**3. Results and Discussion**

This section is organized in two main parts. In the first part, we make use of the principal strain space and focus on material flow and failure by cracking in incremental tube expansion. Results obtained from conventional tube expansion are included for comparison purposes. In the second part, we discusses the application of different integral forms of stress triaxiality in the effective strain vs. stress triaxiality space to solve an apparent contradiction: on one hand, the in-plane strains of incremental tube expansion exceed the threshold admissible values of the FFL, which were previously determined by means conventional tube forming processes, and on the other hand, the FFL is a material property, and therefore, its threshold values cannot be surpassed and must be independent of any type of applied loading.

#### *3.1. Material Flow and Cracking*

Figure 8 shows the experimental in-plane strains along the longitudinal cross sections of the incremental tube expansion specimens in principal strain space. The results were obtained by ARGUS® (refer to the open triangular markers) for three test repetitions consisting of eight forming stages each. Numerical predictions obtained by finite element modelling with DEFORM™-3D are enclosed and confirm that incremental tube expansion subject the material to biaxial stretching conditions.

**Figure 8.** Experimental vs. finite element predicted in-plane strains for conventional tube expansion with a rigid tapered conical punch (red markers) and incremental tube expansion with a single point hemispherical tool after eight forming stages (black markers).

The in-plane strains at the onset of fracture by tension are represented by the solid black triangular marker and its determination involved measuring the tube wall thickness at the vicinity of the incipient cracking zone and calculating the through-thickness strain *ε*<sup>3</sup> *<sup>f</sup>* at fracture to obtain the 'gauge length' strains *ε*1 *f* ,*ε*<sup>2</sup> *<sup>f</sup>* . 3 1 2

This alternative procedure was necessary because neither ARGUS ® nor DEFORM™- 3D could provide the in-plane strains at fracture. In fact, the application of circle grids with very small diameters to obtain the in-plane strains in the cracked region by means of ARGUS ® is not feasible because it creates measurement problems and delivers values that cannot be considered fracture strains due to inhomogeneous material deformation around the cracks. Similar problems exist in finite element modelling with the use of very refined meshes in the regions where the cracks are likely to be triggered, plus the additional difficulty resulting from these results being sensitive to mesh size.

Under these circumstances, the authors had to measure the tube wall thickness in a NICON® SMZ800 optical microscope equipped with a NIS-Elements ® D software version 5.11.01. Figure 9 shows a longitudinal cross-section detail after completion of the incremental tube expansion process with the corresponding evolution of thickness along the longitudinal direction (starting from the upper tube end). As seen, two different regions may be distinguished: (i) a first region (labeled as "I") located near the upper tube end that is characterized by a sharp decrease in wall thickness and (ii) a second region (labelled as "II"), in which the wall thickness progressively increases, as the distance to the upper tube end increases and approaches the undeformed region (not subjected to incremental expansion), along which the tube wall thickness *t*<sup>0</sup> = 2 mm remained unchanged. <sup>0</sup> = 2

**Figure 9.** (**a**) Detail of a tube section after the incremental expansion and (**b**) evolution of the tube wall thickness with the longitudinal distance to the upper tube end.

Two main conclusions can be inferred from Figure 9: (i) failure by cracking is not preceded by necking and (ii) failure by cracking is related to a sharp decrease in the tube wall thickness in a small region "I" subjected to a great amount of straining. As seen, there is no localized thickness reduction in the detail of the tube section after the last forming stage of incremental expansion. This observation combined with the monotonic increase in the strain loading path up to fracture shown in Figure 8 (refer to the black triangular markers) allow concluding that failure occurs without previous necking.

A closer observation of the tube wall thickness within region "I" confirms the existence of micro-cracks along its length, as it was previously stated by Cristino et al. [16], and justifies the reason why the experimental determination of the "gauge length" strains  *ε*1 *f* ,*ε*<sup>2</sup> *<sup>f</sup>* at fracture was made at point "A" (Figure 9) located 1.5 mm away from the upper tube end, in the transition between regions "I" and "II". 1 2

The finite element predicted evolution of the in-plane strains at point "A" for each individual stage of the incremental tube expansion process is shown in Figure 10. As mentioned before, the "gauge length" strains *ε*1 *f* ,*ε*<sup>2</sup> *<sup>f</sup>* at fracture (refer to the black solid triangular marker) were not obtained by finite elements and their determination made use of the tube wall thickness value at point "A" (Figure 9b) to calculate the through-thickness strain *ε*<sup>3</sup> *<sup>f</sup>* at fracture. 1 2 3

$$\pounds\_{\vec{\mathbb{S}}^{\mathcal{Y}}\_{3}} = \pounds\_{\tpharrow}^{\pounds\_{\tpharrow}} \exists\_{\mathsf{h}} \ln{\frac{0.63}{2}} = \text{--1.16} \tag{7}$$

Then, assuming material incompressibility and the final slope *β* of the strain loading path to remain identical to that of the last piecewise linear path obtained by ARGUS® (Figure 8), it was possible to determine the 'gauge length' strains *ε*1 *f* ,*ε*<sup>2</sup> *<sup>f</sup>* at fracture, as follows: 1 2

$$
\mathfrak{gl} = \frac{\mathfrak{e}\_2}{\mathfrak{e}\_1} = 0.54 \quad \leadsto \quad \varepsilon \mathfrak{e}\_{\mathfrak{f}f} = -\frac{\varepsilon \mathfrak{f} \mathfrak{g} \mathfrak{f}}{1 \; \mathfrak{f} \mathfrak{f} \mathfrak{f}} = 0.75 \quad \varepsilon\_2 \mathfrak{e}\_{2/\mathfrak{f}} \mathfrak{f} \mathfrak{e}\_{\mathfrak{f}} \mathfrak{e}\_{1/\mathfrak{f}} = 0.40 \tag{8}
$$

The corresponding effective strain at fracture *ε<sup>f</sup>* = 1.17 was obtained from Equation (4) and defines a dashed ellipse of constant effective strain values in principal strain space (refer to both Figures 8 and 9). ̅

**Figure 10.** Finite element predicted in-plane strains of point A during the eight forming stages of incremental tube expansion.

The comparison of the results obtained for incremental tube expansion against those obtained for conventional tube expansion with a rigid tapered conical punch [9–12] allowed identifying two main differences regarding material flow and cracking. First, incremental tube expansion is performed under biaxial stretching conditions, whereas conventional tube expansion subjects the material to near pure tension. Second, both processes fail by

tensile stresses (opening mode I), but while fracture in incremental tube expansion is not preceded by necking, that is not the case in conventional tube expansion, in which fracture is preceded by localized necking.

Even though all the experimental and theoretical results presented in this section are consistent and compatible, there is a fundamental problem arising from the fact that in-plane strains of incremental tube expansion are far greater than the threshold admissible values given by the FFL. Because the FFL is a material property, whose values cannot be surpassed, Cristino et al. [16] put forward the possibility of the FFL having an upward curvature in the first quadrant of principal strain space to accommodate the values in excess (i.e., to accommodate the in-plane strains located above the straight line falling from left to right), but they did not provide evidence for this type of tube material.

In connection to this, it is worth noticing that recent published works in incremental sheet forming also report the existence of strain paths that go beyond the FFL determined by means of conventional sheet forming tests with proportional strain paths [22].

The following section focuses on this problem and aims at providing an explanation for the reason why the critical strains of incremental tube expansion at fracture are far superior to those of conventional tube expansion that is simultaneously compatible with the FFL being a material property, whose threshold values cannot be surpassed by any type of loading. The explanation will make use of the effective strain vs. stress triaxiality space instead of the principal strain space.
