**5. Results and discussion**

#### **5.1. Tensile test**

Table 5 illustrates mechanical properties of the tested sheet material deduced from the tensile test at room temperature. Tensile tests results on 1.08 mm thick Ti-6Al-4V sheets show the average values for strain-hardening (n) and anisotropy (r) are (0.151, 3.63). As it can be seen Ti alloy has large plastic strain ratio (r) values. Generally higher strain-hardening exponent (n) delays the onset of instability and this delay, enhances the limiting strain (i.e. a better stretchability and formability is achieved with higher (n) value). Also, increasing plastic strain ratio (r) results in a better resistance to thinning in the thickness direction during drawing which in turn increase the formability of sheet material. On the other hand, high planar anisotropy will bring about earring effect in sheet metal forming processes especially in deep drawing process [19].


**Table 5.** Mechanical properties of tested sheet material obtained in tensile test

#### **5.2. Hydroforming bulge test**

**Process parameter Value/Type** Punch diameter (mm) 50 Diameter of die opening (mm) 55 Die profile radius (mm) 8 Punch speed (mm/s) 1 Punch travel Up to rupture Clamping type Draw bead (lock mode)

**Figure 8.** Schematic view of the model used in FE analysis as well as gridded sample shapes

Table 5 illustrates mechanical properties of the tested sheet material deduced from the tensile test at room temperature. Tensile tests results on 1.08 mm thick Ti-6Al-4V sheets show the average values for strain-hardening (n) and anisotropy (r) are (0.151, 3.63). As it can be seen Ti alloy has large plastic strain ratio (r) values. Generally higher strain-hardening exponent (n) delays the onset of instability and this delay, enhances the limiting strain (i.e. a better stretchability and formability is achieved with higher (n) value). Also, increasing plastic strain ratio (r) results in a better resistance to thinning in the thickness direction during drawing which in turn increase the formability of sheet material. On the other hand, high planar anisotropy will bring about earring effect in sheet metal forming processes especially in deep

**Table 4.** Process parameters used for the simulation

94 Titanium Alloys - Advances in Properties Control

**5. Results and discussion**

**5.1. Tensile test**

drawing process [19].

#### **Investigation of bursting pressure**

As discussed in section 3, in order to discern the bursting pressure of Ti-6Al-4V sheet material, at least three specimens were bulged up to bursting point and average bursting pressure for these alloys was obtained (Table 6). After obtaining burst pressure, test samples were bulged up to 90-95% bursting pressure while the bulge height was being monitored by the indicator. The resulted bulging pressure vs. dome height curves were then extrapolated up to burst pressure by using a third order polynomial approximation. Fig.9 shows bulge pressure versus dome height for the tested material. In this figure, experimentally measured curves along with the extrapolated regions are depicted. In Fig.10 tested samples are shown.

**Figure 9.** Experimental bulge pressure versus dome height curve for Ti-6Al-4V alloy (the curve is extrapolated)


**Table 6.** Burst pressure for different samples

Measurements/calculations of thickness at the dome apex

the dome of the bulge.

**Figure 12.** Comparison of calculated and measured bulge radii for Ti-6Al-4V

Eqs.3-5 were used to calculate the sheet thickness at the dome apex and the results were compared with the step-wise experiments. For measurement of the thickness at the dome apex, a 10mm diameter circle was imprinted on the centre of each bulge sample. After each step, the major and minor diameters of the formed circle were measured by using an accurate caliper (0.05 mm accuracy). Subsequently, the effective strain was obtained for each step at the dome apex by replacing the instantaneous sheet thickness obtained from Eq.26 into Eq.7. The instantaneous thickness of the dome apex was extracted from Eq.8 and is shown as follows:

> 0 ( ) *t*

The results show that for calculating the sheet thickness at the dome apex, Kruglov's approach gives best results when compared with step-wise experiments. Fig.13 shows the comparison of calculated sheet thickness at the dome apex with the step-wise measured sheet thickness at

As discussed before (Table 5) higher n-value indicates better stretchability and formabili‐ ty, therefore, for the same bulge height, sheet materials with larger n-values have lower thinning than sheet materials with smaller n-values. Also, when drawing is the deforma‐ tion mode (e.g. tensile test) r-values strongly influence the thinning process of sheet deformation since higher r-values promote in-plane deformation (ε1>0, ε2<0). As a result, Ti-6Al-4V sheet is very resistant to thinning due to the high normal anisotropy during

<sup>+</sup> = (26)

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*t e* q f e e

**Figure 10.** Burst and not burst samples of Ti-6Al-4V alloy

#### **Measurements/calculations of bulge radius**

In order to measure the bulge radius, several photographs of the bulged samples were taken. The camera was stabilized parallel to the blank. By using Solid Works software, 3-point circle was fitted to the bulge geometry and the radius of the bulge was measured. Fig.11 shows photographs of the bulged samples in stepwise approach. Measured and calculated bulge radii were compared. Among the compared approaches, Panknin's calculations for bulge radius yielded values closer to step-wise bulge results for the alloy tested. Fig.12 shows the compar‐ ison of calculated bulge radii with the step-wise measured bulge radius.

**Figure 11.** Photography of the bulged sample in stepwise approach

Measurements/calculations of thickness at the dome apex

**Burst pressure (Bars) Ti-6Al-4V** Sample 1 305 Sample 2 307 Sample 3 308 Sample 4 309 Average 306

In order to measure the bulge radius, several photographs of the bulged samples were taken. The camera was stabilized parallel to the blank. By using Solid Works software, 3-point circle was fitted to the bulge geometry and the radius of the bulge was measured. Fig.11 shows photographs of the bulged samples in stepwise approach. Measured and calculated bulge radii were compared. Among the compared approaches, Panknin's calculations for bulge radius yielded values closer to step-wise bulge results for the alloy tested. Fig.12 shows the compar‐

ison of calculated bulge radii with the step-wise measured bulge radius.

**Table 6.** Burst pressure for different samples

96 Titanium Alloys - Advances in Properties Control

**Figure 10.** Burst and not burst samples of Ti-6Al-4V alloy

**Measurements/calculations of bulge radius**

**Figure 11.** Photography of the bulged sample in stepwise approach

**Figure 12.** Comparison of calculated and measured bulge radii for Ti-6Al-4V

Eqs.3-5 were used to calculate the sheet thickness at the dome apex and the results were compared with the step-wise experiments. For measurement of the thickness at the dome apex, a 10mm diameter circle was imprinted on the centre of each bulge sample. After each step, the major and minor diameters of the formed circle were measured by using an accurate caliper (0.05 mm accuracy). Subsequently, the effective strain was obtained for each step at the dome apex by replacing the instantaneous sheet thickness obtained from Eq.26 into Eq.7. The instantaneous thickness of the dome apex was extracted from Eq.8 and is shown as follows:

$$t = \frac{t\_0}{e^{(\varepsilon\_\theta + \varepsilon\_\phi)}}\tag{26}$$

The results show that for calculating the sheet thickness at the dome apex, Kruglov's approach gives best results when compared with step-wise experiments. Fig.13 shows the comparison of calculated sheet thickness at the dome apex with the step-wise measured sheet thickness at the dome of the bulge.

As discussed before (Table 5) higher n-value indicates better stretchability and formabili‐ ty, therefore, for the same bulge height, sheet materials with larger n-values have lower thinning than sheet materials with smaller n-values. Also, when drawing is the deforma‐ tion mode (e.g. tensile test) r-values strongly influence the thinning process of sheet deformation since higher r-values promote in-plane deformation (ε1>0, ε2<0). As a result, Ti-6Al-4V sheet is very resistant to thinning due to the high normal anisotropy during

Influence of material properties and anisotropy on material deformation

**Figure 13.** Comparison of calculated and measured thickness at the dome apex for Ti-6Al-4V alloy

drawing deformations. On the other hand, when stretching is the deformation mode (e.g. bulge test), then both ε1 and ε2 are positive and thinning has to occur by constancy of volume (in this case the n-value having a strong influence). This experimental conclusion directly validates the numerical finding obtained by Gutscher et al. [2]. Their FE simula‐ tions indicated that anisotropy had very small influence on the correlation between the dome wall thickness at the apex of the dome and the dome height. They also concluded that anisotropy had no significant effect on the radius at the apex of the dome.

indicates, that balanced biaxial bulge test covers larger strain range than tensile test. Under balanced biaxial loading, the theoretical effective strain at instability is twice the instability strain under uniaxial loading. Comparing the data between uniaxial and bulge tests (Fig. 15 and Table 7), it can be seen that strain values obtained in the bulge test are higher than in the tensile test. This is an advantage of the bulge test, especially if the flow stress data is to be used for FE simulation, since no extrapolations is needed as in the case of tensile data. Moreover, in Table 7 it is shown that the percent difference is as high as 504% (for Ti-6Al-4V). This empha‐ sizes the importance of the bulge test because of its capability to provide data for a wider range of strain compared to the traditional tensile test. Also, the constant scaling factor (kb) [11], which transforms biaxial stress-strain relationships into the effective stress-strain curves, is

**Figure 14.** Comparison of measured and calculated flow stress curves for Ti-6Al-4V alloy

Maximum true strain that can be obtained in tensile test (up to uniform elongation) 0.115

Maximum true strain obtained in the bulge test 0.58

Maximum difference between tensile test and bulge test, (%) 504

**Table 7.** Comparison between the maximum true strain in tensile and hydroforming bulge tests

**Ti-6Al-4V**

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0.937 for the tested material.

#### **Determination of flow stress curves**

Several flow stress curves were calculated for Ti-6Al-4V titanium sheet by using several proposed approaches discussed in previous sections. Calculated flow stress curves for titanium alloy were first corrected for anisotropy according to Eqs.9 and 10. Corrected curves are depicted in Fig.14. Step-wise measurements of stress-strain relationships in biaxial test are also shown in the same figure. As it can be seen, step-wise experiments are in good agreement with calculated flow stress curves when Kruglov's and Panknin's approaches are used for dome thickness and bulge radius calculations, respectively.

Fig.15 shows an overall comparison between flow stress curves obtained from tensile (up to instability point) and hydroforming bulge test. A constant scaling parameter was applied to transform biaxial stress-strain curve into effective flow stress curve which can be compared with the uniaxial curve [11]. Kruglov's sheet thickness calculation combined with Panknin's bulge radius calculation was used to obtain these curves. In this figure, tensile curves are depicted along the direction in which the highest elongation was obtained. This comparison

**Figure 14.** Comparison of measured and calculated flow stress curves for Ti-6Al-4V alloy

drawing deformations. On the other hand, when stretching is the deformation mode (e.g. bulge test), then both ε1 and ε2 are positive and thinning has to occur by constancy of volume (in this case the n-value having a strong influence). This experimental conclusion directly validates the numerical finding obtained by Gutscher et al. [2]. Their FE simula‐ tions indicated that anisotropy had very small influence on the correlation between the dome wall thickness at the apex of the dome and the dome height. They also concluded

Several flow stress curves were calculated for Ti-6Al-4V titanium sheet by using several proposed approaches discussed in previous sections. Calculated flow stress curves for titanium alloy were first corrected for anisotropy according to Eqs.9 and 10. Corrected curves are depicted in Fig.14. Step-wise measurements of stress-strain relationships in biaxial test are also shown in the same figure. As it can be seen, step-wise experiments are in good agreement with calculated flow stress curves when Kruglov's and Panknin's approaches are used for

Fig.15 shows an overall comparison between flow stress curves obtained from tensile (up to instability point) and hydroforming bulge test. A constant scaling parameter was applied to transform biaxial stress-strain curve into effective flow stress curve which can be compared with the uniaxial curve [11]. Kruglov's sheet thickness calculation combined with Panknin's bulge radius calculation was used to obtain these curves. In this figure, tensile curves are depicted along the direction in which the highest elongation was obtained. This comparison

that anisotropy had no significant effect on the radius at the apex of the dome.

**Figure 13.** Comparison of calculated and measured thickness at the dome apex for Ti-6Al-4V alloy

**Determination of flow stress curves**

98 Titanium Alloys - Advances in Properties Control

dome thickness and bulge radius calculations, respectively.

Influence of material properties and anisotropy on material deformation

indicates, that balanced biaxial bulge test covers larger strain range than tensile test. Under balanced biaxial loading, the theoretical effective strain at instability is twice the instability strain under uniaxial loading. Comparing the data between uniaxial and bulge tests (Fig. 15 and Table 7), it can be seen that strain values obtained in the bulge test are higher than in the tensile test. This is an advantage of the bulge test, especially if the flow stress data is to be used for FE simulation, since no extrapolations is needed as in the case of tensile data. Moreover, in Table 7 it is shown that the percent difference is as high as 504% (for Ti-6Al-4V). This empha‐ sizes the importance of the bulge test because of its capability to provide data for a wider range of strain compared to the traditional tensile test. Also, the constant scaling factor (kb) [11], which transforms biaxial stress-strain relationships into the effective stress-strain curves, is 0.937 for the tested material.


**Table 7.** Comparison between the maximum true strain in tensile and hydroforming bulge tests

**Figure 15.** Comparison of effective flow stress curves obtained from tensile and bulge tests

#### **5.3. Determination of the FLD**

High resolution photography was employed to measure the diameters of the deformed circles imprinted on the samples (Fig.16). Non-deformed circles were used for calibration of the pictures. As a result, deformed circles were measured using measuring techniques in Solid‐ works software. Ellipses located in the fractured region, were considered as unsafe points. Likewise, ellipses with one row offset from the fractured region were considered as marginal points and ellipses located in other rows of imprinted grids were considered as safe points. Eqs.27 and 28, were used to obtain true major strain (*ε1*) and true minor strain (*ε2*) from the measured diameters considering the approach shown in Fig.17.

$$
\varepsilon\_1^\* = L\iota \frac{d\_1}{d\_0} \tag{27}
$$

BBC2000 yield functions and hardening model was expressed by Swift equation. As it can be observed in Fig.18, although curves predicted using Hill93-Swift model and M-K with BBC2000 are in rather good agreement with the experimental data, the best prediction is obtained when M-K model when the yield surface of Hill93 and initial geometrical defect (*f0*) of 0.955, are used. As it can be observed in Fig.18, for Ti-6Al-4V sheet alloy, curve predicted using Hill-Swift model has small deviation from the experimental data from uni-axial region (*ρ=-1/2*) to equi-biaxial region (*ρ=1).* Moreover, slight difference in stretching region of the FLD between the experimental data points and the theoretical curve can be seen when BBC2000 is

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Note: As discussed by Graf and Hosford [29], applying prestrain in biaxial tension (*ρ=1*) will decrease the formability if followed by plane strain or biaxial tension. Moreover, for uniaxial tension sample (*ρ=-1/2*), if both prestrain and final testing with ε<sup>1</sup> are applied normal to the rolling direction, the FLD will be increased for subsequent plane strain and biaxial regions. Furthermore, for plane strain sample (*ρ=0*) a slight increase of the overall level of the curve is expected when prestrain and final testing with ε1 are applied normal to the rolling direction.

used as the yielding surface for the M-K model.

**Figure 16.** The ruptured tensile and bulge specimens

Influence of n-value and r-value on FLD

**Figure 17.** Deformation of the grid of circles to ellipses.

$$
\varepsilon\_2^\* = \text{L}\boldsymbol{\nu} \frac{d\_2}{d\_0} \tag{28}
$$

#### **Experimental determination and theoretical calculation of the FLD**

One of the most important factors for prediction of FLD through the M-K analysis is the applied constitutive yield model. Fig.18 presents the experimental and numerical forming limits for Ti-6Al-4V titanium alloy. For numerical analysis, yield surfaces were described by Hill93 and BBC2000 yield functions and hardening model was expressed by Swift equation. As it can be observed in Fig.18, although curves predicted using Hill93-Swift model and M-K with BBC2000 are in rather good agreement with the experimental data, the best prediction is obtained when M-K model when the yield surface of Hill93 and initial geometrical defect (*f0*) of 0.955, are used. As it can be observed in Fig.18, for Ti-6Al-4V sheet alloy, curve predicted using Hill-Swift model has small deviation from the experimental data from uni-axial region (*ρ=-1/2*) to equi-biaxial region (*ρ=1).* Moreover, slight difference in stretching region of the FLD between the experimental data points and the theoretical curve can be seen when BBC2000 is used as the yielding surface for the M-K model.

Note: As discussed by Graf and Hosford [29], applying prestrain in biaxial tension (*ρ=1*) will decrease the formability if followed by plane strain or biaxial tension. Moreover, for uniaxial tension sample (*ρ=-1/2*), if both prestrain and final testing with ε<sup>1</sup> are applied normal to the rolling direction, the FLD will be increased for subsequent plane strain and biaxial regions. Furthermore, for plane strain sample (*ρ=0*) a slight increase of the overall level of the curve is expected when prestrain and final testing with ε1 are applied normal to the rolling direction.

**Figure 16.** The ruptured tensile and bulge specimens

**5.3. Determination of the FLD**

100 Titanium Alloys - Advances in Properties Control

High resolution photography was employed to measure the diameters of the deformed circles imprinted on the samples (Fig.16). Non-deformed circles were used for calibration of the pictures. As a result, deformed circles were measured using measuring techniques in Solid‐ works software. Ellipses located in the fractured region, were considered as unsafe points. Likewise, ellipses with one row offset from the fractured region were considered as marginal points and ellipses located in other rows of imprinted grids were considered as safe points. Eqs.27 and 28, were used to obtain true major strain (*ε1*) and true minor strain (*ε2*) from the

> \* 1 1

> \* 2 2

e

e

**Experimental determination and theoretical calculation of the FLD**

0 *<sup>d</sup> Ln d*

0 *<sup>d</sup> Ln d*

One of the most important factors for prediction of FLD through the M-K analysis is the applied constitutive yield model. Fig.18 presents the experimental and numerical forming limits for Ti-6Al-4V titanium alloy. For numerical analysis, yield surfaces were described by Hill93 and

= (27)

= (28)

measured diameters considering the approach shown in Fig.17.

**Figure 15.** Comparison of effective flow stress curves obtained from tensile and bulge tests

Influence of n-value and r-value on FLD

**Figure 17.** Deformation of the grid of circles to ellipses.

Generally, there are two material properties which have significant influence on forming limit diagram; the anisotropy and the work-hardening exponent (n-value). R-values less than one (rave<1), will result in the reduction of limit strains in drawing side of the FLD and lower levels for the FLD in plane strain region is expected. For R-values larger than one (rave>1), the opposite trend is expected

Fig.19 compares the FLDs for 1.08 mm thick Ti-6Al-4V sheet determined in this study and the same sheet investigated by Djavanroodi and Derogar [19] during hydroforming deep drawing process. Frictional effect between the toolset and the sheet were not considered in their study. Moreover, in hydroforming deep drawing process, the strain rate is far different from the bulge and the tensile test method. Consequently, the experimental procedure and the strain rate are

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Figs.20 and 21 show the numerical results generated by *Autoform 4.4.* The mechanical proper‐ ties were input into the software, and the yield surface and FLD were generated as shown in Fig.20. Fig.21 shows that engraved circles were deformed and their shapes were changed to ellipses. The major and minor diameters of the ellipses were measured in the software to simulate the FLDs through the FE method. Fig.22 shows the comparison of the experimental and numerical FLDs. As shown in the figure, employing Hill's yield criterion for Ti sheet will result in better prediction of the FLDs compared to experimental data. Predicted FLDs using the industrial sheet metal forming code showed that the shape of the yield loci will have influence on the level of the FLD. Moreover, Fig.22 shows that necking points predicted by Hill's yield criterion and BBC yield criterion stand in good agreement compared to the

two reasons for this offset between the experimental FLD results for Ti-6Al-4V sheet.

**Figure 19.** Comparison of forming limit diagrams for Ti-6Al-4V alloy.

**5.4. Finite element analysis**

Consequently, among the tested sheets, Ti-6Al-4V severely resists to thinning during the sheet metal forming processes.

For most materials, forming limit curve intersects the major strain axis at the point equivalent to n-value. As n-value decreases, the limit strain level decreases. For Ti-6Al-4V, the major strain values are approximately 0.14 and 0.16 at plane strain region of the FLD.

**Figure 18.** Experimental and calculated forming limit diagram for Ti-6Al-4V alloy.

Fig.19 compares the FLDs for 1.08 mm thick Ti-6Al-4V sheet determined in this study and the same sheet investigated by Djavanroodi and Derogar [19] during hydroforming deep drawing process. Frictional effect between the toolset and the sheet were not considered in their study. Moreover, in hydroforming deep drawing process, the strain rate is far different from the bulge and the tensile test method. Consequently, the experimental procedure and the strain rate are two reasons for this offset between the experimental FLD results for Ti-6Al-4V sheet.

**Figure 19.** Comparison of forming limit diagrams for Ti-6Al-4V alloy.

#### **5.4. Finite element analysis**

Generally, there are two material properties which have significant influence on forming limit diagram; the anisotropy and the work-hardening exponent (n-value). R-values less than one (rave<1), will result in the reduction of limit strains in drawing side of the FLD and lower levels for the FLD in plane strain region is expected. For R-values larger than one (rave>1), the opposite

Consequently, among the tested sheets, Ti-6Al-4V severely resists to thinning during the sheet

For most materials, forming limit curve intersects the major strain axis at the point equivalent to n-value. As n-value decreases, the limit strain level decreases. For Ti-6Al-4V, the major strain

values are approximately 0.14 and 0.16 at plane strain region of the FLD.

trend is expected

metal forming processes.

102 Titanium Alloys - Advances in Properties Control

Comparison of experimental FLD with other works

**Figure 18.** Experimental and calculated forming limit diagram for Ti-6Al-4V alloy.

Figs.20 and 21 show the numerical results generated by *Autoform 4.4.* The mechanical proper‐ ties were input into the software, and the yield surface and FLD were generated as shown in Fig.20. Fig.21 shows that engraved circles were deformed and their shapes were changed to ellipses. The major and minor diameters of the ellipses were measured in the software to simulate the FLDs through the FE method. Fig.22 shows the comparison of the experimental and numerical FLDs. As shown in the figure, employing Hill's yield criterion for Ti sheet will result in better prediction of the FLDs compared to experimental data. Predicted FLDs using the industrial sheet metal forming code showed that the shape of the yield loci will have influence on the level of the FLD. Moreover, Fig.22 shows that necking points predicted by Hill's yield criterion and BBC yield criterion stand in good agreement compared to the experimental marginal points, the overall comparison shows a fair agreement between FE results and data obtained from the experiments. The small deviation between numerical and experimental results may be the conclusion of frictional effects between hemispherical punch (in FE simulation) and the sheet metals. While frictional effects remained as an unknown, in order to define the FLDs for different materials, either procedures without frictional effects should be employed or the effect of friction should be taken into account.

**Figure 21.** Deformed specimens simulated using *Autoform* software

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**Figure 22.** Comparison of the experimental FLD with the ones obtained by FEM for Ti-6Al-4V alloy

**Figure 20.** Numerical flow stress, yield locus and FLD generated in *Autoform* software.

**Figure 21.** Deformed specimens simulated using *Autoform* software

experimental marginal points, the overall comparison shows a fair agreement between FE results and data obtained from the experiments. The small deviation between numerical and experimental results may be the conclusion of frictional effects between hemispherical punch (in FE simulation) and the sheet metals. While frictional effects remained as an unknown, in order to define the FLDs for different materials, either procedures without frictional effects

should be employed or the effect of friction should be taken into account.

104 Titanium Alloys - Advances in Properties Control

**Figure 20.** Numerical flow stress, yield locus and FLD generated in *Autoform* software.

**Figure 22.** Comparison of the experimental FLD with the ones obtained by FEM for Ti-6Al-4V alloy

Although the forming behavior of materials can be well expressed through uni-axial tensile tests, the theoretical prediction of FLD may still lie in large deviations from the experimentally determined FLDs. This finding proves that suitable theoretical approaches depend not only on the thorough understanding of the forming behavior of materials, but also on the suppo‐ sitions for yield surfaces as well as material specifications.

**Nomenclature**

*ta 0* , *tb*

Nomenclature Description *dd* Die diameter *hb* Bulge height

*p* Bulge pressure

*n* Work hardening exponent

*m* Strain rate sensitivity factor

*R* Average normal anisotropy

*Rf* Upper die fillet radius Δ*R* Planar anisotropy

σ¯ *isotropic* Isotropic effective stress σ¯ *anisotropic* Anisotropic effective stress

σ0, σ<sup>90</sup> Yielding stresses obtained from tensile tests

σ¯ Effective stress σ1, σ<sup>2</sup> Principle stresses

σ*<sup>b</sup>* Biaxial yield stress εθ, εϕ, ε*<sup>t</sup>* Principal strains

ε*¯* Effective strain ε<sup>0</sup> Pre-strain ˙ε Strain rate

ε1, ε<sup>2</sup> Major and minor limit strain

*d*ε2*a*, *d*ε2*<sup>b</sup>* Strains parallel to the notch

*f0* Geometrical defect *R*0, *R*45, *R*<sup>90</sup> Anisotropy coefficients *K* Strength coefficient

*Rb* Bulge radius *Rd* Die radius

*t0,t* Initial thickness, instantaneous thickness *c, p, q* Coefficients of Hill'93 yield criterion *a ,b, c, k* Material parameters in BBC2000 criterion

*d ,e, f, g* Anisotropy coefficients of material in BBC2000 yield criterion

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*<sup>0</sup>* Initial thicknesses at homogeneous and grooved region
