*2.2. Formability Limits by Necking and Fracture*

Figure 2 shows the formability limits of the aluminum AA6063-T6 tubes by necking (FLC) and by fracture under tension (FFL) in principal strain space. Determination of the FLC required measuring the strain paths of conventional tube expansion, inversion and bulging by means of digital image correlation (DIC) and combining these results with timedependent and force-dependent methodologies that were specifically developed by the authors for tube materials [11,12]. Determination of the FFL required measuring the wall thickness of the tube cracked regions by optical microscopy (D software version 5.11.01, NIS-Elements, Tokyo, Japan)to obtain the "gauge length" strains at fracture. Information about the different tests, methods and procedures that were used by the authors to determine the FLCs and FFLs of tube materials is given in Magrinho et al. [12].

1 2 − Figure 2 also includes the strain loading path obtained in conventional tube expansion with a rigid tapered conical punch having a semi-angle of 15 ◦ , which was previously obtained by the authors [12] and will be used for reference purposes throughout this paper. As seen, the in-plane strains of conventional tube expansion at the onset of necking (*ε*1*n*,*ε*2*n*) = (−0.25, 0.41) are very close to the FLC, and the in-plane fracture strains (*ε*<sup>1</sup> *<sup>f</sup>* ,*ε*<sup>2</sup> *<sup>f</sup>* ) = (−0.25, 0.71) are exactly on top of the FFL.

1 2 − The formability limits shown in Figure 2 can alternatively be plotted in the effective strain vs. stress triaxiality space (Figure 3). The transformation of the formability limits from principal strain space into this other space can be carried out analytically by assuming linear, proportional strain paths under plane stress loading conditions (*σ<sup>t</sup>* = *σ*<sup>3</sup> ≈ 0). Plane stress loading conditions are commonly assumed in the analytical modelling of sheet and thin-wall tube forming [12,18].

**Figure 2.** Forming limit curve (FLC) and fracture forming limit (FFL) line of the aluminum AA6063- T6 tube in principal strain space. The red line represents the experimental strain loading path of conventional tube expansion with a rigid tapered conical punch having a semi-angle of 15 ◦ (adapted from [12]).

For this purpose, let us consider, for example, the tube material to be isotropic and to follow the von Mises yield criterion, so that its effective stress *σ* and effective strain *dε* are given by:

$$
\overline{\sigma} = \sqrt{\sigma\_1^2 - \sigma\_1 \sigma\_2 + \sigma\_2^2} \tag{1}
$$

$$d\overline{\varepsilon} = \frac{2}{\sqrt{3}} \sqrt{d\varepsilon\_1^2 + d\varepsilon\_1 d\varepsilon\_2 + d\varepsilon\_2^2} \tag{2}$$

� ̅ Then, applying the Levy–Mises constitutive equations, one obtains the following relation between the stress triaxiality ratio *η* = *σm*/*σ* and the slope *β* = *dε*<sup>2</sup> /*dε*<sup>1</sup> of the strain path [9]:

$$\eta = \frac{1+\beta}{\sqrt{3}\sqrt{1+\beta+\beta^2}}\tag{3}$$

̅ √ �<sup>1</sup> <sup>2</sup> 1<sup>2</sup> <sup>2</sup> 2 The above equation together with the following modified version of Equation (2) to include the slope *β* in the effective strain,

$$
\overline{\varepsilon} = \frac{2}{\sqrt{3}} \sqrt{1 + \beta + \beta^2} \varepsilon\_{1\text{ }\prime} \tag{4}
$$

 √ � <sup>2</sup> allows accomplishing the above-mentioned transformation of the FLC from principal strain space into the effective strain vs. stress triaxiality space (Figure 3).

� 2<sup>1</sup>

̅ 2 √3

**Figure 3.** Forming limit curve (FLC) and fracture forming limit (FFL) line of the aluminum AA6063- T6 tube in the effective strain vs. stress triaxiality space, obtained from analytical transformation assuming material isotropy, linear strain paths and plane stress loading conditions.

̅ The transformation of the FFL from principal strain space into the effective strain vs. stress triaxiality space requires consideration of the experimentally observed strain path deviation towards plane strain deformation conditions at the onset of necking (FLC); see, for instance, Martinez-Donaire et al. [19]. In case of the effective strain *ε*, this is realized by modifying Equation (4) to account for the two piecewise linear strain paths involving the initial path (up to necking) with a given slope *β* and the final path (from necking to fracture) with a slope *β* = 0 resulting from strain localization in the tube material:

$$\overline{\varepsilon}\_{f} = \int\_{0}^{\overline{\varepsilon}\_{n}} d\overline{\varepsilon} + \int\_{\overline{\varepsilon}\_{n}}^{\overline{\varepsilon}\_{f}} d\overline{\varepsilon} = \frac{2}{\sqrt{3}} \left[ \varepsilon\_{1f} + \left( \sqrt{1 + \beta + \beta^{2}} - 1 \right) \left( \varepsilon\_{2f} / \beta \right) \right] \tag{5}$$

1 2 ̅ In the above equation, *ε*<sup>1</sup> *<sup>f</sup>* and *ε*<sup>2</sup> *<sup>f</sup>* are the major and minor in-plane strains at fracture, and *ε<sup>f</sup>* is the effective strain at fracture.

 ̅ In case of the stress triaxiality *η*, the transformation is carried out in accordance with Martinez-Donaire et al. [19], who introduced an integral form *η <sup>f</sup>* , named average-stress triaxiality at fracture, that accounts for stress triaxiality in an average sense over the two piecewise linear strain paths:

$$\overline{\eta}\_{f} = \frac{1}{\overline{\varepsilon}\_{f}} \int\_{0}^{\overline{\varepsilon}\_{f}} \frac{\sigma\_{\mathrm{m}}}{\overline{\sigma}} d\overline{\varepsilon} = \frac{1}{\overline{\varepsilon}\_{f}} \left( \int\_{0}^{\overline{\varepsilon}\_{\mathrm{n}}} \frac{\sigma\_{\mathrm{m}}}{\overline{\sigma}} d\overline{\varepsilon} + \int\_{\overline{\varepsilon}\_{\mathrm{n}}}^{\overline{\varepsilon}\_{f}} \frac{\sigma\_{\mathrm{m}}}{\overline{\sigma}} d\overline{\varepsilon} \right) = \frac{\sqrt{3}}{3} \left[ \frac{\varepsilon\_{1f} + \varepsilon\_{2f}}{\varepsilon\_{1f} + \left( \sqrt{1 + \beta + \beta^{2}} - 1 \right) \left( \varepsilon\_{2f} / \beta \right)} \right] \tag{6}$$

The FLC and FFL resulting from the above-mentioned analytical transformation procedure are shown in Figure 3 and are slightly different from those obtained by Magrinho et al. [12] due to the following two main reasons. First, the authors made use of the von Mises yield criterion instead of the Hosford yield criterion that was utilized by Magrinho et al. [12]. Second, Magrinho et al. [12] transformed the FFL from principal strain space into the effective strain vs. stress triaxiality space by replacing the strains at fracture directly on Equations (3) and (4) instead of using the two piecewise linear strain path approach given by Equations (5) and (6), i.e., without considering the kink in the strain loading path from necking towards fracture.

To conclude, it is worth mentioning that the main reason why the Hosford yield criterion was not utilized in this work was due to its unavailability in the commercial finite element computer program utilized by the authors. Hill's 48 yield criterion was not considered as well because of the difficulty in obtaining the Lankford's coefficient at 45 ◦ in a tube, and because Cristino et al. [16] achieved good analytical estimates of material flow neglecting anisotropy.
