**3. Identification process**

6 Metal Forming – Process, Tools, Design

strain ε =<sup>z</sup> d 0 , and then relations (4) and (5) become:

2

3

final geometric data linked to the tube (diameter and wall thickness):

θ

(Equation 2) becomes:

following relation:

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

Taking into account the relations expressing strain tensor increments, the equivalent stress

In the studied case, the tube ends are fixed. As a consequence, the longitudinal increment

 + + <sup>+</sup> σ = <sup>σ</sup> ε = <sup>ε</sup> <sup>+</sup> <sup>+</sup>

The knowledge of the two unknown strain <sup>θ</sup> ε and stress <sup>θ</sup> σ needs the establishment of the

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

<sup>θ</sup> − +γ <sup>ε</sup> = (1 )

Finally, the material characteristics of the tube (base metal) are expressed by the effective

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

stress and effective strain according to the following equation (Swift model):

internal pressure and the thickness at the center of the tube are required.

 ε = <sup>0</sup> <sup>d</sup> ln d

2R 3R 1 1 R d d (1 R) 1 2R

and <sup>θ</sup> σ =

 + σ= + γ + γ σ + ++ γ <sup>2</sup> 2R 1 2R <sup>1</sup> 1R 1RR

θ

θ θ

Pd 2t

<sup>0</sup> t te (8)

σ= ε +ε <sup>n</sup> K( ) <sup>0</sup> (9)

(5)

(7)

(6)

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 tensile specimens.

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

**Figure 3.** Identification process
