**2. Hydroforming process**

For production of low-weight, high-energy absorbent, and cost-effective structural automotive components, hydroforming is now considered the only method in many cases.

The principle of tube hydroforming is shown in Figure 1. The hydroforming operation is either force-controlled (the axial forces vary with the internal pressure) or stroke-controlled (the strokes vary with the internal pressure). Note that the axial force and the stroke are strongly interrelated (see figure 1).

Force-controlled hydroforming is at the focus in (Asnafi et al.,2000), where the constructed analytical models are used to show


The hydroforming operation comprises two stages: free forming and calibration. The portion of the deformation in which the tube expands without tool contact, is called free forming. As soon as tool contact is established, the calibration starts.

**Figure 1.** The principle of tube hydroforming: (a) original tube shape and (b) final tube shape (before unloading).

During calibration, no additional material is fed into the expansion zone by the axis cylinders. The tube is forced to adopt the tool shape of the increasing internal pressure only.

Many studies have been devoted to the mechanical and numerical modeling of the hydroforming processes using the finite element analysis, allowing the prediction of the material flow and the contact boundary evolution during the process. However, the main difficulty in many hydroforming processes is to find the convenient control of the evolution of the applied internal pressure and axial forces paths. This avoids the plastic flow localization leading to buckling or fracture of the tube during the process. In fact, when a metallic material is formed by such processes, it experiences large plastic deformations, leading to the formation of high strain localization zones and, consequently, to the onset of micro-defects or cracks. This damage initiation and its evolution cause the loss of the formed piece and indicate that the forming process itself should be modified to avoid the damage appearance. In principle, all materials and alloys used for deep drawing or stamping can be used for hydroforming applications as well.

#### **2.1. Mechanical characteristic of welded tube behaviour**

4 Metal Forming – Process, Tools, Design

applications as well (Koç et al,2002).

**2. Hydroforming process** 

strongly interrelated (see figure 1).

analytical models are used to show

forming result, and

unloading).

• which are the limits during hydroforming,

experiences large plastic deformations, leading to the formation of high strain localization zones and, consequently, to the onset of micro-defects or cracks. This damage initiation and its evolution cause the loss of the formed piece and indicate that the forming process itself should be modified to avoid the damage appearance (Cherouat et al.,2002). In principle, all materials and alloys used for deep drawing or stamping can be used for hydroforming

This chapter presents firstly a computational approach, based on a numerical and experimental methodology to adequately study and simulate the hydroforming formability of welded tube and sheet. The experimental study is dedicated to the identification of material parameters using an optimization algorithm known as the Nelder-Mead simplex (Radi et al.,2010) from the global measure of displacement and pressure expansion. Secondly, the reliability analysis of the hydroforming process of WT is presented and the numerical results are given to validate the adopted approach and to show the importance of this analysis.

For production of low-weight, high-energy absorbent, and cost-effective structural automotive components, hydroforming is now considered the only method in many cases. The principle of tube hydroforming is shown in Figure 1. The hydroforming operation is either force-controlled (the axial forces vary with the internal pressure) or stroke-controlled (the strokes vary with the internal pressure). Note that the axial force and the stroke are

Force-controlled hydroforming is at the focus in (Asnafi et al.,2000), where the constructed

• how different material and process parameters influence the loading path and the

The hydroforming operation comprises two stages: free forming and calibration. The portion of the deformation in which the tube expands without tool contact, is called free

**Figure 1.** The principle of tube hydroforming: (a) original tube shape and (b) final tube shape (before

During calibration, no additional material is fed into the expansion zone by the axis cylinders. The tube is forced to adopt the tool shape of the increasing internal pressure only.

• what an experimental investigation into hydroforming should focus on.

forming. As soon as tool contact is established, the calibration starts.

Taking into account the ratio thickness/diameter of the tube, the radial stress is considerably small compared to the circumferential <sup>θ</sup> σ and longitudinal stresses σ<sup>z</sup> (see Figure 2). In addition, the principal axes of the stress tensor and the orthotropic axes are considered coaxial. The transverse anisotropy assumption represented through the yield criterion can be written as:

$$\overline{\sigma}^2 = \mathbf{F} \left( \sigma\_x - \sigma\_\theta \right)^2 + \mathbf{G} \sigma\_x^2 + \mathbf{H} \sigma\_\theta^2 \tag{1}$$

with (F,G,H) are the parameters characterizing the current state of anisotropy.

If the circumferential direction is taken as a material reference, the anisotropy effect can be characterized by a single coefficient R and the equation (1) becomes:

$$\overline{\sigma}^2 = \frac{1}{1+\mathcal{R}} \left[ \mathcal{R} \left( \sigma\_x - \sigma\_\theta \right)^2 + \sigma\_x^2 + \sigma\_\theta^2 \right] \tag{2}$$

The assumptions of normality and consistency lead to the following equations:

$$\begin{cases} \mathrm{d}\varepsilon\_{\theta} = \frac{\mathrm{d}\overline{\varepsilon}}{\overline{\mathfrak{G}}} \left( \sigma\_{\theta} - \frac{\mathrm{R}}{1 + \mathrm{R}} \sigma\_{x} \right) \\\\ \mathrm{d}\varepsilon\_{x} = \frac{\mathrm{d}\overline{\varepsilon}}{\overline{\mathfrak{G}}} \left( \sigma\_{x} - \frac{\mathrm{R}}{1 + \mathrm{R}} \sigma\_{\theta} \right) \end{cases} \tag{3}$$

where ε is the effective plastic strain and ( ) <sup>θ</sup> ε ε<sup>z</sup> , are the strains in the circumferential and the axial directions. The effective strain for anisotropic material can be derived from equivalent plastic work definition, incompressibility condition, and the normality condition:

$$\mathrm{d}\overline{\varepsilon} = \frac{\sqrt{1+\mathrm{R}}}{\sqrt{1+2\mathrm{R}}} \sqrt{\mathrm{d}\varepsilon\_x^2 + \mathrm{d}\varepsilon\_\theta^2 + \mathrm{R}\left(\mathrm{d}\varepsilon\_x - \mathrm{d}\varepsilon\_\theta\right)^2} = \left(\sqrt{\gamma^2 + \frac{2\mathrm{R}}{1+\mathrm{R}}\gamma + 1}\right) \frac{1+\mathrm{R}}{\sqrt{1+2\mathrm{R}}} \mathrm{d}\varepsilon\_\theta \quad \text{with} \quad \gamma = \frac{\mathrm{d}\varepsilon\_x}{\mathrm{d}\varepsilon\_\theta} \tag{4}$$

Taking into account the relations expressing strain tensor increments, the equivalent stress (Equation 2) becomes:

$$\overline{\sigma} = \left( \sqrt{1 + \chi^2 + \frac{2\mathcal{R}}{1 + \mathcal{R}}} \gamma \right) \frac{\sqrt{1 + 2\mathcal{R}}}{1 + \mathcal{R} + \mathcal{R}\gamma} \sigma\_{\theta} \tag{5}$$

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 7

The parameters ( ) K, ,n <sup>0</sup> ε are computed in such a way that the constitutive equations associated to the yield surface reproduce as well as possible the following characteristics of the sheet metal. The problem which remains to be solved consists in finding the best combination of the parameters damage which minimizes the difference between numerical forecasts and experimental results. This minimization related to the differences between the m experimental measurements of the tensions and their numerical forecasts conducted on

Due to the complexity of the used formulas, we have developed a numerical minimization strategy based on the Nelder-Mead simplex method. The identification technique of the material parameters is based on the coupling between the Nelder-Mead simplex method (Matlab code) and the numerical simulation based finite element method via ABAQUS/Explicit© of the hydroforming process. To obtain information from the output file

A three dimensional finite element analysis (FEA) has been performed using the finite

element code ABAQUS/Explicit to investigate the hydroforming processes.

of the ABAQUS/Explicit©, we use a developed Python code (see Figure 3).

**3. Identification process** 

tensile specimens.

**Figure 3.** Identification process

**4. Results and discussion** 

In the studied case, the tube ends are fixed. As a consequence, the longitudinal increment strain ε =<sup>z</sup> d 0 , and then relations (4) and (5) become:

$$
\overline{\sigma} = \sqrt{\frac{2\mathbf{R}^2 + 3\mathbf{R} + 1}{\left(1 + \mathbf{R}\right)^3}} \begin{vmatrix} \sigma\_0 & & & \mathbf{d}\overline{\mathbf{e}} = \left(\frac{1 + \mathbf{R}}{\sqrt{1 + 2\mathbf{R}}}\right) \mathbf{d}\mathbf{e}\_0 \end{vmatrix} \tag{6}
$$

The knowledge of the two unknown strain <sup>θ</sup> ε and stress <sup>θ</sup> σ needs the establishment of the final geometric data linked to the tube (diameter and wall thickness):

$$
\varepsilon\_{\theta} = \ln \left( \frac{\text{d}}{\text{d}\_0} \right) \text{ and } \quad \sigma\_{\theta} = \frac{\text{Pd}}{2\text{t}} \tag{7}
$$

where P is the internal pressure, ( ) <sup>0</sup> d,d are the respective average values of the current and initial diameter of the sample and (t) is the current wall thickness obtained according to the following relation:

$$\mathbf{t} = \mathbf{t}\_0 \mathbf{e}^{-(1+\gamma)\mathbf{e}\_\theta} \tag{8}$$

Finally, the material characteristics of the tube (base metal) are expressed by the effective stress and effective strain according to the following equation (Swift model):

$$\overline{\mathfrak{a}} = \mathbf{K} (\varepsilon\_0 + \overline{\varepsilon})^n \tag{9}$$

The values of the strength coefficient *K*, the strain hardening exponent *n*, the initial strain ε<sup>0</sup> and the anisotropic coefficient R in Equations (2) and (9) are identified numerically. For the determination of the stress–strain relationship using bulge test, the radial displacement, the internal pressure and the thickness at the center of the tube are required.

**Figure 2.** Stress state at bulge tip
