Marciniak And Kuczynski Research Articles

Experimental and Numerical Analysis of Prestrain on the Formability of Zn-Cu-Ti Alloy Zinc Sheet

The forming limit diagrams (FLDs) characterizing the formability of sheet metals are usually obtained by applying proportional loadings. Nevertheless, the industrial processes involve strain path changes that can modify the limit-strain values. In addition, for strongly anisotropic sheet metals such as the Zn-Cu-Ti zinc alloy, large differences in forming limit curves (FLCs) with respect to the sheet rolling direction are observed. In the present work, the analysis of the effect of bilinear strain paths on the FLC is addressed by both experimental measurements and numerical simulations. For this purpose, a miniature testing device was used that allows evaluation of the influence of strain path changes on the limit strain on samples at 0°, 45° and 90° with respect to the sheet rolling direction cut from non-standard large samples previously subjected to a prestrain along the RD up to an early deformation of ~0.12. Numerical simulations were carried out using the well-known Marciniak and Kuczynski (MK) theory in conjunction with the viscoplastic self-consistent (VPSC) crystal plasticity model. In order to account for the grain fragmentation process due to the continuous dynamic recrystallization (CDRX) mechanism, an ad hoc short-range interaction effect (SRE) model was included in the simulations. Additionally, the measured and simulated texture evolution of Zn-Cu-Ti alloy sheets at the different stages of the deformations were shown. The capacity of the MK-VPSC-SRE model was validated, and the limitations to simulating the texture development, flow stress and forming limit curves, including a non-proportional strain path, were discussed.

Micromechanical Effect of Martensite Attributes on Forming Limits of Dual-Phase Steels Investigated by Crystal Plasticity-Based Numerical Simulations

This study analyses the effect of martensite grain size and its volume fraction in dual-phase (DP) steel on (1) the formability limit, (2) average global behavior under different loading conditions, and (3) damage initiation. The virtual RVEs (Representative Volume Elements) were constructed using DREAM.3D software with a variation of microstructural attributes. The numerical simulations were carried out using DAMASK, which evaluates the polycrystalline material point behavior and solves versatile constitutive equations using a spectral solver. The simulations were post-processed to obtain global and local stress, strain, and damage evolution in constructed RVEs. The global results were processed to obtain FLDs according to Keeler-Brazier (K-B) and Marciniak and Kuczynski (M-K) criteria. In this work, the capability of microstructure-based numerical simulations to analyze the FLDs has been established successfully. From Forming Limit Diagrams (FLDs), it was observed that formability changes by changing the strain hardening coefficients (n-values), the martensite fraction, and martensite grain sizes of DP steels. The improved formability was observed with lower martensite fraction, i.e., 17%, decreased martensite grain size, i.e., 2.6 µm, and higher strain hardening coefficient. The M-K approach shows the better capability to predict the formability by various loading conditions and clarifies the necking marginal zone of FLD. The damage propagation is also strongly affected by the loading conditions. The current study would be a good guide for designers during the manufacturing and selecting of appropriate DP steels based on the service loading conditions.

Study of forming limit diagram (FLD) prediction of anisotropic sheet metals using Gurson model in M-K method

This study uses the Marciniak and Kuczynski (M-K) method to present an analytical forming limit diagram (FLD) for sheet metals. The procedure for the analytical FLD prediction is described in detail and step-wise manner, and an algorithm is written using MATLAB. First, an appropriate algorithm is determined to establish the theoretical analyses, and various anisotropic yield functions, such as Hill’s 48, Barlat 89, and Hosford, are considered. The predicted FLDs are compared with experiments involving a typical AA6016-T4 aluminum alloy. Second, the Gurson model that considers damage growth is implemented when Hosford is the yield function, as Hosford criterion predicts the best comparable analytical FLD with experiments among the yield functions. Third, a parametric study is performed to investigate the effects of parameters on the FLD prediction. Results indicate that an extremely low value for the initial void volume fraction in the safe and groove zones has minimal effects on the FLD prediction. Lastly, the values of void volume fractions are calculated assuming no geometrical imperfections and the imperfection is because of higher void volume fraction in groove zone than that in safe zone.

Evaluation of Temperature Effects on Forming Limit Diagrams of AA6061-T6 Considering the Marciniak and Kuczynski Model

Abstract In this study, forming limit diagrams (FLDs) of AA6061-T6 have been determined numerically and experimentally at room and elevated temperatures. The computational method has been based on the Marciniak and Kuczynski (M-K) model in conjunction with the Johnson-Cook (J-C) constitutive model. In the present numerical approach, four nonlinear equations have been solved simultaneously via the Quasi-Newton Technique. Moreover, the effects of temperature on formability have been discussed. To verify the theoretical results, out-of-plane Nakazima tests have been performed at the same temperatures, and a good match between the present numerical approach and experimental data has been seen. Both the experimental and numerical results illustrated that as the temperature increased, the FLD moved upward.

A miniaturized device for the measurement of sheet-metal formability using digital image correlation.

A laboratory-scale device to obtain forming-limit diagram data was designed to utilize the Marciniak and Kuczynski (MK) sample geometry. The design uses a high-resolution photographic camera, automatic trigger, and light-emitting diode (LED) lighting to record the time history of deformation calculated with the digital-image correlation technique. Because the testing device was miniaturized, it was possible to halt the forming experiments at intermediate strains and recrystallize the MK carrier blank. This permits large formability strains to be obtained without cracks developing at the carrier blank's central hole, an advantage over full-size specimens and conventional testing rates. A number of initial experiments were performed on a zinc alloy sheet (Zn-Cu-Ti) over the entire forming-limit range (-0.5 ≤ ε2/ε1 ≤ 1), and the strain fields reduced employing the Bragard criterion to obtain limit strains. These results are compared favorably to previous data of this material obtained with a hemispherical, Nakazima, punch and a circle-grid pattern.

A thermal forming limit prediction method considering material damage for 22MnB5 sheet

Hot stamping of high-strength steel is an innovative technology to achieve automobile lightweight and guarantee security simultaneously. However, the formability of high-strength steel is still limited at elevated temperatures by the evolution of void damage inside materials. Thus, the establishment of an efficient forming limit prediction method is urgently demanded. In the present work, the Gurson-Tvergaard-Needleman (GTN) model is extended by containing Hosford anisotropic yield criterion which can characterize void damage in normal anisotropic materials. The flow behavior of matrix material 22MnB5 at different forming temperatures is simulated by the modified Norton-Hoff hardening law. And the damage-related parameters are calibrated through FEM inverse analysis. Furthermore, the prediction method of forming limit combined with void damage is realized within the framework of the Marciniak and Kuczynski (M-K) model. Meanwhile, the high-temperature Nakazima test is conducted to obtain experimental thermal forming limit diagram (TFLD). The comparison of results by theoretical prediction and experimental test shows good consistency. Based on the proposed method, the effects of temperature, void damage, initial thickness imperfection, and anisotropy on the forming limit are analyzed. The formability improved with the increasing of deformation temperature but worse for other factors.

Predicting the Warm Forming Behavior of WE43 and AA5086 Alloys

In the present work, we have studied the formability behaviour of two types of magnesium alloys, WE43 hot rolled and WE43 cold rolled by carrying out uniaxial tensile test at elevated temperatures of 350°C to 500°C both experimentally and numerically at a constant strain rate of 10-3s-1. Finite element (FE) model is simulated in ABAQUS/CAE 6.7-6 using coupled temperature-displacement step at higher temperature considering material's property to be isotropic in nature. The effect of temperature on maximum flow stress and major strain at onset of necking is discussed. The true stress-strain behaviour and necking evolution through strain mapping are predicted from FE model and compared with the experimental results. The results show that with increase in temperature, the maximum flow stress decreases and necking delays with increase in limiting major strain for the Magnesium alloys. The work has been extended to predict the forming limit strains of Al 5086 alloy only on the negative minor strain region using M-K (Marciniak and Kuczynski) concept. An FE model based on M-K concept is simulated at 20°C, 150°C and 200°C using coupled temperature-displacement step considering anisotropic sheet material. A groove is created in the middle of the model with an optimizedf value of 0.99, after much iteration. The forming limit strains from such FE simulations are compared with the available experimental data. The results are encouraging providing scope for further improvements in modelling.

Theoretical FLD Prediction Based on M-K Model using Gurson's Plastic Potential Function for Steel Sheets

The aim of this study is to develop a new analytical method for predicting localized necking in the plastic deformation of sheet metals with internal cavitation. This method is based on the model of Marciniak and Kuczynski (M-K) as well as Gurson's plastic potential function. Stowell's model was used to illustrate void growth behavior during plastic deformation. In order to examine the effect of the voids on localized necking, the void volume fraction was considered in the imperfection factor and the plastic volume constancy principle. The nonlinear system of equations was solved with the modified Newton-Raphson method with globally convergence procedure, using MATLAB software. This new analytical method (M-K-Gurson) was used to predict the forming limit diagram (FLD) of different steel alloy sheets and the results were compared with those of other researchers. The results showed that the M-KGurson method predicted the FLD with good agreement.

Experimental and Numerical Analysis on the Formability of a Heat-Treated AA1100 Aluminum Alloy Sheet

The objective of this work is to experimentally and numerically determine the influence of plastic anisotropy on the forming limit curve (FLC) for a heat-treated (300 °C-1 h) AA1100 aluminum alloy sheet. The FLCs were obtained by the Nakajima test, where the anisotropy effect on the FLC was evaluated using hourglass-type samples taken at 0°, 45°, and 90° with respect to the sheet rolling direction. The effect of crystal orientations on the FLC is investigated using three micro-macro averaging schemes coupled to a Marciniak and Kuczynski (MK) analysis: the tangent viscoplastic self-consistent (VPSC), the tuned strength αVPSC, and the full-constraint Taylor model. The predicted limit strains in the left-hand side of the FLC agree well with experimental measurements along the three testing directions, while differences are found under biaxial stretching modes. Particularly, MK-VPSC predicts an unexpected limit strain profile in the right-hand side of the FLC for samples tested along the transverse direction. Only MK-αVPSC, with a tuning factor of 0.2, predicts satisfactorily the set of FLC measurements. Finally, the correlation of the predicted limit strains with the predicted yield surface by each model was also discussed.

The flow behavior and forming limit modeling of an Al-Cu-Mg aluminum alloy sheet considering the effects of pre-strain and solution heat treatment

Al-Cu-Mg aluminum alloy sheet is widely used for aircraft skin. Multi-stage forming combined with Solution Heat Treatment (SHT) is necessary for some geometrically complex productions in aircraft skin components. However, the effects of pre-strain and natural ageing time on the formability of Al-Cu-Mg aluminum alloy sheet are not fully understood. In this study, the deformation responses of an Al-Cu-Mg aluminum alloy sheet, over a range of pre-strain levels (0% to 6%) and natural ageing time levels (0 h to 2 h) were investigated. A modified Hollomon constitutive model coupling with pre-strain and natural ageing time was proposed to characterize the deformation responses of the Al-Cu-Mg aluminum alloy sheet. The hemispherical dome tests were performed at natural ageing time levels ranging from 0.25 h to 1.5 h. Marciniak and Kuczynski (M-K) theory was used to obtain the forming limit prediction model which takes pre-strain and natural ageing time into account. The theoretical Forming Limit Curves (FLCs) were compared with the experimental results.

Numerical study of the effect of martensite plasticity on the forming limits of a dual-phase steel sheet

The formability prediction of dual-phase steel sheets is highly important in the present automotive industry. In this study, the forming-limit curve (FLC) of a DP-780 steel sheet is predicted based on the well-known Marciniak and Kuczynski (MK) theory using a Visco-Plastic Self-Consistent (VPSC) crystal-plasticity scheme. To calibrate the polycrystal model, the stress–strain curves of the ferritic and martensitic phases are inferred by accounting for three martensitic plastic behaviors. Thus, the effect of martensitic plasticity on the FLC simulation can be analyzed. In addition, two different hardening laws – namely saturation and Voce models – are considered in order to study the effects of the extrapolated hardening behavior on the shape of the predicted FLCs. The best agreement with experimentation is found when the FLCs are calculated using the saturation hardening law and when the martensite deformation is either not allowed or retarded to occur after the point of necking. An analysis of the ferritic/martensitic slip system activity inside and outside the MK instability band suggests that, within the MK-VPSC framework, localization occurs much faster in the ferritic than in the martensitic phase. In addition, it is found that, unlike uniaxial tension, after plane-strain deformation and equi-biaxial stretching there is a strong correlation between the orientation of the ferritic grain and the strain that it accommodates. The predictive capability of the model is also confirmed by comparing the measured and simulated crystallographic textures close to necking.

Study of curvature and pre-damage effects on the edge stretchability of advanced high strength steel based on a new simulation model

Edge cracking is one of the common failures during the stamping of Advanced High Strength Steel(AHSS). The limit strains of the edge are hard to predict by traditional methods because edge formability is influenced by many factors such as edge quality, process parameters, material properties and so on. Hole expansion tests are conducted with DP780 under three hole diameters and two punching clearances to investigate the effects of curvature and pre-damage caused by punching on edge stretchability. The corresponding simulations are carried out meanwhile with a new model based on Marciniak and Kuczynski (M-K) model and Hill’s localized necking theory to predict the edge limit strain. This new model is proposed for the hole edge with pre-damage and improves the predictability of edge cracking in hole-expansion simulation. The problem that the traditional ideal model fails to predict the hole expansion ratio (HER) of pre-damage edge is also successfully solved by the new model with the use of the Damage Zone and Effective Failure Strain Ratio (EFSR). The comparison between the experiments and simulations reveals that this new model can be used in edge forming failure prediction of AHSS.

Experimental and theoretical investigation of strain path change effect on forming limit diagram of AA5083

Loading path changes affect forming processes markedly. This matter is getting more important especially in metal forming process in which engineers face unexpected results after drawing processes. As a result, understanding of formability changes through strain-path changes is highly demanding. This paper investigates the effect of strain path change on the forming limit diagram (FLD) of 5083 aluminium alloy sheet experimentally and theoretically. The experimental method is a new approach for determining the FLD under nonproportional loadings. The aim is to understand the formability of this alloy under various strain paths. For this purpose, out-of-plane formability test method with a hemispherical punch was used. Considering trim operation during forming process will simulate the changes in loading. Furthermore, forming limit diagram was determined theoretically by using the well-known Marciniak and Kuczynski (M-K) method. The results showed that uniaxial prestraining increases and shifts the FLD to the left-hand side of the diagram for both parallel and perpendicular to the rolling direction. Biaxial prestraining shifts the FLD to the right-hand side of the diagram for both directions and also decreases the forming limit curve (FLC) for the specimens parallel to the rolling direction. Theoretical results showed good agreements with experimental observations.

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.

The effect of normal stress on the formability of sheet metals under non-proportional loading

Theoretical and experimental studies have shown that when a compressive stress is applied normal to the sheet surface its formability can be significantly improved. In forming processes such as hydroforming, the normal stress imposed at the surface of the sheet or tube can be very significant in specific locations and it is therefore necessary to account for this stress state when assessing the forming severity of industrial parts. It is also well known that the forming limit curve (FLC) can vary significantly in strain space as a result of non-linear strain paths. In this work, a numerical code based on the Marciniak and Kuczynski (MK) analysis (1967) was developed to predict the FLC of sheet metals by simultaneously accounting for the effects of strain path non-linearity and the normal stress. The FLC predicted with this code can be used to assess the forming severity of parts that are deformed under complex loading conditions in industrial metal forming processes. The predictive FLC model was validated using experimental data in which linear and bi-linear strain paths were applied to different steel and aluminum sheets or tubes under plane stress and three-dimensional stress conditions. This research work has two specific purposes: first, to investigate the influence of the magnitude of the prestrain in bi-linear loading on the sensitivity of the FLC to the normal stress, and second, to study the effect of the normal stress on the path dependence of the FLC. This work showed that, with increasing magnitudes of prestrain the influence of the normal stress on the formability of sheet metal will decrease, regardless of the loading path. Moreover, it was observed that even a significant normal stress will practically not affect the path dependence of the FLC. • Forming Limit Curves (FLC) were predicted in 3D stress states and non-linear loading paths. • The effect of prestrain magnitude on the sensitivity of the FLC to the through-thickness compressive stress was investigated. • The effect of normal stress on the path-dependence of FLC was studied. • It was shown that higher prestrain magnitudes will reduces the sensitivity of FLC to the normal stresses. • In most cases, the normal pressure on the sheet metal surface will not affect the path-dependence of FLC.

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).

Prediction of formability of tailor welded blanks by modification of MK model

A new model is proposed to predict the Forming Limit Diagrams (FLDs) of Tailor Welded Blanks (TWBs). This model is derived by modification of the Marciniak and Kuczynski (MK) analysis and imposing the weld constraints to deformation state of each blank at each strain increment. Based on this model, it is possible to predict the strains of both weaker and stronger blanks once necking occurs. By comparison to the experimental results of two laser welded TWBs, it is demonstrated that this model provides a close prediction of FLD of TWBs, when necking occurs far from the weld line at a region out of HAZ (Heat Affected Zone) and close to it so that it is affected by the weld line constraints.

Experimental and numerical study of the role of crystallographic texture on the formability of an electro-galvanized steel sheet

In this work, the influence of plastic anisotropy on forming-limit strains for a drawing-quality steel sheet was investigated. For this purpose, hourglass-type samples, taken at 0°, 45° and 90° with respect to the sheet rolling direction were tested with a typical punch and die fixturing. Numerical simulations were carried out in order to validate two viscoplastic (VP) polycrystalline models, self-consistent (SC) and full-constraint Taylor-type (FC), in conjunction with the Marciniak and Kuczynski (MK) localization approach. The observed shift to the right in the minimum of the forming limit diagram (FLD), inherent to Nakazima test, was taken into account in the simulations. Keeping the set of adjustable parameters to a minimum in the calibration of the viscoplastic polycrystal model, only the material's initial texture and a power law fit to the tensile data needed to be measured. Without other adjustments to either model, MK-VPSC gives realistic predictions over the entire FLD, while the MK-FC predictions only follow the measured limit curve on the tensile side of plane strain. In the positive biaxial quadrant of the FLD, MK-FC predicts unrealistic high limit values. It was found that, despite these extremely high limit values, the similarity in the measured limit strains for the three sample orientations is captured by both models. However, a consistency in the MK-VPSC predictions indicates that this model seems to be a more suitable tool for describing the role of crystallographic texture on the sheet metal forming processes.

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.

Theoretical Determination of Forming Limit Diagram for Al 2024 T3 sheet when changing strain paths

The effect of changing strain paths on forming limit curve of aluminum alloy Al 2024T3 sheet has been studied by determining forming limit diagrams after prestraining theoretically . The method of calculating the shape of forming limit diagrams used Hosford yield criterion (high exponent) with the Marciniak and Kuczynski (M-K) analysis to include the effects of changing the strain path to several levels in (uniaxial, biaxial, and plane strain tension).In this work it was found that theoretical forming limit curve, calculated using the above analysis and yield function is closed to the experimental curve of complex strain path. Keywords : sheet metal forming, forming limit diagram(FLD), complex strain path