Strain-based Forming Limit Diagram Research Articles

Forming Limit Analysis of Thin-Walled Extruded Aluminum Alloy Tubes under Nonlinear Loading Paths Using an Improved M-K Model

To meet the requirement of lighter weight and better performance in tube hydroforming, one of the most important tasks is to accurately predict the forming limit of thin-walled tubes under nonlinear loading paths. This work established the M-K+DF2012 model, a combination of the M-K model and the DF2012 ductile fracture criterion, for the forming limit prediction of thin-walled tubes under nonlinear loading paths. In this model, the failure of the groove is determined by the DF2012 criterion, and the corresponding strains in the uniform region are the limit strains. The limit strains of an AA6061 aluminum alloy tube under a set of linear loading paths and two typical nonlinear loading paths were tested. Parameter values of the M-K+DF2012 model for the tube were determined based on the experimental limit strains under linear loading paths, and the limit strains under the two nonlinear loading paths were predicted. Then the strain-based forming limit diagram (ε-FLD) and the polar effective plastic strain FLD (PEPS-FLD) of the tube under different pre-strains were predicted and discussed. The results show that the limit strains of the tube are obviously path-dependent, and the M-K+DF2012 model can reasonably capture the limit strains of the tube under both linear and nonlinear loading paths. The predicted ε-FLD shows a strong dependence on the pre-strain, while the predicted PEPS-FLD is weakly strain path-dependent and almost path-independent on the right-hand side for the AA6061 tube.

A Novel Approach for Process and Tool Design During Manufacturing of Industrial Sheet Metal Components

Manufacturing of sheet metal components feeds into several domains such as automotive, aerospace and strategic applications. The optimum process and tool design are crucial in ensuring successful manufacturing of components without developing any defects. The current trend is to do it right the first time during the industrial stamping practice, so that the costly shop floor trials are eliminated or at least minimized. One of the common failures that are observed during manufacturing of industrial sheet metal components is the splitting/cracking. The present work focuses on addressing the design considerations for preventing development of such defects. Traditionally, the strain-based forming limit diagram (FLD) is used to assess if a proposed process and tool design will lead to splitting failures. The approach is to measure or predict the strains that will develop during the stamping process and to infer that splitting failures will occur if these strains are above the FLD. The optimum process and tool design should ensure that the strains developed in the component must be below the FLD. The approach of using the strain-based FLD has serious limitation, as it does not account for the influence of complex strain paths on the forming limit strains. As an alternative, stress-based FLDs have been proposed, which does address the effect of complex strain paths. But, the stress-based FLDs are not intuitive and therefore are not very suitable to use for analysis of industrial sheet metal components. In this work, a novel approach, based on monitoring development of local thickness gradients during manufacturing of sheet metal components, is proposed to assess a given process and tool design for developing splitting/cracking failures. The novel approach is validated for sheets of different metals and thicknesses. The effect of bilinear strains paths on the forming limits is captured and validated using this proposed novel approach. A new approach, based on the novel thickness gradient criteria, is proposed for quantifying the forming severity of industrial sheet metal components.

Forming Limit Diagrams of Low-Carbon Steels Obtained Using Digital Image Correlation Technique and Enhanced Formability Predictions Incorporating Microstructural Developments

In the present work, drawing quality (DQ) and interstitial-free (IF) steel sheets were subjected to limiting dome height tests for determining strain-based forming limit diagrams (FLDs) experimentally and effect of dynamic (variable) material properties, such as work hardening ‘n’ and plastic anisotropy ( $$\bar{r}$$ ), on FLDs has been studied and simulated by PAMSTAMP™ finite element software. Dot prints on the steel specimens and an efficient optical strain measurement system GOM™ which works on the digital image correlation technique principle were used to measure limiting strains accurately instead of conventional circle grid analysis technique and strain measurement by traveling microscope. The dynamic (variable) material properties (n) and ( $$\bar{r}$$ ) were estimated by studying the microstructural developments in terms of changes in grain average misorientation and crystallographic texture, respectively, at different strains and strain paths, during deformation. In our proposed work, a novel technique of incorporating dynamic (variable) material properties (n and $$\bar{r}$$ ) in FE simulations was carried out during FLDs predictions. Though marginal but improved predictions in FLDs were observed in both IF and DQ steels. In addition to strain-based FLDs, stress-based FLDs were also determined for both grades. Interestingly, in both cases it was noticed that IF steel had higher formability than DQ steel.

Improvement in Accuracy of Failure Prediction in Sheet Hydroforming of Square Cups Using Stress-Based Forming Limit Diagram

Prediction of failure in sheet metal forming processes accurately is very important for successful production and optimization of parameters. A major problem of conventional strain-based forming limit diagrams (FLDs) is their inability to predict failure accurately in processes such as sheet hydroforming where there is a change in strain path and mode of deformation. In the present work, a stress-based forming limit diagram has been developed for AA 5182 alloy sheets modifying the analytical procedure, proposed by Stoughton, to determine forming limits in stress space from failure strains incorporating anisotropy using Balart’s yield criterion. The developed stress-based FLD has been used to predict failure in sheet hydroforming of square cups. Results are compared with Hill’s quadratic yield criterion. Significant difference has been found in failure prediction between strain-based and stress-based criteria when they are applied to sheet hydroforming. A change in strain path has been observed at the critical corner regions in hydroforming of square cups due to initial drawing and then biaxial stretching during calibration. The experimental validation clearly showed that accuracy in failure prediction can be improved in sheet hydroforming by using a stress-based forming limit diagram.

Simulation of shear fracture in sheet metal forming of thick plates under triaxial stress states

In industrial sheet forming of high strength steel sheets, shear fracture has been reported to occur in narrow die shoulders where the sheet is subject to excessive bending and stretching. For high strength steel plates, which generally show low ductility, the die curvature becomes relatively large for thin sheets, increasing the risk of fracture. Under these circumstances, failure is caused by the shear stresses induced by the high through-thickness stress gradient. Since shear fracture cannot be predicted by the strain-based forming limit diagram, efforts have been made to overcome this limitation and eventually led to the triaxiality failure diagram. In this study, we investigate shear fracture occurring in the forming of thick high strength steel sheets by means of finite element analysis. To do so, we implement the TFD in the commercial finite element analysis software AFDEX through a subroutine. We compare and analyze the predicted damage values obtained for diverse damage models with those obtained via the subroutine so to gain better insight in the theory and background of shear fracture. Finally, the influence of the mesh quality on the predicted stress triaxiality-dependent damage parameter is investigated.

Experimental study on the formability of AA6016 sheets pre-strained by rolling

The effects of prestrain by rolling on the formability of AA6016 sheets in the biaxial stretching region are studied in this paper. An experimental program including formability experiments has been carried out for as-received and prestrained sheets. Forming limit strains are determined by the digital image correlation technique and then mapped into stress space using an anisotropic plasticity model in order to assess the path-independence of forming limit stresses. Different diagrams for description of formability such as the traditional strain-based forming limit diagram (FLD), the forming limit stress diagram (FLSD) and various equivalent plastic strain based (EPS-based) diagrams are applied in order to study the effect of prestrain. It is shown that prestrain by rolling has the same effect on the forming limit strains as prestrain by plane strain tension. Furthermore, the forming limits of the virgin material nearly coincide with the forming limits of the prestrained material in the FLSD and EPS-based diagrams, serving as a conservative estimate for the prestrained material. Due to the roping phenomenon, the material displays significant anisotropy in forming limit strains. At the same time, the presence of roping does not affect the path-independence of the forming limits in the FLSD and EPS-based diagrams.

Analytical determination of forming limit curve for zirlo and its experimental validation

Zirconium alloy sheets used in the nuclear fuel assembly are subject to high temperature and stress, a corrosive and radiation environment and their performance has thus been of big concern. However, knowledge regarding the properties and the formability of zirconium alloys is still lacking. In this paper, strain-based forming limit diagrams (FLDs) are established for the zirconium alloy Zr1Sn1Nb0.1Fe (also called zirlo). The analytical forming limit curve (FLC) varies with forming limit model and yield criterion. To obtain the right hand side of the FLD, we apply the Swift's diffuse necking model, and for the left hand side, Hill's local necking model. Solutions from the analytical method are compared with experimental FLD established based on limit dome height (LDH) tests. Using the strain hardening exponent of a modified stress–strain relation, we could find excellent agreement of the analytical FLC with the LDH-based FLC.

Modeling of forming limit for multilayer sheets based on strain-rate potentials

In order to evaluate the forming limits of planar anisotropic multilayer sheet materials, the Marciniak–Kuczynski (M−K) model was formulated based on the strain-rate potential. A common practice to predict the forming limits of sheet materials has been based on yield stress potentials defined in the stress field. As an alternative, a method based on plastic stain-rate potentials, which are especially convenient to apply for rigid-viscoplasticity, was considered in this work. The formulation based on the strain-rate potential facilitates the modeling of forming limits for multilayer sheet materials because the number of unknown variables is significantly reduced by assuming the iso-strain condition for each layer without delamination. As for the strain-rate potential, Srp2003-2d, which is the pseudo-conjugate of the yield stress potential Yld2000-2d, was applied along with Hill's 1948 strain-rate potential for comparison. In the approach proposed, rigid-viscoplasticity was formulated according to the incremental deformation theory based on the minimum plastic work path. The rotation of anisotropic symmetry axes in the groove region was also properly accounted for in the formulation. For verification purposes, the predicted forming limit criteria such as the strain-based forming limit diagram (FLD), the stress-based forming limit diagram (FLSD) and the effective strain-based forming limit diagram (x-EPS) were experimentally validated for a monolithic aluminum alloy (AA5182-O) sheet and a three-layer AA5182-O/polypropylene/AA5182-O (AA/PP/AA) sandwich sheet, which confirmed good agreement. In addition, the path-sensitivity of the strain-based FLD and the path-insensitivity of the FLSD and x-EPS were numerically proven based on the M−K model for the three-layer sandwich sheet. Finally, the M−K model was compared with a maximum force model most recently developed.

A hybrid method for detecting the onset of local necking by monitoring the bulge forming load

Strain-based Forming Limit Diagrams (FLD), which are typically obtained under linear or quasi-linear loading conditions, describe the limiting strains in terms of the major and minor in-plane strains before the onset of necking or the final failure (FFD). These strains can be detected by analysing the strain field in the vicinity of necking or cracking defects. It has generally been agreed that the loading versus time signal is not suitable for detecting necking processes. A novel hybrid method of detecting the onset of necking based on the experimental and simulated bulging load is presented in this paper. This method consists mainly in comparing the experimental forming load, i.e., a load showing plastic instability, with the numerical predictions obtained by performing finite element simulation. The simulation of the bulging process does not include any damage or failure criteria. A homogeneous forming load can therefore be simulated without requiring any information about the localization. This method was applied to detecting the onset of local necking in circular and elliptic quasistatic bulge tests on sheet material, with a diameter of 200 mm. Two materials were tested, a 0.8 mm thick DP450 Dual Phase steel sheet and a 1 mm thick AA6016-T4 aluminium sheet. The onset of necking observed with our method was compared with the results obtained by performing Hogstrom’s analysis based on the measured strain field over time and similar necking strains were obtained. Beside, the Bressian Williams Hill (BWH) shear criterion was identified for each test from experimental results. A slight scattering of the shear stress values was observed.

Bifurcation Analysis of Forming Limits for an Orthotropic Sheet Metal

A bifurcation analysis of forming limits for an orthotropic sheet metal is presented in this paper. The approach extends Stören and Rice's (S–R) bifurcation analysis for isotropic materials, with materials following a vertex theory of plasticity at the onset of localized necking. The sheet orthotropy is represented by the Hill’48 yield criterion with three r-values in the rolling (r0), the transverse (r90) and the diagonal direction (r45). The emphasis of the study is on the examination of r-value effect on the sheet metal forming limit, expressed as a combination of the average r-value raverage and the planar anisotropy (Δr). Forming limits under both zero extension assumption and minimum extension assumption as well as necking band orientation evolution are investigated in detail. The comparison between the experimental result and predicted forming limit diagram (FLD) is presented to validate the extended bifurcation analysis. The r-value effect is observed under uniaxial and equal-biaxial loadings. However, no difference is found under plane strain condition in strain-based FLD which is consistent with Hill's theory. The force maximum criterion is also used to analyze FLD for verification.

A different approach to estimate the process parameters in tube hydroforming

An enhanced unfolding inverse finite element method (IFEM) has been used together with an extended strain-based forming limit diagram (FLD) to develop a fast approach to predict the feasibility of tube hydroforming process of concept part and determine where the failure or defects can occur. In tube hydroforming, the inverse IFEM has been used for estimating the initial length of tube, axial feeding and fluid pressure. The already developed IFEM algorithm used in this study is based on the total deformation theory of plasticity. Although the nature of tube hydroforming is three-dimensional deformation, in this article a modeling technique has been used to perform the computations in two-dimensional space. Therefore, compared with conventional forward finite element methods, the present computations are quite fast with no trial and error process. In addition, the solution provides all the components of strain. Using the extended strain-based forming limit diagram, the components of strain can lead us to measure the potentials for failures during the deformation. The extended strain-based FLD based on the Marciniak and Kuczynski (M-K) model has been computed and used to predict the onset of necking during tube hydroforming. The extended strain-based FLD is built based on equivalent plastic strains and material flow direction at the end of forming. This new forming limit diagram is much less strain path dependent than the conventional forming limit diagram. Furthermore, the use and interpretation of this new diagram is easier than the stress-based forming limit diagram. The results of analysis for free bulging and square bulging have been compared with some published experimental data and the results obtained by conventional commercial software. The results indicate that the fluid pressure estimated by this method is 2.7 % greater than the results obtained by the experiment in the square bulging sample. In addition, the fluid pressure estimated in the free bulge sample is 5.6 % greater than the experimental results.

Analytical Method for Forming Limit Diagram Prediction with Application to a Magnesium ZEK100-O Alloy

A significant barrier to broader implementation of magnesium alloys is their poor room temperature formability, a consequence of the anisotropic response of the Mg hexagonal closed-packed (hcp) crystal structure. Additions of rare earth (RE) elements, such as in the ZEK100 alloys, weaken the texture and improve formability. Room temperature forming limit analyses of RE-containing Mg alloys, particularly Mg ZEK100, have not been explored to any significant extent in the literature. In this paper, strain-based forming limit diagrams (FLDs) are derived for an Mg ZEK100-O alloy (Zn1.34Zr0.23Nd0.182, wt.%) using an analytical method that combines the vertex theory of Storen and Rice (J Mech Phys Solids, 23:421-441, 1979), the anisotropic yield criterion of Barlat and Lian (Int J Plast, 5:51-66, 1989), and a hardening law. The method does not rely on assumptions about pre-existing defects, is broadly applicable to sheet alloys exhibiting in-plane anisotropy requiring a higher-order yield criterion, and requires only minimal experimental inputs. Results from the analytical method are compared with experimentally derived FLDs based upon the well-known Nakajima test and tensile deformation, and with predictions from an existing analytical method for FLDs. Close agreement between the experimentally derived FLDs and the present theoretical method was obtained. Sheet materials where the theoretical method does not apply are also discussed.

Comparative analysis between stress- and strain-based forming limit diagrams for aluminum alloy sheet 1060

The forming limit diagram (FLD) and forming limit stress diagram (FLSD) of aluminium alloy 1060 under linear and nonlinear strain paths are investigated. The calculation of FLSD is based on experimental FLD using the method proposed by Stoughton. Different from the FLD that varies with the strain path, the FLSD is not sensitive to the strain path. Therefore, FLSD is convenient as a forming limit criterion for multi-stage sheet forming. The influences of the material's yield criteria on FLSD are also discussed by comparison of the Hill's 48, Hill's 79 and Hosford non-quadratic criterion. The impacts of material hardening laws (Voce and Swift models) on translation of FLD and FLSD are analyzed. The Voce hardening law and the Hosford yield criterion are appropriate for the FLSD calculation of the aluminium alloy 1060. The stress calculation program and display interfaces of FLD and FLSD are developed on MATLAB, where the strain data can be input from experiment measurement or FEM calculations.

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.

Loading Path Dependence of Forming Limit Diagram of a TRIP800 Steel

In this paper, the microstructure-based finite element modeling method is used in investigating the loading path dependence of formability of transformation induced plasticity (TRIP) steels. For this purpose, the effects of different loading path on the forming limit diagrams (FLD) of TRIP steels are qualitatively examined using the representative volume element (RVE) of a commercial TRIP800 steel. First, the modeling method was introduced, where a combined isotropic/kinematic hardening rule is adopted for the constituent phases in order to rightly describe the cyclic deformation behaviors of TRIP steels during the forming process with combined loading paths which may include the unloading between the two consecutive loadings. Material parameters for the constituent phases remained the same as those in the authors’ previous study [1] except for some adjustments for the martensite phase due to the introduction of the new combined hardening rule. Based on the new material parameters and new hardening rule, the predicted deformation behaviors of the TRIP800 steel show quite similar qualitative trends to those reported in other experimental works. Pseudo-forming limit strain diagrams (Pseudo-FLD) for the TRIP800 steel were, then, obtained for various loading paths. The computational results show that, similar to other single phase materials, the TRIP800 steelmore » shows very sensitive loading path dependence in the strain-based forming limit diagrams (strain-FLD), but does not in the stress-based forming limit diagrams (stress-FLD), and that the phase transformation does not have significant effects on the FLD for the TRIP800 steel. From the observations in this study, the current modeling methods can be used in examining the qualitative trends of FLD of TRIP steels under different loading paths/prestrains.« less

Prediction of Defects in Tube Hydroforming Based on Forming Stress Limit

Transformation relation between limit strain and limit stress is derived on the basis of the plastic theory and Hill79 yield criterion. Then, the conventional strain-based forming limit diagram (FLD) of 2008T4 Aluminum alloy sheet is converted into stress-based forming limit stress diagram (FLSD). This FLSD is used in conjunction with LS-DYNA finite element simulation to predict the onset of necking and the limit pressure in T-shaped tube hydroforming. The prediction results using FLSD and FLD are also compared. The research indicates that two predicted forming limits are obviously different. Although the necking occurs at the same location in both prediction results, the limit pressures make a great difference. And the limit pressure predicted by FLSD compares well with the experimental result. In view of the strong path-dependence of the FLD, it will be more convenient and reliable to use FLSD criterion in tube hydroforming analysis.

Development and Application of the Finite Element Analysis Program of the Stress-Based Forming Limit Criterion for Sheet Metal Forming

In order to solve the problem existed in the numerical simulation of sheet metal forming for its use the strain-based forming limit diagram as criterion, which has the flaw of dependence on the strain paths, this paper develops the finite element analysis program based on the stress forming limit criterion applicable to the blank plastic forming technique, which follows the stress-strain transformation relationship when the sheet metal is undergoing plastic deformation, chooses Hill’s quadratic normal anisotropic criterion as computational model and selects the commercial finite element code Dynaform as its development environment. Also it be analyzed the finite element numerical simulation results of two deep drawing parts by the developed program module and realizes the prediction of sheet metal forming limit adopting the FLSD as criterion. The stress-based forming limit criterion for the developed program provides a new means to analyze the forming limit for the multistage sheet metal forming.

A study on fracture behaviour of three different high strength low alloy steel sheets during formation with different strain ratios

This paper deals with the study on fracture behaviour of three different steels, namely, the HSLA, the Micro alloyed and the Carbon–Manganese steels, using stress- and strain-based forming and fracture limit diagrams obtained from the experimentally determined strain values for different strain or stress ratios. With an aim to analyse the fracture behaviour, the strain- and stress-based forming and fracture limit diagrams have been developed for three different steel sheets, namely, the HSLA steel sheet of 1.6 mm thickness, the Micro alloyed steel sheet of 1.2 mm thickness, and the Carbon–Manganese steel sheet of 1.4 mm thickness. The forming and fracture behaviour and size and shape of voids (particularly, the ratio of length to width of voids, ( L/ W)) developed at fracture during the formation of these steels, under different strain ratio conditions, have been correlated with the mechanical properties such as microstructure of steels, minor strain at fracture, and Mohr’s circle shear strains, various combinations of mean strain and effective strain, and material properties, namely, strain hardening exponent ( n)-value, normal anisotropy ( r ¯ ) and the range of localized neckings.

Application of the forming limit stress diagram to forming limit prediction for the multi-step forming of auto panels

The strain-based forming limit diagram (FLD) is generally obtained on the linear (proportional) or near linear loading condition. So it often leads to erroneous assessments on the analysis of metal sheet formability under complex strain paths such as multi-step forming by the FLD criterion. In recent years, forming limit stress diagrams (FLSDs) have been intensively studied and have been considered being path-independent. In this study, it has been carried out that application of the FLSD to forming limit prediction for the multi-step forming of auto panels. Firstly, it was been performed that the forming limit tests of sheet metal for the SPECN steel sheet of thickness of 1.6 mm under the loading conditions considered include both proportional and non-proportional loading. Secondary, the forming limit stress diagram (FLSD) of the sheet metal is gained by the transformation relations between the strain state and stress state in deformation of sheet metal. Afterwards, implementation of the forming limit stress diagrams into finite element numerical simulations of rear hanger of auto panels multi-step forming has been established by DANAFORM program code, the FLSD is applied to forming limit prediction in the stamping finite element analysis. The verification experiment of the simulation results were executed in motorcar manufacturing factory, the predicted results agree well with those determined experimentally.

유한요소법과 FLSD를 이용한 관재 하이드로포밍 공정에서의 성형 한계 예측

Among the failure modes which can occur in tube hydroforming such as wrinkling, bursting or buckling, the bursting by local instability under excessive tensile stresses is irrecoverable phenomenon. Thus, the accurate prediction of bursting condition plays an important role in producing the successfully hydroformed part without any defects. As the classical forming limit criteria, strain-based forming limit diagram (FLD) has widely used to predict the failure in sheet metal forming. However, it is known that the FLD is extremely dependant on strain path throughout the forming process. Furthermore, The application of FLD to hydroforming process, where strain path is no longer linear throughout forming process, may lead to misunderstanding for fracture initiation. In this work, stress-based forming limit diagram (FLSD), which is strain path-independent and more general, was applied to prediction of forming limit in tube hydroforming. Combined with the analytical FLSD determined from plastic instability theory, finite element analyses were carried out to find out the state of stresses during hydroforming operation, and then FLSD is utilized as forming limit criterion. In addition, the approach is verified by a series of bulge tests in view of bursting pressure and shows a good agreement. Consequently, it is shown that the approach proposed in this paper will provide a feasible method to satisfy the increasing practical demands for judging the forming severity in hydroforming processes.