**4.1. Tensile test**

Rectangular specimens are made with the following geometric characteristics: thickness=1.0mm, width=12.52mm and initial length=100mm, were cut from stainless (Figure 4). All the numerical simulations were conducted under a controlled displacement condition with the constant velocity v=0.1mm/s. The predicted force versus displacement curves compared to the experimental results for the three studied orientations are shown in Figure 1. With small ductility (Step 1) the maximum stress is about 360MPa reached for 25% of plastic strain and the final fracture is obtained for 45% of plastic strain. With moderate ductility (Step 3) the maximum stress is about 394MPa reached in 37.2% of plastic strain and the final fracture is obtained for 53% of plastic strain. The best values of the material parameters using optimization procedure are summarized in Table 1. Within these coefficients the response (stress versus plastic strain) presents a non linear isotropic hardening with a maximum stress σ = max 279 MPa reached in ε = <sup>p</sup> 36.8% of plastic strain and the final fracture is obtained for 22 % of plastic strain. The plastic strain map of the optimal case is presented in Figure 1.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 9

In this case, the BM with geometrical singularities found in the WT is supposed orthotropic transverse, whereas its behaviour is represented by Swift model. The optical microscope observation on the cross section of the wall is used to build the geometrical profile of the notch generated by the welded junction. By considering the assumptions relating to an isotropic thin shell (R=1) with a uniform thickness, the previously established relations (6), (7) and (9) allow to build the first experimental hardening model using measurements of internal pressure/radial displacements. This model is then proposed, as initial solution, to solve the inverse problem of required hardening law that minimizes the following objective

> = <sup>−</sup> ξ = p

F i p i 1 exp 1 F F

<sup>2</sup> <sup>m</sup> i i exp num

exp F is the experimental value of the thrust force corresponding to ith nanoindentation

Different flow stress evolutions of isotropic hardening (initial, intermediate and optimal) are proposed in order to estimate the best behavior of the BM with geometrical singularities found in the WT. Figures 5 and 6 show the effective stress versus plastic strain curves and the associate pressure/radial displacement for these three cases. As it can be seen, there is a good correlation between the optimal evolution of Swift hardening and the experimental

**Hardening model** ε<sup>0</sup> K (MPa) n

The anisotropy factor R is determined only for the optimal hardening evolution. In the problem to be solved there is only one parameter which initial solution exists, that it corresponds to the case of isotropic material (R = 1). The numerical iterations were performed on the WT with non-uniformity of the thickness (see Figure 7), and the obtained results are shown in Figure 8. A good improvement in the quality of predicted results is

Initial 0.025 1124.6 0.2941

Intermediate 0.055 692.30 0.2101

Optimal 0.080 742.50 0.2359

results. Table 1 summarizes the parameters of these models.

**Table 2.** Swift parameters of different hardening evolution

noted if R corresponds the value of 0.976.

num F is the corresponding simulated thrust force and mp is the total number of

<sup>m</sup> <sup>F</sup> (10)

**4.2. Welded tube (WT) hydroforming process** 

function:

where <sup>i</sup>

depth Hi, <sup>i</sup>

experimental points.


**Table 1.** Properties of the used material

**Figure 4.** Force/elongation for different optimization steps and plastic strain map

#### **4.2. Welded tube (WT) hydroforming process**

8 Metal Forming – Process, Tools, Design

optimal case is presented in Figure 1.

**Table 1.** Properties of the used material

Rectangular specimens are made with the following geometric characteristics: thickness=1.0mm, width=12.52mm and initial length=100mm, were cut from stainless (Figure 4). All the numerical simulations were conducted under a controlled displacement condition with the constant velocity v=0.1mm/s. The predicted force versus displacement curves compared to the experimental results for the three studied orientations are shown in Figure 1. With small ductility (Step 1) the maximum stress is about 360MPa reached for 25% of plastic strain and the final fracture is obtained for 45% of plastic strain. With moderate ductility (Step 3) the maximum stress is about 394MPa reached in 37.2% of plastic strain and the final fracture is obtained for 53% of plastic strain. The best values of the material parameters using optimization procedure are summarized in Table 1. Within these coefficients the response (stress versus plastic strain) presents a non linear isotropic hardening with a maximum stress σ = max 279 MPa reached in ε = <sup>p</sup> 36.8% of plastic strain and the final fracture is obtained for 22 % of plastic strain. The plastic strain map of the

Step Critical plastic strain K [MPa] ε<sup>0</sup> n 1 25,8% 381,3 0.0100 0.2400 2 29,8% 395,5 0.0120 0.2415 3 37,2% 415,2 0.0150 0.2450 Optimal 36,8% 416,1 0.0198 0.2498

**Figure 4.** Force/elongation for different optimization steps and plastic strain map

**4.1. Tensile test** 

In this case, the BM with geometrical singularities found in the WT is supposed orthotropic transverse, whereas its behaviour is represented by Swift model. The optical microscope observation on the cross section of the wall is used to build the geometrical profile of the notch generated by the welded junction. By considering the assumptions relating to an isotropic thin shell (R=1) with a uniform thickness, the previously established relations (6), (7) and (9) allow to build the first experimental hardening model using measurements of internal pressure/radial displacements. This model is then proposed, as initial solution, to solve the inverse problem of required hardening law that minimizes the following objective function:

$$\mathfrak{E}\_{\rm F} = \frac{1}{\mathbf{m}\_{\rm P}} \sqrt{\sum\_{i=1}^{\mathbf{m}\_{\rm P}} \left( \frac{\mathbf{F}\_{\rm exp}^{i} - \mathbf{F}\_{\rm num}^{i}}{\mathbf{F}\_{\rm exp}^{i}} \right)^{2}} \tag{10}$$

where <sup>i</sup> exp F is the experimental value of the thrust force corresponding to ith nanoindentation depth Hi, <sup>i</sup> num F is the corresponding simulated thrust force and mp is the total number of experimental points.

Different flow stress evolutions of isotropic hardening (initial, intermediate and optimal) are proposed in order to estimate the best behavior of the BM with geometrical singularities found in the WT. Figures 5 and 6 show the effective stress versus plastic strain curves and the associate pressure/radial displacement for these three cases. As it can be seen, there is a good correlation between the optimal evolution of Swift hardening and the experimental results. Table 1 summarizes the parameters of these models.


**Table 2.** Swift parameters of different hardening evolution

The anisotropy factor R is determined only for the optimal hardening evolution. In the problem to be solved there is only one parameter which initial solution exists, that it corresponds to the case of isotropic material (R = 1). The numerical iterations were performed on the WT with non-uniformity of the thickness (see Figure 7), and the obtained results are shown in Figure 8. A good improvement in the quality of predicted results is noted if R corresponds the value of 0.976.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 11

**Figure 7.** Radial displacement for different values of anisotropy coefficient R

Experiment R=0,976 R=0,930 R=1,010

Sheet metal forming examples will be presented in order to test the capability of the proposed methodology to simulate thin sheet hydroforming operation using the fully isotropic model concerning elasticity and plasticity (Cherouat et al.,2008). These results are carried out on the circular part with a diameter of 300mm and thickness of 0.6mm. During hydroforming of the blank sheet, the die shape keeps touching the blank, which prevents the deformed area from further deformation and makes the deformation area move towards the outside. The blank flange is drawn into the female die, which abates thinning deformation of deformed area and aids the deformation of touching the female die and uniformity of deformation. Compared with the experiments done before, the limit drawing

012345 **Radial displacement (mm)**

By considering the assumptions relating to an isotropic thin shell with a uniform thickness, the previously established relations allow to build the first experimental hardening model using measurements of force/displacement. This model is then proposed, as initial solution, to solve the inverse problem of required hardening law that minimizes the following

> <sup>−</sup> <sup>=</sup>

1 P P

m P

error i

E

<sup>m</sup> i i exp num

1 exp

exp P is the experimental value of the thrust pressure corresponding to ith

num P is the corresponding predicted pressure and m is the total number of

2

(11)

**4.3. Thin sheet hydroforming process** 

20

22

24

26

**Internal pressure (MPa)**

28

30

32

ratio of the blank is improved remarkably.

objective function:

displacement δ<sup>i</sup> , <sup>i</sup>

experimental points.

where <sup>i</sup>

**Figure 5.** Stress-strain evolutions for different hardening laws

**Figure 6.** Internal pressure versus radial displacement

**Figure 7.** Radial displacement for different values of anisotropy coefficient R

#### **4.3. Thin sheet hydroforming process**

10 Metal Forming – Process, Tools, Design

**Figure 5.** Stress-strain evolutions for different hardening laws

0 0,1 0,2 0,3 0,4 **Effective plastic strain**

012345 **Radial displacement [mm]**

Experiment Initial Intermediate Optimal

200

300

400

500

**Effective stress [MPa]**

600

700

800

900

Initial Intermediate Optimal

**Figure 6.** Internal pressure versus radial displacement

15 17

27 29 31

**Internal pressure [MPa]**

33 35 Sheet metal forming examples will be presented in order to test the capability of the proposed methodology to simulate thin sheet hydroforming operation using the fully isotropic model concerning elasticity and plasticity (Cherouat et al.,2008). These results are carried out on the circular part with a diameter of 300mm and thickness of 0.6mm. During hydroforming of the blank sheet, the die shape keeps touching the blank, which prevents the deformed area from further deformation and makes the deformation area move towards the outside. The blank flange is drawn into the female die, which abates thinning deformation of deformed area and aids the deformation of touching the female die and uniformity of deformation. Compared with the experiments done before, the limit drawing ratio of the blank is improved remarkably.

By considering the assumptions relating to an isotropic thin shell with a uniform thickness, the previously established relations allow to build the first experimental hardening model using measurements of force/displacement. This model is then proposed, as initial solution, to solve the inverse problem of required hardening law that minimizes the following objective function:

$$\mathbf{E}\_{\text{error}} = \frac{1}{\text{m}} \sqrt{\sum\_{1}^{\text{m}} \left( \frac{\mathbf{P}\_{\text{exp}}^{\text{i}} - \mathbf{P}\_{\text{num}}^{\text{i}}}{\mathbf{P}\_{\text{exp}}^{\text{i}}} \right)^{2}} \tag{11}$$

where <sup>i</sup> exp P is the experimental value of the thrust pressure corresponding to ith displacement δ<sup>i</sup> , <sup>i</sup> num P is the corresponding predicted pressure and m is the total number of experimental points.

The controlled process parameters are the internal fluid pressure applied to the sheet as a uniformly distributed load to the sheet inner surface and is introduced as a linearly increasing function of time with a constant flow from approximately 10 ml/min. The comparator is used to measure the pole displacement. The effect of three die cavities (D1, D2 and D3 see Figure 8) on the plastic flow and damage localisation is investigated during sheet hydroforming. These dies cavities are made of a succession of revolution surfaces (conical, planes, spherical concave and convex). The evolution of displacement to the poles according to the internal pressure during the forming test and sheet thicknesses are investigated experimentally. The profiles of displacements are obtained starting from the deformations of the sheet after bursting. Those are reconstituted using 3D scanner type Dr. Picza Roland of an accuracy of 5μm with a step of regulated touch to 5mm. In addition, two measurement techniques were used to evaluate the thinning of sheet after forming; namely a nondestructive technique using an ultrasonic source of Sofranel mark (Model 26MG) and a destructive technique using a digital micrometer calliper of Mitiyuta mark of precision 10μm (see Figure 9).

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 13

by plastic instability and damage. Figures 10, 11 and 12 present as such, the main results and simulations of all applications processed in this study. The predicted results with cavity dies show that the equivalent von Mises stress reached critical values high and then subjected to a significant decrease in damaged areas. This decrease is estimated for the three die cavities

Comparisons between numerical predictions of damaged areas and the experimental

1. The numerical calculations show that increasing pressure, the growing regions marked by a rise in the equivalent stress followed by a sudden decrease can be correlated with the damaged zones observed experimentally. In this context, the results of the first die cavity D1 show that instabilities are localized in the central zone of the blank, limited by a circular contour of the radius 72mm. The largest decrease in stress is located in the area bounded by two edges of respective radii 51 and 64mm. While the rupture occurred at the border on the flat surface with the spherical one located on a circle of radius 60mm. With the die cavity D2, the largest decrease is between two contours of radii 10 and 19mm, the rupture is observed at a distance of 17mm from the revolution axis of deformed blank. Finally with the die cavity D3, the calculations show that the damaged area is located in a region bounded by two edges of respective radii 54 and 73mm, the rupture occurred in

2. The pressures that characterize the early instabilities are respectively the order of 4.90MPa (for D1), 2.85MPa (for D2) and 5.1MPa (for D3). For applications with die cavities D1 and D3, regions where the beginnings of instability have been identified (see

The results presented in Fig. 9 show that the relative differences between predicted and experiment results of pole displacement are in the limit of 7% while the pressure levels are

D1, D2 and D3, respectively 29%, 14% and 36%.

**Figure 9.** 3D scanner G Scan for reconstitution

Table 2).

observations of fracture zones led us to the following findings:

the connection of the flat surface with the surface spherical one.

below a threshold characterizing the type of application.

**Figure 8.** Geometry of die cavities (D1, D2 and D3)

Experiments results of circular sheet hydroforming are shown in Figure 10 (Die D1), Figure 11 (Die D2) and Figure 12 (Die D3). For the die cavities D1 and D3, fracture appeared at the round corner (near the border areas between the conical and the hemispherical surfaces of the die). For the die cavity D2, the fracture occurred at the centre of the blank when the pressure is excessive. This shows that the critical deformation occurs at these regions. It is noted that the rupture zone depends on the die overflow of the pressure medium from the pressurized chamber and the reverse-bending effect on the die shoulder were not observed in the experiment. In this part we are interested in the comparison between experimental observations of regions where damage occurred and numerical predictions of areas covered by plastic instability and damage. Figures 10, 11 and 12 present as such, the main results and simulations of all applications processed in this study. The predicted results with cavity dies show that the equivalent von Mises stress reached critical values high and then subjected to a significant decrease in damaged areas. This decrease is estimated for the three die cavities D1, D2 and D3, respectively 29%, 14% and 36%.

**Figure 9.** 3D scanner G Scan for reconstitution

12 Metal Forming – Process, Tools, Design

10μm (see Figure 9).

**Figure 8.** Geometry of die cavities (D1, D2 and D3)

The controlled process parameters are the internal fluid pressure applied to the sheet as a uniformly distributed load to the sheet inner surface and is introduced as a linearly increasing function of time with a constant flow from approximately 10 ml/min. The comparator is used to measure the pole displacement. The effect of three die cavities (D1, D2 and D3 see Figure 8) on the plastic flow and damage localisation is investigated during sheet hydroforming. These dies cavities are made of a succession of revolution surfaces (conical, planes, spherical concave and convex). The evolution of displacement to the poles according to the internal pressure during the forming test and sheet thicknesses are investigated experimentally. The profiles of displacements are obtained starting from the deformations of the sheet after bursting. Those are reconstituted using 3D scanner type Dr. Picza Roland of an accuracy of 5μm with a step of regulated touch to 5mm. In addition, two measurement techniques were used to evaluate the thinning of sheet after forming; namely a nondestructive technique using an ultrasonic source of Sofranel mark (Model 26MG) and a destructive technique using a digital micrometer calliper of Mitiyuta mark of precision

Experiments results of circular sheet hydroforming are shown in Figure 10 (Die D1), Figure 11 (Die D2) and Figure 12 (Die D3). For the die cavities D1 and D3, fracture appeared at the round corner (near the border areas between the conical and the hemispherical surfaces of the die). For the die cavity D2, the fracture occurred at the centre of the blank when the pressure is excessive. This shows that the critical deformation occurs at these regions. It is noted that the rupture zone depends on the die overflow of the pressure medium from the pressurized chamber and the reverse-bending effect on the die shoulder were not observed in the experiment. In this part we are interested in the comparison between experimental observations of regions where damage occurred and numerical predictions of areas covered Comparisons between numerical predictions of damaged areas and the experimental observations of fracture zones led us to the following findings:


The results presented in Fig. 9 show that the relative differences between predicted and experiment results of pole displacement are in the limit of 7% while the pressure levels are below a threshold characterizing the type of application.

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 15

**Figure 12.** Experimental and numerical results of hydroforming using die cavity D3

0 1234 56 **Internal pressure (MPa)**

Experiment D1 Simulation D1 Experiment D2 Simulation D2 Experiment D3 Simulation D3

**Figure 13.** Pole displacement versus internal pressure

**Pole displacement (mm)**

**Figure 10.** Experimental and numerical results of hydroforming using die cavity D1

**Figure 11.** Experimental and numerical results of hydroforming using die cavity D2

Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 15

**Figure 12.** Experimental and numerical results of hydroforming using die cavity D3

**Figure 13.** Pole displacement versus internal pressure

**Figure 10.** Experimental and numerical results of hydroforming using die cavity D1

**Figure 11.** Experimental and numerical results of hydroforming using die cavity D2



Hydroforming Process: Identification of the Material's Characteristics and Reliability Analysis 17

In this case a significant design variable for formability of blank during hydroforming

50 D 250mm 20 d 100mm . The experimental final shape is shown in Figure 10a. ≤ ≤ ≤≤ The comparison of the force versus the maximum displacement with the initial and optimized blank shape is present in Figure 10b. Good agreement between the optimum shape and the experimental values. Figures 10c and 10d compare the initial and the optimum blank shape. Successfully decreased the cost function (thinning ratio) from 50% to

Recently, RBDO has become a popular philosophy to solve different kind of problem. In this part, we try to prove the ability of this strategy to optimize loading path in the case of THP where different kind of nonlinearities exist (material, geometries and boundary conditions). The aim of this study is to obtain a free defects part with a good thickness distribution, decrease the risk of potential failures and to let the process insensitive to the input parameters variations. For more detailed description of the RBDO methodology and variety of frameworks the reader can be refer to the following references (Youn et al., 2003; Enevoldsen et al., 1994; El Hami et al., 2011). The RBDO problem can be generally

( ) ( )

subject to P G d,X 0 0 i 1, ,np

where f(d,X) is the objective function, d is the design vector, X is the random vector, and the probabilistic constraints is described by the performance function Gi(X), np, ndv and nrv are the number of probabilistic constraints, design variables and random variables, respectively,

probability of failure and Φ is the cumulative distribution function for standard normal

The process failure state is characterized by a limit state function or performance function G(X), and G(X)=0 denotes the limit state surface. The m-dimensional uncertainty space in thus divided into a safe region ( ) Ω= > <sup>s</sup> {X:G X 0 ( ) } and a failure region

 <sup>i</sup> i t L U ndv nrv

(12)

1

t f P where f P is the

 ≤ − Φ −β ≤ = ≤≤ ∈ ∈

<sup>β</sup>ti is the prescribed confidence level which can be defined as ( ) <sup>−</sup> β = −Φ <sup>i</sup>

d d d , d R and X R

20% is obtained without wrinkling (Figure 10e and 10f) (see Ayadi et al.,2011).

i.e. Cost function:

**5. Reliability analysis** 

formulated as:

distribution.

= <sup>−</sup> <sup>=</sup> <sup>2</sup> <sup>n</sup> <sup>0</sup>

i 1 0 t t <sup>f</sup> t

where 0t is the initial thickness and t the final thickness.

process and the design (D and d) constraints were defined:

( )

Min f d,X

 

( ) Ω= ≤ <sup>f</sup> {X:G X 0 ( ) } (see Radi et al.,2007).


**Figure 14.** Optimisation of complex shape part
