted off from the punch tip, and (b) the specimen is folded over and wrapped around the punch.

Figure 8 .
Figure 8.An illustration of the introduced test cases: (a) the specimen surface is lifted off from the punch tip, and (b) the specimen is folded over and wrapped around the punch.

Figure 9 .
Figure 9. Examples of invalid test cases: (a) the lift-off case of MP980 with 1.2 mm thickness, and (b) the folding-over case of 980-GEN3 with 1.4 mm thickness.

Figure 10 .
Figure10.Test result comparison: major surface fracture strain of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 10 .
Figure10.Test result comparison: major surface fracture strain of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 11 .
Figure 11.Test result comparison: Fracture bend angle of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 12 .
Figure 12.Test result comparison: the (a) major surface fracture strain and (b) fracture bend angle of MP980 at the VDA load threshold using the various punch radii and sheet thicknesses.

Figure 11 .
Figure 11.Test result comparison: Fracture bend angle of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 10 .
Figure10.Test result comparison: major surface fracture strain of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 11 .
Figure 11.Test result comparison: Fracture bend angle of MS1500 at the VDA load threshold using the various punch radii and the sheet thicknesses.

Figure 12 .
Figure 12.Test result comparison: the (a) major surface fracture strain and (b) fracture bend angle of MP980 at the VDA load threshold using the various punch radii and sheet thicknesses.

Figure 12 .
Figure 12.Test result comparison: the (a) major surface fracture strain and (b) fracture bend angle of MP980 at the VDA load threshold using the various punch radii and sheet thicknesses.

Figure 13 .
Figure 13.Mechanical response of DP780 in different gauges (1.0 mm, 1.2 mm, and 1.4 mm) from the JIS tensile and the v-bend tests (using the VDA load drop criterion).

Figure 13 .
Figure 13.Mechanical response of DP780 in different gauges (1.0 mm, 1.2 mm, and 1.4 mm) from the JIS tensile and the v-bend tests (using the VDA load drop criterion).

Metals 2023 , 24 Figure 14 .
Figure 14.An example of test cases with (a) a decreasing trend of the fracture strain for larger punch radii: PHS1500, sheet thickness = 1.55 mm, and (b) noticeable impact of load flattening on the load threshold.

Figure 14 .
Figure 14.An example of test cases with (a) a decreasing trend of the fracture strain for larger punch radii: PHS1500, sheet thickness = 1.55 mm, and (b) noticeable impact of load flattening on the load threshold.

Figure 15 .
Figure 15.Fracture strains of (a) PHS1500 and (b) 1180-GEN3 evaluated by the VDA's load-based analysis, and the time-dependent LBF analysis.

Figure 16 .
Figure 16.(a) An example of a discrepancy between the VDA load threshold and the time-dependent fracture timing, and (b) the corresponding surface condition from DIC images.

Figure 16 .
Figure 16.(a) An example of a discrepancy between the VDA load threshold and the time-dependent fracture timing, and (b) the corresponding surface condition from DIC images.

Figure 17 .
Figure 17.Comparison between the VDA fracture bend angle (a,b), the VDA fracture strain (c,d), the tensile uniform elongation (a,c), and the tensile total elongation (b,d).

Figure 17 .
Figure 17.Comparison between the VDA fracture bend angle (a,b), the VDA fracture strain (c,d), the tensile uniform elongation (a,c), and the tensile total elongation (b,d).Metals 2023, 13, x FOR PEER REVIEW 16 of 24

Figure 18 .
Figure 18.The multiple regression model using tensile test data.The comparison between the predicted and the measured bend angles.

Figure 18 .
Figure 18.The multiple regression model using tensile test data.The comparison between the predicted and the measured bend angles.

Figure 19 .
Figure 19.A multiple regression model using the VDA bend test result from all test groups.The comparison between the predicted and the measured bend angle.

Figure 19 .
Figure 19.A multiple regression model using the VDA bend test result from all test groups.The comparison between the predicted and the measured bend angle.

Metals 2023 ,
13,  x FOR PEER REVIEW 20 of 24 AHSS.The prediction interval does not correspond to the mean, but instead corresponds to the value of a new measurement.

Figure 20 .
Figure 20.Predictions of the regression model (Equation (9)) using a new set of materials in TableA3.The R-sq value is calculated based on the ideal prediction case of y = x.

Table 1 .
Classes and sheet thicknesses of the provided advanced high-strength steels in the study.

Table 2 .
Tensile mechanical properties of the AHSS along the TD obtained using the JIS No. 5 tensile geometry.

Table 3 .
An overview of test conditions with the punch lift-off (L) and material folding-over (F) cases.

Table 5 .
Relevant variables for regression analysis and the corresponding intervals for the tensile regression model.

Table 5 .
Relevant variables for regression analysis and the corresponding intervals for the tensile regression model.

Table 6 .
Relevant variables for regression analysis and the corresponding intervals for the regression model with the VDA bend test result.

Table 6 .
Relevant variables for regression analysis and the corresponding intervals for the regression model with the VDA bend test result.

Table 7 .
Relevant variables for regression analysis and the corresponding intervals for the re-arranged regression model to predict fracture strain.

Table 8 .
Relevant variables for regression analysis and the corresponding intervals for the optimized regression model.

Table A2 .
Plane strain fracture strain limits of AHSS updated with the time-dependent fracture detection method for 980-GEN3, 1180-GEN3, and PHS1500 test cases.The test cases with punch lift-off (L) and folding over (F) were not shown in the chart.

Table A3 .
Input values for the multiple linear regression model evaluation in Section 5 to predict VDA fracture strain from critical variables.