---

One of the most important parameters in prediction of sheet metal forming process is the influence of yield criteria on prediction of forming limit strains. In this paper, the effects of normal anisotropy coefficient on the Hill’s (1948) quadratic, Hosford’s (1979) higher-order, Hill’s (1979) non-quadratic anisotropic yield criteria, with attention to plane strain location is studied. Also the effects of these yield criteria and normal anisotropy coefficient, strain rate sensitivity factor, strain hardening exponent and yield criterion exponent on the forming limit diagram based on the M-K model have been discussed. The different effects of normal anisotropy coefficient on the limit strains of three above criteria have a good agreement with the effect of normal anisotropy coefficient on the yield different surfaces. The comparison has been made between yield criterion exponent and normal anisotropy coefficient effect on the Hill’s non-quadratic yield criterion. The results show that the proper selection of yield criterion have a significant influence on the forming limit diagram.

---

In hydroforming process, applying hydraulic pressure to the inner surface of tube along with axial loads to two ends of tube simultaneously cause the tube to be formed to the die shape. Application of finite element simulation is common practice to predict the geometrical dimensions of the produced part and analysis of probable defects. For finite element simulation, precise mechanical properties of tube material are required. Obtaining these properties from a test similar to the tube hydroforming process is desirable. In this study hydraulic bulge test using T-shape die has been introduced to obtain the stress-strain curve of the tube material. Using hydroforming set-up, several experiments were carried out on C12200 copper samples. Geometrical parameters required to be used in analytical solutions have been identified and the stress-strain curve has been plotted. The results of the proposed experiment have been compared to the results of the tensile test. In addition, the effects of anisotropy on the obtained stress-strain curve of both tests have been determined. The stress-strain curve obtained has been used to plot the forming limit diagram. The bulge test mechanical properties and the forming limit diagram have been applied to simulate the tube bursting and prediction of the final part geometrical dimensions in T-shape tube hydroforming and these results have been compared to the part being experimentally produced by hydroforming. The results show that when stress-strain curve obtained by the proposed experiment is used, there is a good agreement between the simulated hydroformed part and the experimental part.

---

In This paper dynamic forming limit diagram has been investigated as fracture criteria for St13 steel. In fact, effect of various strain rates has been studied. This fracture criterion is based on the Marciniak-Kuczynski (M-K) theory and Solutions of equations have been obtained by applying the Newton -Raphson method. After solution three forming limit diagrams has been created: independent strain rate, dependent strain rate and dynamic forming limit diagram. Dynamic damage criteria investigates forming limit diagram in every strain rate. It is observed that the forming limit is increased by increasing the strain rate, Also for considering the anisotropic and the elastic-plastic behavior of material, the Hill 1948 yield criterion and the Swift hardening rule are used respectively. Also this paper is concerned with the uniaxial tensile properties and formability of sheet metal in relation to the strain rate effects. In order to verification of the results several experiments have been done with a Drop Hammer which is a high speed impact machine. For comparison between quasi static and dynamic damage criterions, all of the stages of experiment were simulated in finite element software Abaqus and results are compared together.

---

Mathematical modeling is an important step in the design and optimization of process parameters for metal forming. Researchers have been concerned the metal forming limit diagram as an efficient tool to optimize the production of components using forming methods. Due to the low ductility of titanium alloys and wide applications of these alloys in advanced industries such as aerospace, researchers have focused on studying the forming behavior of these alloys. Due to the high cost of experimental methods, especially at high temperatures, numerical methods, has attracted the attention of many researchers. The accuracy of the numerical methods is affected by model of elastic-plastic material behavior. Unusual mechanical behavior of Ti-64 titanium alloys such as high in-plane anisotropy/asymmetry of yield stress and hardening response has been observed. In this paper, the Marciniak model with Cazacu and Hill yield criterions has been used for forming limit prediction. It is observed that the prediction of forming limit using the Cazacu criterion is closer to the experimental results. This is due to the better prediction of the behavior of the titanium alloy, specially Lankford and stress anisotropy coefficients by Cazacu criterion. Cazacu and Hill criterions prediction of Lankford coefficients and yield stresses have been compared.

---

Nowadays the forming limit curves is very useful in forming of metal sheets and the effect of yield criteria is one of the most important parameters in prediction of the limit strain especially in anisotropic aluminum sheets. In this paper, first the effects of advanced BBC2008, Soare2008, Plunkett2008 and Yld2011 yield criteria on limit strain calculation and then on forming limit stress diagram will be investigated. Plastic instability model is studied based on Marciniak-Kuczynski model and the non-linear equations are solved by using Newton-Rophson method. These functions are used to evaluate the limit forming predictability of AA2090-T3 aluminum sheet based on the Swift hardening law and is compared with the forming limit curves predicted by Hill’s 1948 classic yield criterion. It was observed that the classic yield functions is not appropriate for anisotropic aluminum sheets forming estimation. Numerical results obtained from the forming limit diagram for AA5754 with Plunkett2008 yield function and Swift hardening law, although the experimental results confirm at close range to plane strain case, but CPB06ex2 yield criterion to predict the behavior of anisotropic aluminum sheets. The limit strain prediction for AA3104-H19 by using Yld2011 yield criterion and Voce hardening law show better conformity with experimental results.

---

In this article, an experimental and theoretical study on the prediction of forming limit diagram (FLD) for aluminum alloy (2024) is developed. To identify and calibrate coefficients of YLD2004-18P, YLD2011-18P, YLD2011-27P and BBC2008-16P advanced yield criteria, tensile tests were performed in seven directions with respect to the rolling direction. Directional yield stresses and anisotropy coefficients were determined. Then, an appropriate error-function was defined and optimized by using Levenberg-Marquardt algorithm. By considering 14, 12, 10 and 8 anisotropy parameters, the effect of number of parameters on the accuracy of yield functions were investigated. The best condition with minimum error can be achieved, when 14 anisotropy parameters are used. To compare the calculated yield stresses and r-values with experimental data, a method presented by Leacock was used. The results have shown that all four criteria give predictions of yield stresses which are nearly close to experimental values. The prediction of yield stresses and anisotropy coefficients by means of YLD2011-27P and YLD2004-18P criteria have more correlation and good agreement with the experimental data, respectively. For obtaining experimental FLD Nakazima test was performed. In order to simulate the necking phenomenon and calculate the limit strains, the modified Marciniak-Kuczynski (MK) model, Swift hardening law and some new yield criteria including YLD2004-18P, YLD2011-18P, YLD2011-27P and BBC2008-16P were utilized. At the right hand side of FLD, YLD2004-18P and YLD2011-27P criteria and also at the left hand side YLD2011-27P criterion have shown better conformity with experimental results.

---

Use of Forming limit diagrams (FLD) in process design of metal forming is a conventional method. Therefore many experimental and theoretical efforts have been carried out in order to investigate the FLDs. Many ways to obtain this FLDs and their effective parameters have been studied. But the stress state at these studies is planar which lead to an untrue model for several metal forming process such as incremental sheet forming. With this technique, the forming limit curve (FLC) appears in a different pattern, revealing an enhanced formability, compared to conventional forming techniques. Therefore, in this study, the effect of through thickness shear stress has been examined on the prediction of the forming limit diagrams (FLDs). Determination of the FLD is based on the Marciniak and Kuczynski (M–K) model with some modifications on the stress states for consideration of the through thickness shear stress effects. Also, the effective range of this stress has been investigated. The results showed that if the through thickness shear stress has a 10 per cent of yield stress value, this stress component has no effect on the FLD.

---

Incremental sheet forming has already provided distinct advantages, such as inexpensive tools and the simplicity of the process, over conventional sheet forming processes. However, the method still has some limitations. Among these limitations, severe thinning has significant effects on the performance of the final product. Also, some parts with high wall angles cannot be formed by single stage incremental forming. To overcome these restrictions, multistage incremental forming can be implemented to achieve the desired wall angle, better thickness distribution, and the lower thinning. In this study, a two-stage incremental forming of an aluminum truncated pyramid with a wall angle of 70° was studied experimentally and numerically in order to improve the achievable minimum thickness. By introducing two-stage forming strategies and achieving their defining parameters using finite element simulation, the sheet thinning was compared to the one in the single-stage forming. Experiments were used to validate the finite element analysis. The results revealed that using the two-stage forming strategy, the minimum thickness can be improved twice than the one in the single-stage forming. A good agreement was observed between the thickness distribution obtained by experiments and predicted by the finite element modeling. Finally, the effect of forming strategies on the strain paths was investigated through the finite element simulation and the experimental fracture forming limit diagram.

---

In hydromechanical deep drawing process, the traditional matrix is replaced by pressurized fluid, and the final shape is determined based on the shape of a rigid punch. It is required to change the fluid pressure within the allowed working zone during the process to prevent the workpiece from rupturing and wrinkling,. Working zone curve represents the range of maximum available drawing ratios without rupture under the highest chamber pressure. In this paper, hydromechanical deep drawing of square cups made of aluminum-steel double layer sheets are studied by experiments and finite element simulations. In order to detect the rupture onset in simulations, experimental forming limit diagrams were obtained using for aluminum/steel double layer sheet. Experimental data were used to validate the finite element model. The effects of process parameters such as thickness of the various layers, prebulge pressure, chamber pressure and the friction coefficient were investigated on the working zone and the process window. The numerical results show that an optimum amount for the drawing ratio exists for each prebulge pressure. Also, with increasing the chamber pressure, shrinkage is reduced on the flange area. With increasing the friction between the sheet and matrix or the sheet and blank-holder, working zone becomes smaller; while with increasing the friction between the sheet and the punch it becomes larger. Experiments were performed for different drawing ratios to evaluate the numerical results, in which a good agreement was observed.

---

Forming limit diagram (FLD) is one of the useful tools in the assessment of the sheet formability for designing industrial products. Experimental methods have been developed to determine FLDs. Costly and time-consuming experiments have led to several studies on the use of analytical methods and finite element softwares for predicting FLDs. In the present study, the necking and fracture forming limit curves of AA2024 aluminum alloy sheet were experimentally and numerically obtained through the hemispherical stretching test. Different geometries of the initial blank were considered to create different strain paths. The commercial finite element code Abaqus/Explicit was utilized to simulate experimental tests. Using theoretical equations and experimental results, fracture properties of the aluminum sheet in terms of the equivalent plastic strain at fracture, the stress triaxiality and the Lode angle parameter were captured and implemented in the Abaqus software. In order to capture necking forming limit strains, a numerical criterion based on the major strain variation in the necking zone has been considered. The comparison of the results shows that the numerical model can predict the forming and fracture limit strains with the maximum error of about 6%.

---

In the new sheet metal forming process as incremental sheet forming and spinning forming, this is not perfectly true in Marciniak-Kuczyinski model to assume that sheet deformation occurs in the plane-stress state indispose there are normal compressive stress and through-thickness stress. In this type of forming processes, the obtained limit strains refer to improving the sheet forming. However, in researches the effects of through-thickness shear stresses, also known as out-of-plane shear, has been studied less. The generalized forming limit diagram is a great curve that includes all six components of the stress tensor. In this paper, the effect of normal comprehensive and through-thickness shear stresses on the limit strain AA6011 aluminum sheet using a modified M-K and the anisotropic Yield function, Hill 48 and by using numerical solutions of nonlinear equations, Newton-Raphson method. The first the forming limit diagram was drawn with the assumption that the through-thickness shear stresses and then the effects of normal comprehensive stress and through-thickness shear stress on the limit strains were proved and the generalized forming limit curves were obtained. The results show that forming limits can be increased significantly by both normal compressive stress and through-thickness shear stresses. Also, the effects of normal stress on increasing the formability of sheet compared with the effects of through-thickness shear stress is greater.

---

Forming limit diagrams (FLDs) are useful tools for prediction of the instability of sheet in metal forming. The goal of this study is to evaluate the formability of 260 brass alloy sheets under various strain rates (particularly at high strain rate). Three types of experimental procedure were developed: Nakazima test (for determination of the FLD at quasi-static condition), hydrodynamic forming (for determination of the FLD at intermediate strain rate) and Electrohydraulic forming (for determination of high strain rate FLD). Electrohydraulic forming (EHF) is a high velocity sheet metal forming process in which two or more electrodes are positioned in a water filled chamber and a high-voltage discharge between the electrodes generates a high pressure to form the sheet. Arbitrary Lagrangian Eulerian (ALE) formulations coupled with fluid–structure interaction (FSI) algorithms (that are available in the advanced finite element code LS-DYNA) were used to the numerical simulation of process and design of sheet metal specimen geometries. It was found that the forming limits of brass 260 in EHF increased more than 11% relative to the quasi-static. In addition, the formability of this material under the hydrodynamic loading is 4% higher than quasi-static values.

---

The sheet metals formability can be restricted by localized necking and internal cavitation. On the one hand, nucleation and growth of cavities during plastic deformation can increase the inhomogeneity of sheet metal and accelerate the localized necking. On the other hand, localized necking at the intervals between the cavities can lead to accelerate the joining and coalescence of the internal cavities. In this paper an analytical model based on Marciniak-Kuczynski (M-K) model and Gurson plastic potential function in order to exert the internal voids effect on localization necking has been developed. 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 using MATLAB software. This new analytical method (M-K-Gurson) was used to predict the forming limit diagram (FLD) of IF steel alloy sheets and the results were compared with those of other researchers. The results showed that the M-K-Gurson method predicted the FLD with better agreement comparing with experimental results. Thereafter, the effects of strain hardening exponent, anisotropy coefficients, geometrical imperfection factor, the void volume fraction and the void growth rate parameter on the FLD were investigated.

---

In this study, special attention has been paid to modeling of the interface between the sheet metals in prediction of forming limit diagram (FLD) of two-layer sheets. In the present work, a two-layer sheet consists of 1.35 mm steel sheet and 0.45 mm copper sheet has been used. This two-layer sheet has been made by explosive welding method. To determine the FLD, numerical method has been used by applying ABAQUS finite element software. For this purpose, the so called Nakazima method has been simulated. The criteria used for determining the failure in steel and copper layers was GTN model. Also, in order to determine the failure in interface between the layers, the traction-separation law was used. For modeling the interface, cohesive elements were used. In order to verify the results, Nakazima tests were performed. The simulations and experimental works were done for both side directions of the sheets. The results indicate that the FLDs obtained by the numerical modeling are in good agreement with the experimental results.

---

Incremental sheet forming is one of the novel processes which is used for rapid prototyping and manufacturing of parts with complex geometries. Forming limit of sheet metal in this process is high compared to other conventional forming processes. In this paper, warm single-point incremental forming process through uniform heating to sheet along with tool heating is studied experimentally and numerically. Formability of sheet is investigated in various process condition based on the straight groove test in experimental approach and numerical simulation using finite element method. Tool heating along with uniform heating to sheet makes tool and sheet isothermal, reduces the heat loss in deformation zone and improves the deformation process. So, attainment of high forming limit is made possible. Comparison of forming limit diagrams obtained from experimental and numerical approaches shows a good agreement between the results. Effects of temperature and feed rate on the forming limit of aluminum 1050 sheet are investigated. Results show that increasing the temperature improves the formability of sheet significantly; but, the temperature is more influential on forming limit in low feed rates. Increasing the feed rate reduces the forming limit slightly; this effect is more evident in higher temperatures.
 

---

---

---

Forming Limit Diagrams (FLDs) are very useful measures for safe forming of sheet metals without failure due to necking or fracture under different loading conditions. This paper uses ductile fracture criteria to predict the formability of low carbon steel sheets to evaluate their accuracy in predicting the FLDs. In addition, the fracture forming limit curves (FFLD) and necking forming limit curves (NFLD) for St12 low-carbon steel have been extracted experimentally and numerically. In the experimental procedure, the Nakazima stretching test was used. In the numerical procedure, by defining six phenomenological ductile fracture criteria in ABAQUS / Explicit finite element software, the failure is predicted and compared with the experimental results. These criteria were calibrated using 6 tests namely as In-plane shear, uniaxial tensile test, circle hole test, notched tension test, plane stress test, and Nakazima stretching test. The results showed that the criteria, which include both the stress triaxiality (η) and Lode parameter (L), provide a more accurate prediction of failure. Also to predict necking during numerical simulation of Nakazima test and also to extract the NFLD, three criteria of the second derivative of major strain, the second derivative of thickness strain and the second derivative of equivalent plastic strain have been used.

---

Forming limit diagrams (FLDs) are very important in predicting the behavior of the sheet. Therefore, predicting and drawing these diagrams by theoretical and experimental methods has been one of the main objectives of this paper. In this paper, the formability behavior of 5083 aluminum sheet was investigated by considering the strain hardening behavior. Tensile tests has performed in seven directions 0°, 15°, 30°, 45°, 60°, 75° and 90° from the rolling direction due to identify and calibrate coefficients of BBC2008 advanced yield criteria. The yield stresses was defined in the plane strain mode, also the anisotropy coefficients and the appropriate error function were extracted; Then the relationships of the plane strain yield stress were added to the error function. The error function was optimized using Genetic Algorithm and limit strains were calculated using yield coefficients. The results showed that if the strain hardening exponent increases by 0.1, the limit strains increase by 30 to 40%. Also the results showed that the initial imperfection factor ( ) has a great effect on determining the FLD and with a very small change, it has a great effect on the FLD; So that by increasing this factor to about 0.016, the values of the limit strains are almost doubled. Using the results of this paper and having sheet properties such as yield strengths and anisotropy coefficients and proper selection of yield criteria, the FLD of different sheets to be theoretically determined with acceptable accuracy.