Investigation on punch force–displacement and thickness changes in the shallow drawing of AA2014 aluminium alloy sheets using finite element simulations

2.1 Characteristics of the material and process variables

Forged Aluminium alloy AA2014 blanks with diameters of 50, 60, and 70 mm and thickness of 1 mm were fabricated from a forged block using a wire-cutting EDM device. Tensile data for the AA2014 forged plate were obtained through experiments, and the material properties are detailed in Table 1 [6]. In Table 1, R0, R45, and R90 represent the ratios of the final width to the initial width of the tensile test specimen along the forging direction (L), 45° with the forging direction (L + 45°), and perpendicular to the forging direction (T) directions, respectively. The drawing process was executed utilizing a press (hydraulic) equipped with the following specifications: 40 ton capacity, displacement of 500 mm, and holding pressure of up to 50 bar. The testing speed was set at 1 mm per min under room temperature, with a holding pressure of 15 bar. Three sample test blanks were used in the drawing process.

2.2 Drawing process: Modelling and simulation

The diagram illustrating the model for drawing in PAM STAMP is depicted in Figure 1(a) in its unmeshed form along with dimensions and displayed with meshing in Figure 1(b). Simulation parameters are as follows: 30 mm punch diameter, 2 mm punch corner radius, 3 mm die corner radius, 31 mm die diameter, coefficient of friction 0.015, 0.05, 0.1, 0.15, and 0.2, mesh size, 1 mm, and blank diameters, 70, 60, and, 50 mm.

Figure 1

(a) PAM STAMP-3D modelling setup of the drawing operation – Geometrical model (without meshing). (b) PAM STAMP- 3D modelling setup of the drawing operation – Finite Element (FE) model (with meshing).

The analysis parameters were determined according to the available experimental range and literature database, as described earlier. An FE code was employed to simulate the drawing process, utilizing shell elements of quadrilateral type and five integration points of Belytschko–Tsay formulation. This element has more degrees of freedom, which provides a more accurate solution. Hollomon’s constitutive power law, i.e., σ = Kε ⁿ was applied to determine the strain hardening exponent, n, and strength coefficient, K. The material model for the blank was constructed using the properties specified in Table 1. The element selection was tailored to the system configurations and problem, with simulation parameters detailed in Table 1. A 15 bar constant blank pressure was applied while the punch descended at a speed of 1 mm/min. Fifteen simulations were conducted, and the coefficient of friction varied across five different levels, maintaining the mesh size constant at one level. The resulting distribution of thickness, displacement, and punch force resulted from the simulations. These findings were integrated into the numerical code using a remeshing adaptive technique, facilitating design engineers in predicting the feasibility of forming the real component based on the design proposed. The constraints considered for the simulation are hardening law, constant coefficient of friction, progression steps up to 100 stages, and punch velocity.

3.1 Experimental evaluation of the punch force

During the drawing operation, the blank holding force (BHF) has a direct impact on the drawing force (DF). As the cup takes shape during deep drawing, the DF increases in tandem with the BHF. Initially, the DF increases from minimum to maximum during the ongoing drawing operation before gradually decreasing (refer to Figure 2(a) and (b)). The maximum thinning of the cup reached 20%, resulting in a 0.8 mm wall thickness reduction, which remained within acceptable limits. Therefore, a press machine with a capacity maximum of 60 tons can be utilized for the necessary operation to prevent errors. It was observed that as the friction coefficient between the sheet blank and forming tools increases to its maximum value, the required punch load increases significantly before decreasing [7]. Table 2 summarizes the punch force–displacement values for three blank sizes obtained from both experimental and simulation results. The variations in punch force vs displacement, as depicted in Figure 2(a) and (b), are evident from both simulation and experimental data.

Figure 2

(a) Punch force vs displacement for different blank diameters by simulation. (b) Punch force vs displacement for different blank diameters by experiments.

When the drawing speed increases, the maximum DF initially decreases before gradually increasing. Excessively high drawing speeds can lead to ductile fractures in the sheet. Optimal drawing performance is achieved at speeds between 12 and 24 mm per min. Moreover, if the radius of the die fillet is very large, the specimen’s edge may become unstable before reaching the mould, forming wrinkles due to compressive tangential stresses [8]. With an increase in deformation magnitude, the punch force progressively increases till the component wall ruptures, and then it starts to decrease. This trend continues until the wall ruptures, after which the force diminishes. The relationship between the punch force and displacement in experimental data often exhibits irregularities and lacks consistent smoothness in numerous instances. Experimental data typically reflect the fluctuation of the punch force with displacement from computer-controlled test systems, which frequently demonstrate non-smooth behaviour. These values are actually obtained experimentally and connected one to one. The abrupt increase in all the punch forces in the case of all the blank diameters could be due to non-homogeneous and localized deformation, which encompasses both plane stress and plane strain conditions. Such inhomogeneous deformation occurs not only in the present Aluminium alloy AA2014 but also in most other work hardenable metallic materials. The key findings indicate that smaller punch radii tend to concentrate localized stresses, which results in deeper forming depths but also a higher risk of material failure. In contrast, larger punch radii distribute stresses more uniformly, leading to shallower forming depths and enhancing the structural integrity of the formed parts [9].

The highest punch force observed for the blank diameter of 70 mm during experimentation measures 77.47 and 78.09 kN for simulations. Similarly, with a blank diameter of 50 mm, the maximum punch force recorded during the experiments was 49.890 kN, while for simulations, it was 52.12 kN. Table 3 presents a comparison of the highest punch force vs displacement between the simulation and experimental results, along with the error percentage. Larger blank sizes result in higher maximum punch forces. During experiments, the 50 mm diameter blank exhibited a 49.89 kN fracture load at a displacement of 13.260 mm. It is noteworthy that the drawn part depth is less than one-half the diameter of the blank, indicating that the material is mainly suited for the process of shallow drawing.

3.2 Numerical evaluation of the punch force

Simulation computations typically involve CAD models, which are then translated into the computational mesh of the CAE model. This process is also applicable to numerical simulations of the deformation of blank. Computational mesh is generated by a graphical pre-processor and defines contact, essential boundary and loading conditions to solve the mathematical problem that represents the actual deformation processes. During this process, the punch moves vertically while the blank holder and die remain stationary. Rigid bodies are used to model the punch, blank holder, and die, while the blank is represented as a shell-type element, with the mesh subdivided into reduced integration elements.

Figure 2(a) illustrates the variation in punch force versus displacement of simulations. To evaluate simulation outcomes, comparisons should be drawn with experimental tests that have been conducted, concentrating on load curves, stroke to failure, and maximum force. The characteristic zones and maximum forces identified during simulation closely correspond to experimental observations, with the error percentage of the highest punch force falling within acceptable levels. The Lou-Huh ductile fracture criterion (DFC) parameters were calibrated using the least-squares method for surface fitting. The resulting DFC curve is then extended to predict fractures for deep drawing. DFC reliability is confirmed by evaluating the location and fracture stroke through testing. The obtained results demonstrated that the DFC curve accurately predicts both the onset and location of fractures, accurately predicting fractures in advance under different BHF conditions [10,11].

FE results usually anticipate a greater forming load compared to actual operations, as illustrated in Figure 2(a). This overestimation, approximately 10%, aligns with other comparisons between the FE and experimental outcomes. Therefore, when selecting the forming machine capacity based on FE results, it is advisable to incorporate a safety factor to ensure successful operations.

3.3 Determination of variation in thickness

The drawn cup from a 50 mm blank, as shown in Figure 3, was cut into two equal parts, and the change in thickness between the cup bottom and the wall’s highest point was gauged. The contact method was used for wall thickness measurements. A standard dial gauge with coned flat tips was employed to record the variations in the wall thickness [12,13]. Figure 4 illustrates the change in the thickness of the sheet relative to the distance calculated from the centre. In deep drawing processes, wall thinning occurs near the cup bottom because the material is stretched the most to fit the die shape. Near the top, the wall may experience slight thickening due to material accumulation and inward flow, as this area is more constrained and undergoes less deformation than the bottom. An objective function was developed to optimize both the consistency of the final product and the maximum decrease in sheet thickness. The optimum parameters for the process variables were determined using the Bees Algorithm [12].

Figure 3

Drawn cup of 50 mm blank.

Figure 4

The variation in sheet thickness vs distance measured from the centre (for a 50 mm diameter blank).

During the deep drawing process, the thickness of the sheet varies across regions due to stretching. The thickness variation in the cup wall starts at the bottom corner, with the thinning parameter predominantly affected by the depth of the drawing. As the drawing depth increases, the reduction in thickness of the drawn component becomes more pronounced, leading to a decrease in the wall thickness. The study aimed to investigate the thinning of the sheet blank in both the wall and flange areas of the cup drawn, considering the specified drawing ratio across different frictional conditions. It was observed that increasing friction led to a greater reduction in the thickness.

Furthermore, the investigation revealed that the bottom plane of the cup remained unstrained, resulting in no thickness variation in the cup itself. It was observed that the die radius has the greatest influence on the process of deep drawing, with the BHF and friction coefficient following closely in importance.

In the 2G PAMSTAMP module, the Keeler theory was used to monitor the sheet necking/failure. The thickness distribution is simulated during Aluminium alloy AA2014 cup formation by employing a 50 mm diameter blank, with a friction coefficient set at 0.015 and a material mesh size of 1 mm. Figures 5(a) and 6(a) display the necking or failure assessment using the Keeler criterion before and after failure, respectively. Figures 5(b) and 6(b) display the forming limit diagrams before and after failure, respectively. It is evident that prior to necking or failure of the sheet, there is minimal change in thickness values, as shown in Figure 5(c). However, in Figure 6(b) (after failure), a noticeable reduction in thickness is observed in the deformed zone. Several factors, including strain hardening exponent (n), strength coefficient (K), yield stress (σ _YS), BHF, punch velocity, coefficient of friction, and other properties, play a significant role in calculating the detailed output process. When the parameters are configured according to setup values during the experiment, there is a noticeable difference between Figures 5(c) and 6(c) in terms of thickness variations.

Figure 5

(a) Necking or failure assessment using the Keeler criterion prior to failure. (b) Forming limit diagrams: Prior to failure. (c) Thickness values-prior to failure.

Figure 6

(a) Necking or failure assessment utilizing the Keeler criterion: Post-failure. (b) Forming limit diagrams: Post-failure. (c) Thickness values post-failure.