Strain-based Forming Limit Curve Research Articles

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.

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.

On the Use of Strain Path Independent Metrics and Critical Distance Rule for Predicting Failure of AA7075-O Stretch-Bend Sheets.

The strain-based forming limit curve is the traditional tool to assess the formability of metal sheets. However, its application should be restricted to proportional loading processes under uniform strain conditions. Several works have focused on overcoming this limitation to characterize the safe process windows in industrial stretch-bend forming processes. In this paper, the use of critical distance rule and two path-independent stress-based metrics are explored to numerically predict failure of AA7075-O stretch-bend sheets with 1.6 mm thickness. Formability limits of the material were experimentally obtained by means of a series of Nakazima and stretch-bending tests at different thickness-over-radius ratios for inducing controlled non-uniform strain distributions across the sheet thickness. By using a 3D calibrated finite element model, the strain-based forming limit curve was numerically transformed into the path-independent stress and equivalent plastic strain polar spaces. The numerical predictions of necking strains in the stretch-bending simulations using the above approaches were successfully compared and critically discussed with the experimental results for different values of the critical distance. It was found that failure was triggered by a critical material volume of around the half thickness, measured from the inner surface, for the both path-independent metrics analyzed.

A simplified stress-based forming limit criterion for advanced high strength steel (AHSS)

A methodology for stress-based forming limit analysis has been developed for advanced high strength steel (AHSS). It was proposed that localized necking occurs when a critical normal stress condition is met. Using a basic, isotropic material model (von Mises, power law hardening), the criterion was applied to various 980 Class AHSS. In most cases the simplified criterion adequately described the experimental strain-based forming limit curve (FLC). For AHSS with substantial volume fractions of metastable austenite, a more sophisticated material model and/or an adapted failure criterion will be required. A strong linear relationship was found between the critical normal stress and the measured true stress at maximum load in tension. This empirical functionality applies over a large range of strength levels and may form the basis for a methodology by which FLCs may be estimated from standard tension tests. Finally, in the context of the proposed failure criterion, the effects of work hardening behavior on the “shape” of the strain-based FLC are explored.

Prediction of complete forming limit diagram from tensile properties of various steel sheets by a nonlinear regression based approach

Experimental Forming Limit Curve (FLC) and tensile properties of various steel grades are collected from literatures to validate the proposed model. The experimental results show that the ultimate tensile stress (σUTS), total elongation (ɛt), coefficient of normal anisotropy (r), tensile strain hardening exponent (n) and sheet thickness (t) are strongly related to the plane strain forming limit (FLC0) values for steel sheets. A nonlinear regression equation is proposed to predict FLC0 from uniaxial tensile properties like σUTS, ɛt, r, n and t. To verify the predictive capability of the proposed equation, predicted FLC0 values for fifty six steel grades in various thickness and strength ranges are compared with experimentally measured FLC0 values. It is observed that the newly developed nonlinear regression equation predicts well the FLC0 values.Left side of the strain-based FLC is calculated from a criterion (a line with slope of –1) which is normally well matched with experimental observation. Right side of the strain-based FLC is calculated from modified Keeler–Brazier power equation. In the original Keeler–Brazier power equation for the right side of the FLC the exponent is considered as 0.5, while experimental finding revels that it varies systematically with FLC0. Complete strain-based FLCs calculated from proposed method are matched well with experimental FLCs for various automotive steel grades like IF, IF 340, DP 780, TRIP 780 and TWIP 940 steels.

Prediction of Forming Limit Curves from Hardness for Steels

This paper presents a method for predicting the strain-based forming limit curve (FLC) for steels using hardness. The stretching side (positive minor strain component) of the FLC was calculated by using a Marciniak-Kuczynski model with a non-quadratic yield function, while the drawing side (negative minor strain component) of the FLC was predicted based on the relationship between the major and minor critical strains, in accordance with the theory of maximum sheet tension for local necking. The requisite parameter that describes the plastic flow behavior (in this case, the strain hardening exponent) was calculated, based on correlations with the measured microhardness. Additionally, the strain rate sensitivity was considered in the model by using a newly developed empirical correlation between hardness and strain rate sensitivity. This hardness-based model was used to predict FLCs that demonstrate good agreement with experimental FLCs of a high-strength low-alloy steel and a dual-phase steel. Equations are provided that enable the calculation of the FLC from given hardness values for different severities of the material inhomogeneity.

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.

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.

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.

Application of an Extended Stress-Based Forming Limit Curve to Predict Necking in Stretch Flange Forming

An extension of the stress-based forming limit curve (FLC) advanced by Stoughton (2000, “A General Forming Limit Criterion for Sheet Metal Forming,” Int. J. Mech. Sci., 42, pp. 1–27) is presented in this work. With the as-received strain-based FLCs and stress-strain curves for 1.6-mm-thick AA5754 and 1-mm-thick AA5182 aluminum alloy, stress-based FLCs are obtained. These curves are then transformed into extended stress-based forming limit curves (XSFLCs), which consist of the invariants, effective stress, and mean stress. By way of application, stretch flange forming of these aluminum alloy sheets is considered. The AA5754 stretch flange displays a circumferential crack during failure, whereas the AA5182 stretch flange fails through a radial crack at the edge of the cutout. It is shown that the necking predictions obtained using the strain- and stress-based FLCs in conjunction with shell element computations are inconsistent when compared with the experimental results. By comparing the results of the shell element computations with those in which the mesh comprises eight-noded solid elements, it is demonstrated that the plane stress approximation is not valid. The XSFLC is then used with results from the solid-element computations to predict the punch depths at the onset of necking. Furthermore, it is shown that the predictions of failure location and failure mode obtained using the XSFLC are in accord with the differences observed between the two alloys/gauges.

Prediction of necking of high strength steel tubes during hydroforming—multi-axial loading

This article studies tubular hydroforming of high strength low alloy (HSLA) and dual phase (DP600) straight tubes under the action of end feeding loads. Experiments demonstrate that higher end feed loads enhance the formability of the tubes and increase the internal fluid pressure for onset of necking and bursting. Because of the action of the internal pressure and the axial compressive load, the onset of localization (necking) is due to a complex three-dimensional state of stress. Using free expansion experiments, approximate upper and lower bound strain-based forming limit curves are determined for the tube materials. These limit curves, in turn, are used to derive upper and lower bound extended stress-based forming limit curves [Simha et al., Prediction of necking in tubular hydroforming using an extended stress-based FLC. Transactions of the ASME Journal of Engineering Materials and Technology 2007;129(1): 36–47]. In conjunction with finite element computations that use solid elements to model the tube, these stress-based limit curves are used to predict upper and lower bound necking pressures under the action of end feed loading. These predictions of necking pressures, when an appropriate coefficient of tube-die friction is used, are found to bracket the experimentally measured necking pressures. Computations using plane stress shell elements to model the tubes are shown to give erroneous results, since the plane stress approximation is not valid when tubes are hydroformed in a die.

Inverse Analysis to Formability Design in a Deep Drawing Process

Deep drawing process is an important process adding values to flat sheet metals in many industries. An important concern in the design of a deep drawing process generally is formability. This paper aims to present the connection between formability and inverse analysis (IA), which is a systematical means for determining an optimal blank configuration for a deep drawing process. In this paper, IA is presented and explored by using a commercial finite element software package. A number of numerical studies on the effect of blank configurations to the quality of a part produced by a deep drawing process were conducted and analyzed. The quality of the drawing processes is numerically analyzed by using an explicit incremental nonlinear finite element code. The minimum distance between elemental principal strains and the strain-based forming limit curve (FLC) is defined as tearing margin to be the key performance index (KPI) implying the quality of the part. The initial blank configuration has shown that it plays a highly important role in the quality of the product via the deep drawing process. In addition, it is observed that if a blank configuration is not greatly deviated from the one obtained from IA, the blank can still result a good product. The strain history around the bottom fillet of the part is also observed. The paper concludes that IA is an important part of the design methodology for deep drawing processes.