Experiments with two different materials were carried out. Medium strength steel grade TH330 is double reduced and tin coated sheet, specifically intended for round or rectangular cans. It is the most common material used in can industry [14]. The alternative is aluminium alloy AA5352, used for packaging production processes, but also in automotive and electrotechnical industry. This aluminium alloy has good formability and high hardening rate. Due to the presence of magnesium, it is susceptible to oxidation and has to be heat treated [15]. In order to determine input data required for FEM simulation, basic mechanical and technological tests were carried out. In the present study, the material properties were characterized by both tensile and hydraulic bulge tests. The tensile test was carried out according to standard ISO 6892-1:2016. From the test, basic mechanical properties as yield point, ultimate tensile stress, the strain-hardening exponent, uniform elongation, Lankford’s coefficients and others material parameters were obtained. The mechanical properties obtained from the tensile test are shown in .

Table 1 Basic mechanical properties of studied materials

Thin sheet metal usually splits at a strain of approximately 10% for aluminium and 20% for steel in a uniaxial tensile test. In order to obtain the material data at higher strains, hydraulic bulge test is commonly used to extend the tensile test data. The use of this test contributes to reducing discrepancies on data extrapolation of hardening curves too. Scheme of the hydraulic bulge test is shown in Figure 1. Following equations were used to calculate the equivalent stress:

Figure 1 Scheme of the hydraulic bulge test

$$\begin{array}{r}{\sigma}_{b}=\frac{pR}{2t}\end{array}$$(1)$$\begin{array}{r}t={t}_{0}\ast \mathrm{e}\mathrm{x}\mathrm{p}\left({-\epsilon}_{t}\right)\end{array}$$(2)where *σ*_{b} is effective stress [MPa], *p* is hydraulic pressure [MPa], *R* is the radius of curvature [mm], *ε*_{t} is current thickness strain, *t* and *t*_{0} are actual and initial thicknesses [mm].

Using values from hydraulic bulge test the strain-hardening curve can be established. The parameters for determination of hardening curve and results of hydraulic bulge test are shown in .

Table 2 Material properties obtained from hydraulic bulge test and Krupkowski hardening curve

In sheet metal forming, the FLD is commonly used to predict material failure in forming operations. The failure of sheet material can be induced by strain paths ranging from uniaxial tension to plane strain and biaxial tensile loading (different ratios between major and minor strains). Forming limit curve (FLC) represents the major and the minor strains in the plane of a sheet and corresponds to the maximum admissible local strains achievable just before the occurrence of visible defects in the sheet metal like fracture or necking.

FLC’s were determined according to the ISO 120042:2008 for both sheet materials. The universal sheet metal testing machine Erichsen 145-60 was used for performing the tests. Experimental samples with different radii were cut and Nakajima test with punch diameter of 100 mmwas applied. The machine can provide controllable slide velocity, and punch speed was set to 1.5 mm/s. The test was measured by 3D optical system Aramis M5. The experimental device is shown in Figure 2.

Figure 2 Device for estimating of FLC by Nakajima test

The experiment consisted of three operations. Cup was formed in a two-stage process of drawing and reverse redrawing using a combination tool (Figure 3), which occur sequentially during a single stroke of the drawing operation punch. The formed cup was subsequently subjected to die expansion operation. The diameter of the blank was Ø162.96 for both materials. Die extension operation was realized in the separate tool (Figure 4). Inner die diameter in deep drawing operation was Ø116.84 mm. Cup without flange with an average height of 27.5 mm was drawn. After cup was fully drawn followed redrawing operation. As it is shown in Figure 3, in the combination tool the punch from the first operation was used as die in the second operation. Inner die diameter was Ø87.85 mm and cup was drawn without failure with approximately 52.5 mm of height.

Figure 3 Combination tool for deep drawing and redrawing operations

Figure 4 Experimental tool for expansion operation

After drawing operations, the die expansion operation was performed. From practical knowledge we know, that force required to expand cup up to failure will cause the material flow to the bottom of the cup. This cause wrinkling in the cup bottom and reduce cup height. To avoid this, the bottom of the cup was clamped by inner clamp plate attached by a bolt to cup support tool. The preformed cup was placed between inner clamp plate and cup support tool as shown in Figure 4. In the centre of the bottom of cup is placed hole with Ø9 mm diameter. Clamping plate was screwed to cup support tool (Figure 4). Bleed at punch prevents accumulation of air. The same clamp arrangement is used for both materials - steel and aluminium alloy. Some process parameters defined in FEM code are shown in .

Table 3 Parameters defined in Pam-Stamp 2G software

FEM software with the explicit time-integration scheme was used. Parameters defined in numerical simulation are shown in .

Table 4 Parameters defined in FEM software

The tools geometry together with material input data were imported into FEM software. Krupkowski hardening curve was used for both materials because it shows the best correlation with experimental curves. For each material, different yield function was used. Hill48 yield function best describes the behaviour of the conventional steel materials during plastic deformation. This material law is most frequently used for steel materials and therefore been used in this case for steel tinplate TH330. However, this yield function is not suitable for aluminium alloys [16]. Therefore, more sophisticated yield function e.g. Barlat2000 is needed [16]. This law requires determining both, the exponent *m* and eight parameters used in the model. Exponent *m* is defined on the basis crystalline structure of the material. For metals with so-called BCC unit cell like steel is *m* = 6, while for FCC metals like aluminium alloys is exponent *m* = 8. These eight coefficients are required for this function and they have been gained numerically, based on the mechanical tests referred to [17]. For anisotropic yield models, the biaxial *r* value (*rb*) is required and obtained from disk compression tests.

Table 5 Summary of experimental data used in calibration of the Yld2000-2d model

Krupkowski hardening curve is defined as [17]:

$$\begin{array}{r}\sigma =K\ast {\left({\epsilon}_{p}+{\epsilon}_{o}\right)}^{n}\end{array}$$(3)where *εp* is plastic strain [-], *ε*_{o} is offset strain [-], *n* is strain hardening exponent [-] a *K* is material constant [MPa].

For plane stress conditions Hill48 yield function may be written as [16]:

$$\begin{array}{r}{\sigma}_{K}^{2}={\sigma}_{1}^{2}-\left(\frac{2\ast {r}_{0}}{1+{r}_{0}}\right)\ast {\sigma}_{1}\ast {\sigma}_{2}+\frac{{r}_{0}}{{r}_{90}}\ast \left(\frac{1+{r}_{90}}{1+{r}_{0}}\right)\ast {\sigma}_{2}^{2}\end{array}$$(4)where *σ*_{K} is yield stress [MPa], *σ*_{1} is major principal stress [MPa], *σ*_{2} is minor principal stress [MPa] and *r*_{0}*, r*_{45}*, r*_{90} are Lankford’s coefficients [-].

Barlat2000 (Yield2000 2^{nd}) yield function can be written as [18]:

$$\begin{array}{r}{\varphi}^{\prime}={\left|{S}_{1}^{\prime}-{S}_{2}^{\prime}\right|}^{M}{;}^{\u2033}={\left|{2S}_{2}^{\u2033}+{S}_{1}^{\u2033}\right|}^{M}+{\left|{2S}_{1}^{\u2033}+{S}_{2}^{\u2033}\right|}^{M}\end{array}$$(5)$$\begin{array}{r}\varphi ={\left|{S}_{1}^{\prime}-{S}_{2}^{\prime}\right|}^{M}+{\left|{2S}_{2}^{\u2033}+{S}_{1}^{\u2033}\right|}^{M}+{\left|{2S}_{1}^{\u2033}+{S}_{2}^{\u2033}\right|}^{M}={2\sigma}_{y}^{M}\end{array}$$(6)The ${S}_{i}^{{}^{\prime}}$and ${S}_{i}^{{}^{\u2033}}$are the principal values of stress tensors *S′* and *S′′*, which can be expressed by:

$${\stackrel{~}{S}}_{1,2}=\frac{1}{2}\left({\stackrel{~}{s}}_{xx}+{\stackrel{~}{s}}_{yy}\pm \sqrt{{\left({\stackrel{~}{s}}_{xx}-{\stackrel{~}{s}}_{yy}\right)}^{2}+4{\stackrel{~}{s}}_{xy}^{2}}\right)$$(7)The ${S}_{ij}^{{}^{\prime}}$and ${S}_{ij}^{{}^{\u2033}}$are the linear functions of the stress deviators.

$$\begin{array}{r}\left(\begin{array}{c}{s}_{xx}^{{}^{\prime}}\\ {s}_{yy}^{{}^{\prime}}\\ {s}_{xy}^{{}^{\prime}}\end{array}\right)=\left[\begin{array}{ccc}{L}_{11}^{{}^{\prime}}& {L}_{12}^{{}^{\prime}}& 0\\ {L}_{21}^{{}^{\prime}}& {L}_{22}^{{}^{\prime}}& 0\\ 0& 0& {L}_{66}^{{}^{\prime}}\end{array}\right]\left(\begin{array}{c}{s}_{xx}\\ {s}_{yy}\\ {s}_{xy}\end{array}\right)\end{array}$$(8)$$\begin{array}{r}\left(\begin{array}{c}{s}_{xx}^{{}^{\u2033}}\\ {s}_{yy}^{{}^{\u2033}}\\ {s}_{xy}^{{}^{\u2033}}\end{array}\right)=\left[\begin{array}{ccc}{L}_{11}^{{}^{\u2033}}& {L}_{12}^{{}^{\u2033}}& 0\\ {L}_{21}^{{}^{\u2033}}& {L}_{22}^{{}^{\u2033}}& 0\\ 0& 0& {L}_{66}^{{}^{\u2033}}\end{array}\right]\left(\begin{array}{c}{s}_{xx}\\ {s}_{yy}\\ {s}_{xy}\end{array}\right)\end{array}$$(9)$$\begin{array}{r}\left(\begin{array}{c}{L}_{11}^{{}^{\prime}}\\ {L}_{12}^{{}^{\prime}}\\ {L}_{21}^{{}^{\prime}}\\ {L}_{22}^{{}^{\prime}}\\ {L}_{66}^{{}^{\prime}}\end{array}\right)=\left[\begin{array}{ccc}2/3& 0& 0\\ -1/3& 0& 0\\ 0& -1/3& 0\\ 0& 2/3& 0\\ 0& 0& 1\end{array}\right]\left(\begin{array}{c}{\alpha}_{1}\\ {\alpha}_{2}\\ {\alpha}_{7}\end{array}\right)\end{array}$$(10)$$\begin{array}{r}\left(\begin{array}{c}{L}_{11}^{{}^{\u2033}}\\ {L}_{12}^{{}^{\u2033}}\\ {L}_{21}^{{}^{\u2033}}\\ {L}_{22}^{{}^{\u2033}}\\ {L}_{66}^{{}^{\u2033}}\end{array}\right)=\left[\begin{array}{ccccc}-2& 2& 8& -2& 0\\ 1& -4& -4& 4& 0\\ 4& -4& -4& 4& 0\\ -2& 8& 2& -2& 0\\ 0& 0& 0& 0& 9\end{array}\right]\left(\begin{array}{c}{\alpha}_{3}\\ {\alpha}_{4}\\ {\alpha}_{5}\\ {\alpha}_{6}\\ {\alpha}_{8}\end{array}\right)\end{array}$$(11)The coefficients ${L}_{ij}^{{}^{\prime}}$ and ${L}_{ij}^{{}^{\u2033}}$ are described by relationships of a set of eight coefficients *αk* as shown in (10) and (11). The coefficients *α*_{1} − *α*_{8} are reduced to one coefficient if an isotropic case is considered.

Even if the aim of this section is to failure prediction during expansion operation, other important values were analysed too. Overall, there are four parameters evaluated in this work:

(i)

Ear profile after deep drawing,

(ii)

Wall thickness after deep drawing in 25 mm and 45 mm distance from cup bottom,

(iii)

Punch force,

(iv)

The position of crack.

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