Stress-based Forming Limit Curves Research Articles

An extended stress-based forming limit diagram focusing on the wrinkling phenomenon and the effect of the normal pressure on clamped surfaces

A novel wrinkling limit representation following the pattern of the conventional forming limit curves (FLCs) and the stress-based forming limit curves (SFLCs) as well as the application of the assumed criterion in finite element modelling are discussed in this manuscript. FLCs can partially refer to the wrinkling potential in the area left to the uniaxial tension line, but just like the SFLCs, cannot characterize the limits of the material behavior in a deeper sense, if negative in-plane stress and normal pressure act together on the sheet. This study predicts the wrinkling risk of clamped surfaces with a stress-based criterion through solving the analytical equations of the critical compressive stress causing wrinkling, and the corresponding blank holder pressure. The critical values were calculated based on the Wang and Cao’s theory using Hill48 anisotropic yield function coupled with the Swift hardening law. The results draw a novel wrinkling limit curve representation methodology, in which the minor stress responsible for wrinkling and its ratio to the major stress are distinguished in the function of the applied blank holder pressure. The applicability of the calculated curves was investigated using finite element simulations, which showed that this method provides the opportunity to quantitatively interpret how close a component is to the wrinkling limit. The calculated wrinkling tendencies were verified by standard cup drawing tests supplemented by round shape error measurements on three different automotive steel sheets. It can be stated that the obtained numerical conditions of wrinkling were fitted to the experiments fairly well.

A general instability framework for ductile metals in complex stress states from diffuse to acute localization

Adoption of stress-based forming limit curves (FLC) and their equivalent strain representations were introduced to overcome the path dependence of the conventional strain-based FLCs. The stress state and its instantaneous magnitude govern formability such that the triaxial stress state, induced through tool contact, can no longer be neglected. The present work reveals that in addition to the composition of the stress state, the boundary conditions of how the stress is applied to the material determine the onset of plastic instability. To this end, the relatively unknown generalized instability model of Hillier, derived from a stable bifurcation of the stress state under neutral stability, is revisited. Analytical solutions for instability in triaxial plane strain loading were derived for boundary conditions of a proportional, constant, or evolving contact pressure that enables critical evaluation of phenomenological plane stress mapping criteria proposed in the literature. As part of this work, the Hillier instability framework is extended to model the localization process under the constraint of quasi-stable plastic flow in arbitrary principal stress states to form the Generalized Incremental Stability Criterion (GISC). It is shown that the Modified Maximum Force Criterion (MMFC) emerges as one of many special cases of the GISC when restricted to plane stress proportional stressing with a prescribed minor load. Finally, the GISC is applied to Marciniak and Nakazima formability tests of a DP980 steel for identification of appropriate boundary conditions.

Investigating the incremental forming capabilities of extra deep drawn steel

Incremental Sheet Forming (ISF) is a sheet metal forming process, which relies on minimum part-specific tooling. Extra Deep Drawn (EDD) steel has been used for making door inners, dash panels, bodyside inners, etc. due to its good weldability and relatively low yield strength. However, to date very limited work is reported on ISF of EDD steel. Besides, no work, which exhaustively discusses the practicability of EDD steel for effectual incremental forming of components, is found to be reported. The present work discusses the ISF of EDD steel sheets and examines the limitations of ISF in forming the parts out of 1.0 mm thick EDD steel sheets. Two geometries, i.e., ellipsoidal cone (with varying wall angle) and truncated cone (with constant wall angle) were used as test cases to evaluate the formability of EDD steel sheets, in terms of Critical Wall Angle (CWA). Further, the formability of EDD steel with ISF is evaluated using both strain-based and stress-based forming limit curves. The thickness distribution, geometrical accuracy, forming forces, and surface roughness and hardness are evaluated in the formed components. The work, through numerical simulations and experimental analyses, investigates the process capabilities of ISF to form the EDD steel sheets in terms of formability, thickness distribution, geometrical accuracy, forming forces, and surface roughness and hardness to ascertain the scope of the proposed process.

Analysis of the pre-straining effects on the limit strains of interstitial-free steel: experiments and elasto-plastic modeling of strain and stress-based forming limit curves

In this work, the limit strains under as-received and pre-strained conditions of an interstitial-free steel sheet were investigated by mechanical tests and numerical modeling. An original testing procedure was proposed to reproduce two equivalent pre-straining amounts under equibiaxial stretching (EB-PS) and uniaxial tension (UT-PS) deformation modes. The limit strains were defined from Nakajima’s hemispherical punch tests using digital image analysis according to the ISO 12004:2 standard. The plastic behavior of the interstitial-free steel sheet was evaluated by uniaxial tensile, disk compression, and hydraulic bulge tests. Moreover, an elasto-plastic framework based on the Marciniak–Kuczynski (M–K) model is proposed to forecast both the strain-based and stress-based forming limits. The parameters of the Yld-2000-2d yield criterion were identified using the plastic anisotropy coefficients with the flow-stress values determined from two distinct amounts of plastic-work per unit volume. For both deformation modes and pre-straining levels defined by the von Mises effective plastic-strain (4.8% and 9.0% EB-PS; 5.0% and 10% UT-PS), the transformed polar effective plastic-strain (PEPS) and forming limit stress (FLSC) results shown to be more strain-path insensitive than the conventional strain-based forming limit curve (FLC). The proposed elasto-plastic M–K model successfully forecasted the linear strain-path limit strains defined using the FLC, PEPS, and FLSC representations. The M–K model also provided quite conservative predictions for the pre-strained PEPS and FLSC. The Yld-2000-2d parameters identified using the flow-stress values defined from an upper-level of the plastic-work per unit volume proved to be the best choice in the present M–K modeling framework.

Formability prediction of advanced high-strength steel sheets by means of combined experimental and numerical approaches

Abstract In this work, a group of four capable prediction tools—Classical Forming Limit Curve (FLC), Forming Limit Stress Curve (FLSC), Damage Curve (DC) associating stress-triaxiality with plastic strain and Nonlinear Strain Path Forming Limit Curve (nonlinear strain path FLC) based on an innovative punch originally designed by the Institute for Metal Forming Technology (IFU), University of Stuttgart—are employed to evaluate formability of DP590 advanced high-strength steel (AHSS). The results are subsequently compared. A classical FLC is created with restriction to the Nakajima stretching-test guideline, in which a 100-Tonne universal tensile testing machine is operated to press 1-mm thick DP590 sheets. A nonlinear strain path FLC is generated with an application of an IFU-designed punch to realise nonlinear strain paths under a particular standardised Nakajima testing environment. A stress-based FLC is computationally obtained by transforming the classical FLC, realised in a major-minor strain space, into an FLSC in a principal-stress space using the Hill’48 yield criterion equipped with the Swift strain-hardening law. A DC is arithmetically determined through a combination of the plastic flow rule and Hill’48 yield criterion. To verify and compare those gained curves, both hole-expansion and Fukui stretch-drawing tests are conducted both experimentally and numerically. Regarding the simulation model for numerical investigation, material flow behaviour and deformation anisotropy are modelled by the Swift hardening law and Hill’48 yield criterion, respectively. This research has once again proved that FLSC, DC and nonlinear strain path FLC can obviously identify the forming limit of DP590 AHSS more realistically than the classical FLC can.

Prediction of forming limit curves for nonlinear loading paths using quadratic and non-quadratic yield criteria and variable imperfection factor

Industrial sheet metal forming processes often involve complex deformation modes and it is necessary to consider nonlinear loading path effects when predicting forming limit curves. Moreover, the yield criterion plays a critical role in the accuracy of predicted forming limits.In this work the MK analysis was modified to relate the initial imperfection factor to a physical property such as the surface roughness, and the orientation of the imperfection was also allowed to vary. This model was used to predict the strain-based and stress-based forming limit curves (FLC and SFLC) of sheet materials that are subject to either linear or non-linear strain paths.Two different yield criteria were employed in this study, Hill's 1948 quadratic yield criterion and Hosford's 1979 non-quadratic yield criterion. The theoretical model was validated by comparing predicted FLC and experimental FLC curves obtained from the literature. FLCs and SFLCs predicted with these two yield criteria were compared for both linear and nonlinear loading paths.Results showed that both the quadratic and non-quadratic yield criteria predict the FLC with acceptable accuracy however on the whole the non-quadratic yield criterion generally provides a slightly better correlation with experimental data, especially on the right side of the FLC.

An approach to predicting the forming limit stress components from mechanical properties

Forming limit curves (FLCs) are used to determine the amount of deformation that can be applied to a sheet metal before the onset of a localized neck. Most FLCs are shown in strain space, and stress-based FLCs have advantages because they are often strain-path independent. The current study develops a method to calculate a stress-based forming limit curve. The necessary data for this calculation can be obtained from a uniaxial tensile test. The calculations depend on the Z parameter, which can be considered to be the point of instability during a tensile test. With the use of the Keeler–Brazier equation, the effective stress in plane strain at the forming limit is shown to be a function of the Z parameter and thickness. Data from 4 experimental studies are shown to be consistent with this function. With the generally accepted observation that the left side of the strain-based FLC is a line with slope of –1 and an appropriate constitutive model for the stress-strain behavior of the material, the stress-based FLC corresponding to the left side of the strain-based FLC can be calculated. Comparison of the calculated stress-based FLC for three steels with the stress-based FLC determined directly from the strain-based FLC shows good agreement. The calculated stress-based FLC is 15–20MPa below the data generated directly from the strain-based FLC.

Effect of through-thickness normal stress on forming limits under Yld2003 yield criterion and M-K model

Forming limit curve considering through-thickness normal stress is theoretically obtained based on Yld2003 yield criterion and M-K model with the assumption that hydrostatic stress has no effect on plastic deformation. The suitability of Yld2003 yield criterion to describe the yield surface and forming limit under plane-stress condition for aluminum alloy AA6111-T43 is verified by the experimental data obtained from the literature. Then the effect of through-thickness normal stress on forming limit is analyzed. Different forms of forming limit presentation including traditional strain-based forming limit curve (FLC), forming limit effective strain curve (eFLC), stress-based forming limit curve (FLSC) and extended stress-based forming limit curve (XSFLC) are investigated in detail. It is found that forming limit curves in strain-space enhance with increased through-thickness normal stress under both linear and nonlinear strain paths with different pre-strains. However, the variation of forming limit curves in stress-space does not show single trend but depends on the variables of the coordinate plane. A new form of forming limit presentation named correcting FLSC is proposed. The correcting major stress enhances with through-thickness normal stress increase, and the floating range of the correcting FLSC due to through-thickness normal stress is much less than that of the traditional FLSC. The correcting FLSC can be taken as a steadier criterion for sheet metal forming limit prediction under three-dimensional stress state.

A methodology for determination of extended strain-based forming limit curve considering the effects of strain path and normal stress

In this paper, an approach based on the modified Marciniak-Kuczynski (M-K) method for computation of an extended strain-based forming limit curve (FLC) is presented. An extended strain-based FLC is built based on equivalent plastic strains and material flow direction at the end of forming. This curve has some advantages in comparison with other necking criteria such as the traditional FLC and also the stress-based FLC. This new criterion is much less strain path dependent than the conventional FLC. Furthermore, the use and interpretation of this new curve is easier than the stress-based FLC. The effect of strain path on the predicted extended strain-based FLC is reexamined. For this purpose, two types of pre-straining on the sheet metal have been imposed. Moreover, the plane stress state assumption is not adopted in the current study. The verifications of the theoretical FLCs are performed by using some available published experimental data.

Forming limits of an age hardenable aluminum sheet after pre-straining and annealing

Abstract A two-stage stamping with an intermediate annealing method was developed to improve formability of age hardenable aluminum alloys, which are more challenging than the non-age hardenable aluminum alloys because of the effects that intermediate annealing will have on dissolution of solutes, precipitation of strengthening particles, and the subsequent mechanical behavior of the materials. The process could readily be applied locally to highly strained areas of a complex panel for 6000 series aluminum alloys in order to form a good part without fracture; however, there will be a decrease of precipitation strengthening of those locally treated, highly strained areas. In this study, the strain-based forming limit curves (FLC) of as-received AAx610-T4PD alloy sheets were shown to increase after pre-straining to ~15% and annealing at 425 °C for 10 s. However, since strain-based FLCs are very sensitive to the pre-strain history, both stress-based FLCs and polar-effective-plastic-strain (PEPS) FLCs, which are path-independent, were used to evaluate the forming limits after pre-straining and annealing. Due to the change of mechanical behavior during annealing, this paper describes the procedure to calculate the stress-based FLC in which a residual-effective-plastic-strain (REPS) is determined by overlapping the hardening curve of the pre-strained and annealed material with that of material heat treated without pre-strain. After converting the strain-based FLCs using the constant REPS method, it was found that the stress-based FLCs and the PEPS FLCs of the post-annealed materials are quite similar and both tools are applicable for evaluating the forming limits of Al–Mg–Si alloys for the two-step stamping process with intermediate annealing heat treatment.

Analysis of the extended stress-based forming limit curve considering the effects of strain path and through-thickness normal stress

In this study, an approach based on the modified Marciniak–Kuczynski (M–K) method for computation of an extended stress-based forming limit curve (FLC) is presented. The extended stress-based FLC is built based on equivalent plastic stress versus mean stress. This curve has some advantages in comparison with the conventional FLC. This new criterion is much more strain path independent than the conventional FLC. The effect of strain path on the predicted extended stress-based FLC is reexamined. For this purpose, two types of pre-straining on the sheet metal have been loaded. Moreover, the plane stress state assumption is not adopted in the current study. The influence of a through-thickness compressive normal stress is also investigated theoretically. The verifications of the theoretical FLCs are performed by using some available published experimental data.

Forming limit analysis for two-stage forming of 5182-O aluminum sheet with intermediate annealing

Stress-based forming limit diagrams for a two-stage forming technique with an intermediate annealing step were generated for aluminum alloy 5182-O with a new experimental/theoretical methodology. It was demonstrated that unlike strain-based forming limits, stress-based forming limits are independent of pre-strain levels, pre-strain paths and annealing conditions, and converge to a single forming limit curve that is close to the stress forming limit of the as-received material. For this purpose, AA5182-O specimens were pre-strained to various levels in uniaxial, plane strain, and equibiaxial tension; smaller shapes were extracted, annealed, and tested in limiting dome height (LDH) tests. Strain fields for the forming limit diagrams were measured with stereo digital image correlation during each LDH test, where the onset of localized necking was identified to construct the forming limits. In the calculation of stress-based forming limit curves, only the as-received material properties were required, while the effects of pre-strain and annealing were accounted for with a calculated constant “effective plastic strain.” An analytical method is described to develop a new experimental/theoretical methodology effectively predicting the pre-strain and annealing effects. This enables reliable prediction of formability in multi-stage forming processes interrupted by annealing treatment employed to recover additional stretchability in critical areas of a formed part.

On the use of effective limit strains to evaluate the forming severity of sheet metal parts after nonlinear loading

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).

Effect of Element Types on Failure Prediction Using a Stress-Based Forming Limit Curve

Strain-based forming limit diagrams (FLDs) are the traditional tool used to characterize the formability of materials for sheet metal forming processes. However, this failure criterion exhibits a significant strain path dependence. Alternatively, stress-based FLDs have been proposed and shown to be less sensitive to the deformation path. The stress-based failure criterion can be conveniently implemented in numerical simulations. However, for reliable numerical modeling, the sensitivity of the models to the selection of discretization parameters, in particular, the element type must be assessed. In this paper, Marciniak tests have been numerically simulated to investigate failure prediction using three different element types (shell, solid, and solid-shell). Seven different specimen geometries were modeled in order to vary the loading paths. The results show that despite differences in stress calculation assumptions, shell, solid, and solid-shell elements do not provide differences in failure prediction when a stress-based failure criterion is used.

Formability evaluation of dimple forming process based on numerical and experimental approach

Exhaust gas recirculation (EGR) cooler consists of a number of heat transfer tubes that have relatively larger net area than that of flat type tubes. The surface of the tubes is made up with lots of grooves and protrusions for enlarging the net heat transfer area. Most tubes are manufactured through forming processes, such as bending, spinning, roll forming, stamping and so on. Therefore, a series of fracture or defect can occur during the various forming processes. In this study, the manufacturing process of a dimple-type rectangular heat transfer tube used for an EGR cooler system is investigated based on the numerical simulation and the experimental approach. A prototype of the tube is designed and modified to a newly designed tube considering the conservation of the net heat transfer area based on the numerical and analytical approach. Formability evaluation of the tube sheet is carried out by using forming limit curves based on the plastic instability conditions. Strain- and stress-based forming limit curves are utilized to ensure the strain path independence. The newly designed tube having a number of dimples on the both sides are manufactured by the press forming process. Thickness distributions for the principal cross-sections are observed from both the simulation and the experiment and compared each other. From the results, it is confirmed that the forming process is robust to manufacture the dimple type rectangular tubes with the comparison of thickness, and application of the forming limit curves.

Anisotropic responses, constitutive modeling and the effects of strain-rate and temperature on the formability of an aluminum alloy

Finite deformation anisotropic responses of AA5182-O, over a wide range of strain-rates (10 −4 to 10 0 s −1) and temperatures (293–473 K) are presented. The plastic anisotropy parameters were experimentally determined from tensile experiments using specimens from sheet material. Using the experimental results under plane stress conditions, the anisotropy coefficients for Barlat’s yield function (YLD96) were calculated at different strain-rates and temperatures. The correlations obtained from YLD96 are in good agreement with the observed experimental results. The strain-rate sensitivity of AA5182-O alloy changed from negative at 293 K to positive at 473 K. Khan–Huang–Liang (KHL) constitutive model is shown to correlate the observed strain-rate and temperature dependent responses reasonably well. The material parameters were obtained from the experimental responses along the rolling direction (RD) of the sheet. Marciniak and Kuckzinsky (M–K) theory was used to obtain the theoretical strain and stress-based forming limit curves (FLCs) at different strain-rates and temperatures. The experimental result from the published literature is compared with the FLCs from the current study.

A new approach for failure criterion for sheet metals

The interpretation of sheet forming simulations relies on failure criteria to define the limits of metal deformation. The common requirements for these criteria across a broad range of application areas have not yet been satisfied or fully identified, and a single criterion to satisfy all needs has not been developed. Areas where existing criteria appear to be lacking are in the comprehension of the effects of non-proportional loading, general non-planar and triaxial stress loading, and process and material mechanisms that differentiate between necking and fracture. This study was mainly motivated to provide an efficient method for the analysis of necking and fracture limits for sheet metals. In this paper, a model for the necking limit is combined with a model for the fracture limit in the principal stress space by employing a stress-based forming limit curve (FLC) and the maximum shear stress (MSS) criterion. A new metal failure criterion for in-plane isotropic metals is described, based on and validated by a set of critical experiments. This criterion also takes into consideration of the stress distribution through the thickness of the sheet metal to identify the mode of failure, including localized necking prior to fracture, surface cracking, and through-thickness fracture, with or without a preceding neck. The fracture model is also applied to the openability of a food can for AA 5182. The predicted results show very good agreement with the experimentally observed data.

Investigation on the strain-path dependency of stress-based forming limit curves

Path-dependent forming limits have been computed for sheet metals undergoing various combinations of plane stress loading conditions. This paper presents a theoretical model for prediction of stress-based forming limit curves (SFLC) based on the Marciniak and Kuczynski (MK) model. Acceptable agreement was observed between calculated forming limit curves (FLC) and experimental data for AISI-1012 steel (Molaei 1999) and AA-2008-T4 alloys (Graf and Hosford Metallurgical Trans 24A:2503–2512, 1993). In this paper, the path dependency of SFLCs predicted for different non-proportional loading histories has been investigated. For a range of prestrain values in different bilinear loading paths, the SFLC remains practically unchanged. However, some strain path dependency is observed for large values of prestrain (\( \bar{\varepsilon } \geqslant 0.35 \) for AISI-1012 steel) and for abrupt changes in strain path. Nevertheless, the SFLC remains a good failure criterion for virtual forming simulations because the path dependency of SFLCs is much less significant than that of strain-based FLCs.

Advanced modelling of failure mechanisms in aluminium sheet forming simulation

In this paper novel and efficient approaches to model localized necking and ductile fracture in sheet forming simulation using the finite element method are presented. The stress-based forming limit curve is used to detect localized necking. Post-necking is captured using a new concept called accelerated plastic thinning. A simple ductile fracture model was developed. Various application examples demonstrate the capabilities of the modelling framework.

Orientation and Path Dependence of Formability in the Stress- and the Extended Stress-Based Forming Limit Curves

This article analyzes the formability data sets for aluminum killed steel (Laukonis, J. V., and Ghosh, A. K., 1978, “Effects of Strain Path Changes on the Formability of Sheet Metals,” Metall. Trans. A., 9, pp. 1849–1856), for Al 2008-T4 (Graf, A., and Hosford, W., 1993, “Effect of Changing Strain Paths on Forming Limit Diagrams of Al 2008-T4,” Metall. Trans. A, 24A, pp. 2503–2512) and for Al 6111-T4 (Graf, A., and Hosford, W., 1994, “The Influence of Strain-Path Changes on Forming Limit Diagrams of Al 6111 T4,” Int. J. Mech. Sci., 36, pp. 897–910). These articles present strain-based forming limit curves (ϵFLCs) for both as-received and prestrained sheets. Using phenomenological yield functions, and assuming isotropic hardening, the ϵFLCs are transformed into principal stress space to obtain stress-based forming limit curves (σFLCs) and the principal stresses are transformed into effective stress and mean stress space to obtain the extended stress-based forming limit curves (XSFLCs). A definition of path dependence for the σFLC and XSFLC is proposed and used to classify the obtained limit curves as path dependent or independent. The path dependence of forming limit stresses is observed for some of the prestrain paths. Based on the results, a novel criterion that, with a knowledge of the forming limit stresses of the as-received material, can be used to predict whether the limit stresses are path dependent or independent for a given prestrain path is proposed. The results also suggest that kinematic hardening and transient hardening effects may explain the path dependence observed in some of the prestrain paths.