**Table 3.**

*Process parameter data.* 

 The stretch forming was carried out using a 101.6 mm hemispherical diameter punch. The design parameters are given in **Table 1**. The blanks used are 175 mm × 175 mm, 175 mm × 100 mm and a modified blank design of 25 mm width which ensures all modes of deformations, such as biaxial stretching, plain strain and compression-tension region.

All tools were considered to be rigid bodies and the blank was taken as a deformable body and it has been meshed as quadrilateral shell elements. The default shell element formulation is BELYTSCHKO-TSAY, it is generally used in similar simulations. Draw bead is defined as line bead and was locked at the die. The 'formingone-way' type contact interface is used for defining the contact. The arrangement of meshed tools and blank is shown in **Figure 2**.

The details of material properties and process parameters required for simulation were experimentally carried out and are shown in **Tables 2** and **3** respectively.

#### **3. Simulation results**

 As shown in **Figure 3**, the simulation results of the stretch forming for different blank sizes width varied from 175 to 25 mm. The forming limit diagram and the failure is predicted based on the correlation already presented in the post processor of LS DYNA.

**Figure 3.**  *Simulation results of stretch forming from different blank widths.* 

#### **4. Experimental determination of forming limit diagrams**

 As suggested by Hecker [6], samples of varying width were deformed using a 101.6 mm hemispherical punch. The width was varied to obtain all possible deformation modes i.e. biaxial, plane strain condition and compression-tension. The schematic diagram of the arrangements of tools used in experiments is shown in **Figure 4**.

 Samples of different width varying from 100 to 175 mm with constant length (175 mm) and a hourglass shaped specimen with widths having 25, 50 and 75 mm were prepared for punch stretching experiments (**Figure 5**). These blanks were cut with length perpendicular to the rolling direction. These metal blanks were laser marked with 5 mm diameter circles. These grid-marked specimens were deformed with a hydraulic press of 100-tonne capacity having double action. The LDH tests on samples of different widths were done on the hydraulic press using the experimental setup. The experiment was not continued when a visible necking or fracture was obtained on the samples, deformed specimen are shown in **Figure 5**.

Major strain and minor strains have been calculated by calculating major and minor length of ellipses on the deformed samples. A travelling microscope having at least a count of 0.001 mm was used to calculate major and minor length of ellipses for strain calculations. The major strain and minor strains in the necking region were measured and FLCs were drawn such that strains at necking/fracture lie above

*Prediction of Failure Using Forming Limit Curve in FE Analysis of Aluminium Alloy DOI: http://dx.doi.org/10.5772/intechopen.81083* 

**Figure 4.**  *Setup for LDH test.* 

*Specimens before and after experiment. (a) Specimen before deformation and (b) specimen after deformation.* 

**Figure 6.**  *Experimental forming limit diagram.* 

the line as shown in **Figure 6** [11–14]. The limited dome height (LDH) of all the samples at the point of necking was measured using a vernier height gauge.

#### **5. Theoretical method for predicting the FLC using Sing and Rao's method**

Sing and Rao [13] have proposed another method for predicting the FLC by means of tensile testing results. This method uses the material properties determined by simple tension.

 The Sing and Rao method for developing the FLC was programmed in MATLAB. This method uses tensile properties like UTS, n, K and R for predicting the FLC.

Using the n value, the critical strain for localised neck was calculated by the equation:

$$\mathcal{E}\_{\rm 1L} = \mathbf{2n}, \mathcal{E}\_{\rm 2L} = \mathbf{0} \tag{1}$$

 The equivalent strain at this localised strain can be calculated using the equation:

$$
\begin{split}
\boldsymbol{\mathcal{E}}^{\rm C} &= \left[ \left( \frac{\mathbf{1} + \mathbf{R}}{\left( \mathbf{1} + \mathbf{2}^{\rm M - 1} \mathbf{R} \right)} \right)^{\frac{1}{\left( \mathbf{M} - \mathbf{1} \right)}} \left( |\boldsymbol{\mathcal{E}}\_{1}|^{\mathbf{M}} + |\boldsymbol{\mathcal{E}}\_{2}|^{\mathbf{M} - 1} \right) \\\\
& \quad \times \left( \frac{\mathbf{1} + \mathbf{R}}{2} \right)^{\frac{1}{\left( \mathbf{M} - \mathbf{1} \right)}} \left( \mathbf{1} - \left( \frac{\mathbf{1}}{\mathbf{1} + \mathbf{2}^{\rm m - 1} \mathbf{R}} \right)^{\frac{1}{\left( \mathbf{M} - \mathbf{1} \right)}} \right) \left( |\boldsymbol{\mathcal{E}}\_{1} + \boldsymbol{\mathcal{E}}\_{2}|^{\mathbf{M} - 1} \right)^{\frac{\mathbf{M} - 1}{\mathbf{M}}} \end{split}
\tag{2}
$$

 From the equivalent localised necking strain, the localised neck stress was determined with this equation:

$$
\Box \overline{\sigma L} \tag{3}
$$

$$
\Box \overline{K} \text{ = } K \text{(}\overline{\varepsilon}L\text{)}\text{''}
$$

**Figure 7.**  *FLC using Sing and Rao method.* 

*Prediction of Failure Using Forming Limit Curve in FE Analysis of Aluminium Alloy DOI: http://dx.doi.org/10.5772/intechopen.81083* 

The mechanical properties and the localised neck stress of the aluminium alloys were used for the Sing and Rao method.

From this, a limit yield stress curve was plotted using Hill's yield criterion.

$$\left( \left| \sigma\_1^{M} \right| + \left| \sigma\_2 \right|^{M} \right) + \mathbf{R} \left| \sigma\_1 - \sigma\_2 \right|^{M} = \left( \mathbf{1} + \mathbf{R} \right) \sigma^{M} \tag{4}$$

Using a linear regression method, this FLC was converted into a straight line (linear FLC). Depending on the material properties, the limit yield stress curve will change accordingly. The value of stress corresponding to each point on the linear FLC can be calculated by the equation. FLC is shown in **Figure 7**.

#### **6. Comparison of different methods for predicting FLCs**

A comparison of the FLCs of simulation, experimental and Sing and Rao method is shown in **Figure 8**. It is clear that the FLC generated from the post processor using the existing correlations predicts the FLC much higher as compared to the actual experimental FLC.

The FLCs predicted by Sing and Rao's method are compared with the FLCs predicted with the developed correlation and with the experimentally determined FLC. The minimum value of major strain did not appear exactly at the plain strain condition i.e. at FLC0. Using this method, limit strains with higher negative minor strains could not be predicted due to nature of the yield locus and the material properties. **Figure 7**, it is seen that the Sing and Rao method also overestimates the limit strains compared to the experimental curve and in most cases, the FLCs generated by the developed correlation are closer to the experimental curves.

**Figure 8.**  *Comparision FLC different methods.* 

#### **7. Conclusions**

The following conclusions are drawn:

1.FLCs generated from the post processor using the existing correlations predict much higher limit strains when compared to the actual experimental FLC.

#### *Proceedings of the 4th International Conference on Innovations in Automation...*

