Forming limit diagram of Al-Cu two-layer metallic sheets considering the Marciniak and Kuczynski theory

Abstract

The aim of this research was to introduce a computational approach for prediction of the forming limit diagram of Al-Cu two-layer metallic sheets. The computational approach was based on the modified Marciniak and Kuczynski theory. In this study, the forming limit diagrams of aluminum–copper two-layer metallic sheets were obtained through the modified Marciniak and Kuczynski theory and experimental investigations. In the present modified Marciniak and Kuczynski theory, there existed four nonlinear equations which were solved simultaneously. The Quasi-Newton Method was applied for a solution to the system of equations. To verify the theoretical predictions, the experimental works were accomplished on the Al-Cu two-layer metallic sheets and a good agreement between the proposed method and experimental works was observed.

Introduction

Two-layer metallic sheets are progressively utilized in industrial and domestic parts. Their advantages include an increased corrosion and wear resistance, sound and vibration damping, uniform temperature distribution due to the different heat expansion coefficients of the layers, decreased wrinkling and spring back, and finally reduced cost and weight. ¹ Forming limit curves (FLCs) are applied to determine the sheet metals formability. The laboratory test results have demonstrated that the FLCs are affected by several factors containing strain rate,^2–6 strain hardening index and anisotropy coefficients,^7–9 heat treatment,^10,11 grain size and microstructure,^12,13 strain path changes^14–16 and sheet thickness. ¹⁷

Keeler and Backofen ¹⁸ presented the FLCs concept for the first time. After that, many researchers such as Kami et al., ¹⁹ Hashemi et al., ²⁰ Safikhani et al., ²¹ Situ et al., ²² Mohebbi and Akbarzadeh, ²³ Barlat and Lian, ²⁴ Banabic et al. ²⁵ and Zhang et al. ²⁶ tried to develop some numerical and analytical models to determine the sheet metal formability. However, a few attentions have been paid to study the behavior of multi-layer sheets. In 1979, Semiatin and Piehler ²⁷ performed the first research on multi-layer materials. Yoshida and Hino ²⁸ investigated the laminated sheet formability both numerically and experimentally. They found that the FLC of the laminates lied between the FLC of its components. Weiss et al. ²⁹ tried to determine the laminated sheet formability in different temperatures to specify its effect on the formability of this kind of sheets. Aghchai et al. ³⁰ determined the formability of two-layer Al/St sheet both theoretically and experimentally. Their results showed that the two-layer sheet formability was between its components formability. Bagherzadeh et al. ³¹ developed analytical models to evaluate instability condition and stress analysis of cylindrical St-Al two-layer cups in the hydro-mechanical deep drawing. Liu et al. ³² developed an Al/polyethylene/Al sandwich sheet and investigated its formability. Their study demonstrated that by enhancing the polyethylene core thickness, the formability of this kind of sandwich sheet could be better. Parsa et al. ³³ studied the FLC of Al/polypropylene/Al sandwich sheets both numerically and experimentally. They used finite element (FE) analysis to determine the FLCs of a sandwich sheet in their numerical investigations. They studied the influences of several parameters such as shape constraints and thickness of polymer layer on the FLCs in their research. Karajibani et al. ³⁴ investigated the formability of two-layer (aluminum-st12 steel) sheets in the deep drawing process through numerical simulations and experiments. Their research illustrated that the formability of two-layer sheet was dependent on the process parameters.

Although the fracture occurs by previous necking in several sheet metal forming operations, there are operations or simply operating conditions where fracture can develop without previous necking. ³⁵ The scope of this research is limited to situations where necking occurs before failure by the fracture.

According to the literature review, it could be seen that many efforts were spent to determine the forming limit diagram (FLD) in one-layer sheets and a few efforts were spent to study the two-layer metallic sheet formability. In this research, the FLDs of aluminum–copper two-layer metallic sheets were obtained through the modified Marciniak and Kuczynski (M-K) theory. Barlat and Lian’s ²⁴ 1989 anisotropic yield function was applied. In the present modified M-K theory, there existed four nonlinear equations which were solved simultaneously. The Quasi-Newton Method was applied for a solution to the system of equations. To verify the analytical results, a stretch-forming process with a hemispherical punch was used to obtain the experimental FLD. The predicted forming limits correlate well with experimental data.

Experimental procedure

Materials

The two-layer sheets (aluminum–copper) were used in this study. The explosive welding was utilized for an excellent joining of aluminum and copper sheets in order to fabricate the Al-Cu two-layer sheets. ³⁶ The total thickness of aluminum and copper sheet was both 1 mm and the two-layer blank was made of 0.37 mm Cu and 0.63 mm Al sheets. An STM-50 (SANTAM Company) electronic tensile machine was employed to accomplish the tensile tests. The mechanical and material properties of each layer were determined by standard test using specimens which were prepared according to ASTM-E8 specification at a constant crosshead speed of 2 mm/min. ³⁷ The tensile test samples and dimension of the specimens used in this research are shown in Figure 1. The mechanical and material properties of each layer are illustrated in Table 1.

Figure 1.

The dimensions of the standard specimens.

Table 1.

The material and mechanical properties of Cu and Al sheets from tensile tests.

Material	Specific gravity (kg/m3)	Young’s modulus, E (GPa)	Yield strength, YS (MPa)	Strength coefficient, K (MPa)	Strain hardening index, n
Cu	8940	115	306	540	0.11
Al	2710	69	122	232	0.12
Al-Cu	−	87	226	390	0.12

The plastic strain ratios ( $r_0$ , $r_{45}$ and $r_{90}$ ) that describe the anisotropy of the material in three rolling directions were obtained using specimens made according to ASTM-E8M specification by wire electro-discharge machining (EDM) cutting machine. Measured values of the plastic strain ratios are presented in Table 2.

Table 2.

The plastic strain ratios.

	r 0	r 45	r 90	R = (r0 + 2 × r45 + r90)/4
Al	0.657	0.644	0.679	0.656
Cu	0.643	1.382	0.631	1.0095
Al-Cu	0.651	0.870	0.643	0.7585

Punch stretching test

The punch stretching tests were carried out with a hemispherical punch of 100-mm diameter according to the procedure suggested by Nakazima et al. ³⁸ on a 30 ton hydraulic press (Figure 2). Circular blanks of polyethylene foil with 0.05-mm thick were used for lubrication in the experiments. The specimens with different geometries were used to obtain the FLC in this study according to ISO 12004-2 standard. ³⁹ Figure 3 shows the geometries of the used specimens to obtain the FLC. ⁴⁰

Figure 2.

The experimental FLC setup.

Figure 3.

The different specimen geometries to obtain the FLC (all the dimensions in mm). ⁴⁰

To measure the principal strains after performing tests to determine the FLC, the circular grids were marked on the blanks by the electro-chemical etching method. In previous research, ⁴¹ it has been observed that, when the layer with a lower formability was in contact with the punch and the layer with a higher strength and formability was on the die side, the formability of the two-layer sheet would be improved. Thus, the circular grids were marked on the Cu side (the layer with higher formability) of the aluminum–copper two-layer sheet which was on the die side in order to obtain more formability. The circular grids were deformed to elliptic shapes during the tests. For each specimen, the major and minor limited strains were measured from the major and minor axes of the ellipse that was located at the nearest distance to the localized necking zone. Mylar tape was used to measure directly the major and minor strains in the deformed circle (Figure 4).

Figure 4.

Mylar transparent tape. ⁴⁰

The M-K theory

In the modified M-K theory represented in this study, the necking was ascribed to a geometrical inhomogeneity in the form of a weak region which existed before loading began. In the zone with geometrical inhomogeneity, sheet thickness was a little smaller than the safe zone. As the result of the applied force, this region originated and changed its direction with respect to initial coordinates. Finally, necking began from the zone with geometrical inhomogeneity. A geometrical inhomogeneity coefficient was defined as $f_0=t^b/t^a$ . The modified representation of the M-K model is shown in Figure 5.

Figure 5.

Schematic of the modified M-K approach for two-layer sheet.

It was assumed that in the safe zone (e.g. zone “a”), strain path was linear. The strain and stress calculations were carried out for this zone and then for the grooved zone (e.g. zone “b”). Barlat and Lian’s ²⁴ 1989 anisotropic yield function and Holloman’s hardening law were used in this work.

To start the analysis, a small value for $d{\overline{\epsilon }}^a$ (the effective strain increment in the safe zone) was applied. The effective strain in each step would be calculated by

$${{\overline{\epsilon }}^a}_{\mathrm{new}}={{\overline{\epsilon }}^a}_{\mathrm{old}}+d{\overline{\epsilon }}^a$$

(1)

Now applying hardening law, the effective stress could be found

$${\overline{\sigma }}^a=k({\overline{\epsilon }}^a{)}^n(\stackrel{·}{\overline{\epsilon }}/{\stackrel{·}{\overline{\epsilon }}}_0{)}^{m^a}$$

(2)

Having the effective stress, the stress ratio $(\alpha ={\sigma }_2/{\sigma }_1)$ and the yield criteria, ${\sigma }_1^a$ and ${\sigma }_2^a$ , were computed and the stress tensor in principal coordinates was presented as follows

$$[{\sigma }^a{]}_{\mathrm{xyz}}=\left[\begin{array}{cc}{\sigma }_1^a\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\sigma }_2^a\hfill \end{array}\right]$$

(3)

To determine the stress tensor in the groove coordinates which was shown in Figure 5, rotation matrix was used

$$\begin{array}{c}\multicolumn{1}{c}{T=\left[\begin{array}{cc}\cos \theta \hfill & \sin \theta \hfill \\ \multicolumn{1}{c}{-\sin \theta }& \cos \theta \hfill \end{array}\right]}\\ \multicolumn{1}{c}{[{\sigma }^a{]}_{\mathrm{ntz}}=T\times [{\sigma }^a{]}_{\mathrm{xyz}}\times T^T=\left[\begin{array}{cc}{\sigma }_{\mathrm{nn}}^a\hfill & {\sigma }_{\mathrm{nt}}^a\hfill \\ \multicolumn{1}{c}{{\sigma }_{\mathrm{nt}}^a}& {\sigma }_{\mathrm{tt}}^a\hfill \end{array}\right]}\end{array}$$

(4)

Applying incompressibility condition and flow rule, the zone “a” was completely solved and then all strains and stresses in zone “b” should be calculated. In this zone, unknown parameters were ${\sigma }_{\mathrm{nn}}^b,{\sigma }_{\mathrm{nt}}^b,{\sigma }_{\mathrm{tt}}^b,{\epsilon }_{\mathrm{nn}}^b,{\epsilon }_{\mathrm{nt}}^b,{\epsilon }_{\mathrm{tt}}^b$ and ${\epsilon }_3^b$ , but applying incompressibility condition and flow rule, they reduced to ${\sigma }_{\mathrm{nn}}^b,{\sigma }_{\mathrm{nt}}^b,{\sigma }_{\mathrm{tt}}^b,d{\overline{\epsilon }}^b$ . The four equations required to calculate these unknowns were the strain compatibility equation, plastic work equivalence principle and two force equilibrium equations ⁹

$$\begin{array}{c}\multicolumn{1}{c}{\frac{d{\epsilon }_{\mathrm{tt}}^b}{d{\epsilon }_{\mathrm{tt}}^a}-1=0}\\ \multicolumn{1}{c}{\frac{d{\epsilon }_{\mathrm{nn}}^b{\sigma }_{\mathrm{nn}}^b+d{\epsilon }_{\mathrm{tt}}^b{\sigma }_{\mathrm{tt}}^b+2d{\epsilon }_{\mathrm{nt}}^b{\sigma }_{\mathrm{nt}}^b}{d{\overline{\epsilon }}^b{\overline{\sigma }}^b}-1=0}\\ \multicolumn{1}{c}{f\frac{{\sigma }_{\mathrm{nn}}^b}{{\sigma }_{\mathrm{nn}}^a}-1=0}\\ \multicolumn{1}{c}{f\frac{{\sigma }_{\mathrm{nt}}^b}{{\sigma }_{\mathrm{nt}}^a}-1=0}\end{array}$$

(5)

where f changed according to

$$f=\exp ({\epsilon }_3^b-{\epsilon }_3^a)$$

(6)

Newton–Raphson method was used to solve this set of equations. ⁷ Consider N functions $F_i$ involving variables $x_i,i=1,2,\dots ,N$ as follows^42–44

$$F_i(x_1,x_2,\dots ,x_N)=0i=1,2,\dots ,N$$

(7)

In the neighborhood of x, each of functions $F_i$ can be expanded in Taylor series

$$F_i(\mathrm{x}+\delta \mathrm{x})=F_i\left(\mathrm{x}\right)+\sum _{j=1}^N\frac{\partial F_i}{\partial x_j}\delta x_j+O\left(\delta {\mathrm{x}}^2\right)$$

(8)

In matrix notation, equation (8) is reduced to

$$F(\mathrm{x}+\delta \mathrm{x})=F\left(\mathrm{x}\right)+\mathbf{J}·\delta \mathrm{x}+O\left(\delta {\mathrm{x}}^2\right)$$

(9)

The matrix of partial derivatives appearing in equation (9) is the Jacobian matrix J

$$J_{\mathrm{ij}}\equiv \frac{\partial F_i}{\partial x_j}$$

(10)

Terms of order $\delta {\mathrm{x}}^2$ and higher are neglected and by setting $F(x+\delta x)=0$ , we get a set of equations

$$\mathbf{J}·\delta \mathrm{x}=-F\mathrm{or}\delta \mathrm{x}=-{\mathbf{J}}^{-1}·F$$

(11)

The variable $\delta \mathbf{x}$ is then added to the solution vector

$${\mathrm{x}}_{\mathrm{new}}={\mathrm{x}}_{\mathrm{old}}+\delta \mathrm{x}$$

(12)

and the process is iterated to the convergence. For obtaining the unknown parameters in the groove zone, matrix F is defined as

$$F=[\begin{array}{cccc}{F}_1\hfill & F_2\hfill & F_3\hfill & F_4\hfill \end{array}{]}^T$$

(13)

where

$$F_1=\frac{d{\epsilon }_{\mathrm{nn}}^b{\sigma }_{\mathrm{nn}}^b+d{\epsilon }_{\mathrm{tt}}^b{\sigma }_{\mathrm{tt}}^b+2d{\epsilon }_{\mathrm{nt}}^b{\sigma }_{\mathrm{nt}}^b}{d{\overline{\epsilon }}^b{\overline{\sigma }}^b}-1=0$$

(14)

$$F_2=\frac{d{\epsilon }_{\mathrm{tt}}^b}{d{\epsilon }_{\mathrm{tt}}^a}-1=0$$

(15)

$$F_3=f\frac{{\sigma }_{\mathrm{nn}}^b}{{\sigma }_{\mathrm{nn}}^b}-1=0$$

(16)

$$F_4=f\frac{{\sigma }_{\mathrm{nt}}^b}{{\sigma }_{\mathrm{nt}}^b}-1=0$$

(17)

And the unknown parameter x, is defined as

$$\mathrm{x}=[\begin{array}{cccc}{\sigma }_{\mathrm{nn}}^b\hfill & {\sigma }_{\mathrm{tt}}^b\hfill & {\sigma }_{\mathrm{nt}}^b\hfill & d{\overline{\epsilon }}_b\hfill \end{array}{]}^T$$

(18)

In this study, equations (14)–(17) have been used simultaneously as set of nonlinear equations. After $d{\overline{\epsilon }}^b$ was obtained in each step, it was compared to $d{\overline{\epsilon }}^a$ and if $d{\overline{\epsilon }}^b/d{\overline{\epsilon }}^a>10$ the localized necking began and ${\epsilon }_1^a$ , ${\epsilon }_2^a$ were saved; otherwise, a greater value for $d{\overline{\epsilon }}^a$ was assumed and the process repeated. This procedure was performed for different values of stress ratios and groove angles until the whole diagram was determined. Groove orientation is function of strain increments in homogeneous zone and can be used from following formula ⁴⁵

$$\tan (\theta +d\theta )=\exp \left[(1-\rho )d{\epsilon }_1^a\right]\tan \left(\theta \right)$$

(19)

where

$$\rho =\frac{d{\epsilon }_2^a}{d{\epsilon }_1^a}$$

(20)

For any constant value of $\alpha (0\le (\alpha ={\sigma }_2/{\sigma }_1)\le 1)$ , by changing groove angle $\theta$ , between $0^{\circ }$ and ${90}^{\circ }$ , minimum value of major strain can be found. This minimum value and its corresponding minor strain were defined as point in the FLC.

Constitutive equations

Barlat and Lian’s ²⁴ 1989 yield criterion and Holloman hardening law were used in plastic calculations of FLC. Barlat and Lian’s ²⁴ 1989 yield criterion was the first orthotropic yield function proposed by Barlat and his coworkers. It was proved that this yield criterion could describe the behavior of orthotropic sheet metals exhibiting planar anisotropy and it was written as follows

$${\overline{\sigma }}_y={\left(\frac{a{|K_1+K_2|}^m+a{|K_1-K_2|}^m+c{|2K_2|}^m}{2}\right)}^{1/m}$$

(21)

where

$$K_1=\frac{{\sigma }_1+h·{\sigma }_2}{2},K_2=\sqrt{{\left(\frac{{\sigma }_1-h·{\sigma }_2}{2}\right)}^2+p^2·{\sigma }_{12}^2}$$

(22)

This yield criterion contained five coefficients $(m,a,c,h,p)$ where m was a material parameter associated with the crystal structure of the sheet metal. For body-centered cubic (BCC) and face center cubic (FCC) metals, the values of m equaled to 6 and 8, respectively. The anisotropic material constants $a,c$ and h were obtained from the Lankford coefficients r-value corresponding to 0°, 45° and 90° orientations $(r_0,r_{45}\mathrm{and}{r}_{90})$ according to the following equations

$$a=2-2\sqrt{\frac{r_0}{1+r_0}·\frac{r_{90}}{1+r_{90}}},c=2-a,h=\sqrt{\frac{r_0}{1+r_0}·\frac{1+r_{90}}{r_{90}}}$$

(23)

The last anisotropic parameter p could be determined implicitly by some r values in the following form

$$r_{\varphi }=\frac{2m{\sigma }_y^m}{(\frac{\partial \varphi }{\partial {\sigma }_{\mathrm{xx}}}+\frac{\partial \varphi }{\partial {\sigma }_{\mathrm{yy}}})·{\sigma }_{\varphi }}-1$$

(24)

Assuming $\varphi =45$ and the value of p could be determined by an iterative operation.

In this work, the Holloman law was used to describe the effective stress ${\overline{\sigma }}_Y$ as a function of effective strain $\overline{\epsilon }$ , strain hardening exponent n and strength coefficient K. This relation was defined as

$${\overline{\sigma }}_Y=K(\overline{\epsilon }{)}^n$$

(25)

Results and discussion

In this research, the results of the modified M-K approach for FLDs of Al-Cu two-layer metallic sheets were presented. The predicted FLDs were compared with the FLD obtained experimentally for Al-Cu two-layer metallic sheets (Figure 6). It could be concluded from Figure 6 that this method was in good agreement with the experimental test results for Al-Cu two-layer sheets.

Figure 6.

The FLDs for the Al-Cu two-layer metallic sheets.

Figure 6 showed that there was a low difference between the results of the modified M-K approach and experiment for the FLD₀ (i.e. major strain in plane strain state). This difference could be due to the errors in strain measurement by the conventional “circle grid analysis” method. Therefore, it could be deduced that the theoretical results were in fairly good agreement with experimental test results. Also typical deformed test samples in experimental works have been illustrated in Figure 7.

Figure 7.

Typical deformed test samples.

Figure 8 illustrates the influence of inhomogeneous factor f₀ on FLD₀ (FLD₀ is the major strain in plane strain state). Sensitivity of the FLD to variations of f₀ is very high, and if it is assumed wrongly, the resulting FLD will differ greatly from the experimental data. By increasing this parameter, the FLD level shifted up. Thus, to determine any specific FLD, at first, the suitable value of this parameter must be adopted. For this purpose, the value of this parameter decreases from the greatest value, that is, 1, until predicted FLD₀ has good agreement with corresponding experimental value.

Figure 8.

The influence of inhomogeneous factor f₀ on FLD₀.

Conclusion

In this article, a methodology already developed for prediction of strain FLD (e.g. for single layer sheets) has been extended and applied for determination of the FLD of the Al-Cu two-layer metallic sheets. The methodology was based on the modified M-K theory with Barlat’s 1989 anisotropic yield function. The numerical results for the FLDs were verified by comparing them with experimental test results. Results from the present approach were in fairly good agreement with experimental investigations.