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The present study investigates the use of both Marciniak and Nakazima tests to generate the forming limit curves (FLCs) for a power-law hardening dual phase steel, DP980, and an aluminum-magnesium alloy, AA5182, that exhibits dynamic hardening and saturation-type behavior. The recently proposed methodology of Min et al. (2016) to compensate for the so-called process effects of non-linear strain paths (NLSP) and contact pressure was evaluated and applied to the Nakazima FLCs to enable comparison with the Marciniak FLCs and the formability predictions of the Modified Maximum Force Criterion (MMFC) of Hora et al. (1994, 2013) for in-plane stretching. An emphasis was placed upon the experimental determination of the constitutive model to plastic strains in excess of 0.5 using tensile and shear tests. A flexible variant of the Hockett-Sherby (1975) hardening model for large strain levels along with constraints upon the hardening model calibration are proposed. It is demonstrated that the choice of hardening model, test data, and the calibration procedure can have a marked influence on the Nakazima process corrections and the predicted FLC using the MMFC model. An extension to the Linear Best Fit (LBF) limit strain detection methodology by Volk and Hora (2011) was developed to account for bend severity and the material hardening response and is compared with the ISO 12004-2 limit strains. If the hardening model is accurately calibrated to large strain levels, the analytical MMFC predictions were in excellent agreement with the process-corrected Nakazima and Marciniak FLCs for DP980. The results for the AA5182 were found to be strongly dependent upon the choice of hardening model that influences both the MMFC and the contact pressure correction.

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... The contact stress was approximated from a homogenized model, equivalent to the solution of Ma et al. [16], and the corrected limit strains were in reasonable agreement although no verification of the predicted contact stresses was provided. Noder and Butcher [17] also reported good agreement between the corrected Nakazima and Marciniak FLCs for 5182 aluminum and DP980 steel using the methodology of Min et al. [9]. The potential for non-physical overcorrections of the Nakazima limit strains due to the contact pressure was documented as a result of its strong coupling with the hardening rate. ...

... The transition of the strain path across stress states from uniaxial tension to plane strain tension and equibiaxial stretching is demonstrated in Figure 9 for the DP980 AHSS of Noder and Butcher [17]. This causes the formation of a localization band in which the strain rate also increases and provides additional hardening for stability in a positive rate-sensitive material. ...

... Major strain (b) (a) Figure 9: Evolution of the strain path and transition towards plane strain tension during the localization process in Nakazima formability tests of the DP980 AHSS studied in Noder and Butcher [17]. ...

... In this method, the lack of localized necking up to very large strains in the shear specimens is exploited, and shear stresses and strains are converted to tensile values using the shear-to-tensile stress ratio and plastic-work equivalence. In this work, as per a suggestion by Noder and Butcher [46], the stress ratios at the diffuse-necking (UTS) points were adopted for converting the shear stresses (until the peak shear stress) to the tensile stresses beyond the UTS points. The work-conjugate equivalent plastic strains (ε p eq ) were then computed using Equation (14) [47], considering the deviation of stress and strain principal axes from each other during simple-shear loading: ...

... To evaluate the tensile-plus-converted shear flow curves (shown in Figure 11), they were first fit to a modified Hockett-Sherby hardening model (Equation (A6) [46]) and then extended beyond the experimental ranges using the calibrated models. Next, the tensile tests were modelled in the finite-element (FE) software of LS DYNA (R12.0.0, ...

... To evaluate the tensile-plus-converted shear flow curves (shown in Figure 8b), they were first fit to a modified Hockett-Sherby hardening model (Equation (A6) [46]) and then extended beyond the experimental ranges using the calibrated models. ...

Emerging grades of press-hardening steels such as Ductibor® 1000-AS are now commercially available for use within tailor-welded blanks (TWBs) to enhance ductility and energy absorption in hot-stamped automotive structural components. This study examines the constitutive (hardening) response and fracture limits of Ductibor® 1000-AS as functions of the as-quenched microstructure after hot stamping. Three different microstructures consisting of bainite and martensite were obtained by hot stamping with die temperatures of 25 ◦C, 350 ◦C, and 450 ◦C. Mechanical characterization was
performed to determine the hardening curves and plane-stress fracture loci for the different quench conditions (cooling rates). Uniaxial-tension and shear tests were conducted to experimentally capture the hardening response to large strain levels. Shear, conical hole-expansion, plane-strain notch tension, and Nakazima tests were carried out to evaluate the stress-state dependence of fracture. A mean-field homogenization (MFH) scheme was applied to model the constitutive and fracture behavior of the mixed-phase microstructures. A dislocation-based hardening model was adopted for the individual
phases, which accounts for material chemistry, inter-phase carbon partitioning, and dislocation evolution. The per-phase fracture modelling was executed using a phenomenological damage index based upon the stress state within each phase. The results revealed that the 25 ◦C hot-stamped material condition with a fully martensite microstructure exhibited the highest level of strength and the lowest degree of ductility. As bainite was formed in the final microstructure by quenching at higher die temperatures, the strength decreased, while the ductility increased. The predicted constitutive and fracture responses in the hot-stamped microstructures were in line with the measured data. Accordingly, the established numerical strategy was extended to predict the mechanical behavior of Ductibor® 1000-AS for a broad range of intermediate as-quenched microstructures.

... The true stress-strain data obtained from the UT tests are used to identify parameters of the before-mentioned hardening laws. For this purpose, the common and constrained curve fitting methods [44] are detailed in the next subsection. ...

... A previous study pointed out the link between the accuracy of a hardening law's prediction for ε * u and its usefulness in estimating the forming limit curve (FLC) of sheet metals [11,44]. Pham et al. [11] suggested increasing the weight factor of the points surrounding ε * u in the fitting routine. ...

... Pham et al. [11] suggested increasing the weight factor of the points surrounding ε * u in the fitting routine. Furthermore, Noder and Butcher [44] proposed enforcing the maximum force criterion as a constraint of the fitting procedure. The constraint is expressed as follows: ...

Constitutive modeling of sheet metals involves building a system of equations governing the material behavior under multi-axial stress states. In general, these equations require a hardening law that describes the stress-strain relationship. This study provides a thorough examination of the existing phenomenological hardening laws in the literature. Based on their ordinary differential equations, special efforts were made to discuss the degree of flexibility of these hardening laws. Four new phenomenological hardening laws were proposed during the discussions to capture the stress-strain relationship of automotive sheet metals, such as aluminum alloy and steel sheets. Then, applications of 18 hardening laws for fitting the uniaxial tensile stress-strain data of 12 automotive sheet metals were thoroughly compared. The comparisons reveal that the proposed hardening laws capture well the experimental stress strain data of all examined materials. Compared to several combined hardening laws, the proposed functions have comparable flexibility but require fewer parameters.

... The higher temperature was chosen because it is expected to represent an upper limit of the temperature rise due to adiabatic deformation at 15 % plastic strain. Finally, a constitutive model is developed using a novel rate dependent term, a modified Hockett-Sherby (MHS) model [24] to capture the high hardening response at low strains, and a tension-compression asymmetry term to model the material response of ASTM A1035CS Grade 120 steel. ...

... A constitutive model was developed to model the hardening, thermal softening, strain rate dependent, and tension-compression asymmetric material response of A1035CS steel. The model was developed using a novel strain rate dependent term, a modified Hockett-Sherby (MHS) model [24], and incorporating Lode angle to capture the tension-compression asymmetry of A1035CS steel. Details of the model formulation and a comparison between the experiments and the model are presented in the following subsections. ...

... Typically, Ludwik strength and hardening model is used, however it is insufficient in capturing the significant hardening in the initial stages of plastic deformation for the A1035CS steel. Therefore, a modified Hockett-Sherby (MHS) model from Noder and Butcher [24], an extension to the original Hockett and Sherby [43] material model, was used to capture the non-linear hardening behavior of this material: ...

The mechanical response of an advanced high strength and corrosion resistant 10 % Cr nanocomposite steel (ASTM A1035CS Grade 120) is measured under uniaxial tension and compression at the strain rates of 10-4 s-1 , 10-2 s-1 , 10 0 s-1 , 700 s-1 , and 3000 s-1. The experiments are performed at 22 °C as well as 80 °C to investigate the material behavior at the expected temperature rise due to adiabatic deformation at 15 % strain. Additionally, different compression-shear hat-shaped specimens are tested at quasi-static and dynamic strain rates to investigate the localization behavior of this material. The material exhibits small strain rate sensitivity (SRS) during quasi-static loading, but a pronounced SRS between quasi-static and dynamic strain rates. Tension-compression asymmetry is also observed at both temperatures. Experiments at 80 °C reveal a decrease in flow stress in both tension and compression indicating the material is sensitive to thermal softening due to adiabatic heating. Load-Unload-Reload (LUR) and strain rate jump experiments are performed to investigate the reasoning behind the approximate rate insensitivity of ASTM A1035CS steel during quasi-static strain rates. A new constitutive model is also developed using a novel rate dependent material model with a modified Hockett-Sherby (MHS) hardening model and incorporating Lode angle dependence to capture the tension-compression asymmetry. The model is also used to predict the LUR and strain rate jump experiments. Finally, reasoning behind the unique rate dependent thermo-mechanical behavior of ASTM A1035CS steel is discussed in regards to adiabatic heating, strain-partitioning, and phase transformation.

... Fig. 13 illustrates the schematic of the tool set up for two different punch geometries. Nakazima test (Nakazima et al., 1968;Noder and Butcher, 2019) is conducted with a hemispherical punch (diameter of 101.6 mm), and the Marciniak tests (Marciniak et al., 1973;Marciniak and Kuczynski, 1967;Noder and Butcher, 2019) is conducted with a cylindrical punch (diameter of 101.6 mm and a profile radius of 12 mm). Various types of strain states (equi-biaxial to plane strain to uniaxial tension) can be generated for both punch geometries by changing the specimen width. ...

... Fig. 13 illustrates the schematic of the tool set up for two different punch geometries. Nakazima test (Nakazima et al., 1968;Noder and Butcher, 2019) is conducted with a hemispherical punch (diameter of 101.6 mm), and the Marciniak tests (Marciniak et al., 1973;Marciniak and Kuczynski, 1967;Noder and Butcher, 2019) is conducted with a cylindrical punch (diameter of 101.6 mm and a profile radius of 12 mm). Various types of strain states (equi-biaxial to plane strain to uniaxial tension) can be generated for both punch geometries by changing the specimen width. ...

... The initial nonlinearity of deformation paths in Nakazima test is one of the causes for the small deviation between FLCs obtained from two different punch geometries. For this reason (Noder and Butcher, 2019), noticed the plane strain forming limit in 101.6 mm width specimen for Nakazima test and 127 mm width specimen for Marciniak test. Tharett and Stoughton (2003) reported that necking would start in sheet metal during stretching superimposed with bending according to the concave-side rule (CSR). ...

Sheet metal forming industry widely adopted the forming limit curve (FLC) as a limiting criterion in sheet metal forming. Regardless of its commercial importance, controlling factors of FLC are not systematically investigated up to date. The present paper comprehensively discussed the effect of different influencing factors on FLC, including limiting strain determination method, punch geometry, microstructure, pre-straining path, strain rate, and temperature. Different tensile properties and their correlations with FLC are also investigated. The introduction of microstructural features such as reducing void nucleation sites i.e., non-coherent particles, evenly distribution of fine hard phases, the introduction of twin and transformation induced plasticity (TRIP & TWIP), are favourable for higher FLC.

... Quasi-static tensile tests in the transverse direction were performed for the 270 Mild steel, 590R and all three 3rd Gen. alloys adopting the JIS no. 5 tensile geometry and fullfield stereoscopic DIC measurements in the Vic-3D® software from Correlated Solutions Inc. The DP980 and PHS1500 constitutive response of the same lots of material was previously performed by Noder and Butcher [17] and ten Kortenaar [12], respectively. The engineering stress-strain response is illustrated for one representative test in Fig. 2a where an extensometer length of 50 mm was utilized for all studied alloys except for the PHS1500 by ten Kortenaar [12] who utilized a sub-size ASTM tensile geometry with a virtual extensometer length of 12.5 mm. ...

... For computation of the hardening rate in Fig. 2b, the material stress-strain response was calibrated utilizing the Modified Hockett-Sherby model of Noder and Butcher [17] in Eq. (1). The Considère criterion for the onset of diffuse necking was enforced as a constraint in the calibration to ensure that the hardening model accurately predicts the experimental uniform elongation. ...

... The stress metric of Eq. (15) can now be evaluated under three conditions: (i) constant thickness assumption (ii) DIC thickness measurement and (iii) using the empirical correlation of Eqs. (16)- (17). A 1% failure threshold was arbitrarily selected for quantitative comparison; a detailed discussion on identification of a failure threshold is presented in Section 5.1. ...

Background
The VDA 238–100 tight radius V-bend test can be used to efficiently characterize the bendability and fracture limits of sheet metals in severe plane strain bending. Material performance in plane strain bending is critical for the selection of advanced high strength steels for energy absorbing structural components.Objective
The detection of failure based upon a reduction in the punch force can lead to erroneous predictions of failure for ductile or thin gage alloys in the VDA 238–100 test. New failure criteria were proposed and evaluated across a range of automotive steels.Methods
Four detection methods in the V-bend test were evaluated based upon the load drop, bending moment, novel stress metric and the strain rate for seven steels with strength levels from 270 to 1500 MPa. The appropriate failure threshold was identified from visual inspection of the surface during bending.ResultsThe vertical punch force will decrease as a consequence of the mechanics in the V-bend test at intermediate bend angles even without fracture. The novel stress-based metric accounts for sheet thinning and could successfully identify “false positives” and punch lift-off when considering the strain-rate evolution.Conclusions
Failure detection using the VDA load threshold method may significantly under-report the bend performance of alloys with intermediate-to-high bendability or thin gauges. The proposed stress-based metric can reliably detect fracture for bend angles in excess of 160° and be readily calculated using the existing data. The VDA load threshold for failure can work well for materials that exhibit significant cracking.

... The plane stress mapping procedure in Eq. (12) can be quite sensitive to the hardening behavior as discussed by Noder and Butcher (2019). The influence of the contact stress increases non-linearly with the hardening rate of the material. ...

... Experimental and numerical studies by Matin and Smith (2005) and Allwood and Shouler (2008) reported that the mapping criteria of Smith et al. (2003) appears to overestimate the influence of contact stress. However, the mapping criteria has worked very well for the correction of Nakazima FLC data for a multi-phase steel, MP980 by Min et al. (2016) using two different punch radii, as well as a dual-phase steel, DP980, and AA5182 aluminum by Noder and Butcher (2019). The process-corrections were shown to be relatively insensitive to the choice of yield function by Min et al. (2016). ...

... Rectangular shims of 1.4 mm in height and 75 mm in length were employed to mitigate lockbead cracking in the DP1180. The number of shims was found to influence the strain path so additional Marciniak tests with widths of 152.4 mm and 165.1 mm were performed with only two shims instead of placing 4 shims around the perimeter of these wider geometries as also done by Noder and Butcher (2019) for the same die set. The punch velocity for all formability tests was 0.25 mm/s. ...

Analytical formability models based upon in-plane proportional plane stress loading such as the Marciniak–Kuczynski (MK) model are commonly evaluated against forming limit curves (FLCs) obtained using Nakazima tests with out-of-plane deformation, non-linear strain paths and tool contact. The present study has conducted a comprehensive experimental characterization of a DP1180 advanced high strength steel with relatively low formability to demonstrate that the use of Nakazima FLC data can lead to a physically inconsistent MK model. Although good agreement can be obtained using the MK model with Nakazima data, it is attributed to calibration bias in the hardening model and selection of the imperfection factor. Marciniak tests should be performed to characterize the FLC or process-corrections be applied to the Nakazima FLC to then calibrate the MK imperfection factor. An analysis of the MK framework reveals a false dependence of the imperfection factor on the hardening model. When the Considère constraint for the uniform elongation is imposed during the hardening model calibration, the MK imperfection factor becomes effectively independent of the choice of hardening model. The accuracy of the FLC predicted by the MK model, when appropriately constrained and calibrated in a deterministic manner, was in a reasonable agreement with the experimental data only after accounting for material anisotropy. In contrast, the deterministic Modified Maximum Force Criterion (MMFC) model provided a very close estimate of the experimental forming limit strains of the mildly anisotropic DP1180 using both isotropic and anisotropic yield functions.

... According to the procedure described in Min et al. [24], a numerical subroutine was developed to correct the FLCs for the tested materials. The new FLC obtained after compensation of the process-dependent effects are analyzed in the light of works already published on thicker steel sheets [24,33]. ...

... Figure 15 displays that the pressure correction causes a moderate variation of the data points that do not affect the global tendency given by the linear regressions. The obtained results are in good agreement with those reported in [24,33] in the case of the AA5182-O alloy and an advanced high strength steel. Finally, the correction method adopted in this work to obtain linear deformation paths up to failure is relevant for ultra-thin sheets, since the strain paths obtained in the Nakazima tests exhibited non-linear strain paths. ...

... This could be confirmed by microforming Marciniak tests [35], but these tests are generally performed on a much lower scale than this one (punch of 1.5 mm compared to 35 mm), which is likely to exacerbate other local and scale effects. The corrected FLCs for the pure Cu and CuBe2 alloy in Figure 15 are in good agreement with the results presented by Noder et al. [33] for the study of the formability of an AA5182 alloy 1.55 mm thick and DP980, with a sheet thickness of 1.2 mm. These resulting FLCs may be considered as the true forming limits of the selected materials, for perfectly linear strain paths in the absence of normal pressure through the thickness. ...

The objective is to propose an accurate method for determining the forming limit curves (FLC) for ultra-thin metal sheets which are complex to obtain with conventional techniques. Nakazima tests are carried out to generate the FLCs of a pure copper and a copper beryllium alloy with a thickness of 0.1 mm. Because of the very small thickness of the sheets, the standard devices and the know-how of this test are no longer valid. Consequently, new tools have been designed in order to limit friction effect. Two different methods are used and compared to estimate the necking: the position-dependent measurement method (ISO Standard 12004-2), and the time-dependent method based on the analysis of the derivatives of the planar strain field. It is shown that the ISO standard method underestimates the forming limit curves. As the results present non linear strain paths, a compensation method is applied to correct the FLCs for the tested materials, which combines the effects of curvature, nonlinear strain paths and pressure. The curvature effect for such thickness and punch diameter on the FLCs is weak. The results show that this procedure enables to obtain FLCs that are close to those determined by the reference Marciniak method, leading to a minimum in major strain that converges to the plane strain state.

... When the MK model was calibrated deterministically using only the Marciniak FLC0 and the constraint of Soare [10] on the imperfection factor for diffuse necking in plane strain tension, the predictions were not as accurate as the MMFC model that contained no calibration parameters. Noder and Butcher [18] have also reported that the MMFC model predictions can be in very good agreement with a DP980 steel and the MP980 of Min et al. [19] when Marciniak test data, or process-corrected Nakazima data was used. ...

... Extensions to the MMFC model and BW models were proposed to enhance the predictive accuracy while not introducing any new calibration parameters. Finally, to evaluate the proposed framework for FLC prediction for a larger range of automotive sheet metals, a DP980 and aluminum AA5182 alloy are also considered that were characterized for in-plane formability by Noder and Butcher [18]. ...

... The validity of the experimental hardening curve obtained using the tensile and shear tests is evaluated in Section 2.4 using 3D simulations of the tensile test where large strains develop due to localization. Recent work by Noder and Butcher [18] highlighted the importance of the calibration procedure used in the hardening model and how it can significantly influence analytical formability predictions. A constrained calibration must be performed to enforce the Considère criterion [24] for the beginning of diffuse necking in uniaxial tension, which occurs when the hardening rate is equivalent to the major principal stress such that ...

The objective of the current study is to develop a practical, deterministic approach to the prediction of the in-plane formability of two third generation advanced high-strength steels (AHSS) of 980 and 1180 MPa ultimate tensile strength using only quasi-static mechanical property data. The hardening response to large strains was experimentally measured with the use of simple shear and tensile tests and validated in tensile simulations. The process-corrected limit strains in the Nakazima and Marciniak tests were compared to various analytical Forming Limit Curve (FLC) models for in-plane stretching. It was observed that the widely-used Marciniak–Kuczynski model can adequately predict the experimental FLC in biaxial stretching but significantly underestimated the limit strains in uniaxial stretching for both third generation AHSS. The observed through-thickness shear fracture mode in biaxial stretching was reasonably well-captured by the Bressan–Williams (BW) instability model for the 1180 MPa steel. A proposed extension of the BW model to uniaxial tension by adoption of the maximum in-plane shear stress criterion (BWx model) provided superior experimental correlation relative to the zero-extension model of Hill that was too conservative. Finally, a linearized version of the modified maximum force criterion (MMFC) was proposed that markedly improved the correlation with the process-corrected FLC for in-plane stretching of AHSS. The developed framework for FLC prediction was then applied to a DP980 AHSS and an AA5182 aluminum alloy from the literature. The DP980 corroborated the observed trend for the two third generation AHSS whereas the MK and the BWx models performed best for the AA5182 with its saturation-type hardening behavior and non-quadratic yield surface.

... The mechanical properties, documented in Table 3, were characterized in quasi-static tensile tests adopting the JIS No. 5 tensile geometry. The hardening behavior was characterized using tensile and simple shear tests and modelled using the Modified Hockett-Sherby (MHS) model of Noder and Butcher (2019). ...

... K = 0.836 and L = 357.71 MPa, and the plastic uniform elongation (UE-p) was enforced as a constraint, as discussed in Noder and Butcher (2019). Fig. 17. ...

... Les éprouvettes de cisaillement simple avec entailles (type c) comme proposées dans la norme ASTM B831 par exemple, sont étudiées notamment par Wierzbicki et al. [58], Roth et al. [59], Zhang et al. [60] Noder et al. [61]. Les éprouvettes de type c partagent avec les géométries a et d d'avoir un bord libre en extrémité de zone cisaillée, avec pour conséquence de réduire l'homogénéité des champs mécaniques le long de la zone de cisaillement. ...

... Les éprouvettes de type c partagent avec les géométries a et d d'avoir un bord libre en extrémité de zone cisaillée, avec pour conséquence de réduire l'homogénéité des champs mécaniques le long de la zone de cisaillement. Il est possible de les utiliser pour identifier le comportement mécanique en cisaillement comme le montrent Yin et al. [52] et Noder et al. [61]. Cependant, ces éprouvettes présentent un rapport l'élancement réduit souvent de l'ordre de 3 à 5, et leur usage est souvent mis en avant pour leur capacité à permettre une sollicitation en cisaillement jusqu'à la rupture, surpassant sur ce point les éprouvettes classiques de type a. Ainsi, Roth et al. [59], grâce à une étude numérique très vaste, proposent trois géométries différentes selon le niveau de ductilité de la tôle métallique. ...

La protection du matériel électronique embarqué est devenue un enjeu majeur pour assurer la sécurité du combattant. On peut citer divers exemples tels que la protection des piles à hydrogène dans les véhicules ou dans la batterie embarquée d'un soldat. C'est dans ce contexte que s'inscrit la thèse, où une étude est menée sur une protection de type multi-couches, sollicitée par des projectiles de type fragment, de quelques kilogrammes, allant à des vitesses de l'ordre de 10 m/s. Afin d'assurer la mise en service de telles protections, des essais et des simulations doivent être menés sur un large champ de sollicitations. La littérature montre que les structures multi-couches offrent un bon compromis entre capacité à absorber l’énergie d’impact et légèreté. Le complexe étudié pour cette thèse est composé d’une couche métallique, acier ou aluminium, et d’une couche de polymère. La première partie de cette thèse est consacrée à la caractérisation des plaques métalliques étudiées pour cette thèse : acier DP450 et aluminium AA2024-T3. Un nouvel essai de cisaillement séquencé est proposé afin d’identifier le comportement de la tôle en grande déformation. L’essai de traction à déformation plane est adapté pour identifier la déchirure des tôles en dynamique, jusqu’à des vitesses de déformation de l’ordre de 200/s. La deuxième partie est consacrée à l’identification complète d’une nouvelle résine polydicyclopentadiene (PDCPD) appelée Nextene. Dans la dernière partie, différentes structures multicouches sont sollicitées par des impacts à l’aide d’une catapulte qui projette des projectiles de 2.5 kilogrammes à une vitesses de 10 m/s. Leurs comportements respectifs à l’impact sont comparés et simulés numériquement.

... Using plastic work equivalence, the experimental shear-to-equivalent stress ratio was used to convert the shear stress versus plastic work data to a work-conjugate equivalent stress versus plastic strain. Noder and Butcher (2019) refined the methodology to first use the tensile data until the plastic work at the onset of diffuse necking to avoid uncertainty with initial inhomogeneous yielding of the shear specimen while the hardening curve at large strains was determined by shear data. A limitation of the methodologies of Rahmaan et al. (2017) and Noder and Butcher (2019) is that shear anisotropy was assumed to be marginal over the range of strain considered so that the shear-to-equivalent stress ratio remains constant despite rotation of the material frame. ...

... Noder and Butcher (2019) refined the methodology to first use the tensile data until the plastic work at the onset of diffuse necking to avoid uncertainty with initial inhomogeneous yielding of the shear specimen while the hardening curve at large strains was determined by shear data. A limitation of the methodologies of Rahmaan et al. (2017) and Noder and Butcher (2019) is that shear anisotropy was assumed to be marginal over the range of strain considered so that the shear-to-equivalent stress ratio remains constant despite rotation of the material frame. For materials with shear anisotropy, it was left to the analyst to determine an appropriate cut-off strain for the shear conversion based upon rotation of the material axes. ...

Simple shear tests are an increasingly attractive method to determine the equivalent stress-strain response of sheet metals. Unlike uniaxial tensile tests, shear tests can reveal the hardening behaviour of materials to large strains without stress state deviations triggered by tensile instability. However, there has been some uncertainty surrounding the interpretation of the shear response of anisotropic materials due to the definition of appropriate equivalent strain measures and the development of normal stresses. In the present study, the development of normal stresses during simple shear of anisotropic materials is analyzed and are found to be negligible relative to the magnitude of the applied shear stress. It is demonstrated that erroneous normal stresses may arise as a consequence of calibration of anisotropic yield functions. An experimental methodology was then proposed consisting of shear tests in multiple orientations to characterize shear anisotropy and account for rotation of the material frame on the hardening response. The methodology considers non-linear interpolation using either a calibrated yield function using both shear and tensile data or from a simplified phenomenological form calibrated using only the shear data. A range of automotive alloys were considered including DP980 and DP1180 advanced high strength steel alloys, an aluminum-magnesium alloy, AA5182-O, and an AA6063-T6 aluminum extrusion with severe anisotropy. It is demonstrated that for relatively isotropic materials such as the DP steels, accounting for material frame rotation results in an approximately 2% difference in the extracted hardening data compared to the case when the material rotation is neglected. This variation is expected to be within the experimental uncertainty. For materials with more pronounced anisotropy such as AA5182-O sheet and AA6063-T6 extrusions, the change in the hardening response is more significant and can reach up to 5% and 15%, respectively.

... Various methods can be used to reproduce the experimental FLD, such as analytical methods (Marciniak and Kuczyński, 1967;Shahzamanian and Wu, 2021;Noder and Butcher, 2019;Lei, 2021) and the FEM (Entesari, 2017;Chung et al., 2014;Jalali, 2022;Simoncini, 2021;Yang, 2021;Zhu, 2021;Habibi, 2018) as well as a hybrid method combines the experimental and artificial intelligence methods (Zhang et al., 2023;Jia, 2023). One of the famous analytical methods to predict FLD is the Marciniak and Kuczynski (M-K) (Marciniak and Kuczyński, 1967;Marciniak et al., 1973;Ma, 2023). ...

... In Fig. 3c, the major strain ε 1 is plotted against the minor strain ε 2 during the whole deformation process based on zone BZ. The strain path is nearly linear during the firsthalf of the deformation with a strain ratio β = dε 2 / dε 1 = − 0.5, showing that a uniaxial tension state is achieved; the strain path deviates from a straight line in the later stages of deformation, which is consistent with findings from other formability assessments performed at room temperature [11,12]. Several factors may contribute to the deviation from linearity, such as the geometry of the specimen and the experiment conditions. ...

Medium-Mn (MMn) steels with low austenitisation temperatures have recently attracted much research attention because of their suitability for hot stamping to save cost and improve productivity. However, research on their formability under conditions that mimic industrial hot stamping is still lacking. In this study, an innovative method for biaxial testing, which includes a cruciform specimen design, is used on a Gleeble 3800 simulator to obtain the forming limit curves (FLCs) and fracture forming limit curves (FFLCs) of a representative MMn (8Mn) steel. The localised necking strain and fracture strain are measured using a newly developed spatio-temporal method. The FLCs and FFLCs are constructed using strain limits in different loading states, e.g. uniaxial, plane strain and equi-biaxial stretching. The results show that the loading state conditions are acceptably simulated using the biaxial test system, and the FLCs and FFLCs are well-constructed, displaying a V shape. The data will contribute to the design of forming process. In addition, the forming strain limits of the MMn steel are much higher than those of a conventional boron steel (22MnB5) under the same hot stamping conditions, suggesting that the MMn steel has great potential and benefit as an alternative in hot stamping.KeywordsMedium Manganese SteelForming Limit CurveHot StampingBiaxial Tensile Test

... The modified Hockett-Sherby (Hockett and Sherby [35]) model of Noder and Butcher [36] was adopted to describe the hardening behavior: (12) in which A, B, C, D, and E are the coefficients of the model that were determined based on uniaxial tensile tests data up to the onset of diffuse necking. Table 3 lists the coefficients of the model and Figure 8 compares the stress-strain curves derived from the modified Hockett-Sherby model versus experiments in which a good agreement is evident for both materials in different directions. ...

The yield strength of materials under plane strain deformation is often not characterized
experimentally due to difficulties that arise in interpreting the results of plane strain tensile tests. The strain and stress fields in the gauge region of these tests are inhomogeneous, making it challenging to extract the constitutive response from experimental measurements. Consequently, the plane strain yield stress is instead predicted using phenomenological plasticity models calibrated using uniaxial and biaxial tension data. To remove this uncertainty, a simple finite-element based inverse
technique is proposed to determine the arc of the associated yield locus from uniaxial-to-plane strain tension using a constrained form of Vegter’s anisotropic yield criterion to analyze a notch tensile test.
The inverse problem is formulated under associated deviatoric plasticity and constrained such that only a single parameter, the major principal yield stress under plane strain deformation, needs to be identified from the finite-element simulations. The methodology was applied to two different automotive steel grades, an ultra-high strength DP1180 and a DC04 mild steel. The predictive accuracy of the constitutive models was then evaluated using an alternate notch geometry that provides an intermediate stress state between uniaxial and plane strain tension. By performing notch tensile tests in three sheet orientations, three arcs of the yield surface were obtained and employed to
calibrate the widely used Yld2000 yield function. The study shows that for DP1180, the normalized plane strain yield stress was in the range of 1.10 to 1.14 whereas for DDQ steel, the normalized plane strain yield stress was notably stronger, with values ranging from 1.22 to 1.27, depending on the orientation. The proposed methodology allows for a wealth of anisotropic plasticity data to be obtained from simple notch tests while ensuring the plane strain state is accurately characterized, since it governs localization and fracture in many forming operations

... The AA5182-O sheet was supplied in the annealed condition while the AA6013 and AA7075 were provided in the T4 and peak-aged T6 tempers respectively. The plasticity response characterization of the same lot of AA5182-O and AA7075-T6 was performed in prior work by the authors [7,8]. holding force and strip removal was achieved via a punch stripping system with steel springs designed for compact space requirements as shown in Figure 1 (b). ...

High strength 6000- and 7000-series aluminum alloys have significant potential for automotive light weighting due to their high strength-to-weight ratios. Alternate thermal processing routes and warm forming operations with rapid heating to 150 - 300°C are being explored to expand their forming process windows. While the constitutive and formability behavior of these alloys at elevated temperatures is an active area of research, limited data is available on their sheared edge stretchability, particularly under elevated temperature forming conditions. To this end, the influence of cutting clearance on the edge fracture limits of two precipitation hardening alloys, AA6013-T4 and AA7075-T6, and a non-heat treatable AA5182-O were first investigated at room temperature using the conical hole expansion test. A 12% clearance was found to be suitable for the three materials and used to process the hole expansion samples prior to warm forming. Microhardness tests were used to characterize the depth and severity of the shear affected zone (SAZ) for the AA7075-T6 before and after the warm forming cycle. The forming temperature was found to increase the hole expansion ratio by approximately 400%, 150%, and 520% for the AA5182-O, AA6013-T4, and AA7075-T6 respectively relative to the room temperature tests. The edge stretchability during W-temper forming of the AA6013 and AA7075 at room temperature was also assessed. It was observed that W-temper forming negatively influenced the hole expansion of AA6013 by approximately 30% relative to the T4 condition. Conversely, the hole expansion ratio for AA7075 when formed in W-temper increased by over 35% with respect to the T6 temper.

... Based on the results of photogrammetric measurements, the tool trajectory can be modified to ensure the desired uniform thinning of the sheet along the transverse profile of the ribs. An assessment of possible defects and failures is carried out on the basis of the location of the points corresponding to the actual material deformation in relation to the Forming Limit Curve (FLC) [49,50]. The FLC, as the key feature of the Forming Limit Diagram (FLD), records some pairs of in-plane limit strains (major and minor) and defines the boundary between the safe zone (no necking) and dangerous zone (necking and splitting), which is above the FLC. ...

In the sheet metal forming processes, the transformed induced plasticity (TRIP) steel of RAK 40/70 grade exhibits a so-called springback effect, which is governed by strain recovery of material after the load removal. In this study, the numerical simulation of the bending process was performed and compared with experimental data. Springback can be related to many parameters, including forming conditions, tool geometry, sheet thickness, yield stress, work hardening, strain rate sensitivity, and elastic modulus. The effects of the above parameters on the springback effect were evaluated for the V-shaped sheet metal part made of TRIP steel with a thickness of 0.75 mm and a bending angle of 90°. In the numerical simulation, Hill's and Barlat’s yield criterions were used in combination with Ludwik’s and Swift’s hardening models. The numerical simulation data were compared with the experimental results, showing their close correlation.

... Cleveland and Ghosh [24] computed the linear slope between the maximum and minimum load, which was noted by Govik et al. [27] to not necessarily correlate with the maximum strain prior to unloading and thus effect the magnitude of the chord modulus. Instead, Govik et al. [27] selected the two data points corresponding to the maximum and minimum strain whereas Sun and Wagoner [28] adopted the first data point after unloading for computation of the linear slope. To better capture the nonlinear unloading behavior, Yoshida et al. [29] averaged the chord modulus over different stress ranges. ...

While the third generation of advanced high-strength steels (3rd Gen AHSS) have increasingly gained attention for automotive lightweighting, it remains unclear to what extent the developed methodologies for the conventional dual-phase (DP) steels are applicable to this new class of steels. The present paper provides a comprehensive study on the constitutive, formability, tribology, and fracture behavior of three commercial 3rd Gen AHSS with an ultimate strength level ranging from 980 to 1180 MPa which are contrasted with two DP steels of the same strength levels and the 590R AHSS. The hardening response to large strain levels was determined experimentally using tensile and shear tests and then evaluated in 3D simulations of tensile tests. In general, the strain rate sensitivity of the two 3rd Gen 1180 AHSS was significantly different as one grade exhibited larger transformation-induced behavior. The in-plane formability of the three 1180 MPa steels was similar but with a stark contrast in the local formability whereas the opposite trend was observed for the 3rd Gen 980 and the DP980 steel. The forming limit curves could be accurately predicted using the experimentally measured hardening behavior and the deterministic modified Bressan–Williams through-thickness shear model or the linearized Modified Maximum Force Criterion. The resistance to sliding of the three 3rd Gen AHSS in the Twist Compression Test revealed a comparable coefficient of friction to the 590R except for the electro-galvanized 3rd Gen 1180 V1. An efficient experimental approach to fracture characterization for AHSS was developed that exploits tool contact and bending to obtain fracture strains on the surface of the specimen by suppressing necking. Miniature conical hole expansion, biaxial punch tests, and the VDA 238-100 bend test were performed to construct stress-state dependent fracture loci for use in forming and crash simulations. It is demonstrated that, the 3rd Gen 1180 V2 can potentially replace the DP980 steel in terms of both the global and local formability.

... The equal-biaxial R-value was then determined as R b = TD ∕ RD . The hardening behavior of both materials in the rolling (RD), transverse (TD) and diagonal (DD) directions was described using the flexible modified Hockett-Sherby (MHS) model of Noder and Butcher [59] previously presented in Eq. (14). The coefficients C 1 -C 5 were calibrated to the experimental data from standard uniaxial tension tests shown in Fig. 16 up to the start of diffuse necking. ...

Background
Although accurate knowledge of material behavior in plane strain tension is important for the modelling of sheet metal forming processes, it is often overlooked in yield function calibration because of experimental characterization challenges. Plane strain notch tensile tests, though experimentally convenient, are subject to stress and strain gradients across the gauge width that complicate the analysis.ObjectiveA novel experimental integration methodology was developed to exploit these stress and strain gradients to locally calibrate the arc of an anisotropic yield surface from uniaxial-to-plane strain tension.Methods
Constraining the anisotropic yield surface at the plane strain point, to be consistent with pressure-independent plasticity, enables the local arc to be governed by a single parameter. The arc shape is largely independent of the choice of yield function and can be optimized using a cutting line approach and full-field optical strain measurements. The accuracy of the method was evaluated using finite-element simulations of isotropic and anisotropic materials with different hardening behaviors.ResultsThe methodology was applied to a dual phase DP1180 steel and AA5182-O aluminum alloy in the rolling, transverse, and diagonal directions. Data along each of the three locally calibrated arcs was included in calibrations of Yld2000 and Yld2004 yield surfaces.Conclusions
The plane strain yield strength and arc shape had significant implications on the calibration of advanced anisotropic yield criteria. The yield exponent of the DP1180 agreed with the common value of six for BCC metals while the AA5182 yield surface approximated a Tresca-shape with local yield exponents in excess of 20.

... They calibrated their model experimentally and investigated the localized deformation. 31 Studies on the effect of texture have been fully developed by previous researchers, [32][33][34][35] but, at the best knowledge of the authors, the influence of grain geometry, their aspect ratio, and heterogeneity on FLD and stress-strain relation has not been investigated until now. In this article, considering the mentioned parameters on the tension behavior, the FLD of aluminum 2024 has been obtained by the finite element method using the crystal-plasticity model. ...

Experimental and numerical study regarding the uniaxial tensile test and the forming limit diagram are addressed in this paper for AL2024 with the face-centered cube structure. First, representation of a grain structure can be obtained directly by mapping metallographic observations via scanning electron microscopy approach. Artificial grain micro-structures produced by Voronoi Tessellation method are employed in the model using VGRAIN software. By resorting to the finite element software (ABAQUS) capabilities, the constitutive equations of the crystal plasticity were utilized and implemented as a user subroutine material UMAT code. The hardening parameters were calibrated by a trial and error approach in order to fit experimental tensile results with the simulation. Then the effect of the changing grain size, the heterogeneity factor, and the grain aspect ratio were studied for a uniaxial tensile test to emphasize the importance of the microstudy behavior of grains in material behavior. Furthermore, the polycrystal plasticity grain distribution was employed in the Nakazima test in order to obtain the forming limit diagram. The crystal plasticity-driven forming limit diagram reveals more accurate strains, taking into account the involving the micromechanical features of the grains. An innovative approach is pursued in this study to discover the necking angle, both in tensile test or Nakazima samples, showing a good agreement with the experiment results.

... In the former study [21], the constitutive response of the material up to large strains was obtained via the calibration of a modified Hockett-Sherby model (Eq. (4)) [43] using tensile and shear data. In the latter study [22], the fracture locus was acquired through the calibration of the modified Mohr-Coulomb (MMC) fracture function (Eq. ...

Hot-stamped tailor-welded blanks (TWBs) comprised of steel sheets with different chemical compositions are of interest within the automotive sector to enhance the crash performance of vehicles via imparting local ductile regions to ultra-high strength structural components. This study examines the fracture behavior of hot-stamped TWBs composed of mono- and multi-gauge laser-welded sheets of Ductibor® 500-AS and Usibor® 1500-AS, which are under consideration for use in crash-safety members. Microstructural and microhardness investigations were conducted across the weld line of the TWBs to address the variations of mechanical properties. A fracture characterization campaign via a series of coupon tests, including uniaxial tension, equibiaxial tension Nakazima dome, and V-bend, was conducted on specimens with a range of orientations of the weld line with respect to the loading direction. Numerical models of the tensile and Nakazima dome tests with and without considering the weld region in the TWBs were developed using the LS-DYNA finite-element method software based on mapping constitutive properties and fracture-limit curves across the weld and in parent metals as functions of local hardness. The fracture characterization tests revealed that weld fracture occurred only in the V-bend tests for which the weld line was parallel to the punch (0° V-bend), while in all of the other mechanical tests considered, fracture was initiated in one of the parent metals. The Ductibor® 500-AS region was the site of fracture initiation in the transversely-welded tensile and centrally-welded Nakazima dome samples, whereas fracture initiation occurred in the Usibor® 1500-AS region for the longitudinally-welded tensile and 45° and 90° V-bend samples. The predictions revealed that the inclusion of the narrow weld region of the laser-welded blanks within the hardness-mapped TWB models resulted in an improvement of the predictions in terms of the load–displacement response and fracture locations compared to the “without weld” scenario. Therefore, the correlation of variations in the flow and fracture response of the hot-stamped TWBs of Ductibor® 500-AS and Usibor® 1500-AS with the trend of changes in hardness across the weld can be a more suitable strategy for modelling the mechanical behavior of such TWBs in crash-safety simulations.

... The mini shear tests were conducted using a 100 kN MTS Criterion 45 servo-electric tensile frame at a strain rate of 0.01 s À1 . For the hole-expansion (Pathak et al., 2016) and Nakazima tests (Nakazima et al., 1971;Noder and Butcher, 2019), an MTS Model 866.02 hydraulic press equipped with conical (60°angle) and hemispherical punches was utilized, respectively. The punch speed and binder force were 0.25 mm/s and 660 kN, respectively, and the binder and die included lock beads. ...

Ductibor® 500-AS is a hot-stamping steel that has been designed to introduce local ductile regions within hot-stamped tailor-welded blanks (TWBs) used for energy-absorbing sections of automotive structures. This study investigates the flow and fracture behavior of a range of the microstructures of this alloy, corresponding to mostly ferritic (∼ 96% ferrite plus 4% martensite), intermediate ferritic + martensitic (∼ 57% ferrite plus 43% martensite), and mostly martensitic (∼ 10% ferrite plus 90% martensite), obtained by applying various quench rates after austenitization. A series of mechanical tests in various stress states, ranging from simple shear to biaxial tension, was performed to characterize fracture for the developed microstructures. A mean-field homogenization (MFH) technique was then used to predict the flow and fracture response as a function of the microstructure. An MFH scheme designated as the interpolative Samadian-Butcher-Worswick 1 (INSBW1) within the first-order secant-based linearization method was considered to predict the macroscopic hardening behavior of the ferritic-martensitic microstructures. In the MFH modelling, the flow response of the ferritic and martensitic phases was modelled based on a dislocation-based hardening model, taking into account the chemical composition, carbon partitioning between phases, and dislocation generation and annihilation. Damage accumulation and the onset of fracture were predicted using a phenomenological damage indicator defined within each constituent phase given their calculated fracture loci and instantaneous stress and strain states. The experimental results showed that the mostly-martensitic microstructure had the lowest ductility and highest strength, and ductility and strength increased and decreased, respectively, as the martensite volume fraction in the microstructure decreased. The predicted flow curves and fracture strains as well as the fracture-limit curves for all of the microstructures agreed well with the measured data and interpolated Modified-Mohr-Coulomb (MMC) fracture loci.

... Cleveland and Ghosh [24] computed the linear slope between the maximum and minimum load, which was noted by Govik et al. [27] to not necessarily correlate with the maximum strain prior to unloading and thus effect the magnitude of the chord modulus. Instead, Govik et al. [27] selected the two data points corresponding to the maximum and minimum strain whereas Sun and Wagoner [28] adopted the first data point after unloading for computation of the linear slope. To better capture the nonlinear unloading behavior, Yoshida et al. [29] averaged the chord modulus over different stress ranges. ...

div class="section abstract"> The superior formability and local ductility of the emerging class of third generation of advanced high-strength steels (3rd Gen AHSS) compared to their conventional counterparts of the same strength level offer significant advantages for automotive lightweighting and enhanced crash performance. Nevertheless, studies on the material behavior of 3rd Gen AHSS have been limited and there is some uncertainty surrounding the applicability of developed methodologies for conventional dual-phase (DP) steels to this new class of AHSS. The present paper provides a comprehensive study on the quasi-static and dynamic constitutive behavior, formability characterization and prediction, and the fracture behavior of two commercial 3rd Gen AHSS with an ultimate strength of 1180 MPa that will be contrasted with a conventional DP1180. The hardening response to large strain levels was determined experimentally using tensile and shear tests and then validated with 3-D simulations of tensile tests. In general, the strain rate sensitivity of the two 3rd Gen AHSS was significantly different as one grade exhibited larger transformation-induced behavior. The in-plane formability of the three 1180 MPa steels determined using Marciniak tests was similar but with a stark contrast in the local formability for the 3rd Gen AHSS. The forming limit curves could be accurately predicted using the experimentally measured hardening behaviour and the modified Bressan-Williams through-thickness shear model. An efficient experimental approach to fracture characterization for AHSS was developed that exploits tool contact and bending to obtain fracture strains on the surface of the specimen by suppressing necking. Miniature conical hole expansion and biaxial punch tests are used along with the VDA 238-100 bend test.
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... Based on the results of photogrammetric measurements, the tool trajectory can be modified to ensure the desired uniform thinning of the sheet along the transverse profile of the ribs. An assessment of possible defects and failures is carried out on the basis of the location of the points corresponding to the actual material deformation in relation to the Forming Limit Curve (FLC) [49,50]. The FLC, as the key feature of the Forming Limit Diagram (FLD), records some pairs of in-plane limit strains (major and minor) and defines the boundary between the safe zone (no necking) and dangerous zone (necking and splitting), which is above the FLC. ...

Stringer-stiffened panels made of aluminium alloys are often used as structural elements in the aircraft industry. The load-carrying capacity of this type of structure cannot relieve the reduction in strength in the event of local buckling. In this paper, a method of fabrication of rib-stiffened panels made of EN AW-2024-T3 Alclad and EN AW-7075-T6 Alclad has been proposed using single point incremental forming. Panels made of sheets of different thickness and with different values of forming parameters were tested under the axial compression test. A digital image correlation (DIC)-based system was used to find the distribution of strain in the panels. The results of the axial compression tests revealed that the panels had two distinct buckling modes: (i) The panels buckled halfway up the panel height towards the rib, without any appreciable loss of rib stability, and (ii) the rib first lost stability at half its height with associated breakage, and then the panel was deflected in the opposite direction to the position of the rib. Different buckling modes can be associated with the character of transverse and longitudinal springback of panels resulting from local interaction of the rotating tool on the surface of the formed ribs.

... As light alloy with high strength and excellent formability [1], aluminum alloys are widely used in transportation manufacturing. In automobiles, trains, ships, and aircrafts, lots of aluminum alloy profiles are formed into different parts [2]. ...

Improving mechanical properties and maintaining shape precisions are both significant for the further developments of aluminum alloy profiles. A new kind of ECAP with rubber mandrels can be innovatively used to form these alloy profiles in recent years. The profiles can be maintained as their initial shapes after pressing through the channel. At the same time, their mechanical properties can be improved. In this paper, analytical method is firstly used to find the mechanical mechanism during this process. Both finite element simulation and physical experiment are used to verify these results obtained by the analytical model. Except for the head and tail parts of the profiles, the shape of middle parts can be maintained well. Based on the stress delivered, severe plastic deformation can be applied in forming process of complex profiles. Better mechanical properties and good shape precisions can be creatively realized in complex profiles at the same time.

Inherent features of the Nakajima test to determine forming limit curves (FLC) are inevitable bending strains and pre-strains originating from evolving punch contact in the early stage. Both effects are very annoying and bias the resulting FLC considerably, so that a direct comparison of materials with different gauges is impossible. In the present work, a simple computational procedure is developed to remove both effects from FLC data. Application examples demonstrate its capabilities.

Biaxial tensile test methods using cruciform specimens have been drawing attention as an effective approach to characterize sheet metal formability. The cruciform specimens usually adopt a thickness reduction at the central area to achieve a relatively large strain gradient for localized necking and fracture investigation. For conventional tests, many criteria have been proposed to determine the forming limits at necking, however, none has been widely accepted and applied to the cruciform specimens with non-uniformed thickness reduction. Therefore, in this work, different existing necking criteria are applied and discussed for numerical procedure in order to define an appropriate necking criterion for the thickness reduced cruciform specimens. AA6061-T4 sheet with a thickness of 2mm is used as a target material. The predicted forming limit curves are compared for different criteria, and the feasibility of each criterion is discussed in detail. Thanks to two dedicated specimens proposed by the authors, forming limits are determined for a wide range of strain states.

Dans ce travail, on s'intéresse à la caractérisation et à la modélisation de plusieurs opérations élémentaires de mise en forme rencontrées dans les outils à suivre, à savoir l'emboutissage inverse, le pliage, aux conditions et aux outils numériques qui permettent d'assurer le sucées de ces opérations. Néanmoins, l’utilisation de ces procédés est limitée par le niveau de formabilité du matériau. Les Courbes Limites de Formage (CLF) sont souvent utilisées pour caractériser cette dernière. Cependant, plusieurs variables de l'essai peuvent influer directement sur les limites de formage qui peuvent être atteintes par un matériau lors de sa mise en forme. Parmi ces paramètres, la courbure des tôles, l'effet des chemins de déformation non linéaires et la pression de contact affectent les limites de striction et peuvent contribuer à les surestimer. Dans ce contexte, ce travail vise tout d’abord à introduire une procédure de compensation afin d'éliminer les effets combinés de la courbure, du trajet de déformation non linéaire et de la pression qui se produisent pendant les essais d'emboutissabilité. Des simulations numériques du comportement à la rupture du cuivre pur et d'un alliage de cuivre Béryllium sont menées sur des essais de Nakajima pour différentes configurations d’éprouvettes et validées sur des essais d'emboutissage inverse. Différents critères de rupture macroscopiques, implantés via une subroutine utilisateur dans le code de calcul Abaqus/Standard, sont utilisés. Le dernier objectif de ces travaux est de prédire le phénomène de retour élastique se produisant dans les procédés industriels utilisant des tôles ultra fines.

Ductile fracture during biaxial sheet stretching for two types of ferrous alloys and one type of nonferrous pure metal was predicted using the ellipsoidal void model proposed by one of the authors. First, fracture forming limit obtained experimentally by the Nakajima test scarcely differed from that obtained experimentally by the Marciniak test. Next, fracture forming limit obtained by the experiment scarcely depended on the magnitude of prestrain, which was applied to the sheet by rolling, and scarcely depended on the type of prestrain. Since the number of material constants on ductile fracture is only one, the material constant was determined so that the material thickness at fracture calculated from the simulation agreed with that obtained by the experiment. Fracture forming limit calculated using the ellipsoidal void model agreed closely with that obtained by the experiment, whereas fracture forming limit calculated using conventional ductile fracture criteria differed from that obtained by the experiment. Furthermore, fracture forming limit calculated using the ellipsoidal void model and that calculated using the conventional ductile fracture criteria changed drastically when the strain ratio changed across one for simulating biaxial stretching using a prestrained sheet.

This study presents a method for finite element (FE) simulation of a deep drawing process of a cold-rolled carbon steel (SPCC) sheet material based on the graphical method. First, uniaxial tensile specimens were prepared and experimental tests were conducted to determine the flow stress curves. The calculation of the fracture points at special strain modes (plane strain, uniaxial tensile strain, and biaxial tensile strain) was presented using the modified maximum force criterion (MMFC). After that, the graphical method was adopted for the estimation of the forming limit curve (FLC) based on several hardening laws. FE models for a deep drawing process of the SPCC sheet were then built using the calculated FLCs. Using FE simulations, the fracture heights of cylinder cups formed by the deep drawing process were finally determined and compared with those from experiments. The results showed a good agreement between simulated and measured fracture height with a maximum of 3.6 % deviation. Additionally, simulations and corresponding experiments were performed to investigate the effects of the blank holder force, punch corner radius, and drawing ratio on the fracture height of cylinder cups.

The stress–strain relationship of a sheet metal is conventionally evaluated from the experimental data obtained from the uniaxial tensile test before the onset of localized necking. The identification of the hardening behavior beyond the neck or post-necking behavior is essential for studying material fracture. This study presents a new procedure of identifying the flow stresses of sheet metals at large strains. In this procedure, the experimental data obtained from the uniaxial tensile test and the measured forming limit at a plane–strain tension mode FLC0 are adopted in the curve fitting procedure for calibrating the parameters of different hardening laws. The measured FLC0 is introduced in the cost function of the fitting to provide an additional constraint for the hardening parameters in large strain ranges. Using the uniaxial tensile test data ensures the accuracy of the pre-necking description of the hardening. The proposed method decreases the variation of the predicted flow stresses beyond the uniform elongation for different hardening laws. For validation purposes, the developed approach is applied to calibrate the parameters of the Kim–Tuan and the linear combined Swift/Voce hardening models for three sheet metals tested in Numisheet (2014) benchmarks. The hardening laws with newly identified parameters are implemented into the finite element software ABAQUS/Explicit to simulate the standard uniaxial tensile and reverse drawing tests. The comparative results show that the simulated tensile force in the uniaxial tensile tests and the simulated punch force in the reverse drawing tests agree well with the experiments, thereby validating the accuracy and the efficiency of the proposed identification procedure of large strain hardening in sheet metals.

Ductibor® 500-AS steel is used in hot-stamping applications within tailor-welded blanks to introduce more ductile, energy-absorbing regions within ultra-high strength structural components. This study aimed to explore the sensitivity of the microstructure and mechanical properties of Ductibor® 500-AS to changes in cooling rate. For this purpose, the microstructures and hardening behavior of Ductibor® 500-AS steel sheet were examined for a range of quench conditions, including austenitization followed by air, die, or water quenching. Optical and scanning electron microscopy investigations revealed a wide range of microstructures, comprising either mostly ferritic, mixed ferritic-martensitic, or mostly martensitic with some tempered martensite/bainite. Tensile and shear tests conducted on the as-quenched samples exhibited a high degree of sensitivity to cooling rate as well as strong mechanical anisotropy. The flow curves of the different material conditions obtained using a shear-conversion technique and fit to a modified Hockett-Sherby model were validated using the finite-element method (FEM) simulations of the uniaxial-tension tests in LS-DYNA, considering isotropic von Mises and/or anisotropic Barlat Yld2000 yield criteria. The predicted load–displacement response and strain localization patterns using the isotropic and anisotropic models were shown to agree with the measured results. However, only the anisotropic model was able to predict the local strain histories consistent with the experiments. Consequently, it is important to account for both of the quench-rate sensitivity and plastic anisotropy of Ductibor® 500-AS in crash-safety simulations of structural components utilizing this steel.

Springback simulation of stamped sheet metal components using finite element method depends on the accuracy of appropriate material models and consideration of appropriate experimental strategies. In this work, tension-compression tests with different strategies, e.g. tension-compression, compression-tension up to various strain levels and multicycle compression-tension tests were conducted to determine parameters of the Yoshida-Uemori (Y-U) nonlinear dynamic hardening model using optimization analysis software LS-OPT. Finite element simulations with LS-DYNA were performed to predict springback behavior of both the advanced high strength steel MP980 (a 980 MPa grade multiphase steel) and aluminum alloy 6022-T4, which was then compared to measurements of stamped U-channel specimens. Results suggest that although the various tension-compression testing strategies can significantly affect the determined values of Yoshida-Uemori model parameters, springback prediction accuracy with this model does not depend on the associated variation of model parameters, at least for the two-dimensional sidewall curl of a U-channel shape. For materials (e.g. MP980) exhibiting a clear Bauschinger effect but insignificant texture anisotropy, the selection of suitable yield criteria (e.g. Hill48), the consideration of elastic modulus degradation combined with the Y-U model can obviously increase the accuracy of springback prediction. In contrast, materials (e.g. AA6022-T4) that exhibit little Bauschinger effect but have significant texture anisotropy, the use of a yield criterion that accounts for anisotropy (e.g. YLD2000-2D) is more important for improving the accuracy of springback prediction.

The accuracy of the forming limit diagram (FLD) determined through Nakazima test is often influenced by non-linear strain path (NLSP) and through-thickness stress. The influence of these factors on measurement accuracy can be weakened or avoided in Marciniak test. But meanwhile, Marciniak test is difficult in operation and possesses poor repeatability. Therefore, the Nakazima test eliminating the influencing factors of accuracy is still an attractive method due to the easy operability and good repeatability. In the present research, two FLDs of aluminium alloy 6014-T4 that were established through Nakazima and Marciniak tests were compared. In order to obtain an accurate FLD in plane-stress state and linear strain paths via Nakazima test, a modified compensation method was applied in Nakazima test to eliminate the interferences of NLSP and through-thickness stress on measuring forming limits. The ultimate strains measured in Nakazima test were transformed to stress-based forming limits through constitutive models to remove the influence of through-thickness stress. Then, the corrected ultimate stresses were transformed back to the limit strains along linear strain paths. Compensated results of Nakazima test showed a good agreement with the experimental results from Marciniak test, proving the validity of this method. Thus, through the easy-operate Nakazima test, an FLD with comparable accuracy to that based on Marciniak test was established.

The forming limit behaviour of AA6013-T6 aluminium alloy sheet was characterized under isothermal conditions at room temperature (RT) and 250°C using limiting dome height (LDH) tests. Full field strain measurements were acquired throughout testing using in situ stereoscopic digital image correlation (DIC) techniques. Limit strain data was generated from the resulting full field strain measurements using two localized necking criteria: ISO12004- 2:2008 and a time and position dependent criterion, termed the "Necking Zone" (NZ) approach in this paper, introduced by Martinez-Donaire et al. (2014). The limit strains resulting from the two localization detection schemes were compared. It was found that the ISO and NZ limit strains at RT are similar on the draw-side of the FLD, while the NZ approach yields a biaxial major limit strain 14.8% greater than the ISO generated major limit strain. At 250°C, the NZ generated major limit strains are 31-34% greater than the ISO generated major limit strains for near uniaxial, plane strain and biaxial loading conditions, respectively. The significant variance in limit strains between the two methodologies at 250°C highlights the need for a validation study regarding warm FLC determination.

This work describes a new procedure to remove the differences in measured forming limits obtained from Marciniak and Nakazima tests, which are the two most frequently used testing methods to obtain necking limits for forming limit diagrams (FLD) used in formability analysis of sheet metal stamping processes. The procedure compensates for the combined effects of curvature and nonlinear strain path that occur during these tests, using measurements recorded by digital image correlation (DIC) throughout the deformation history of the point on the test specimen that eventually necks. The severity of forming is then determined by presenting the critical forming conditions in a stress diagram in order to account for the effects of through-thickness pressure that influences the onset of localized necking in the Nakazima test. These stress-based forming limits are then transformed back to the familiar strain limits (or FLD), but now representing the limits under the restriction of in-plane perfectly linear stretching and plane-stress conditions. Accounting for the effects of nonlinear strain path is particularly sensitive to the detection of the actual onset of localized necking, so this work also recommends the use of realistic methods to detect the actual onset of localized necking. The method adopted in this work is based on a new method described by Min et al. (2016a,b), in which a change in the surface curvature is used to detect a geometric effect associated with the onset of localized necking. In addition to demonstrating that the standard Marciniak and Nakazima tests (punch diameter of 101.6 mm) produce essentially identical limit curves for a 980 MPa grade multi-phase high strength steel after correction for curvature, nonlinear strain path, and pressure, the method is also applied to the analysis of data from a non-standard Nakazima test with a smaller punch diameter of 50.8 mm, where the severity of these processing conditions are significantly enhanced. This additional test is further proof of the validity and comprehensive coverage of the corrections for the different processing conditions involved in measurement of forming limit curves.

This study describes the forming behaviour of work hardening AA5182 sheet alloy at various low temperatures. The material parameters including flow curves, Lankford parameters and forming limit curves were obtained by experimental testing in the temperature range from room temperature to-196 °C, which were then used for isothermal forming simulations using the finite element method (FEM). It was found that strength, elongation and forming limits increase with decreasing temperature. The finite element model was validated using deformed Nakajima strips. Numerical results presented in this study demonstrate the potential of cryogenic forming for manufacturing of complex automotive components.

The goal of numerical simulation in sheet metal forming processes is mainly the prediction of failure. The mostly used criteria in the FE-simulation are based on the forming limit curves (FLC) originally proposed by Keeler (1965) and Goodwin (1968). The experimental determination is usually done only for nearly linear paths. Although the linear FLC-criterion is easy to implement within FE-codes, there is a well known lack of accuracy, if non linear strain paths occur. The influence of the non linear strain path has been experimentally proved in different works for two-stage deformation-paths, ref. s. /7,9,11/ and theoretically described by Rasmussen (1981) within the framework of the Marciniak-Kuczynski model. In the presented paper, the strain path dependancy will be discussed within the framework of the modified force maximum criterion proposed by Hora and Tong, (1994). The description bases on the plane stress assumption. The mathematical criterion allows a determination of FLC for arbitrary strain paths as well as for anisotropic materials. The criterion can be implemented directly into FE-codes. FAILURE MODEL The modified force criterion is based on the assumption of the plane strain deformation state during the localized necking. Due to the related stress change, there is an additional hardening effect, which 'hesitates' out the localized necking, s. Fig. 1. This dependancy has been earlier mentioned by Sowerby and Duncan and discussed by Hosford (1988). Mathematical Formulation In the following, the criterion takes the expanded form

The ISO standard 12004-2:2008E for the determination of forming limit curves based on the section method was approved in 2008. About 4 years of measuring experience in different laboratories has shown advantages and weaknesses of the standard and is leading to some minor changes in the specification. In the years from the development of this standard until today a further technical development of the optical measuring devices occurred, so that it is now possible to determine forming limit curves using the time history of the test. This procedure of determination is referred to a time dependent technique and could be the basis of the ISO 12004 part 2 proposal worked out by the work group Erweiterung FLC ISO 12004 of the German group of the IDDRG.
This publication recapitulates existing work which was carried out from the IDDRG work group regarding the determination of forming limit curves for sheet metal materials. On one hand known issues with the current section based approach are discussed and on the other hand it deals with a comparison of different algorithms to determine the FLC from the time history of the Nakajima test using strategies to identify the instant of onset of instable necking. The different time dependent algorithms [ utilised are automatically selecting the area where necking is leading to fracture and then analyze the time history of such points using the first or the second time derivative of the true major strain, or of the true thinning strain using methods like: correlation coefficient (modified method based on [2]), gliding correlation coefficient, linear best fit (modified method based on [3]) and gliding difference of mean to median. The resulting experimental FLC points are compared with the results from the section technique described in ISO 12004 part 2 and with the maximum strain values measured in each test. Further a large number of forming limit curves were determined and used for a comparison of these different methods to define the most promising time dependent algorithm, which was selected as a suggestion for the working group defining the new proposed ISO standard 12004 part 2.

The standard uniaxial tensile test is the widely accepted method to obtain relevant properties of mechanical characterization of sheet metal materials. However the range of strain obtained from tensile test is limited. The bulge test is an alternative to obtain ranges of deformation, higher than tensile test, thus permitting a better characterization for material behaviour. This paper presents a sensitivity analysis for some influencing variables used in bulge measurements, thus giving some guidelines for the evaluation of the stress-strain curve from experimental results using a developed experimental mechanical system. Additionally, using bulge test up to fracture shall give material information regarding damage, which in turn may be used to evaluate and calibrate damage models. A methodology is presented to be used for evaluation and calibration of Ito-Goya damage model of damage prediction.

In order to perform the theoretical evaluation of Forming Limit Curves (FLC), the Modified Maximum Force Criterion (MMFC) has been proposed. This paper investigates the mechanism of the fracture of ductile sheet metals and introduces the MMFC model. The evaluation process and the simplified formulations are presented. The influences of hardening behavior and the yield loci are discussed as well. Comparisons with the experimental data of different materials showed generally satisfactory agreement.

Few shear test techniques exist that cover the range of strain rates from static to dynamic. In this work, a novel specimen geometry is presented that can be used for the characterisation of the shear behaviour of sheet metals over a wide range of strain rates using traditional tensile test devices. The main objectives during the development of the shear specimen have been 1) obtaining a homogeneous stress state with low stress triaxiality in the zone of the specimen subjected to shear and 2) appropriateness for dynamic testing. Additionally, avoiding premature specimen failure due to edge effects was aimed at. Most dimensional and practical constraints arose from the dynamic test in which the specimen is loaded by mechanical waves in a split Hopkinson tensile bar device. Design of the specimen geometry is based on finite element simulations using ABAQUS/Explicit. The behaviour of the specimen is compared with the more commonly used simple shear specimen with clamped grips. Advantages of the new technique are shown. The technique is applied to Ti6Al4V sheet. During the high strain rate experiments high speed photography and digital image correlation are used to obtain the local shear strain in the specimen. Comparison of experimental and numerical results shows good correspondence.

The paper presents an analysis of a recently proposed failure criterion for thin sheets. According to Aretz [1], this criterion becomes numerically unstable for yield surfaces with locally constant exterior normal fields. Here we make more precise statements about the nature of this instability, asses the predictive capabilities of the criterion, and introduce a fitting parameter for its plane strain calibration.

In the paper the authors propose an improved version of the Modified Maximum Force Criterion (MMFC) initially developed by
Hora and his co-workers. The most appealing characteristic of this model is the fact that no assumptions related to the existence
of voids or other defects are needed. The MMFC model is based on the hypothesis that a plane-strain state develops during
the localized necking. Due to the related stress change, there is an additional hardening effect, which ‘hesitates’ out the
localized necking. The improvement proposed in the frame of this paper consists in adding two material constants to the MMFC
model. Their values are determined by enforcing a couple of experimental constraints on the model. In this way, the quality
of the theoretical predictions can be improved. As proven by the numerical tests, the most appropriate constraints are related
to the necking occurrence in plane strain and biaxial tension, respectively. By controlling these points of the FLD, the discrepancies
between Hora’s model and experiments can be minimised. If no experimental data is available, the constraints could be generated
by using some other model for the FLD determination (e.g., empirical formulas or the model developed by Marciniak and Kuczynski).
The performances of the new model are evaluated by comparing its predictions with Hora’s necking criterion and experimental
FLD’s corresponding to different sorts of sheet metals.

Plane-strain forming limit strain (also known as FLD0) is an important data point on a forming limit diagram (FLD). The effects of friction coefficients and material parameters on the specimen width associated with the FLD0 (\(W_{{{\text{FLD}}_0}}\)) in Marciniak test were studied by finite element simulation. \(W_{{{\text{FLD}}_0}}\) was expressed as a function of the Lankford coefficients, n-value, k-value and sheet thickness and validated with various sheet materials. The determination of \(W_{{{\text{FLD}}_0}}\) is of significance not only to reduce iterative attempts to accurately obtain FLD0, but also to obtain a full valid FLD with the least number of test specimens, which largely increases the efficiency and reduces cost to experimentally measure valid FLDs.

Experimental and numerical investigations on the forming limit diagram (FLD) of a ferritic stainless steel were performed in this study. The FLD of this material was obtained by Nakajima tests. Both the Marciniak-Kuczynski (MK) model and the modified maximum force criterion (MMFC) were used for the theoretical prediction of the FLD. From the results of uniaxial tensile tests along different loading directions with respect to the rolling direction, strong anisotropic plastic behaviour was observed in the investigated steel. A recently proposed anisotropic evolving non-associated Hill48 (enHill48) plasticity model, which was developed from the conventional Hill48 model based on the non-associated flow rule with evolving anisotropic parameters, was adopted to describe the anisotropic hardening behaviour of the investigated material. In the previous study, the model was coupled with the MMFC for FLD prediction. In the current study, the enHill48 was further coupled with the MK model. By comparing the predicted forming limit curves with the experimental results, the influences of anisotropy in terms of flow rule and evolving features on the forming limit prediction were revealed and analysed. In addition, the forming limit predictive performances of the MK and the MMFC models in conjunction with the enHill48 plasticity model were compared and evaluated.

There is significant confusion surrounding the appropriateness of the logarithmic (Hencky) strain measure to describe simple shear deformation for finite strain. In simple shear loading of plastically deforming materials, the principal stress and strain directions do not remain coaxial which has led to conflicting derivations of the equivalent strain throughout the literature. The source of this confusion is attributed to a misapplication of the formulas for the von Mises equivalent strain and its increment that are only valid for proportional coaxial loading. In this work, a closed-form solution for the work-conjugate equivalent strain for an arbitrary yield function was derived for simple shear loading that is readily amenable to experimental characterization and is entirely consistent with the logarithmic strain measure. An analytical stress and strain integration of an elastic-plastic material was performed using the logarithmic objective rate to demonstrate that the stress, logarithmic strain, and principal directions are correctly determined within a hypo-elastic-plastic framework to finite strains. It was demonstrated that the integrated equivalent plastic strain is work-conjugate and the logarithmic strain measure is appropriate for finite simple shear. A review of the recent experimental literature for shear characterization has found that the misapplication of the von Mises equivalent strain formula for coaxial loading in simple shear loading is pervasive. The coaxial effective strain formula is the default measure in commercial digital image correlation (DIC) software and may significantly underestimate the equivalent strain in the simple shear loading condition. If the major principal strain in a simple shear test is lower than 50%, the error between the coaxial and work-conjugate equivalent strains is negligible. Otherwise, the error grows in a hyperbolic manner. Within the context of a finite element simulation of a simple shear test, if a hypo-elastic-plastic formulation is employed as in most commercial codes, the equivalent strain will be correctly computed from work conjugacy but the cumulative and principal logarithmic strain tensors will be incorrect at finite strains unless the logarithmic rate is employed.

Rare-earth magnesium alloys such as ZEK100-O offer improved ductility over other conventional magnesium alloys at room temperature; however, they exhibit significant anisotropy and complex yield behaviour. In this work, a systematic investigation of anisotropy of ZEK100-O rolled sheet in shear loading was conducted at room temperature under quasi-static conditions, as the shear deformation of these alloys is not well understood. Furthermore, uniaxial tensile and compressive characterization of the material was performed to provide context for its behaviour under shear loading. It was revealed that ZEK100-O exhibits strong anisotropy in shear which is markedly different than tensile anisotropy with unique trends that suggest the activation of different deformation mechanisms. To characterize shear anisotropy in HCP materials such as ZEK100-O, the shear response of the material should be investigated in three orientations of 0° (or 90°), 45°, and 135° with respect to the rolling direction. The selection and analysis of these directions is discussed in terms of the principal stress directions and activation of different deformations mechanisms. In order to further investigate this behaviour, the microstructure of the deformed specimens was studied using Electron Backscattered Diffraction (EBSD) analysis to quantify the active twinning systems in different test orientations. Moreover, the CPB06 yield criterion with two linear transformations was calibrated with experimental data to describe the complex anisotropic behaviour of ZEK100-O. It was established that the material exhibits asymmetry not only in tension-compression regions represented by the 1st and 3rd quadrants of yield locus but also in shear regions represented by the 2nd and 4th quadrants. Finally, the strain rate sensitivity of ZEK100-O was studied in shear tests at elevated strain rates of 10 s⁻¹ and 100 s⁻¹, at which positive rate sensitivity was observed.

Shear tests were performed at strain rates ranging from quasi-static (0.01 s⁻¹) to elevated rates (600 s⁻¹) considering DP600 steel and AA5182-O aluminum alloy sheet at room temperature. The shear specimen due to Piers et al. [J. Peirs, P. Verleysen, J. Degrieck, Experimental Mechanics, 52 (7), pp. 729-741, 2012] was scaled (reduced in size) to perform high strain rate shear testing. In situ digital image correlation (DIC) techniques were employed to measure the strains in the experiments and methods are proposed for characterizing the local strain within shear bands. Using the DIC strain measurements along with finite-strain theory and the logarithmic objective stress rate, a simple methodology was developed to obtain the work hardening response to large strain levels using only shear and tensile experiments. At lower strains, the DP600 shows positive rate sensitivity while the AA5182 exhibited limited sensitivity as strain rate increases. At equivalent strains greater than approximately 20%, the DP600 and AA5182 alloys demonstrated a reduced work hardening rate at elevated strain rates. For both alloys, the strain to localization (using the Zener-Holloman criterion) and subsequent fracture strain, measured using the DIC technique, decreased with strain rate in shear loading, but increased under uniaxial tensile loading. Microscopic assessment of fractured DP600 specimens was also performed using measurement of grain boundary rotation to determine local strains at fracture corresponding to length scales below the DIC measurements. The local strains at final failure are much higher and reveal an increase in the shear fracture strain with strain rate, in contrast to the trends based on the DIC analysis.

A method using surface curvatures to detect the onset of localized necking in forming limit tests [Min J, Stoughton TB, Carsley JE, Lin J. A Method of Detecting the Onset of Localized Necking Based on Surface Geometry Measurements. Exp Mech 2016; DOI:10.1007/s11340-016-0232-4.] is improved to be more robust and operator-independent. The improved method is referred to as the 3D curvature method, whereby the curvature is fitted on the basis of a small narrow surface encompassing the neck band, and the onset of localized necking is still detected by the sudden increase of the fitted curvature according to the neck expansion theory proposed by Min et al. [Min J, Stoughton TB, Carsley JE, Lin J. A Method of Detecting the Onset of Localized Necking Based on Surface Geometry Measurements. Exp Mech 2016; DOI:10.1007/s11340-016-0232-4.] Appropriate widths and lengths of the small narrow area on the specimen surface are determined by investigation of their effects on the curvature fitting procedure and on the curvature increase that signals the onset of localized necking. In addition, the 3D curvature method is successfully extended to detect the onset of localized necking in Nakazima tests, where the deformation is characterized as out-of-plane, as opposed to the nearly in-plane deformation of Marciniak testing. As a result, forming limit strains associated with the onset of localized necking can be obtained from both commonly used Marciniak and Nakazima tests on an equivalent basis of geometrical change of specimen surface. This is significant for the reconciliation of forming limits from the two tests after compensation for the effects of non-linear strain path, bending and contact pressure described by Min et al. [Min J, Stoughton TB, Carsley JE, Lin J. Compensation for process-dependent effects in the determination of localized necking limits. Int J Mech Sci 2016; 117: 115–134.]

To more accurately capture the onset of localized necking and obtain necking limit strains, this paper proposes a method of detecting the onset of localized necking in the Marciniak test (i.e. under in-plane deformation). The method is merely based on the measured surface geometry of the test specimen using digital image correlation (DIC) techniques. It was inspired by the observation of a sudden increase of the surface curvature obtained from 2D curvature fits along the direction across the surface of the sheet. This increase of the surface curvature is detected just before the dimples, which form the final localized neck, become obvious in the DIC measurements. The appearance of this signal is explained by a neck expansion theory defined by propagation of the instability along the direction of the neck, which is a physical behavior of materials.

Two different shear sample geometries were employed to investigate the failure behaviour of two automotive alloy rolled sheets; a highly anisotropic magnesium alloy (ZEK100) and a relatively isotropic dual phase steel (DP780) at room temperature. The performance of the butterfly type specimen (Mohr and Henn Exp Mech 47:805–820, 16; Dunand and Mohr Eng Fract Mech 78:2919-2934, 17) was evaluated at quasi-static conditions along with that of the shear geometry of Peirs et al Exp Mech 52:729-741, (27) using in situ digital image correlation (DIC) strain measurement techniques. It was shown that both test geometries resulted in similar strain-paths; however, the fracture strains obtained using the butterfly specimen were lower for both alloys. It is demonstrated that ZEK100 exhibits strong anisotropy in terms of failure strain. In addition, the strain rate sensitivity of fracture for ZEK100 was studied in shear tests with strain rates from quasi-static (0.01 s−1) to elevated strain rates of 10 and 100 s−1, for which a reduction in fracture strain was observed with increasing strain rate.

Various factors affecting the prediction of limit strains in biaxially-stretched sheets are studied. Time-independent material behavior is assumed, and both the flow theory of plasticity as well as a finite-strain version of deformation theory are considered. A localization-band bifurcation analysis is first carried out. The influence of geometric imperfections is then analyzed using the long-wavelength approximation treated in Part I. We also discuss the predicted forming limit curves and comment on their relation to published experimental data. The main emphasis in this Part, however, is on comparisons between the corresponding predictions of flow theory and deformation theory.

Application of the inverse analysis to the interpretation of hot tensile tests is the main objective of the work. Tensile tests were performed for carbon-manganese steel samples at 1000°C and at three ram velocities. The measured temperature distribution was used as boundary condition in the finite element model of the direct problem. Recorded loads and elongation of the sample were used as input for the inverse analysis. The flow stress, recalculated for isothermal conditions and constant strain rate deformation, was determined as a function of strain up to the maximum deformations achieved in the neck. Validation of the flow stress model based on comparison of the measured loads with the finite element predictions for the developed Rheological model, confirmed reasonably good accuracy of the procedure. Additionally, sensitivity of the measured parameters (loads and radius of the neck at certain elongation) with respect to the Rheological parameters has been performed.

Mechanical properties of steel sheets governing the press formability were determined and the lubricating effect in stretch forming was made clear. Forming limit determined in model experiment was shown to be useful as the criterion for evaluating the forming severity in practical forming and also for finding countermeasures in forming conditions in order to improve the forming severity and for selecting appropriate steel sheet.

The mechanical behavior of the commercial aluminum alloy AA5182-O is investigated at temperatures ranging from −120 to 150 • C and strain rates from 10 −6 to 10 −1 s −1. The strain rate sensitivity parameter is determined as a function of temperature and plastic strain, and the strain rate and temperature range in which dynamic strain aging leads to negative strain rate sensitivity is mapped. The effect of dynamic strain aging on ductility and strain hardening is investigated. The sensitivity of the measured quantities to the experimental method employed and their dependence on grain shape are discussed. The experimental data are compared with the predictions of a model constructed based on a recently proposed mechanism for dynamic strain ageing. The mechanism is based on the effect solute clustering at forest dislocations has on the strength of dislocation junctions. The model is shown to reproduce qualitatively the experimental trends.

This paper presents the results of an investigation into the rate sensitivity of DP600, TRIP780 and AA5182-O sheet metal alloys. The effect of strain rate on both the flow stress and anisotropy characteristics was also examined. Tensile experiments were performed at room temperature under conditions ranging from quasi-static to high strain rate loading on the three alloys in three orientations (0°, 45°, and 90°) with respect to the rolling direction. The longitudinal and width strains were measured with a biaxial extensometer for the quasi-static experiments and using Digital Image Correlation (DIC) methods for the elevated strain rate experiments. The DP600 and TRIP780 steels showed moderate rate sensitivity while AA5182-O showed relatively low or even negative rate sensitivity that was a function of strain rate. The Portevin-Le Châtelier (PLC) effect was observed in AA5182-O up to a strain rate of 1 s-1 and was considered to be the cause of the negative rate sensitivity. The hardening behavior of the TRIP780 at a strain rate of 1000 s-1 differed from that observed at lower strain rates. This difference is attributed to the effect of the adiabatic temperature rise on the transformation-induced plasticity effect (TRIP). The Lankford coefficients and the variation in flow stress with material orientation were relatively insensitive to strain rate for all three alloys. A number of constitutive fits were considered and validated successfully through finite element analysis simulations of the tensile experiments. The performance of the constitutive models was evaluated through comparison of the predicted and measured engineering stress-strain responses and the area reduction at the measured elongation at fracture.

Plastic deformation in sheet metal consists of four distinct phases, namely, uniform deformation, diffuse necking, localized necking, and final rupture. The last three phases are commonly known as nonuniform deformation. A proper forming limit diagram (FLD) should include all three phases of the nonuniform deformation. This paper presents the development of a unified approach to the prediction of FLD to include all three phases of nonuniform deformation. The conventional method for predicting FLD is based on localized necking and adopts two fundamentally different approaches. Under biaxial loading, the Hill's plasticity method is often chosen when α (=∈2/∈1) < 0. On the other hand, the M-K method is typically used for the prediction of localized necking when α > 0 or when the biaxial stretching of sheet metal is significant. The M-K method, however, suffers from the arbitrary selection of the imperfection size, thus resulting in inconsistent predictions. The unified approach takes into account the effects of micro-cracks/voids on the FLD. All real-life materials contain varying sizes and degrees of micro-cracks/voids which can be characterized by the theory of damage mechanics. The theory is extended to include orthotropic damage, which is often observed in extensive plastic deformation during sheet metal forming. The orthotropic FLD model is based on an anisotropic damage model proposed recently by Chow and Wang (1993). Coupling the incremental theory of plasticity with damage, the new model can be used to predict not only the forming limit diagram but also the fracture limit diagram under proportional or nonproportional loading. In view of the two distinct physical phenomena governing the cases when α (=∈2/∈1) < or α > 0, a set of instability criteria is proposed to characterize all three phases of nonuniform deformation. The orthotropic damage model has been employed to predict the FLD of VDIF steel (Chow et al., 1996) and excellent agreement between the predicted and measured results has been achieved as shown in Fig. 1. The damage model is extended in this paper to examine its applicability and validity for another important engineering material, namely aluminum alloy 6111-T4.

Strain localization is one of the main sources of failure in sheet forming processes. State of the art forming limit curves allow the prediction of localization for linear strain paths but fall short in case of non-proportional loading. The aim of this contribution is to revisit the Modified Maximum Force Criterion (MMFC) and extend it to accommodate distortional hardening models. This is accomplished by uncoupling its formulation from any particular yield function and thus enabling its use as a generic framework for the prediction of forming limits under arbitrary loading conditions. Furthermore a novel approach is proposed for considering strain rate sensitivity, which substantially improves the predictive capabilities of the model under plane strain tension conditions. The method is applied to steel and aluminum materials and the role of phenomena such as Bauschinger effect, latent hardening and cross-loading contraction on localization are discussed.

The work-hardening of the solution-strengthened aluminium alloy AA5182-O is studied as a function of strain rate at room temperature. The main aim is to characterize the influence of strain rate and dynamic strain ageing on the work-hardening of the material. Tensile tests with constant nominal strain rates in the range from 10(-6) to 1 s(-1) were carried out; additionally, strain-rate jump tests were performed. The material exhibits a sharp yield point, a Luders plateau, and serrated yielding within the actual range of strain rates. This indicates that the steady-state strain rate sensitivity is negative as a result of dynamic strain ageing. The serrations in the stress-strain curves and the work-hardening are markedly reduced with increasing strain rates. In most cases, the jump tests exhibit the standard behaviour, i.e., after a transient period following the jump in strain rate, the stress-strain curve converges to the steady-state stress-strain curve at the new strain rate.

A rate dependent Taylor type crystal plasticity formulation is employed to study the effects of post-necking hardening behavior on the Marciniak–Kuczynski (M–K) based forming limit diagram predictions. Three uniaxial stress–strain curves, identical up to the necking region, with three different post-necking hardening behaviors are generated using crystal plasticity formulation. The three curves are obtained by curve fitting the simulated uniaxial true stress–strain curve to the experimental curve for the CC AA5754. In the first curve the true stress strain curve saturates after necking while the other two curves demonstrate post-necking hardening. A representative volume element obtained from Electron Backscatter Diffraction (EBSD) data is considered for the analysis. Numerical simulations of forming limit diagrams based on the classical M–K approach are performed with the same initial texture and the three different stress–strain curves. Analysis with the numerical model demonstrates that the post-necking behavior significantly affects the predicted FLDs. A new calibration approach is presented to accurately capture the post-necking behavior of metals. Predicted FLDs with this new approach are compared to the experimental FLD for the CC AA5754. In order to further characterize the process of finding the accurate post-necking hardening behavior, a method based on using equivalent stress–strain curve is presented. It is shown that the FLD, calculated using experimental uniaxial tensile data as input, is comparable with a FLD arising from equivalent stress–strain data.

The standard ISO 12004-2:2008 provides a position-dependent methodology to estimate the limit strains in Nakazima- and Marciniak-type tests. However, this method is not applicable, at least in its current formulation, when there are significant strain gradients across the sheet thickness, such as when using relatively small punch radii or in stretch-bending operations very commonly in industrial practice. This paper analyses two physically-based methodologies, a time-dependent method and a time-position-dependent method (called here flat-valley method), to detect the onset of necking and to evaluate the limit strains under significant strain gradients through the sheet thickness. The digital image correlation (DIC) technique is used to compute the displacement and strain evolutions at the outer surface of the tested specimens using the commercial software ARAMIS®. A detailed analysis and validation of both methodologies, and comparison with other local methods recently published in the literature, are carried out in the light of a series of Nakazima tests and stretch-bending tests for different cylindrical punch radii in AA7075-O.

According to a recent (original) model, when hardening properties and the ratio of through-thickness normal stress to the first principal stress (γ≡σ3/σ1) are held constant, sheet metal formability can be increased dramatically through the introduction of a compressive through-thickness normal stress, σ3. In practice, however, both the hardening properties and γ evolve with the progression of deformation. To manage most efficiently the evolution of the hardening properties and γ, the original model is cast into a more compact form and presented as a proposed alternative form (proposed model). When the evolution of the hardening properties and γ is considered, the proposed model is shown to be in very good agreement with observed data; the influence of through-thickness normal stress on sheet metal formability is quite small for all practical purposes. Because the structure of the original model is similar to that of the proposed model, the original model is also validated. Ultimately, it is verified that although the theory of the original and proposed model may be acceptable, the implications of such theories are less profound than those first proposed when practical limitations are considered. This work serves as a useful basis for: (1) further understanding the limitations of the influence of compressive through-thickness normal stress on sheet metal formability and (2) exploring opportunities for improving sheet metal formability.

A generalized isotropic yield criterion of the form,
(σ 1−σ 2) ⁿ+(σ 2−σ 3) ⁿ+(σ 1−σ 3) ⁿ2 1/n
= Y
where σ1 ≥ σ2 ≥ σ3 and 1 ≤ n ≤ ∞, is proposed. The corresponding flow rules, Lode variables, and effective strain functions are presented. Experimental and theoretical data on yielding under combined stresses can be described by a single parameter, n.

The validity of using an 'equivalence criterion' to describe the plastic properties of polycrystals at large strains is addressed here from the point of view of the microscopic properties of the constituent grains. Proper account is taken of the grain reorientation as deformation proceeds and of the different amounts of hardening experienced by each grain. The averaged grain behavior is compared with the results of experiments on OFHC copper in three deformation modes: torsion, compression and diametrically controlled large strain tension. An attempt to explain the three different stress/strain curves on the basis of a unique microscopic hardening law using the above theory was only partially successful, and it is concluded that the microscopic hardening mechanisms depend on the deformation mode. Physical mechanisms for such a dependence are discussed and evaluated according to the observed statistics of grain behavior.

The forming limit diagram (FLD) is a convenient tool for classification of sheet metals’ formability in the finite element analysis as well as in the press shop. The FLD indicates the maximum strain values which can be applied on a material without failure as a function of the strain condition. In contrast to the standardized evaluation method described in the standard ISO 12004-2 a new time dependent analysis method is presented. Using a regression analysis the onset of necking can be detected automatically independent of the strain state. Results will be presented and discussed in contrast to the existing standard procedure.

A development of the theory of the formation of a groove, which appears before fracture in sheet metal subject to tension is presented, and is based on the assumption of initial non-homogeneity of the material in order to facilitate the determination of the limit strain of sheet metal subject to biaxial tension. Additional considerations in the present paper include strain-rate sensitivity, plane anisotropy and the difference in the values of the fracture strain which depends on the direction of the fracture with reference to that of rolling. The influence of the above properties of the material on the limit strain curve is analysed over the entire range of biaxial tension.Theoretical curves of limit strain are compared with experimental results for sheet metal in biaxial tension under conditions which eliminate the influence of friction and which ensure uniform strain distribution over the entire surface of the test piece, except in places of strain concentration. The discrepancies between the experimental and theoretical limit strain curves are analysed.

The forming limit diagram (FLD) is used in sheet metal forming analysis to determine how close the sheet metal is to tearing when it is formed into a product shape in a stamping process. The strain-path dependent nature of the FLD causes the method to become ineffective in the analysis of complex forming process, especially restrikes, flanging operations, hydroforming, and even first draw dies with deep pockets or embossments. Experimental evidence for a path-independent stress-based FLD has been reported in the literature, suggesting that the path dependency of the strain-based approach arises from the path dependent constitutive laws governing the relationship between the stress and strain tensors. This paper reviews several theoretical models of sheet metal forming instability, including bifurcation analyses of diffuse and through-thickness neck formation, the M-K model and microscopic void damage models. The equations governing the deformation at the instant of the bifurcation is shown to be independent of path in all of these models, providing a solid theoretical bases for the stress-based approach. The stress-based FLD can now be used equally well for all forming processes, without concern for path effects.

Large strain compression data (true strains to about −3.0) are presented for polycrystalline α U and α Fe at room temperature. The results, together with other published data at low homologous temperatures (≈0.2 Tm), where Tm is the absolute melting temperature, suggest that a steady-state flow stress σs is approached after extensive strain-hardening, α U exhibits a very high strain-hardening rate, with σs ≈ 2900 MPa (420 ksi) indicating that cold-working is a very potent method of strengthening this metal. All the data evaluated can be fit by the stress-strain relation σ = σs− exp (−(Nε)p)(σs− σy), where σy is the yield stess, p is a constant equal to a for the metals analyzed, N is a constant associated with the strain-hardening characteristics of a material, σ is true stress, and ε is true strain.

This article addresses a theoretical analysis of the 'modified maximum force criterion' (MMFC) introduced by Hora et al (1996 A prediction method for ductile sheet metal failure in FE-simulation Proc. Numisheet'96 Conf. pp 252–6). In this work a computational framework for the MMFC is given and it will be shown that the MMFC contains an important singularity which emerges if the yield locus used in the necking simulation exhibits straight line segments and leads to singularities in the predicted forming limit curve or even in total failure in forming limit diagram prediction. It is shown that the singularity emerges in commercial sheet alloys and, therefore, the general reliability of the MMFC appears questionable. It is thus concluded that an extension of the MMFC is required in order to fix the singularity problem.

In this paper, an anisotropic damage model for predicting FLDs is extended to predict the sheet metal formability under non-proportional loading involving changing directions of principal strain plane and damage plane. The material chosen for the prediction of FLDs is AL6111-T4 aluminum alloy. A satisfactory agreement between the model predictions and test results has been achieved.

Plastic instabilities associated with the Portevin–Le Chatelier (PLC) effect are investigated in an Al-based alloy of the 5000 series, in a wide range of temperatures and loading rates. The domains of occurrence of the PLC effect are explored and compared for constant stress rate and constant strain rate tensile testing. In both cases, experimental results show that the PLC domains are bounded towards high temperatures and loading rates. The critical strain for the onset of repeated yielding increases with increasing loading rate at low temperatures and high loading rates (normal behaviour), while it decreases at high temperatures and low loading rates (inverse behaviour), with different types of serration associated with the normal and the inverse behaviour. A simple model is proposed to explain this relation between the PLC type and the behaviour of the critical strain. Moreover, the specimen surface finish is found to have a significant influence on the stress–strain curves and the band propagation velocities. Finally, the experimentally observed PLC domain is shown to be in good agreement with a physical model of the dynamic strain ageing associated with the PLC effect.