**4. Numerical approach**

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

92 Titanium Alloys - Advances in Properties Control

**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

**Specimens no. A (mm) B (mm) C (mm) D (mm) E (mm) F (mm) G (mm)** 150 17 75 12 28 R20 - 150 40 45 16 52.5 R26 - 150 40 12 12 69 R6 8

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 tensiontension side of the FLD (Fig.8).

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 after each time step to evaluate the numerical FLD.


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


**Property**

Yield stress, σy (MPa) 544 558 571 Ultimate tensile stress, σuts (MPa) 632 607 629 Work-hardening exponent, n 0.151 0.134 0.167 Hardening coefficient, K (MPa) 975 912.5 1022 Total elongation, δ(%) 30.7 27.2 28.0 Anisotropy factor, r 2.4644 4.1218 3.8292

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

Normal anisotropy, Rave 3.6343 Poisson's ratio, υ 0.342

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

**5.2. Hydroforming bulge test**

**Investigation of bursting pressure**

**Ti-6Al-4V 0º 45º 90º**

Formability Characterization of Titanium Alloy Sheets

http://dx.doi.org/10.5772/55889

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

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