**2. Theoretical approaches**

**1.2. Forming Limit Diagrams (FLD)**

82 Titanium Alloys - Advances in Properties Control

changes [28-29], and heat treatment [30].

bulge test is comparatively preferred [41].

As mentioned before, due to increased demand for light weight components in aerospace, automotive and marine industries, recently, titanium sheet alloys have gained ever more interests in production of structural parts. In order to better understand the cold formability of these alloys, their behavior in sheet metal forming operations must be determined both experimentally and theoretically. Sheet metal formability is often evaluated by forming limit diagram. The concept of FLD was first introduced by Keeler [16] and Goodwin [17]. Forming limit diagram provides the limiting strains a sheet metal can sustain whilst being formed. Laboratory testing has shown that the forming limit diagrams are influenced by several factors including strain hardening exponent and anisotropy coefficients [18-20], strain rate [21-23], temperature [24], grain size and microstructure [25-26], sheet thickness [27], strain path

In recent years many experimental techniques have been developed to investigate the FLDs from different aspects [31-34]. These studies were based on elimination of frictional effects resulted from toolsets and materials, the uniformity of the blank surface and mechanical

The available tests for the determination of FLDs include: hydraulic bulge test [35], Keeler punch stretching test [36], Marciniak test [37], Nakazima test [38], Hasek test [39] and the biaxial tensile test using cruciform specimen [40] (in short cruciform testing device). From previous studies [36-39], it is widely acknowledged that friction remains an unknown factor yet to be effectively characterized and understood. Thus, the list of available tests is greatly reduced to only two options - hydraulic bulge test and cruciform testing device. Further analysis shows that due to simplicity of equipment and specimen (i.e. less costly), hydraulic

On the other hand, several researchers have proposed a number of analytical models to predict FLD. Hill's localized instability criterion [42], combined with Swift's diffused instability criterion [43] was the first analytical approach to predict FLDs. It was shown that forming limit curves are influenced by material work hardening exponent and anisotropy coefficient. Xu and Wienmann [44] showed that for prediction of the FLD, the shape of the used yield surface had a direct influence on the limit strains. They used the Hill'93 criterion to study the effect of material properties on the FLDs. The M-K model [45] predicts the FLD based on the assumption of an initial defect in perpendicular direction with respect to loading direction. The assumption made for this non-homogeneity factor is subjective and hence, the forming limit diagram is directly influenced by it. This method

Several researchers used the ductile fracture criteria for forming limit predictions in hydro‐ forming process [48], deep drawing process [49], bore-expanding [50] and biaxial stretching [51]. Fahrettin and Daeyong [52] and Kumar et al. [53] proposed the thickness variations and the thickness gradient criterion respectively. These criteria are limited because they require precise measurements of thickness. Bressan and Williams [54] used the method of shear instability to predict the FLDs. Consequently, experimental techniques are widely accepted

properties of sheet materials deduced from the conventional tensile testing.

was then developed considering the material properties [46-47].

### **2.1. Hydroforming bulge test**

Hydroforming bulge test is one of the most commonly used balanced biaxial tests in which a circular sheet metal fully clamped between two die surfaces is drawn within a die cavity by applying hydrostatic pressure on the inner surface of the sheet. The die cavity diameter (*dd =2Rd*), the upper die fillet radius (*Rf* ) and initial sheet thickness (*t0*) are constant parameters of any hydroforming bulge testing. Instantaneous variables of biaxial test are: bulge pressure (*p*), dome height (*hb*), bulge radius (*Rb*) and sheet thickness at the dome apex (*t*). The schematic view of hydroforming bulge test is shown in Fig.1.

**Figure 1.** Scheme of the hydroforming bulge test

In order to obtain flow stress curve, first, a combination of constant and variable parameters are introduced to several equations proposed by the other researchers to calculate the instan‐ taneous bulge radius [6, 57] and the sheet thickness [6, 58-59] at the dome apex. Subsequently, by making the assumption that during the bulging process the sheet metal behavior is the same as thin-walled structure and by implementing the classical membrane theory, the flow stress curves are obtained. Eqs.1 and 2 represent the theories for calculating the bulge radius proposed by Hill [6] and Panknin [57], respectively:

$$R\_b = \frac{d\_d + 4h\_b}{8h\_b} \tag{1}$$

( ) ( )

*R R*

*Sin R R* é ù = ê ú ê ú ë û

/ / *d b d b*

The second stage for calculating the flow stress curves for sheet material is implementing the classical membrane theory. Due to very low ratio of thickness to radius of the sheet (*t/Rd* <<0.1), the stress component in perpendicular direction to sheet surface is not considered (*σz=0*). By considering Tresca's yield criterion, Gutscher et al. [2], proposed an equation to evaluate the

( 1)

= + (6)

Formability Characterization of Titanium Alloy Sheets

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 j*t t*

 eeé ù = - +- +- ê ú ë û (7)

> *t t*

1/2 <sup>2</sup> <sup>2</sup>

*R R*

1/2

= + =- = (8)

2 *b*

*p R t*

Principle strains at dome are: *εθ*, *εϕ* and *εt*. Assuming Von-Mises yield criterion and equality

( ) ( ) ( ) <sup>2</sup> 2 2 <sup>2</sup>

 q

It is known that due to the principle of volume constancy (*εθ* + *εϕ* + *ε<sup>t</sup>* =0), plastic deformation

 e

Since due to rolling conditions, sheet metal properties differ in various directions with respect to rolling direction (anisotropy), effective stress and effective strain components should be corrected for anisotropy. In Equations 6 and 8, no anisotropy correction was introduced. Consequently, assuming Hill'48 yield criterion in conjunction with plane stress assumption, Smith et al. [5], proposed equations 9 and 10 for determination of effective stress and effective

<sup>0</sup> ln *isotropic <sup>t</sup>*

strains for sheet metals considering the average normal anisotropy (*R*), respectively.

( 1) *anisotropic isotropic*

*anisotropic R R*

<sup>=</sup> - +

é ù = - ê ú ë û <sup>+</sup>

2 (2 (2 / 1)) *isotropic*

e

 s

q f

 ee  ee

*isotropic*

qj

 ee

s

0 1

*t t*

effective stress resulted from the hydroforming bulge test:

*εθ* =*εϕ*, the effective strain can be calculated as:

e

does not yield any volume change [2, 60]:

9 *isotropic*

e

s

e

2

(5)

85

(9)

(10)

$$R\_b = \frac{\left(R\_d + R\_f\right)^2 + h\_b^2 - 2R\_f h\_b}{2h\_b} \tag{2}$$

Eqs.3 to 5 represents the theories for calculating the instantaneous sheet thickness at the dome apex proposed by Hill [6], Chakrabarty et al. [58] and Kruglov et al. [59], respectively. In this chapter, Eqs.1 to 5 was used in the theoretical approach.

$$t = t\_0 \left(\frac{1}{1 + \left(h\_b / R\_d\right)^2}\right)^2\tag{3}$$

$$t = t\_0 \left(\frac{1}{1 + \left(h\_b / R\_d\right)^2}\right)^{2-n} \tag{4}$$

Formability Characterization of Titanium Alloy Sheets http://dx.doi.org/10.5772/55889 85

$$t = t\_0 \left[ \frac{\left(R\_d / R\_b\right)}{Sin^{-1}\left(R\_d / R\_b\right)} \right]^2 \tag{5}$$

The second stage for calculating the flow stress curves for sheet material is implementing the classical membrane theory. Due to very low ratio of thickness to radius of the sheet (*t/Rd* <<0.1), the stress component in perpendicular direction to sheet surface is not considered (*σz=0*). By considering Tresca's yield criterion, Gutscher et al. [2], proposed an equation to evaluate the effective stress resulted from the hydroforming bulge test:

$$
\overline{\sigma}\_{isentropic} = \frac{p}{2} (\frac{R\_b}{t} + 1) \tag{6}
$$

Principle strains at dome are: *εθ*, *εϕ* and *εt*. Assuming Von-Mises yield criterion and equality *εθ* =*εϕ*, the effective strain can be calculated as:

**Figure 1.** Scheme of the hydroforming bulge test

84 Titanium Alloys - Advances in Properties Control

proposed by Hill [6] and Panknin [57], respectively:

*b*

*R*

chapter, Eqs.1 to 5 was used in the theoretical approach.

*t t*

*t t*

In order to obtain flow stress curve, first, a combination of constant and variable parameters are introduced to several equations proposed by the other researchers to calculate the instan‐ taneous bulge radius [6, 57] and the sheet thickness [6, 58-59] at the dome apex. Subsequently, by making the assumption that during the bulging process the sheet metal behavior is the same as thin-walled structure and by implementing the classical membrane theory, the flow stress curves are obtained. Eqs.1 and 2 represent the theories for calculating the bulge radius

> 4 8 *d b*

<sup>+</sup> <sup>=</sup> (1)

(3)

(4)

+ +- <sup>=</sup> (2)

*d h <sup>R</sup> h*

( )<sup>2</sup> <sup>2</sup> <sup>2</sup>

2 *d f b fb*

*R R h Rh*

*h*

*b*

Eqs.3 to 5 represents the theories for calculating the instantaneous sheet thickness at the dome apex proposed by Hill [6], Chakrabarty et al. [58] and Kruglov et al. [59], respectively. In this

( )

*h R*

( )


*h R*

*b d*

0 2 1

ç ÷ <sup>=</sup> ç ÷ <sup>+</sup> è ø

1 /

æ ö ç ÷ <sup>=</sup> ç ÷ <sup>+</sup> è ø

0 2 1 1 / *b d*

2

2

*n*

*b*

*b*

$$\overline{\varepsilon}\_{isentropic} = \sqrt{\frac{2}{9} \left[ \left( \varepsilon\_{\theta} - \varepsilon\_{\varphi} \right)^{2} + \left( \varepsilon\_{\theta} - \varepsilon\_{t} \right)^{2} + \left( \varepsilon\_{\varphi} - \varepsilon\_{t} \right)^{2} \right]} \tag{7}$$

It is known that due to the principle of volume constancy (*εθ* + *εϕ* + *ε<sup>t</sup>* =0), plastic deformation does not yield any volume change [2, 60]:

$$
\varepsilon \overline{\varepsilon}\_{isotropic} = \varepsilon\_{\theta} + \varepsilon\_{\phi} = -\varepsilon\_{t} = \ln \frac{t\_0}{t} \tag{8}
$$

Since due to rolling conditions, sheet metal properties differ in various directions with respect to rolling direction (anisotropy), effective stress and effective strain components should be corrected for anisotropy. In Equations 6 and 8, no anisotropy correction was introduced. Consequently, assuming Hill'48 yield criterion in conjunction with plane stress assumption, Smith et al. [5], proposed equations 9 and 10 for determination of effective stress and effective strains for sheet metals considering the average normal anisotropy (*R*), respectively.

$$
\overline{\sigma}\_{anisotropic} = \overline{\sigma}\_{isropic} \left[ 2 - \frac{2R}{(R+1)} \right]^{1/2} \tag{9}
$$

$$\overline{\mathcal{E}}\_{\text{anisotropic}} = \frac{\mathcal{D}\varepsilon\_{\text{isotropic}}}{\left(\mathcal{Q} - \{\mathcal{D}\mathcal{R} / \mathcal{R} + 1\}\right)^{1/2}} \tag{10}$$

#### **2.2. Forming Limit Diagrams**

#### **Hill93 and BBC2000 constitutive models**

In this paper Hill 93 and BBC2000 constitutive models were used to predict the FLDs. Eq.11 represents Hill'93 yield criterion [55]:

$$\frac{\sigma\_1^2}{\sigma\_0^2} - \frac{c\sigma\_1\sigma\_2}{\sigma\_0\sigma\_{90}} + \frac{\sigma\_2^2}{\sigma\_{90}^2} + \left[ (p+q) - \frac{\left(p\sigma\_1 + q\sigma\_2\right)}{\sigma\_b} \right] \frac{\sigma\_1\sigma\_2}{\sigma\_0\sigma\_{90}} = 1\tag{11}$$

**Theoretical prediction of the FLD**

is characterized by Eq.15, where *t*

**Figure 2.** Schematic description of M-K model

grooved region, respectively.

assumed.

The simulation of plastic instability is performed using M–K and Hill-Swift analysis. The rigid plastic material model with isotropic work hardening and the plane stress condition were

A detailed description of the theoretical M–K analysis, schematically illustrated in Fig.2, can be found in [45]. The M-K model is based on the growth of an initial defect in the form of a narrowband perpendicular to the principal axis. The initial value of the geometrical defect (*f0*)

> 0 0 0

*b a t f t*

*<sup>0</sup>* are the initial thicknesses in the homogeneous and

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87

= (15)

*a 0* and *tb*

$$\begin{aligned} p &= \left[ \frac{2R\_0(\sigma\_b - \sigma\_{90})}{(1 + R\_0)\sigma\_0^2} - \frac{2R\_{90}\sigma\_b}{(1 + R\_{90})\sigma\_{90}^2} + \frac{c}{\sigma\_0} \right] \frac{1}{\frac{1}{\sigma\_0} + \frac{1}{\sigma\_{90}} - \frac{1}{\sigma\_b}}\\ \frac{c}{\sigma\_0 \sigma\_{90}} &= \frac{1}{\sigma\_0^2} + \frac{1}{\sigma\_{90}^2} - \frac{1}{\sigma\_b^2} \end{aligned} \tag{12}$$
 
$$\frac{1}{\sigma\_b} \frac{1}{\sigma\_b} = \left[ \frac{2R\_{90}(\sigma\_b - \sigma\_0)}{(1 + R\_{90})\sigma\_{90}^2} - \frac{2R\_0\sigma\_b}{(1 + R\_0)\sigma\_0^2} + \frac{c}{\sigma\_{90}} \right] \frac{1}{\frac{1}{\sigma\_0} + \frac{1}{\sigma\_{90}} - \frac{1}{\sigma\_b}} \tag{12}$$

Where *c, p* and *q* are Hill'93 coefficients and can be calculated using five mechanical parameters obtained from two uni-axial tensile tests and an equi-biaxial tension (Eq.12).

Banabic et. al [56] proposed a new yield criterion called BBC2000 for orthotropic sheet metals under plane stress conditions. The equivalent stress is defined as:

$$\overline{\sigma} = \left[ a \left( b \Gamma + c \Psi \right)^{2k} + a \left( b \Gamma - c \Psi \right)^{2k} + \left( 1 - a \right) \left( 2c \Psi \right)^{2k} \right]^{\frac{1}{2k}} \tag{13}$$

Where *a, b, c* and *k* are material parameters, while *Γ* and *Ψ* are functions of the second and third invariants of a fictitious deviatoric stress tensor and can be expressed as explicit dependencies of the actual stress components:

$$\begin{aligned} \Gamma &= (d+e)\sigma\_{11} + (e+f)\sigma\_{22} \\ \Psi &= \sqrt{\left[\frac{1}{2}(d-e)\sigma\_{11} + \frac{1}{2}(e-f)\sigma\_{22}\right]^2 + g^2 \sigma\_{xy}^2} \end{aligned} \tag{14}$$

Where *d,e, f* and *g* are anisotropy coefficients of material and *k*-value is set in accordance with the crystallographic structure of the material (k=3 for BCC alloys and k=4 for FCC alloys). For the BBC2000 yield model, the detailed description can be found in [56].

#### **Theoretical prediction of the FLD**

**2.2. Forming Limit Diagrams**

86 Titanium Alloys - Advances in Properties Control

**Hill93 and BBC2000 constitutive models**

2 2

 s

 s

0 90 0 90 90 0 <sup>0</sup>

 ss

s s

represents Hill'93 yield criterion [55]:

s

s

22 2

 s

111

ss

s

of the actual stress components:

*c*

s s

In this paper Hill 93 and BBC2000 constitutive models were used to predict the FLDs. Eq.11

*b*

s

2 2 0 0 90 90 0

2 2 1

+ + ë û + -

2 2 1

+ + ë û + -

s

2 2 90 90 0 0 90

 s s s

 ss

1 1 111

s

s

 s

s

1 1 111

 s

s

é ù + - + + +- ê ú = ë û

1

0 90

ss

*b*

 s

*b*

 s

0 90

ss

1

(11)

(12)

(14)

( ) ( ) 2 2 1 12 2 1 2 1 2

> s s

*c p q p q*

0 0 90 90 0 90

( )

*<sup>p</sup> R R*

s

( )

s

*b b b*

s s

=+ - é ù - = -+ ê ú

obtained from two uni-axial tensile tests and an equi-biaxial tension (Eq.12).

under plane stress conditions. The equivalent stress is defined as:

( ) ( )

*de e f*

s

G= + + +

1 1

the BBC2000 yield model, the detailed description can be found in [56].

s

( ) ( )

é ù Y= - + - + ê ú ë û

11 22

*<sup>q</sup> R R*

( ) ( )

0 90 90

*b b*

é ù - = -+ ê ú

*R R c*

*R R c*

( ) ( )

Where *c, p* and *q* are Hill'93 coefficients and can be calculated using five mechanical parameters

Banabic et. al [56] proposed a new yield criterion called BBC2000 for orthotropic sheet metals

Where *a, b, c* and *k* are material parameters, while *Γ* and *Ψ* are functions of the second and third invariants of a fictitious deviatoric stress tensor and can be expressed as explicit dependencies

11 22

*de e f g*

 s

2 2 *xy*

Where *d,e, f* and *g* are anisotropy coefficients of material and *k*-value is set in accordance with the crystallographic structure of the material (k=3 for BCC alloys and k=4 for FCC alloys). For

22 2 <sup>2</sup> 1 2 *kk k <sup>k</sup>*

*ab c ab c a c* é ù = G+ Y + G- Y + - Y ê ú ë û (13)

2

ss

2 2

( ) ( ) ( )( )

The simulation of plastic instability is performed using M–K and Hill-Swift analysis. The rigid plastic material model with isotropic work hardening and the plane stress condition were assumed.

A detailed description of the theoretical M–K analysis, schematically illustrated in Fig.2, can be found in [45]. The M-K model is based on the growth of an initial defect in the form of a narrowband perpendicular to the principal axis. The initial value of the geometrical defect (*f0*) is characterized by Eq.15, where *t a 0* and *tb <sup>0</sup>* are the initial thicknesses in the homogeneous and grooved region, respectively.

**Figure 2.** Schematic description of M-K model

$$f\_0 = \frac{t\_0^b}{t\_0^a} \tag{15}$$

The x, y, z-axes correspond to rolling, transverse and normal directions of the sheet, whereas 1 and 2 represent the principal stress and strain directions in the homogeneous region. The set of axis bound to the groove is represented by *n, t, z*-axes where 't' is the longitudinal one. The plastic flow occurs in both regions, but the evolution of strain rates is different in the two zones. When the flow localization occurs in the groove at a critical strain in homogeneous region, the limiting strain of the sheet is reached. Furthermore, the major strain is assumed to occur along the X-axis. M–K necking criterion assumes that the plastic flow localization occurs when the equivalent strain increment in imperfect region (b) reaches the value ten times greater than in homogeneous zone (a) (*dεb>10dεa*). When the necking criterion is reached the computation is stopped and the corresponding strains (ε<sup>a</sup> xx, ε<sup>a</sup> yy) obtained at that moment in the homogeneous zone are the limit strains. For the model, equation expressing the equilibrium of the forces acting along the interface of the two regions could be expressed as follow:

$$
\sigma\_{1a} t\_a = \sigma\_{1b} t\_b \tag{16}
$$

standard [64]. Tensile test was carried out under constant strain rate of 1×10-3 s-1 at room

Although r-value is introduced as the ratio of width strain *εw* to thickness strain *εt*, the thickness strain in thin sheets can not be accurately measured. Hence, by measuring longitudinal *ε<sup>l</sup>*

width strains and also by implementing the principle of volume constancy (Eq.20), the

0 *lwt*

( ) *t lw*

The strain ratio (r-value) was calculated for all the materials at different direction (0, 45 and 90˚ to the rolling direction) (Eq.22). Subsequent to that, normal anisotropy *R* (Eq.23) as well as

> , , *w x*

e

e

*t x*

0 45 90 2 4 *R RR*

0 90 45 2 2 *RR R*

The experimental apparatus used to conduct the hydraulic bulge test is composed of a tooling set, a hydraulic power generator and measurement devices. For the assembled toolset, maximum forming pressure can reach 500bars. To avoid any oil leakage during the forming process, a rubber diaphragm was placed between the conical part of the die and the conjunctive disc. A pressure gage and a dial indicator were used to measure the chamber pressure and bulge height, respectively during the bulging process. The indicator used in the experiments was delicate and could not withstand impact loads as the specimen bursts. Hence, for bulge testing of titanium sheets, at least three samples were burst in the absence of the indicator to discern the bursting pressures. Other samples were tested up to 90-95% of bursting pressure while the indicator was used to measure the bulge height during the process. In order to ensure pure stretching, the pre-fabricated draw bead was implemented at the flange area of the bulge

+ -

 ee

 e

+ += (20)

Formability Characterization of Titanium Alloy Sheets

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=- + (21)

<sup>=</sup> (22)

+ + <sup>=</sup> (23)

D = (24)

ee

planar anisotropy *ΔR* (Eq.24) were calculated according to ASTM E517-00 [64].

*x*

*R*

*R*

*R*

Where: *x* is the angle relative to the rolling direction (0˚, 45˚, 90˚).

**3.2. Hydroforming bulge test**

e

and

89

temperature.

thickness strain was obtained as follows (Eq.21):

Strains parallel to the notch are equal in both regions:

$$d\varepsilon\_{2a} = d\varepsilon\_{2b} \tag{17}$$

In addition, the model assumes that the strain ratio in zone *a,* is constant during the whole process:

$$d\varepsilon\_{2a} = \rho d\varepsilon\_{1a} \tag{18}$$

Detailed FLD calculation using BBC2000 yield criterion is presented in [61]. Swift workhardening model with strain rate sensitivity factor was used:

$$
\overline{\sigma} = K \left( \varepsilon\_0 + \overline{\varepsilon} \right)^n \dot{\varepsilon}^m \tag{19}
$$

In this chapter, in addition to M-K analysis, the Swift analysis [43] with Hill'93 yield criterion was used to predict the FLDs [62-63].

#### **3. Experimental work**

#### **3.1. Tensile test**

Tensile test specimens were cut according to ASTM E8 standard. At least two samples at each direction (0˚,45˚,90˚) with respect to rolling directions were tested according to ASTM E517-00 standard [64]. Tensile test was carried out under constant strain rate of 1×10-3 s-1 at room temperature.

Although r-value is introduced as the ratio of width strain *εw* to thickness strain *εt*, the thickness strain in thin sheets can not be accurately measured. Hence, by measuring longitudinal *ε<sup>l</sup>* and width strains and also by implementing the principle of volume constancy (Eq.20), the thickness strain was obtained as follows (Eq.21):

$$
\varepsilon\_l + \varepsilon\_w + \varepsilon\_t = 0 \tag{20}
$$

$$
\varepsilon\_t = -\left(\varepsilon\_l + \varepsilon\_w\right) \tag{21}
$$

The strain ratio (r-value) was calculated for all the materials at different direction (0, 45 and 90˚ to the rolling direction) (Eq.22). Subsequent to that, normal anisotropy *R* (Eq.23) as well as planar anisotropy *ΔR* (Eq.24) were calculated according to ASTM E517-00 [64].

$$\mathcal{R}\_x = \frac{\mathcal{E}\_{w,x}}{\mathcal{E}\_{t,x}} \tag{22}$$

$$R = \frac{R\_0 + 2R\_{45} + R\_{90}}{4} \tag{23}$$

$$
\Delta R = \frac{R\_0 + R\_{90} - 2R\_{45}}{2} \tag{24}
$$

Where: *x* is the angle relative to the rolling direction (0˚, 45˚, 90˚).

#### **3.2. Hydroforming bulge test**

The x, y, z-axes correspond to rolling, transverse and normal directions of the sheet, whereas 1 and 2 represent the principal stress and strain directions in the homogeneous region. The set of axis bound to the groove is represented by *n, t, z*-axes where 't' is the longitudinal one. The plastic flow occurs in both regions, but the evolution of strain rates is different in the two zones. When the flow localization occurs in the groove at a critical strain in homogeneous region, the limiting strain of the sheet is reached. Furthermore, the major strain is assumed to occur along the X-axis. M–K necking criterion assumes that the plastic flow localization occurs when the equivalent strain increment in imperfect region (b) reaches the value ten times greater than in homogeneous zone (a) (*dεb>10dεa*). When the necking criterion is reached the computation is

xx, ε<sup>a</sup>

1 1 *aa bb*

2 2 *a b d d* e

2 1 *a a d d* e

 re

( ) <sup>0</sup>

 e ee

 e

In addition, the model assumes that the strain ratio in zone *a,* is constant during the whole

Detailed FLD calculation using BBC2000 yield criterion is presented in [61]. Swift work-

In this chapter, in addition to M-K analysis, the Swift analysis [43] with Hill'93 yield criterion

Tensile test specimens were cut according to ASTM E8 standard. At least two samples at each direction (0˚,45˚,90˚) with respect to rolling directions were tested according to ASTM E517-00

*n m*

 s

acting along the interface of the two regions could be expressed as follow:

s

zone are the limit strains. For the model, equation expressing the equilibrium of the forces

yy) obtained at that moment in the homogeneous

*t t* = (16)

= (17)

= (18)

= + *K* & (19)

stopped and the corresponding strains (ε<sup>a</sup>

88 Titanium Alloys - Advances in Properties Control

process:

Strains parallel to the notch are equal in both regions:

hardening model with strain rate sensitivity factor was used:

was used to predict the FLDs [62-63].

**3. Experimental work**

**3.1. Tensile test**

s

The experimental apparatus used to conduct the hydraulic bulge test is composed of a tooling set, a hydraulic power generator and measurement devices. For the assembled toolset, maximum forming pressure can reach 500bars. To avoid any oil leakage during the forming process, a rubber diaphragm was placed between the conical part of the die and the conjunctive disc. A pressure gage and a dial indicator were used to measure the chamber pressure and bulge height, respectively during the bulging process. The indicator used in the experiments was delicate and could not withstand impact loads as the specimen bursts. Hence, for bulge testing of titanium sheets, at least three samples were burst in the absence of the indicator to discern the bursting pressures. Other samples were tested up to 90-95% of bursting pressure while the indicator was used to measure the bulge height during the process. In order to ensure pure stretching, the pre-fabricated draw bead was implemented at the flange area of the bulge samples. Consequently, pure stretching of the sheet material was obtained during the bulging process. Measuring devices were also calibrated before the test to ensure precise measure‐ ments. For bulge testing of sheet materials a die set was used. For bulge testing of Ti-6Al-4V, a large die set was designed and manufactured in order to reach the bursting pressure through available hydraulic pressure unit. Table 1 shows dimensions of the die set in addition to specifications of hydroforming bulge test apparatus. Hydroforming die used for bulge testing is shown in Fig.3.


**Table 1.** Specifications of hydroforming bulge apparatus.

**Figure 3.** Hydroforming bulge test apparatus

#### **3.3. Forming limit diagrams**

In order to evaluate the FLDs, different strain paths, which cover full domain of the FLD, were examined and shown in Fig.4. These paths are spanned between the uniaxial tension region (ε1=-2ε2) and the equi-biaxial stretching (ε1=ε2). The linear strain paths are described through the strain ratio (Eq.25) parameter representative of the strain state.

$$
\rho = \frac{d\varepsilon\_2}{d\varepsilon\_1} \tag{25}
$$

**Figure 4.** Different types of specimen representative of the linear strain paths

shown in Figures 5-7 and the dimensions are listed in Table 2.

parallel and perpendicular, respectively, to the rolling direction of the sheet.

which were the conclusions of deformed circles were measured precisely.

Three different shapes of tensile specimens were prepared using wire EDM. According to Fig. 4, a non-grooved specimen was prepared to simulate the strain path#1, two distinctive grooved specimens were used to obtain draw points representing the strain paths#2 and 3. The nongrooved and grooved specimens were then drawn under a linear load. The specimens are

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91

In order to obtain the tension-tension side of the FLD (pure stretching region), the bulge test specimens were prepared. The bulge samples for simulation of the strain path#4, were 160mm in diameter and the cavity diameter of circular dies was 90mm (as shown in Table 1). LASER imprinting technique was used to print a grid of circles on the surface of the tension and the bulging samples. The circles were 2mm in diameter. The diameter of the circles of the grids have been measured before and after the deformation throughout the major and minor principal directions taking as reference a perpendicular axis system placed in the geometric centre of each circle or in the centre of the tension specimen. These principal directions are

Tensile specimens were tested up to the fracture point. Likewise, bulge testing of the bulge samples were carried out to reach the bursting point. Subsequent to that, diameters of ellipses

**3.4. Preparation of the specimens**

**Figure 4.** Different types of specimen representative of the linear strain paths

#### **3.4. Preparation of the specimens**

samples. Consequently, pure stretching of the sheet material was obtained during the bulging process. Measuring devices were also calibrated before the test to ensure precise measure‐ ments. For bulge testing of sheet materials a die set was used. For bulge testing of Ti-6Al-4V, a large die set was designed and manufactured in order to reach the bursting pressure through available hydraulic pressure unit. Table 1 shows dimensions of the die set in addition to specifications of hydroforming bulge test apparatus. Hydroforming die used for bulge testing

In order to evaluate the FLDs, different strain paths, which cover full domain of the FLD, were examined and shown in Fig.4. These paths are spanned between the uniaxial tension region (ε1=-2ε2) and the equi-biaxial stretching (ε1=ε2). The linear strain paths are described through

> 2 1

e

*d d* e

r

the strain ratio (Eq.25) parameter representative of the strain state.

Bulge diameter (2*Rd*) 90 mm (3.54 in)

Die fillet radius 6 mm (0.236 in)

Maximum chamber pressure 50 MPa (7400 psi)

Flow rate 2.5 lit/min

**Table 1.** Specifications of hydroforming bulge apparatus.

**Figure 3.** Hydroforming bulge test apparatus

**3.3. Forming limit diagrams**

**Die set specification**

<sup>=</sup> (25)

is shown in Fig.3.

90 Titanium Alloys - Advances in Properties Control

Three different shapes of tensile specimens were prepared using wire EDM. According to Fig. 4, a non-grooved specimen was prepared to simulate the strain path#1, two distinctive grooved specimens were used to obtain draw points representing the strain paths#2 and 3. The nongrooved and grooved specimens were then drawn under a linear load. The specimens are shown in Figures 5-7 and the dimensions are listed in Table 2.

In order to obtain the tension-tension side of the FLD (pure stretching region), the bulge test specimens were prepared. The bulge samples for simulation of the strain path#4, were 160mm in diameter and the cavity diameter of circular dies was 90mm (as shown in Table 1). LASER imprinting technique was used to print a grid of circles on the surface of the tension and the bulging samples. The circles were 2mm in diameter. The diameter of the circles of the grids have been measured before and after the deformation throughout the major and minor principal directions taking as reference a perpendicular axis system placed in the geometric centre of each circle or in the centre of the tension specimen. These principal directions are parallel and perpendicular, respectively, to the rolling direction of the sheet.

Tensile specimens were tested up to the fracture point. Likewise, bulge testing of the bulge samples were carried out to reach the bursting point. Subsequent to that, diameters of ellipses which were the conclusions of deformed circles were measured precisely.

**4. Numerical approach**

tension side of the FLD (Fig.8).

after each time step to evaluate the numerical FLD.

**Table 3.** Dimensions of different FLD samples prepared for FE approach

More recently, several researchers [19, 65-67] have investigated the forming limit diagrams through finite element codes. In this chapter, *Autoform Master 4.4* was employed for FE analysis of forming limit diagrams. The setting of the numerical simulation is based on the hemispher‐ ical punch and different shapes of specimens, as shown in Fig.8. Descriptions of the specimen dimensions and the geometrical model used in the simulation are shown in Table 3 and 4, respectively. The tensile properties of sheet metal were then input into the program and forming limit diagram were generated in *Autoform 4.4* software using Keeler method [16]. *Autoform 4.4* software automatically generates yield surface proposed by Banabic (BBC yield surface) and Hill for sheet materials when anisotropy coefficients and elasto-plastic behavior of sheet are imported. In *Autoform* the use of the shell element for the element formulation is mandatory, and therefore default, for the process steps Drawing, Forming, Bending and Hydroforming. Moreover, since for titanium and ultra high strength steels more complex material laws (for example Barlat or Banabic) are used, *Autoform* uses the implicit integration

algorithm which contribution to the total calculation time is substantially smaller.

In this approach, CAD data were modeled in CATIA software first and then imported into *Autoform 4.4* environment. In order to cover full range of the FLD, different specimens with different groove dimensions were modelled to simulate the tension-compression to tension-

For the FE simulation, the punch, holder and die were considered as rigid parts. A displace‐ ment rate of 1mm/s was assumed for the hemispherical punch while for the clamping a draw bead with lock mode was selected to ensure pure stretching of the sheet into die cavity. Friction coefficient was taken to be 0.15 between the surfaces. The virtual samples were engraved with the gridded pattern of 3mm diameter circles (Fig.8). Major and minor strains were recorded

> **Sample # A(mm) B(mm)** 1 100 5 2 100 12 3 100 20 4 100 30 5 100 40 6 100 50 7 100 60

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8 100 100

**Figure 5.** Uniaxial tensile specimen representative of strain path#1 (no.1).

**Figure 6.** Grooved specimen representative of strain path#2 (no.2).

**Figure 7.** Grooved specimen representative of strain path#3 (no.3).


**Table 2.** Dimensions of the tensile specimens
