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The precise determination of the forming limit strains of a sheet metal is necessary in order to accurately predict the onset of failure in sheet metal forming processes. The method used to detect the onset of necking (i.e. the necking criterion) is a most important factor in formability analysis as it significantly affects the forming limit strains. Many different necking criteria have been developed, which take advantage of the flexibility and high resolution of digital image correlation (DIC) strain measurements. Several time-dependent and time/geometry-dependent necking criteria were carefully investigated in order to evaluate their ability to reliably and consistently detect the onset of necking of TRIP780 sheet specimens that were stretch-formed over a hemispherical punch. It was found that the geometry-dependent, surface slope criterion was the most robust and consistent criterion of those evaluated.

The formability of electro-hydraulically formed AA5182-O aluminium sheet was investigated by means of experimental testing and numerical modelling. The experimental results were compared with quasi-static, as-received forming limit curve (FLC) and were used to calibrate a finite element model of EHF. It is found that the formability improvement of AA5182-O sheets was insignificant when electro-hydraulically formed without a die. However, when specimens were electro-hydraulically formed with sufficient energy into 34 or 40° conical dies, the effective strain measured in safe grids increased by 40 and 70% respectively when compared with a conservative quasi-static FLC. Finite element simulation results suggested that a combination of different mechanical parameters contributed to the formability improvement. The increased strain rate in areas close to the apex of the speimen, the negative stress triaxiality just before specimens contact the die, and significant compressive through-thickness stress generated by high-velocity impact all contributed to the improvement of formability in electro-hydraulic die forming.

Dual phase (DP) steel sheets are increasingly being used for automotive applications due to relatively high strength and formability. In the current work, Marciniak formability tests were carried out to determine the forming limit curve (FLC) of DP600 steel sheet, and an extended version of Rousselier’s ductile damage model, which accounts for void nucleation, growth and coalescence was used to simulate the tests and predict strain localization and failure for three different strain paths: uniaxial tension, plane strain and biaxial tension. In addition, a combination of flat rolling and uniaxial tension tests were used to generate the extended flow curve of the material. Damage evolution in terms of Rousselier scalar damage variable and void volume fraction was assessed for each simulation condition. The FLC as well as neck and fracture morphologies and geometries were obtained from finite element simulations of the Marciniak tests and compared to experimental results. The sensitivity and dependency of the predicted necking limits, damage distribution and geometry predicted by the Rousselier damage model to the type of hardening model, strain path, void nucleation function and void coalescence criterion are discussed. The modified Rousselier model was shown to successfully predict the FLC, damage distribution and the final damage geometry of DP600 sheets.

The objectives of this paper are to quantify the increase in formability of DP600 steel sheets in electrohydraulic die forming (EHDF). The conventional (quasi-static) and high strain rate forming limits of this sheet material were experimentally obtained from specimens deformed using Marciniak tests and EHDF tests, respectively. A numerical model of EHDF was developed in order to determine the strain path of sheet metal experiencing large strains. Forming limits obtained from EHDF specimens formed in plane strain into a V-shaped die indicated that, locally near the top edge of the specimen, more than 120% formability improvement was achieved. Moreover, a formability enhancement of more than 60% was achieved in EHDF in biaxial tension using a conical die, provided sufficient discharge energy was applied. It was found that the significant increases in formability in EHDF are associated with the high-velocity impact against the die wall and the consequent high effective strain rate.

In this study, damage mechanisms in AA5182-O aluminium sheets are investigated using quasi-static (QS) Marciniak tests and high strain-rate electro-hydraulic forming (EHF) process. The results confirm that void nucleation, growth and coalescence are the main damage mechanisms of AA5182-O at both high and low strain rates. The EDS analysis suggests that cracking of Al 3 (Fe-Mn) intermetallic particles is the main source of void nucleation, whereas Mg 2 Si particles do not majorly influence void formation. Void growth analysis suggests that specimens deformed under QS conditions contained more voids in areas away from the sub-fracture surface but specimens deformed at high strain rate exhibit more significant rate of void growth close to sub-fracture areas. Void formation is suppressed by increasing the applied energy in EHF. And more significantly, the growth of voids is suppressed due to the high-velocity impact of the sheet against the die which plays an important role in increasing formability of AA5182-O aluminium sheet in EHF process. When the void percentage increase remains less than about 0.5% AA5182-O can be formed safely. However, when the void percentage increases beyond 0.6–0.8% fracture becomes inevitable.

Electrohydraulic forming is a pulsed metal forming process that uses the discharge of electrical energy across a pair of electrodes submerged in fluid to form sheet metal at high velocities. Pulsed metal forming processes, including electrohydraulic forming, have been shown to increase the formability of sheet metals. Although significant formability enhancement has been reported for electrohydraulic die forming, there have been conflicting reports about the formability in electrohydraulic free forming (EHFF). Numerical modeling was used to design sheet metal specimen geometries to generate data for specific regions of the EHFF forming limit curve. The electrohydraulic free forming specimens were formed with the precise amount of input energy to cause a neck at the center of the gauge section. The quasi-static and EHFF forming limit curves for both AA5182-O and DP600 sheets were determined in accordance with the conventional North American formability evaluation method to allow for direct comparison. It was found that the forming limits in EHFF increased by approximately 5% major strain for DP600 and 8% major strain for AA5182, relative to their respective as-received FLC.

The objectives of this work are to describe the nucleation and growth of voids in DP600 dual phase steel sheets formed along different strain paths, and to investigate the correlation between the micro-mechanisms of damage and the forming limits. DP600 sheet specimens were stretch-formed in uniaxial, plane strain and biaxial tension using the Marciniak–Kuczynski formability test. The evolution of damage in the microstructure was correlated to the formability in each strain path by quantitatively analyzing void density, void area fraction, void aspect ratio and mean void size. Results showed that DP600 steel sheets have less formability in plane strain compared to uniaxial and biaxial tension due to the more rapid elongation and growth of voids. Moreover, the sheets safely deformed to much greater effective strains in biaxial tension due to the smaller mean void size and slower void growth. As proposed by Gurson Gurson (1977) , Tvergaard and Needleman Tvergaard and Needleman (1984) , void volume fraction is an important damage parameter. However, this work shows that void aspect ratio and mean void size also significantly influence the evolution of damage and the formability of dual phase steels. Keywords: Dual phase steel, Microstructure, Strain path, Voids, Forming limit curve

Electrohydraulic forming (EHF) is a high energy rate forming process in which the strain rate in the sheet metal can vary from 5 × 10 2 to 10 5 s −1 depending on various factors. Several mechanisms have been reported to cause an improvement in formability in EHF such as material deformation mechanisms, inertial effects and the dynamic impact of the sheet against the die. EHF is a complex high speed forming process and experimental work alone is not sufficient to properly understand this process. To understand the variation of some influential variables in EHF, electrohydraulic die-forming (EHDF) and free-forming (EHFF) of DP590 dual phase steel were simulated in ABAQUS/Explicit by considering the fluid/structure interactions. Three-dimensional finite element simulations were conducted by modelling the water with Eulerian elements with a view to investigating the effect of released energy on the sheet deformation profile history, strain distribution, loading path and damage accumulation type. The Johnson–Cook constitutive material model was used to predict the sheet behaviour and the parameters in this model were calibrated based on experimental test results available for DP590 at various strain rates. The Johnson–Cook phenomenological damage model was also used to predict the ductile failure (damage accumulation) in both EHDF and EHFF. Predicted final strain values and damage accumulation type showed good agreement with the experimental observations.

Theoretical and experimental studies have shown that when a compressive stress is applied normal to the sheet surface its formability can be significantly improved. In forming processes such as hydroforming, the normal stress imposed at the surface of the sheet or tube can be very significant in specific locations and it is therefore necessary to account for this stress state when assessing the forming severity of industrial parts. It is also well known that the forming limit curve (FLC) can vary significantly in strain space as a result of non-linear strain paths. In this work, a numerical code based on the Marciniak and Kuczynski (MK) analysis (1967) was developed to predict the FLC of sheet metals by simultaneously accounting for the effects of strain path non-linearity and the normal stress. The FLC predicted with this code can be used to assess the forming severity of parts that are deformed under complex loading conditions in industrial metal forming processes. The predictive FLC model was validated using experimental data in which linear and bi-linear strain paths were applied to different steel and aluminum sheets or tubes under plane stress and three-dimensional stress conditions. This research work has two specific purposes: first, to investigate the influence of the magnitude of the prestrain in bi-linear loading on the sensitivity of the FLC to the normal stress, and second, to study the effect of the normal stress on the path dependence of the FLC. This work showed that, with increasing magnitudes of prestrain the influence of the normal stress on the formability of sheet metal will decrease, regardless of the loading path. Moreover, it was observed that even a significant normal stress will practically not affect the path dependence of the FLC.

Hydropiercing is an efficient way of piercing holes in mass produced hydroformed parts with complex geometries. By driving piercing punches radially into a hydroformed and fully pressurized tube, holes will be pierced and extruded into the tube-wall. Recent experimental studies have shown that the formability of advanced high strength steel (AHSS) tubes can be increased with the application of internal pressure. In this study, three-dimensional finite element simulations of a tube hydropiercing process of a dual phase steel (DP600) were performed in LS-DYNA, using phenomenological, micromechanical and combined damage criteria. Damage was included in the numerical analysis by applying constant equivalent plastic strain (CEPS), the Gurson-Tvergaard-Needleman (GTN), and the Extended GTN (GTN+JC) model. In order to calibrate the parameters in each model, a specialized hole-piercing fixture was designed and piercing tests were carried out on non-pressurized tube specimens. Of the various ductile fracture criteria, the results predicted with the GTN+JC model, such as the punch load-displacement, the roll-over depth, and the quality of the clearance zone correlated the best with the experimental data.

Under certain conditions, strain rate sensitive materials such as dual phase steels, show formability improvement under high strain rate forming which is known as hyperplasticity. In this research, two commercial dual phase steel sheets, DP500 and DP780, were formed under quasi-static conditions using the Nakazima test and under high strain rate conditions by electrohydraulic forming (EHF) into a conical die. Macro-strains, measured from electro-etched circle grids with an FMTI analyzer, showed remarkable formability improvement in EHF specimens. Micro-strains, i.e., the strains in the ferrite and the martensite, were calculated by quantitative metallography of more than 7000 ferrite grains and 10,500 martensite islands. The goal was to investigate the deformation improvement of the constituents under EHF. Around 20 and 100% deformation improvements were observed in ferrite and martensite, respectively. Furthermore, as a micro-mechanical modeling technique, correlation of the micro-strains with the macro-strains was investigated by applying the mixture rule. Results showed a reasonable correlation between the macro and micro-scale strains; however in banded microstructures, the strain in the martensite should be determined precisely for more accuracy.

The conventional forming limit curve (FLC) is significantly strain path-dependent and therefore is not valid for formability evaluation of sheet metal parts that undergo nonlinear loading paths during the forming process. The stress-based forming limit curve (SFLC) is path-independent for all but very large prestrains and is a promising tool for formability evaluation. The SFLC is an ideal failure criterion for virtual forming simulations but it cannot be easily used on the shop floor as there is no straightforward experimental method to measure stresses in stamped parts. This paper presents a theoretical basis for predicting the effective limit strain curve (ELSC) using the Marciniak and Kuczynski (MK) analysis (Int J Mech Sci 9:609–620, 1967, Int J Mech Sci 15:789–805, 1973). Since the in-plane strain components are sufficient to calculate the effective strain, the ELSC can easily be determined from strains measured in the stamping plant, and therefore it is a better alternative to the SFLC for formability evaluation. This model was validated using experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 aluminum alloys Graf and Hosford (Metall Trans 24A:2503–2512, 1993). Predicted results showed that, similar to SFLC, the ELSC remains practically unchanged for a significant range of prestrain values under various bilinear loading paths, but some strain-path dependence can be observed for significant magnitudes of the effective prestrain (ee ≥ 0.37 for AISI-1012 steel and ee ≥ 0.25 for AA-2008-T4 aluminum).

Pre-bending is a critical process required prior to hydroforming. The bending has an effect on the tube thickness and strain which will use up a portion of the formability of the as-received tube. To compensate for this loss of formability, a multi-objective optimization method was applied to improve the hydroforming process after pre-bending. A multi-objective genetic algorithm (MOGA) and Kriging surrogate model were used to optimize the loading path. The Kriging model was used to replace the finite element simulation in constraint handling. The optimal loading parameters in the hydroforming process were obtained for a tube that was previously bent 90°, and showed an improvement in reducing the corner radii of the part at the extrados and intrados of the bend (8.73 mm and 11.24 mm for the extrados and intrados of the bend, respectively). The corresponding corner fill expansion (CFE) was improved by 16.7% (or 1.79 mm) compared to the maximum expansion of 10.73 mm obtained experimentally.

The production of a tubular hydroformed part often requires a combination of internal pressure and axial force at the tube ends to fully form the tube to its specified geometry. A successful hydroforming process requires not only achieving a part that conforms to the design specifications, but also ensures that the part has a reasonably uniform thickness distribution and is free of defects, such as wrinkles, severe thinning, or fractures. The load path design (pressure vs. end feed history) largely determines the robustness of the process and the quality of the finished parts. In this paper, a hybrid constrained optimization method was proposed to solve this type of multi-objective problem by coupling a multi-objective genetic algorithm and a local search. The load path design procedure was developed by considering five objectives: four formability objectives (i.e., to minimize the risk of wrinkling, global and local thinning, and fracture) and a geometric objective (to minimize the corner radius). A Kriging predictor was used to accelerate the computation of genetic operations and generate new feasible solutions. Finite element simulations of the hydroforming process were also used after each generation to accurately evaluate the objectives of the offspring, and solutions with rank 1 were retained throughout all generations. Once the Pareto solutions were obtained by multi-objective genetic algorithm, a local search was carried out in the regions of interest with the assistance of visualization. This optimization method was applied to the hydroforming of a straight tube to create a part with an expanded region with a square cross section; the optimum load path produced a very safe part with a corner radius of only 9.115 mm and a maximum thinning of only 23.9%.

The prediction of the forming limits of sheet metals typically assumes plane stress conditions that are really only valid for open die stamping or processes with negligible out-of-plane stresses. In fact, many industrial sheet metal forming processes lead to significant compressive stresses at the sheet surface, and therefore the effects of the through-thickness stress on the formability of sheet metals cannot be ignored. Moreover, predictions of forming limit curves (FLC) that assume plane stress conditions may not be valid when the forming process involves non-negligible out-of-plane stresses. For this reason a new model was developed to predict FLC for general, three-dimensional stress states. Marciniak and Kuczynski (Int J Mech Sci 9:609-620, 1967) first proposed an analytical method to predict the FLC in 1967, known as the MK method, and this approach has been used for decades to accurately predict FLC for plane stress sheet forming applications. In this work, the conventional MK analysis was extended to include the through-thickness principal stress component (σ3), and its effect on the formability of different grades of sheet metal was investigated in terms of the ratio of the third to the first principal stress components (\( \beta = {{{{\sigma_3}}} \left/ {{{\sigma_1}}} \right.} \)). The FLC was predicted for plane stress conditions (β = 0) as well as cases with different compressive through-thickness stress values (β ≠ 0) in order to study the influence of β on the FLC in three-dimensional stress conditions. An analysis was also carried out to determine how the sensitivity of the FLC prediction to the through-thickness stress component changes with variations in the strain hardening coefficient, in the strain rate sensitivity, in plastic anisotropy, in grain size and in sheet thickness. It was found that the out-of-plane stress always has an effect on the position of the FLC in principal strain space. However, the analysis also showed that among the factors considered in this paper, the strain hardening coefficient has the most significant effect on the dependency of FLC to the through-thickness stress, while the strain rate sensitivity coefficient has the least influence on this sensitivity.

The forming limit curve (FLC), a plot of the limiting principal surface strains that can be sustained by sheet metals prior to the onset of localized necking, is useful for characterizing the formability of sheet metal and assessing the forming severity of a drawing or stamping process. Both experimental and theoretical work reported in the literature has shown that the FLC is significantly strain-path dependent. In this paper, a modified Marciniak and Kuczynski (MK) approach was used to compute the FLC in conjunction with two different work-hardening models: an isotropic hardening model and a mixed isotropic–nonlinear kinematic hardening model, which is capable of describing the Bauschinger effect. Predictions of the FLC using the MK analysis have been shown to be dependent on the shape of the initial yield locus and on its evolution during work hardening; therefore the hardening model has an influence on the predicted FLC. In this investigation, published experimental FLCs of AISI-1012 low carbon steel and 2008-T4 aluminum alloy sheets that were subjected to various nonlinear loading paths were compared to predictions using both hardening models. The predicted FLCs were found to correlate quite well with experimental data and the effects of strain path changes and of the hardening model on predicted FLCs are discussed.

Several researchers have reported that dual phase steel sheets exhibit hyperplasticity, that is, a significant formability improvement under certain high strain rate forming conditions. Hyperplastic behavior of dual phase steels formed using an electrohydraulic forming (EHF) process was previously investigated by the authors at both macro- (Golovashchenko et al., 2013, “Formability of Dual Phase Steels in Electrohydraulic Forming,” J. Mater. Process. Technol., 213, pp. 1191–1212) and microscales (Samei et al., 2013, “Quantitative Microstructural Analysis of Formability Enhancement in Dual Phase Steels Subject to Electrohydraulic Forming,” J. Mater. Eng. Perform., 22(7), pp. 2080–2088). A relative deformation improvement of approximately 20% in ferrite grains and 100% in martensite islands was reported in the EHF specimens compared to specimens formed under quasi-static conditions. In this paper, the remarkable deformation improvements of the constituents are discussed in terms of metallurgical mechanisms of deformation. The nucleation and multiplication of dislocations in ferrite and deformation twinning in martensite were found to be the principal mechanisms responsible for the significant improvements of deformation in EHF. In addition, these mechanisms enhance the plastic compatibility between the two phases which reduces the risk of decohesion and delays the onset of fracture in EHF specimens.

Another factor that has a very significant effect on sheet formability is the loading path. Indeed, non-linear strain paths cause the forming limit curve (FLC) to translate in principal strain space and therefore the actual FLC for a given location on a part may be quite different from the as-received FLC. Consequently, the virtual design of a forming process will only be reliable if the forming limits used to assess the forming severity of the part were determined with due consideration of the (non-linear) strain paths in each critical location of the part. The FLC of sheet metals have often been predicted using the MK method which was proposed by Marciniak and Kuczynski (1967) and which models the inhomogeneity of the sheet material by a very shallow groove. Predictions of theoretical FLC using the MK method generally assume plane stress conditions, but in this work the MK analysis was modified to consider both the nonlinearity of the loading path and the presence of a through-thickness stress, simultaneously. Different magnitudes of prestrain were applied to a virtual sheet so as to simulate bi-linear strain paths and different values of the normal stress were applied for the prediction of the FLC. Published experimental FLC data obtained for different bilinear loading conditions were used to validate the theoretical work in both plane stress and general three-dimensional stress conditions. It was found that this modified MK model predicts FLC that correlate well with the limited experimental data.

In most predictions of forming limits for the evaluation of sheet metal forming severity, it is generally assumed that plane stress conditions apply and consequently the effects of the normal stress component on sheet metal formability are not considered. However, many industrial metal forming processes, such as tubular hydroforming, involve significant through‐thickness stresses in the tube wall while the part is being formed.In this work, the well‐known MK analysis originally proposed by Marciniak and Kuczynski was modified to account for general three‐dimensional stress states with a view to predicting the forming limit curve (FLC) and the stress‐based forming limit curve (SFLC) of sheet materials in the presence of a non‐negligible through‐thickness stress. This analysis was formulated in a way that the FLC/SFLC can be predicted for given ratios of the normal stress component to the major in‐plane stress component (β = σ3/σ1). The FLC and SFLC were then predicted for both plane‐stress conditions (β