Insight into the Influence of Punch Velocity and Thickness on Forming Limit Diagrams of AA 6061 Sheets—Numerical and Experimental Analyses

Abstract

In this article, the forming limit diagram (FLD) for aluminum 6061 sheets of thicknesses of 1 mm and 3 mm was determined numerically and experimentally, considering different punch velocities. The punch velocity was adjusted in the range of 20 mm/min to 200 mm/min during the Nakazima test. A finite element (FE) simulation was carried out by applying the Johnson–Cook material model into the ABAQUS^TM FE software. In addition, a comparison between the simulation and the experimental results was made. It was observed that by increasing the punch velocity, the FLD also increased for both thicknesses, but the degree of the improvement was different. Based on these results, we found a good agreement between numerical and experimental analyses (about 10% error). Moreover, by increasing the punch velocity from 20 mm/min to 100 mm/min in 1 mm-thick specimens, the corresponding FLD increased by 3.8%, while for 3 mm-thick specimens, this increase was 5.2%; by increasing the punch velocity from 20 mm/min to 200 mm/min in the 3 mm-thick sheets, the corresponding FLD increased by 9.3%.

1. Introduction

Today, the manufacturing of many industrial products has become relevant to sheet metal forming processes. To reduce waste materials, energy, and time costs, much attention has been focused on the forming limits and the number of the required steps in the forming process. Studying the effects of new sheet metal forming methods such as high-rate processes on forming limit diagrams (FLDs) is necessary. Among the defects developed in sheet forming processes, tearing, wrinkling, and springback are the most problematic, so they are investigated [1,2,3]. In the sheet metal forming processes, which are often tensile, when the tensile stresses applied to the sheet surpass the sheet’s resistance, plastic instability happens first, followed by tearing [4]. Wrinkling can often be solved by increasing the clamping force and using drawbeads. To improve the sheet forming process, it is necessary to predict sheet necking and tearing. More recently, a novel methodology of residual stress estimate based on an FE model has been developed by Aminzadeh and his research team [5]. Moreover, some research focused on determining the optimum input parameters during the formation of Tailor welded blanks (TWBs) and proposed a new FEM model for the sheet metal forming process [6,7,8]. Sheet metal forming is usually studied using FLDs [9]. These diagrams show the combination of strains that a sheet can resist and provide a critical range for localized necking. The FLDs separate safe areas and tearing areas from each other. Since the 1960s, many research studies have been conducted on the influence of different parameters on sheet metal forming. Nevertheless, the effect of the forming speed on FLDs has been less considered in comparison to other parameters. This indicates that sheet metal forming in special conditions like in the presence of high temperatures and high rates of deformation is hugely influenced by the forming rate. Various researchers have investigated the effect of the strain rate on FLDs, especially for the AA 6061 alloy. Djavanroodi and Derogar experimentally and numerically examined the FLD of AA 6061 and found a good agreement between the experimental and the numerical data [10]. They also used an artificial neural network (ANN) to predict the FLD and adapt it to the experimental diagrams [11]. In addition to the FLD, the forming limit stress diagram (FLSD) for this alloy was also checked numerically [12]. As mentioned, strain rate and temperature are two very influential factors that were not considered in these studies. Chu et al. determined the forming limit diagrams of AA5086 under various strain rates (0.02, 0.2, and 2 s⁻¹) and temperatures (20, 150, and 200 °C) using the Marciniak test [13]. Their research showed that both strain rate and temperature negatively and positively affect the FLDs of AA5086, respectively. Based on experimental results, they showed that by increasing the temperature and decreasing the forming speed, the FLD improved. Naka et al. [14] investigated the influences of temperature and forming speed on the FLDs of AA5083-O using stretch-forming tests at various temperatures (20–300 °C) and speeds (0.2–200 mm/min). According to this research, forming limit diagrams increased considerably when reducing the speed for any loading path at elevated temperatures ranging from 150 to 300 °C. In contrast, at ambient temperature, the FLDs were not sensitive to the forming speed. The research of Wang et al. [15] also indicated that punch velocity and temperature had a substantial effect on the forming limits of AA2024 according to the tensile test and the cup punch test. Their results showed that temperature up to 450 °C gradually increased the elongation of tensile specimens, which then decreased sharply. Lee et al. [16] examined the influences of strain rate and temperature on the square cup drawing of an AZ31 alloy sheet using finite element simulation and experiments. Their tests were carried out by a hemispherical punch to study the forming limits for various loading paths. Their results showed differences in uniform and total elongation changes in the range from 200 to 400 °C.

The influences of strain rate sensitivity on necking were investigated by Hutchinson and Neale [17] under uniaxial tension. Naka and Yoshida [18] concluded through cylindrical deep drawing tests that the limiting drawing ratio of AA5083-O decreased with an increasing forming velocity and improved with the increase of the die temperature. Seth et al. [19] studied the forming limits at high velocity of cold-rolled sheet steel as developed in impact with a curved punch. They found that the formability of steels with low quasi-static ductility increased considerably. In contrast, the steel forming limit with already high quasi-static ductility did not increase at very high strain rates. Gerdooei et al. [20] studied the effect of the forming speed on the forming limit diagrams of AISI 1045 and AA 6061-T6 sheets experimentally. They showed that significant enhancements in the forming limits could be obtained at a high forming speed. Verleysen [21] studied the strain rate’s effect on the formability of four sheets of steel (DC04, S235, CMnAl TRIP, and AISI 409). The resulting FLDs indicated a non-negligible influence on the strain rate. The decreased formability at higher strain rates was reflected in an unfavorable downward shift of the FLDs of the S235 and DC04steel grades. On the other hand, for the AISI 904 steel, the improved strain rate did not substantially influence the FLD. For the TRIP steel, the forming limits increased if the strain rate increased.

In this article, the FLD was determined numerically and experimentally, considering different punch velocities for aluminum 6061-T6 sheets with thicknesses of 1 mm and 3 mm. The major novelty of this work is the study of the effect of velocity (strain rate) in the range of 20–200 mm/min on ductility at ambient temperature. As mentioned, little research has been carried out to investigate the effect of strain rate on ductility and FLDs at ambient temperature, and most research on high temperature resulted in high ductility due to the superplastic effect.

The Nakazima test was conducted. Finite element (FE) simulation was carried out by entering the Johnson–Cook model in ABAQUS^TM R21 FE software (Dassault Systemes Company, Waltham, MA, USA). A comparison between the simulation and the experiment was performed. It was observed that by increasing the strain rate, the FLD also increased for both thicknesses, but the degree of improvement was different. By increasing the punch velocity from 20 mm/min to 100 mm/min in 1 mm-thick specimens, the corresponding FLD increased by 3.8%, whereas for the 3 mm-thick specimens, this increase was 5.2%. In addition, by increasing the punch velocity from 20 mm/min to 200 mm/min in 1 mm- and 3 mm.thick sheets, the corresponding FLD increased by 17.8 and 9.3%, respectively.

2. Materials and Methods

In this laboratory research, FLDs for AA 6061 with thickness of 1 mm at two speeds of 20 and 100 mm/min and thickness of 3 mm at three speeds of 20, 100, and 200 mm/min were obtained by the Nakazima test with a hemispherical punch. The number of samples for the FLD test was 6; the samples had different widths (45–110 mm) and constant lengths of 110 mm [22]. Three repetitions were performed. The sample dimensions needed for the test are presented in Figure 1. For preparation and sample cutting, a wire cut machine (Mitsubishi Corporation, Tokyo, Japan) was used. For the experimental tests, the samples were stretched on a hydraulic press with a hemispherical punch of 100 mm diameter (MÜLLER, Germany) according to the procedure suggested by Nakazima. During the Nakazima test, stress concentration in the matrix hole was too high, so to prevent the tearing in this area, edge notches were applied [23]. The edge notches’ radiuses were constant in all samples, and corresponded to 25 mm. Circular mesh grading with a radius of 2.54 mm according to ISO12004 was performed on the sheet using an electrolytic marking machine (ÖSTLING Marking Systems GmbH, Solingen, Germany).

This maximum punch velocity was 200 mm/min, and it was adjusted at 20, 100, 200 mm/min for our tests. Tensile and pressure loadings awee both possible using this machine by controlling speed and force. It was also possible to control the process manually and digitally at any time using the machine monitoring system. The force diagram decreased sharply when necking happened, so that the testing stopped. The uniaxial tensile test was performed according to the standard of ASTM-B557M to obtain the general mechanical properties. The sheet’s mechanical and physical properties are shown in

Table 1. Mechanical properties of aluminum alloy 6061.

Quantity	Mechanical Properties
2.75 (gr/cm3)	Density
69 (GPa)	Modulus of Elasticity
0.33	Poisson Ratio
275 (MPa)	Yield Stress
400 (MPa)	Ultimate Strength

.

In order to read the FLDs data, the diameters of grading circles were measured in the tearing zone by a traveling microscope (National Analytical Corporation, Mumbai, India), as shown in Figure 2. Changes in the diameters ere calculated and used to determine the major and minor strains according to Equations (1) and (2) and then draw the FLDs. The symbols “a” and “b” are, respectively, the large and the small diameter of the ellipse converted from a circle with a diameter of “d”.

$$e_{max}=\frac{a-d}{d}\times 100$$

(1)

$$e_{min}=\frac{b-d}{d}\times 100$$

(2)

3. Finite Element Simulation

In this research, a 2D tensile process simulation was carried out to determine aluminum sheet FLD using ABAQUS^TM FE package (Dassault Systemes, Waltham, MA, USA). All configurations (die components, including punch, matrix, and sheet holder), mesh refinement, and boundary conditions were designed based on previous studies [24,25]. A quarter of the die and sheet was modeled in the simulation because of geometric symmetry and to reduce the analysis time. The mechanical properties of the sheet such as density, elastic properties, modulus of elasticity, and poison ratio, and its plastic properties (Johnson–Cook factors) were entered according to Table 1 and

Table 2. Johnson–Cook model constants for AA 6061-T6 [26].

Quantity	Johnson-Cook Model Constants
275 (MPa)	A
393.16 (MPa)	B
0.011	C
1.34	m
0.441	n
650 (°C)	Melting point

. Johnson and Cook presented their experimental model to explain metals’ behavior by considering the effects of temperature for high strain rates. In this model, stress is defined by Equation (3).

$$\mathsf{\sigma }=\left[A+B{\epsilon }^n\right][1+C\mathit{ln}(\frac{\dot{\epsilon }}{\dot{{\epsilon }_0}})][1-{\left(\frac{T-T_0}{T_m-T_0}\right)}^m]$$

(3)

The effective plastic strain rate $\dot{\epsilon }$ is normalized with a reference strain rate $\dot{{\epsilon }_0}$. Moreover, $T_m$ is the melting temperature of the material, and $T_0$ is room temperature. While the parameter m models the thermal softening effect, the parameter n considers the strain hardening effect, and C represents the sensitivity to the strain rate. The parameter A is the initial yield strength of the material at room temperature. This model has five constants, A, B, n, C, and m, determined experimentally. The most usual method for determining these constants is the Hopkinson test, which is performed in different ways, considering pressure, tensile, and torsion. Because of the difficulty of performing the uniaxial tensile test at very high strain rates and the limited availability of fixed parameters for the Johnson–Cook model for AA 6061, Johnson–Cook data were extracted from other references [26,27,28,29] Furthermore, the contact between the punch sheet, matrix sheet, and holder sheet was defined as Coulomb friction, and the friction factor was considered 0.16.

The simulation process was carried out in two steps. In the first step, the holder tangent to the sheet surface was first lowered by 2.5 mm holding the sheet on the matrix quite firmly so that the sheet remained entirely still during the process. In the second step, the punch moved downwards with the adjusted speed and formed the sheet. In both steps, applying force to the punch was avoided, and the process was defined entirely according to the practical procedure with movement control and constant speed. In Figure 3, the symmetry boundary condition in the assembled model is shown.

According to the forming process, the S4R shell element was used for meshing, and the R3D4 element was used for rigid components. The sheet elements were considered small enough to ensure the best accuracy during the simulation. For this purpose, we used the mesh sensitivity diagram for all models to choose the best element size. The maximum amounts of acceleration of the thickness strain, major strain, and plastic strain were equivalent. The acceleration of the major strain was first reported by Situ, based on the second derivative in the sheet [30].

4. Results

After performing the experimental tests and measuring the circles’ diameters around the tearing zone, major and minor strains were calculated. The FLDs for the sheets were drawn using measured strains, so that the vertical axis indicated the major strain, and the horizontal axis indicated the minor strain. Before reporting and using the results of finite element modeling, a simulation must be verified. The load–movement curve, the tearing location, and the FLD resulting from experimental work and simulation must be compared, and if they match, the simulation is approved. The criterion of the maximum amount of acceleration of the major strain was used to determine the limit strains. Therefore, according to Figure 4, by extracting the strain history for all elements, the element with the maximum amount of strain was considered the necking location. This was deteremind by trial and error and curve fitting using power equations in After performing the experimental tests and measuring the circles’ diameters around the tearing zone, major and minor strains were calculated. The FLDs for the sheets were drawn using measured strains, so that the vertical axis indicated the major strain, and the horizontal axis indicated the minor strain. Before reporting and using the results of finite element modeling, a simulation must be verified. The load–movement curve, the tearing location, and the FLD resulting from experimental work and simulation must be compared, and if they match, the simulation is approved. The criterion of the maximum amount of acceleration of the major strain was used to determine the limit strains. Therefore, according to Figure 4, by extracting the strain history for all elements, the element with the maximum amount of strain was considered the necking location. This was deteremind by trial and error and curve fitting using power equations in MATLAB^TM software (2017b, The Mathworks, Inc., Natick, Massachusetts, USA). After fitting the curve and obtaining the strain rate and strain acceleration curves, according to Figure 4, the moment of necking when the strain acceleration reached its maximum value was considered, and large and small strains were selected as limit strains. By using the same method for different strain paths while changing the width of the specimens, the limit strains can be determined for the entire range of the forming limit diagram [31]. Figure 5 also shows the strain paths to obtain the FLD that covered almost all loading modes (biaxial, uniaxial, and plane strain), with different widths for each condition.

Regarding FE verification, Figure 6 shows the experimental and numerical load–displacement curves for Al sheets with 1 mm thickness and at 20 mm/min punch velocity. As shown in Figure 6a, which reports the curves for experimental samples with different widths, the amount of force increases when increasing the width of the samples. It should also be noted that the test was stopped as soon as the force dropped. Figure 6b compares two similar samples (width and punch velocity) obtained by numerical and experimental testing and shows a good agreement between the two load–displacement curves. The reason for fluctuations and distortions in the numerical data is the more limited data extraction. To validate the numerical results, three parts of the force–displacement diagram, rupture, necking location, and FLD diagrams were used. Due to the high number of samples and the similar process, for the first two parts, only one sample is presented, as suggested by one of the referees. In the FLD section, which was the main focus of the article, all numerical and experimental data are compared, and at the end, the average data error of each diagram was determined.

In Figure 7, the comparison between the PEEQ (term for the ABAQUS parameter for the equivalent plastic strain) output of the software and the necking position of the experimental tests is presented. As shown, it was confirmed that the numerical model predicted the rupture site with acceptable accuracy.

The FLDs for the sheets mentioned were obtained at different speeds using the finite element simulation and the criterion of maximum major strain acceleration. In Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, the FLDs from the experiment and the simulation are presented for a sheet with a thickness of 1 mm at speeds of 20 and 200 mm/min and for a 3 mm-thick sheet at speeds of 20, 100, 200 mm/min. By comparing the experimental and simulation diagrams, it was observed that the simulation data were close to the experimental data, with acceptable accuracy; therefore, the data can be trusted. As shown in the diagrams, the modeling results predicted a lower diagram minimum point. The average errors between the experimental and the simulation diagrams for all the tests are shown in

Table 3. Average errors between the experimental and the simulation diagrams.

Thickness (mm)	Punch Velocity (mm/min)	Error (%)
1	20	6.2
	200	10.7
3	20	7.3
	100	10.2
	200	12.5

. Based on the results, there was good agreement between them.

By approving the data, the FLDs were drawn for sheets of 1 mm and 3 mm thickness at their respective testing speeds; the diagrams are shown in Figure 13 and Figure 14. The results obtained indicated that the points moved toward the diagram’s left-hand side when decreasing the samples’ width. For the square-shaped sample, strains appeared in the biaxial tensile limited area; this means that the grading circles around the tearing zone transformed to circles with larger diameters. By decreasing the sample’s width, the reticular circles on the samples around the tearing zone deformed to an ellipse. By decreasing the sample’s width, the ratio between the ellipse’s big diameter and small diameter became more significant.

By increasing the punch velocity for both sheets with thicknesses of 1 mm and 3 mm, the FLDs improved, indicating the direct effect of the strain rate parameter on sheet ductility, which the inertia moment phenomenon can explain. Due to the inertia moment phenomenon during the forming process at higher velocity, the dynamic stress component in the defect area grew. As a result, at any moment with less stress concentration in the defect area, the equilibrium at the interface of the two necking and healthy areas was satisfied. For this reason, inertia is considered a phenomenon that spreads the necking band and improves ductility. The increasing formability of thicker sheets can be explained by a more significant effect on the grids around the neck and by a larger range of strain changes due to the fixed body of the mold and the effect of sheet thickness on the sheet bending phenomenon. In addition, the alloying element increases the work hardening during the punching, therefore decreasing the plasticity and consequently reducing the forming limit. The increase in the FLDs for different thicknesses and speeds is presented in

Table 4. Average increasing rate of FLD.

Thickness (mm)	Change in Velocity (mm/min)	FLD Increasing Rate (%)
1	20 to 100	5.2
	20 to 200	17.8
3	20 to 100	3.8
	20 to 200	9.3

. In Figure 15, all of the obtained diagrams and their differences are shown. Briefly, this increase can have different reasons, such as the effect of inertia, by which inertia causes deformation in the workpiece and increases the deforming stability against necking growth [32]. According to Equation (4), during local necking creation by deformation concentration on a narrow spline, the speed profile changes quickly.

$${\mathrm{F}}_{\mathrm{in}}=\underset{0}{\overset{\mathrm{l}}{{\displaystyle \int }}}\mathsf{\rho }\mathrm{S}\left(\frac{∆\mathrm{v}}{∆\mathrm{t}}\right)\mathrm{dx}$$

(4)

5. Conclusions

The tearing location and the FLDs obtained from the simulation were compared with those from the experimental results to validate the experimental results. The tearing locations obtained from experimental samples and modeling were compared, and the tearing zone was found to be approximately the same. The obtained diagrams showed similar test speeds with acceptable errors. According to this, the experimental data can be trusted. In order to study the effect of punch velocity on FLDs for 1 mm- and 3 mm-thick AA 6061 sheets, FLDs were drawn at the respective testing speeds. The experiments showed that by increasing punch velocity, FLD also increased for both sheets, but the degree of improvement was different. In fact, by increasing the punch velocity from 20 mm/min to 100 mm/min in 1 mm-thick sheets, FLD increased by 3.8%. In contrast, for 3 mm-thick sheets, this increase was 5.2%; in addition when increasing the punch velocity from 20 mm/min to 200 mm/min in 3 mm sheets, FLD increased by 9.3%. By a uniform increase of the punch velocity, the diagrams for 1 mm- and 3 mm-thick sheets improved. Nevertheless, the amount of increase was different. Depending on the thickness and on the increase of the punch velocity for constant thicknesses, the diagrams are improved. By further increasing the punch velocity, the effect of the strain rate increases.