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This paper presents results of experimental studies of forming limit curves (FLC) for sheet forming under complex strain paths. The Nakazima-type formability tests have been performed for the as-received steel blank and for the blank pre-strained by13%. Prestraining leads to abrupt change of strain path in the blank deformation influencing the form...
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... formability tests consisting in stretching of the blank over a hemispherical punch have been carried out for the steel grade DC04 1 mm thick. Figure 1 shows the geometry and set-up of the tools. The tests have been per- formed for the as-received sheet supplied by the manufacturer and for the blank pre-stretched in uniaxial tension conditions by13% along the rolling direction. ...
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... this way point C p in polar coordinates is obtained. Applying this procedure to a number of points defining the FLC in the Cartesian coordinates ( Fig. 9(a)) we can obtain the FLC in the polar coordinates ( Fig. 9(b)). Transformation of the FLC for the complex strain paths is explained in Fig. 10. The transformation procedure is modified by taking the angle defining the strain path at the last stage before fracture as the polar coordinate θ and accumulative evaluation of the equivalent plastic ...
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... FLCs for different pre-strain coincide with good accuracy in the polar coordinates. This allowed them to take a certain average curve in the polar coordinates as the strain-path independent FLC. Now, the transformation procedure from the Cartesian to the polar coordinates will be applied to the set of our own experimental FLCs presented in Fig. 11. The set includes the as-received FLC and two FLCs for different pre-strains. The FLCs considered are also presented in Fig. 8. One of the FLCs in Fig. 11 have been obtained by swapping coordinates in one of the FLCs presented in Fig. 8. Applying the transformation described earlier a set of FLCs in the polar coordinates shown in Fig. ...
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... as the strain-path independent FLC. Now, the transformation procedure from the Cartesian to the polar coordinates will be applied to the set of our own experimental FLCs presented in Fig. 11. The set includes the as-received FLC and two FLCs for different pre-strains. The FLCs considered are also presented in Fig. 8. One of the FLCs in Fig. 11 have been obtained by swapping coordinates in one of the FLCs presented in Fig. 8. Applying the transformation described earlier a set of FLCs in the polar coordinates shown in Fig. 12 is obtained. Keeping the angle constant and av- eraging the radius in the polar diagrams an average FLC is obtained which can be used as the strain-path ...
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... in Fig. 11. The set includes the as-received FLC and two FLCs for different pre-strains. The FLCs considered are also presented in Fig. 8. One of the FLCs in Fig. 11 have been obtained by swapping coordinates in one of the FLCs presented in Fig. 8. Applying the transformation described earlier a set of FLCs in the polar coordinates shown in Fig. 12 is obtained. Keeping the angle constant and av- eraging the radius in the polar diagrams an average FLC is obtained which can be used as the strain-path in dependent FLC. Analysing the set of transformed FLCs in the polar co- ordinates ( Fig. 12) we observed a larger discrepancy than we expected. However, such discrepancy can be ...
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... Fig. 8. Applying the transformation described earlier a set of FLCs in the polar coordinates shown in Fig. 12 is obtained. Keeping the angle constant and av- eraging the radius in the polar diagrams an average FLC is obtained which can be used as the strain-path in dependent FLC. Analysing the set of transformed FLCs in the polar co- ordinates ( Fig. 12) we observed a larger discrepancy than we expected. However, such discrepancy can be understood since similar disagreement can be noticed for some cases in [21]. This motivated us to check an alternative transformation to the polar coordinates. Instead of the equivalent plastic strain we propose to take the absolute value of the ...
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... such discrepancy can be understood since similar disagreement can be noticed for some cases in [21]. This motivated us to check an alternative transformation to the polar coordinates. Instead of the equivalent plastic strain we propose to take the absolute value of the thickness strain |ε 3 | obtained from the condition of constant volume as Fig. 12. Set of FLCs after transformation to the polar coordinates according to [21,20] ...
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... idea is consistent with the shift of the FLC along the line of constant thickness proposed by Hosford and Caddell in [10], which was discussed above. Applying this transformation to the set of the FLCs shown in Fig. 11 we obtain the polar FLCs plotted in Fig. 13. Much better coincidence of the transformed FLCs can be seen in comparison with the polar FLCs shown in Fig. 12. This indicates that the new concept of the polar FLCs is worth further studies and ...
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... idea is consistent with the shift of the FLC along the line of constant thickness proposed by Hosford and Caddell in [10], which was discussed above. Applying this transformation to the set of the FLCs shown in Fig. 11 we obtain the polar FLCs plotted in Fig. 13. Much better coincidence of the transformed FLCs can be seen in comparison with the polar FLCs shown in Fig. 12. This indicates that the new concept of the polar FLCs is worth further studies and ...
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... with the shift of the FLC along the line of constant thickness proposed by Hosford and Caddell in [10], which was discussed above. Applying this transformation to the set of the FLCs shown in Fig. 11 we obtain the polar FLCs plotted in Fig. 13. Much better coincidence of the transformed FLCs can be seen in comparison with the polar FLCs shown in Fig. 12. This indicates that the new concept of the polar FLCs is worth further studies and ...
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... However, in complex forming processes where the loading path characteristic is nonlinear, the FLD cannot accurately predict the formability of sheet metals [3]. The polar effective plastic strain-forming limit diagram (PEPS-FLD) is one of the forming limit diagrams that has been applied to estimate the formability of sheet metal forming in place of the conventional FLC [4]. Because the PEPS-FLD has a number of advantages, like not being susceptible to history of deformation and being independent of hardening law selection [5], having no significant/noticeable path dependency, independence with the stress-strain relation, and a similar shape to the conventional FLC [6], the transformation of the FLD to the PEPS diagram can be done without much difficulty using the assumption of proportional loading for each source point selected on the FLC [7]. ...
In this work, experimental forming limit curve (FLC) and polar effective plastic strain -forming limit diagram (PEPS-FLD) are developed in order to predict the formability of dual phase (DP) steel grade 440. The Forming Limit Diagrams (FLD) of the examined steel are first obtained using the Nakajima stretch-forming test with a decreased hemispherical punch diameter of 50 mm in combination with Auto-Grid Vialux strain measurement system. To achieve a PEPS-FLD, an experimental FLC data must be completely projected onto the polar space using the results of FLC, Hill's 48 anisotropic yield criterion, and PEPS algorithm through plasticity calculation. Finally, a two-step forming test is conducted both experiments and simulation to demonstrate the practical applicability of the PEPS-FLD. It really would appear that the PEPS-FLD could predict the formability in non-linear strain path and can be utilized to enhance formability by creating non-linear strain path of DP440 more accurately than FLD.
... The uniformity of strain in the centre region is not related to fracture of the specimen but it defines the quality of biaxial testing. (3) The third equation [23] allows to compare all samples taking into account both previously introduced factors: FR and COV. This equation is called the Cost Function (CF) and it uses wf coefficient which indicates, which of the first two coefficients is more important. ...
The paper contributes to the global research of the best shape of cruciform samples for biaxial tensile tests. Based on literature research, one of the proposed samples’ shape comparison methodology was employed to achieve the ultimate results. A variety of specimens with different shapes were compared and modelled with the use of Finite Element Method (FEM). Afterwards, five the most promising sample shapes were examined on a tests stand to validate the simulation results. Based on the obtained research results a new parameter defining the ability of the cruciform sample to reach the possible achievable strain in the centre area, was introduced. The obtained data shows, that sample with the highest Relative Cost Function parameter (RCF) presented the highest strain in the gauge region and the sample with the lowest RCF reached the lowest strain in the center. The rest of the samples remained within the general trend showing a compatibility between the RCF parameter and the obtained strain.
... Most of industrial sheet metal forming processes involves the existence of complex non-linear strain paths during the manufacturing operations, making the conventional FLC unsuitable tool for assessing formability (Graf and Hosford, 1993;Barata-Rocha et al., 1985;Rojek et al., 2013). To overcome this limitation, the proposal of pathindependent forming limit curves has been intensively evaluated in the research community. ...
Conventional sheet metal forming processes usually involve the simultaneous action of stretching and bending, inducing strain/stress gradients across the sheet thickness and complex strain paths, which
decisively influence the onset and development of the failure mechanism by necking. Therefore, an accurate assessment of formability requires the use of strain path independent tools that take into account the bending effects on the sheet failure. The use of the theory of critical distance (TCD), widely used in fatigue and fracture mechanics to account the effect of the stress/strain gradients in
metal failure, has been satisfactorily analysed by the authors to predict failure by necking under stretch-bending conditions. The current work develops, using and experimental and numerical
analysis, two failure models for strain/stress gradient situations which combine both the use of path-independent forming limit diagrams, particularly the forming limit stress diagram (FLSC) and the
equivalent plastic strain based forming limit curve (epFLC), along with the concepts of critical distance to account for the effect of bending in the onset of necking. Both developed models are used to predict numerically the failure in a series of stretch-bending tests over H240LA steel sheets. Their predictive capabilities are discussed and compared in the light of the experimental results.
... Prestraining the material in plane strain condition improves the formability if the preloading and the loading direction of experimental tests for the determination of FLD are made in the same direction as seen on an aluminum alloy [26]. Nevertheless, a decrease in the formability is observed if prestraining and FLD angle tests are different as published on an aluminum alloy by Graf and Hosford [9] and on a steel by Rojek et al. [21]. Therefore, this zone is predicted fractured by the stress criterion and not by the conventional forming limit curve [15]. ...
Experimental and numerical cup drawing process has been investigated on 0.65 mm zinc sheets. The cup exhibits anisotropic earrings due to the material microstructure. The material formability is studied through elliptical bulge tests in the rolling, diagonal and transverse direction. High anisotropy of the formability is observed. The numerical simulation of cup drawing is then made and demonstrates the correct fitting with experimental results. A stress formability criterion developed by Jansen et al. [14] is then implemented into a finite element method software and applied to predict the material rupture observed for some process conditions. The risk zone of the cup is subjected to some strain path changes according to the simulation whereas the strain value does not explain the rupture according to the experimental formability measured by the bulge tests. It has been shown that the rupture is due to some critical stresses, which are reached in the risk zone of the cup. The use of the stress criterion and its non-dependence on the strain path change allows the fracture prediction. Finally, the numerical fracture propagation by the “kill element method”, as briefly discussed by Bouchard et al. [4], is used and shows a good similarity with the experience.
... Zhao et al. (1996) provide analytical methods to calculate the strain-based FLC for linear, bilinear and trilinear strain paths. Rojek et al. (2013) showed that strain-based FLCs will change if the strain path is complex. Specifically, these studies show that the applicability of normal strain-based FLCs is restricted to proportional loading. ...
Forming limit curves (FLCs) are used to determine the amount of deformation that can be applied to a sheet metal before the onset of a localized neck. Most FLCs are shown in strain space, and stress-based FLCs have advantages because they are often strain-path independent. The current study develops a method to calculate a stress-based forming limit curve. The necessary data for this calculation can be obtained from a uniaxial tensile test. The calculations depend on the Z parameter, which can be considered to be the point of instability during a tensile test. With the use of the Keeler-Brazier equation, the effective stress in plane strain at the forming limit is shown to be a function of the Z parameter and thickness. Data from 4 experimental studies are shown to be consistent with this function. With the generally accepted observation that the left side of the strain-based FLC is a line with slope of -1 and an appropriate constitutive model for the stress-strain behavior of the material, the stress-based FLC corresponding to the left side of the strain-based FLC can be calculated. Comparison of the calculated stress-based FLC for three steels with the stress-based FLC determined directly from the strain-based FLC shows good agreement. The calculated stress-based FLC is 15-20 MPa below the data generated directly from the strain-based FLC.
The forming limit diagram is one of the efficient tools to investigate the plastic deformation ability of sheet metals. The forming limits for rate-dependent ideal-orientation crystalline materials are anticipated for the entire variety of linear and nonlinear strain paths and the stress-based forming limit diagrams are also studied. For the first time, crystal plasticity forming limit strains and stresses on the combination of different strain paths such as two-step linear paths, nonlinear strain paths, and the prestrain for an ideal orientation is examined. While the forming limit is very sensitive to the strain path, the strain path has a minor effect on stress forming limit diagrams. A rate-dependent crystal plasticity method is used in combination with the Marciniak–Kuczyniski (M–K) method to anticipate forming limits in face-centered cubic crystalline material with Goss, Cube, and Brass orientations. The results show that strain paths greatly affect the forming limits. It is further shown that in the case of an ideal orientation, forming limit stress curves are not entirely path independent.
This paper reviews the most recent models for description of the anisotropic plastic behavior and formability of sheet metals. After a brief review of classic isotropic yield functions, recent advanced anisotropic criteria for polycrystalline materials of various crystal structures and their applications to cup drawing are presented. Next, the discussion focuses on novel formulations of anisotropic hardening. A brief review of the experimental methods used for characterizing and modeling the anisotropic plastic behavior of metallic sheets and tubes under biaxial loading is presented. The experimental methods and theoretical models used for measuring and predicting the limit strains, development of new tests for determining the Forming Limit Curves (FLC), as well as on studying the influence of various material or process parameters on the limit strains are presented.
This paper presents an investigation on the detection of strain localization in numerical simulation of sheet metal forming. Two methods to determine the onset of localized necking have been compared. The first criterion, newly implemented in this work, is based on the analysis of the through-thickness thinning (through-thickness strain) and its first time derivative in the most strained zone. The limit strain in the second method, studied in the authors’ earlier works, is determined by the maximum of the strain acceleration. The limit strains have been determined for different specimens undergoing deformation at different strain paths covering the whole range of the strain paths typical for sheet forming processes. This has allowed to construct numerical forming limit curves (FLCs). The numerical FLCs have been compared with the experimental one. Mesh sensitivity analysis for these criteria has been performed for the selected specimens. It has been shown that the numerical FLC obtained with the new criterion predicts formability limits close to the experimental results so this method can be used as a potential alternative tool to determine formability in standard finite element simulations of sheet forming processes.
Materials used for the production of aluminum beverage cans must meet strict strength and plastic requirements. Meeting these requirements by manufacturers ensures minimization of defects in optimized production line, trouble-free operation and good efficiency of production. Depending on some factors, such as: chemical composition of the 3104 aluminum alloy, sheet rolling method, lubricant layer and cooling method, the final product may be characterized by different mechanical parameters. In this study properties of six types of sheets from four different suppliers have been measured. Thickness distributions have been analyzed both on the width and length of the supplied coils. Sets of samples have been prepared for the research, cut out from the coil in three directions 0°, 45° and 90° of the rolling direction. On the basis of the uniaxial tension tests, yield strength, tensile strength, elastic modulus and elongation have been determined. The results have been summarized taking into account their variable values depending on sampling directions. Forming limit curves have been defined for the examined sheets. In order to identify the effect of selected data on wall thickness of both final and semi-products, thickness measurements have been conducted for the drawn cups and cans on different forming stages.
