Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback

The constitutive model for the unusual asymmetric hardening behavior of magnesium alloy sheet presented in a companion paper (Lee, M.G., Wagoner, R.H., Lee, J.K., Chung, K., Kim, H.Y., 2008. Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheet, Int. J. Plasticity 24(4), 545–582) was applied to the springback prediction in sheet metal forming. The implicit finite element program ABAQUS was utilized to implement the developed constitutive equations via user material subroutine. For the verification purpose, the springback of AZ31B magnesium alloy sheet was measured using the unconstrained cylindrical bending test of Numisheet (Numisheet ’2002 Benchmark Problem, 2002. In: Yang, D.Y., Oh, S.I., Huh, H., Kim, Y.H. (Eds.), Proceedings of 5th International Conference and Workshop on Numerical Simulation of 3D Sheet Forming Processes, Jeju, Korea) and 2D draw bend test. With the specially designed draw bend test the direct restraining force and long drawn distance were attainable, thus the measurement of the springback could be made with improved accuracy comparable with conventional U channel draw bend test. Besides the developed constitutive models, other models based on isotropic constitutive equations and the Chaboche type kinematic hardening model were also considered. Comparisons were made between simulated results by the finite element analysis and corresponding experiments and the newly proposed model showed enhanced prediction capability, which was also supported by the simple bending analysis adopting asymmetric stress–strain response.     

Comparison of the work-hardening of metallic sheets using tensile and shear strain paths

This work deals with the characterization of the kinematic work-hardening of a bake-hardening steel. A shear test device has been designed and its use for the characterization of the work-hardening of sheet metals is described. Two main results are presented. Firstly, a local strain measurement, based on the following of three dots drawn on the gauge area, gives the evolution of the strain tensor eigenvalues during the test. It is shown, by comparing the theoretical kinematics of simple shear with a slightly perturbated one, that the strain state is close to the ideal one in the center of the gauge area. Secondly, reversal of the shear direction is performed after several prestrain and the evolution of the kinematic work-hardening with the equivalent plastic strain has been identified using an anisotropic elasto-viscoplastic model of Hill 1948 type. Isotropic and kinematic contributions of the work-hardening are also calculated from loading–unloading tensile tests and are compared to those obtained from the simple shear tests. The results show a discrepancy between both identification for the isotropic and the kinematic hardening. However, they are in agreement concerning the evolution of the global work-hardening.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

The shear fracture of dual-phase steel

Unexpected fractures at high-curvature die radii in sheet forming operations limit the adoption of advanced high strength steels (AHSS) that otherwise offer remarkable combinations of high strength and tensile ductility. Identified as “shear fractures” or “shear failures,” these often show little sign of through-thickness localization and are not predicted by standard industrial simulations and forming limit diagrams. To understand the origins of shear failure and improve its prediction, a new displacement-controlled draw-bending test was developed, carried out, and simulated using a coupled thermo-mechanical finite element model. The model incorporates 3D solid elements and a novel constitutive law taking into account the effects of strain, strain rate, and temperature on flow stress. The simulation results were compared with companion draw-bend tests for three grades of dual-phase (DP) steel over a range of process conditions. Shear failures were accurately predicted without resorting to damage mechanics, but less satisfactorily for DP 980 steel. Deformation-induced heating has a dominant effect on the occurrence of shear failure in these alloys because of the large energy dissipated and the sensitivity of strain hardening to temperature increases of the order of 75 °C. Isothermal simulations greatly overestimated the formability and the critical bending ratio for shear failures, thus accounting for the dominant effect leading to the inability of current industrial methods to predict forming performance accurately. Use of shell elements (similar to industrial practice) contributes to the prediction error, and fracture (as opposed to strain localization) contributes for higher-strength alloys, particularly for transverse direction tests. The results illustrate the pitfall of using low-rate, isothermal, small-curvature forming limit measurements and simulations to predict the failure of high-rate, quasi-adiabatic, large-curvature industrial forming operations of AHSS.     

Multi-scale finite element analyses of sheet metals by using SEM-EBSD measured crystallographic RVE models

In this study, two multi-scale analyses codes are newly developed by combining a homogenization algorithm and an elastic/crystalline viscoplastic finite element (FE) method (Nakamachi, E., 1988. A finite element simulation of the sheet metal forming process. Int. J. Numer. Meth. Eng. 25, 283–292; Nakamachi, E., Dong, X., 1996. Elastic/crystalline viscoplastic finite element analysis of dynamic deformation of sheet metal. Int. J. Computer-Aided Eng. Software 13, 308–326; Nakamachi, E., Dong, X., 1997. Study of texture effect on sheet failure in a limit dome height test by using elastic/crystalline viscoplastic finite element analysis. J. Appl. Mech. Trans. ASME(E) 64, 519–524; Nakamachi, E., 1998. Elastic/crystalline viscoplastic finite element modeling based on hardening–softening evaluation equation. In: Proc. of the 6th NUMIFORM, pp. 315–321; Nakamachi, E., Hiraiwa, K., Morimoto, H., Harimoto, M., 2000a. Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification. Int. J. Plasticity 16, 1419–1441; Nakamachi, E., Xie, C.L., Harimoto, M., 2000b. Drawability assessment of BCC steel sheet by using elastic/crystalline viscoplastic finite element analyses. Int. J. Mech. Sci. 43, 631–652); (1) a “semi-implicit” finite element (FE) code and (2) a “dynamic explicit” FE code. These were applied to predict the plastic strain induced yield loci and the formability of sheet metal in the macro scale, and simultaneously the crystal texture and hardening evolutions in the micro scale. The isotropic and kinematical hardening laws are employed in the crystalline plasticity constitutive equation. For the multi-scale structure, two-scales are considered. One is a microscopic polycrystal structure and the other a macroscopic elastic plastic continuum. We measure crystal morphologies by using the SEM-EBSD apparatus with a unit of about 3.8 μm voxel, and define a three dimensional (3D) representative volume element (RVE) for the micro polycrystal structure, which satisfy the periodicity condition of crystal orientation distribution. A “micro” finite element modeling technique is newly established to minimize the total number of finite elements in the micro scale. Next, the “semi-implicit” crystallographic homogenization FE code, which employs the SEM-EBSD measured RVE, is applied to the 99.9% pure-iron uni-axial tensile problem to predict the texture evolution and the subsequent yield loci in the various strain paths. These “semi implicit” results reveal that the plastic strain induced anisotropy in the micro and macro levels can be predicted by our FE analyses. The kinematical hardening law leads a distinct plastic strain induced anisotropy. Our “dynamic-explicit” FE code is applied to simulate the limit dome height (LDH) test problem of the mild steel DQSK, the high strength steel HSLA and the aluminum alloy AL6022 sheet metals, which were adopted as the NUMISHEET2005 Benchmark sheet metals (Smith, L.M., Pourboghrat, F., Yoon, J.-W., Stoughton, T.B., 2005. NUMISHEET2005. In: Proc. of 6th Int. Conf. Numerical Simulation of 3D Sheet Metal Forming Processes, PART A and B(Benchmark), pp. 409–451) to estimate formability. The “dynamic explicit” results reveal that the initial crystal orientation distribution has a large affects to a plastic strain induced texture and anisotropic hardening evolutions and sheet formability.     

Complex unloading behavior: Nature of the deformation and its consistent constitutive representation

Complex (nonlinear) unloading behavior following plastic straining has been reported as a significant challenge to accurate springback prediction. More fundamentally, the nature of the unloading deformation has not been resolved, being variously attributed to nonlinear/reduced modulus elasticity or to inelastic/“microplastic” effects. Unloading-and-reloading experiments following tensile deformation showed that a special component of strain, deemed here “Quasi-Plastic-Elastic” (“QPE”) strain, has four characteristics. (1) It is recoverable, like elastic deformation. (2) It dissipates work, like plastic deformation. (3) It is rate-independent, in the strain rate range 10⁻⁴-10⁻²/s, contrary to some models of anelasticity to which the unloading modulus effect has been attributed. (4) To first order, the evolution of plastic properties occurs during QPE deformation. These characteristics are as expected for a mechanism of dislocation pile-up and relaxation. A consistent, general, continuum constitutive model was derived incorporating elastic, plastic, and QPE deformation. Using some aspects of two-yield-function approaches with unique modifications to incorporate QPE, the model was implemented in a finite element program with parameters determined for dual-phase steel and applied to draw-bend springback. Significant differences were found compared with standard simulations or ones incorporating modulus reduction. The proposed constitutive approach can be used with a variety of elastic and plastic models to treat the nonlinear unloading and reloading of metals consistently for general three-dimensional problems.     

Analytical springback model for lightweight hexagonal close-packed sheet metal

Experiments have shown that magnesium alloy sheet a common hexagonal close-packed metal, exhibits mechanical behavior unlike that of sheets made of cubic metals (X.Y. Lou et al., 2007, Int. J. Plasticity, 24, 44). The unique stress–strain response includes a strong asymmetry in the initial yield and subsequent plastic hardening. In other words, the stress–strain curves in tension and compression are significantly different. A proper representation of the constitutive relationships is crucial for the accurate evaluation of springback, which occurs due to the residual moment distribution through the sheet thickness after bending. In this paper, we propose an analytical model for asymmetric elasto-plastic bending under tension followed by elastic unloading in order to evaluate the bending moment, which is equivalent to the springback amount. To simplify the calculations, the experimentally measured stress–strain curve of the magnesium alloy sheet was approximated with discrete linear hardening in each deformation region, and the material properties were characterized according to several simplifying assumptions. The bending moment was calculated analytically using the approximate asymmetric stress–strain relationship up to the prescribed curvature corresponding to the radius of the tool in sheet metal forming operations. A numerical example showed an unusual springback increase, even with an increase in the applied force; this is an unexpected result for conventional symmetric materials. We also compared the calculated springback amounts with the results of physical measurements. This showed that the proposed model predicts the main trends of the springback in magnesium alloy sheets reasonably well considering the simplicity of the analytical approach.     

A reliability study of springback on the sheet metal forming process under probabilistic variation of prestrain and blank holder force

This work deals with a reliability assessment of springback problem during the sheet metal forming process. The effects of operative parameters and material properties, blank holder force and plastic prestrain, on springback are investigated. A generic reliability approach was developed to control springback. Subsequently, the Monte Carlo simulation technique in conjunction with the Latin hypercube sampling method was adopted to study the probabilistic springback. Finite element method based on implicit/explicit algorithms was used to model the springback problem. The proposed constitutive law for sheet metal takes into account the adaptation of plastic parameters of the hardening law for each prestrain level considered. Rackwitz-Fiessler algorithm is used to find reliability properties from response surfaces of chosen springback geometrical parameters. The obtained results were analyzed using a multi-state limit reliability functions based on geometry compensations.     

An evaluation of a combined isotropic-kinematic hardening model for representation of complex strain-path changes in dual-phase steel

This paper aims at evaluating an elastoplastic constitutive model which accounts for combined isotropic-kinematic hardening for complex strain-path changes in a dual-phase steel, DP800. The capability of the model to reproduce the transient hardening phenomena under two-stage non-proportional loading has been assessed through numerical simulations of sequential uniaxial tension and notched tension/shear tests. Finite element simulations with shell elements were performed using the explicit non-linear FE code LS-DYNA. Numerical predictions of the stress–strain response were compared to the corresponding experimental data. The results from the experiments demonstrated that prior plastic deformation has certainly influenced the subsequent work-hardening behaviour of the material under biaxial or shear deformation modes. Furthermore, the numerical simulations from the two-stage uniaxial tension–notched tension and uniaxial tension–shear tests predicted the general trends of the experimental results such as transitory hardening and overall work hardening. However, some discrepancies were found in accurately describing the transient hardening behaviour subsequent to strain path changes between the experiments and numerical simulations.     

On the use of a reduced enhanced solid-shell (RESS) element for sheet forming simulations

A recently proposed reduced enhanced solid-shell (RESS) element [Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2005. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part I – Geometrically Linear Applications. International Journal for Numerical Methods in Engineering 62, 952–977; Alves de Sousa, R.J., Cardoso, R.P.R., Fontes Valente, R.A., Yoon, J.W., Grácio, J.J., Natal Jorge, R.M., 2006. A new one-point quadrature enhanced assumed strain (EAS) solid-shell element with multiple integration points along thickness: Part II – Nonlinear Applications. International Journal for Numerical Methods in Engineering, 67, 160–188.] is based on the enhanced assumed strain (EAS) method with a one-point quadrature numerical integration scheme. In this work, the RESS element is applied to large-deformation elasto-plastic thin-shell applications, including contact and plastic anisotropy. One of the main advantages of the RESS is its minimum number of enhancing parameters (only one), which when associated with an in-plane reduced integration scheme, circumvents efficiently well-known locking phenomena, leading to a computationally efficient performance when compared to conventional 3D solid elements. It is also worth noting that the element accounts for an arbitrary number of integration points through thickness direction within a single element layer. This capability has proven to be efficient, for instance, for accurately describing springback phenomenon in sheet forming simulations. A physical stabilization procedure is employed in order to correct the element’s rank deficiency. A general elasto-plastic model is also incorporated for the constitutive modelling of sheet forming operations with plastic anisotropy. Several examples including contact, anisotropic plasticity and springback effects are carried out and the results are compared with experimental data.     

Investigation of advanced strain-path dependent material models for sheet metal forming simulations

Sheet metal forming processes often involve complex loading sequences. To improve the prediction of some undesirable phenomena, such as springback, physical behavior models should be considered. This paper investigates springback behavior predicted by advanced elastoplastic hardening models which combine isotropic and kinematic hardening and take strain-path changes into account. A dislocation-based microstructural hardening model formulated from physical observations and the more classical cyclic model of Chaboche have been considered in this work. Numerical implementation was carried out in the ABAQUS software using a return mapping algorithm with a combined backward Euler and semi-analytical integration scheme of the constitutive equations. The capability of each model to reproduce transient hardening phenomena at abrupt strain-path changes has been shown via simulations of sequential rheological tests. A springback analysis of strip drawing tests was performed in order to emphasize the impact of several influential parameters, namely: process, numerical and behavior parameters. The effect of the two hardening models with respect to the process parameters has been specifically highlighted.     

Strain localization analysis using a large deformation anisotropic elastic–plastic model coupled with damage

Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic–plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice’s localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic–plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results – e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.     

Forming limit stresses predicted by phenomenological plasticity theories with anisotropic work-hardening behavior

Forming limit stresses of sheet metals subjected to linear and combined stress paths are analyzed using the M-K model in conjunction with two anisotropic work-hardening models: a work-hardening model which is capable of describing Bauschinger and cross-hardening effects, and a work-hardening model which cannot predict the cross-hardening effect. It is found that the forming limit stress is path-independent when the stress–strain curves for the linear and combined stress paths agree well with each other. On the other hand, the forming limit stress for the combined stress path depends on the strain path when the prestrain changes the subsequent stress–strain relation. We conclude that the stress-based forming limit criterion is efficient only for a material with a work-hardening behavior that is not affected by strain path change. The influence of the work-hardening behavior on the forming limit stress is discussed in detail.     

Elasto-visco-plastic modeling of mild steels for sheet forming applications over a large range of strain rates

A physically based elasto-visco-plastic constitutive model is presented and compared to experimental results for three different mild steels. The experiments consist of tensile tests ranging from quasi-static conditions up to strain rates of 10³ s^?1 as well as quasi-static simple and reverse shear tests at different amounts of pre-strain. Additional two-step sequential mechanical tests (Bauschinger and orthogonal effects) have been performed to further evaluate the ability of the model to describe strain-path changes at moderate/large strains. The model requires significantly fewer material parameters compared to other visco-plasticity models from the literature, while being able to describe some of the main features of the strain-rate sensitivity of mild steels. Accordingly, the parameter identification is simple and intuitive, requiring a relatively small set of experiments. The strain-rate sensitivity modeling is not restricted to a particular hardening law and thus provides a general framework in which advanced hardening equations can be adopted.     

A general analytic solution for plane strain bending under tension for strain-hardening material at large strains

A new analytic solution for plane strain bending under tension of a sheet is proposed for elastic-plastic, isotropic, incompressible, strain-hardening material at large strains. Numerical treatment is only necessary to calculate ordinary integrals and solve transcendental equations. No restriction is imposed on the hardening law. All governing equations and boundary conditions are exactly satisfied. The only exception is that the actual stress distribution over the ends of the sheet is replaced with a concentrated force and a concentrated bending moment. The through-thickness distribution of residual stresses and a measure of springback are also found. The range of validity of the solution is determined. An illustrative example is provided for Swift’s hardening law.     

Finite element modeling of plastic anisotropy induced by texture and strain-path change

Consideration of plastic anisotropy is essential in accurate simulations of metal forming processes. In this study, finite element (FE) simulations have been performed to predict the plastic anisotropy of sheet metals using a texture- and microstructure-based constitutive model. The effect of crystallographic texture is incorporated through the use of an anisotropic plastic potential in strain-rate space, which gives the shape of the yield locus. The effect of dislocation is captured by use of a hardening model with four internal variables, which characterize the position and the size of the yield locus. Two applications are presented to evaluate the accuracy and the efficiency of the model: a cup drawing test and a two-stage pseudo-orthogonal sequential test (biaxial stretching in hydraulic bulging followed by uniaxial tension), using an interstitial-free steel sheet. The experimental results of earing behavior in the cup drawing test, maximum pressure and strain distribution in bulging, and transient hardening in the sequential test are compared against the FE predictions. It is shown that the current model is capable of predicting the plastic anisotropy induced by both the texture and the strain-path change. The relative significance of texture and strain-path change in the predictions is discussed.     

A plastic constitutive equation incorporating strain,strain-rate,and temperature

An empirical plasticity constitutive form describing the flow stress as a function of strain, strain-rate, and temperature has been developed, fit to data for three dual-phase (DP) steels, and compared with independent experiments outside of the fit domain. Dubbed the “H/V model” (for “Hollomon/Voce”), the function consists of three multiplicative functions describing (a) strain hardening, (b) strain-rate sensitivity, and (c) temperature sensitivity. Neither the multiplicative structure nor the choice of functions (b) or (c) is novel. The strain hardening function, (a), has two novel features: (1) it incorporates a linear combination coefficient, α, that allows representation of Hollomon (power law) behavior (α = 1), Voce (saturation) behavior (α = 0) or any intermediate case (0 < α < 1, and (2) it allows incorporation of the temperature sensitivity of strain hardening rate in a natural way by allowing α to vary with temperature (in the simplest case, linearly). This form therefore allows a natural transition from unbounded strain hardening at low temperatures toward saturation behavior at higher temperatures, consistent with many observations. Hollomon, Voce, H/V models and others selected as representative from the literature were fit for DP590, DP780, and DP980 steels by least-squares using a series of tensile tests up to the uniform strain conducted over a range of temperatures. Jump-rate tests were used to probe strain rate sensitivity. The selected laws were then used with coupled thermo-mechanical finite element (FE) modeling to predict behavior for tests outside the fit range: non-isothermal tensile tests beyond the uniform strain at room temperatures, isothermal tensile tests beyond the uniform strain at several temperatures and hydraulic bulge tests at room temperature. The agreement was best for the H/V model, which captured strain hardening at high strain accurately as well as the variation of strain hardening with temperature. The agreement of FE predictions up to the tensile failure strain illustrates the critical role of deformation-induced heating in high-strength/high ductility alloys, the importance of having a constitutive model that is accurate at large strains, and the implication that damage and void growth are unlikely to be determinant factors in the tensile failure of these alloys. The new constitutive model may have application for a wide range of alloys beyond DP steels, and it may be extended to larger strain rate and temperature ranges using alternate forms of strain rate sensitivity and thermal softening appearing in the literature.     

Deformation of thermosetting resins at impact rates of strain. Part 2: constitutive model with rejuvenation

The constitutive responses of three glassy thermoset polymers at impact rates of strain and slower, together with measurements of adiabatic heating, were reported earlier by the authors. The results are interpreted here in the context of a constitutive model proposed previously for amorphous polymers, expanded to incorporate strain-softening and the adiabatic heating deficit. In terms of the model, both features are a natural consequence of strain-induced evolution of the glass structure, as represented by Tool's “fictive temperature”—the phenomenon of structural rejuvenation. A representation is proposed for the evolution of fictive temperature with plastic strain, motivated by an approximate treatment of the kinetics of physical ageing/rejuvenation. Formulated in this manner, the model agrees reasonably well with experimental results across the wide range of strain rates of the previous experiments, 10⁻³ to , and across most of the range of strain to failure in compression. At the highest strains, however, an additional adiabatic heating deficit appears that is not predicted by the model, either suggesting the onset of structural breakdown possibly associated with the appearance of cracks or reflecting a need for better physical understanding of large deformations in glassy polymers.     

Elastic-plastic behaviour of dual-phase, high-strength steel under strain-path changes

The elastic-plastic behaviour of dual-phase, high-strength steel sheets under two-stage strain-path changes has been investigated. Three different loading sequences, namely monotonic, 45° tensile path changes and orthogonal tensile path changes complied by sequences of simple uniaxial tensile tests, were analysed at room temperature. From the experiments, it was found that there is a considerable reduction of the initial flow stress over the strain-path changes. The transient softening phenomenon is observed to be a function of orientation, and the period of the transient behaviour following the strain-path change is lengthened with the amount of pre-strain. A constitutive model is adopted that includes combined isotropic and kinematic hardening and is capable of describing the marked transient softening behaviour after the pre-straining. The experimental stress–strain behaviour subsequent to the strain path change is predicted with reasonable accuracy, while the model fails to accurately describe the transient, deformation-induced anisotropy in the plastic flow.     

Review of theoretical models of the strain-based FLD and their relevance to the stress-based FLD

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.