*3.2. FFL and Stress Triaxiality under Non-Proportional Paths*

Damage accumulation associated to growth and coalescence of voids subjected to tensile normal stresses (Mode I) accounts for the dilatational effects related to stress triaxiality *η* = *σm*/*σ*, in the form of a weighted integral form of the effective plastic strain [23–25]. ⁄�

$$D = \int\_0^{\overline{\varepsilon}} \frac{\sigma\_m}{\overline{\sigma}} d\overline{\varepsilon} \tag{9}$$

The critical damage *Dcrit* at the onset of fracture (FFL) corresponds to the maximum admissible accumulated value of effective strain *ε* = *ε<sup>f</sup>* for a given strain path. ̅ ̅

The accumulation of damage *D* in principal strain space often distinguishes between two different types of strain paths: (i) linear, proportional strain paths (Figure 11a) and (ii) non-proportional strain paths, which are often discretized through a series of piecewise linear strain paths for calculation purposes (Figure 11b). 

**Figure 11.** (**a**) Linear, proportional strain path in principal strain space, (**b**) non-proportional strain path discretized through a series of piecewise linear strain paths in principal strain space, (**c**) representation of the strain paths (**a**,**b**) in the effective strain vs. stress triaxiality space.

 <sup>2</sup> ⁄<sup>1</sup> As shown in Figure 11c, the application of Equations (3) and (4) to linear, proportional strain paths, characterized by a constant slope *β* = *dε*<sup>2</sup> /*dε*<sup>1</sup> (Figure 11a), gives rise to vertical lines *η* = *η<sup>p</sup>* in the effective strain vs. stress triaxiality space. In contrast, the application of Equations (3) and (4) to non-proportional, piecewise linear strain paths with different slopes *β<sup>i</sup>* , gives rise to piecewise linear evolutions *ε* = *f*(*ηi*) (hereafter referred to as *ηpiecewise* based evolutions) in the effective strain vs. stress triaxiality space.

The experimental strain paths disclosed in Figure 8 allow concluding that tube expansion by a rigid tapered conical punch subject the material to linear, proportional (or near proportional) strain paths, whereas incremental tube expansion subjects the material to non-proportional strain paths. The picture inserts of Figure 11a,b are drawn in accordance with this conclusion.

However, the strain paths determined by CGA using the automatic measurement system ARGUS® must be seen as static results obtained at the end of the incremental tube expansion process (Figure 8), or at the end of each intermediate forming stage (Figure 10). Full characterization of the non-proportional strain paths of incremental tube forming with detailed information on the cyclic oscillations from shearing to biaxial stretching, as the single point hemispherical tool approaches, contacts and moves away from a specific location of the incrementally expanded tube surface can only be obtained through finite element modelling.

Figure 12 provides a schematic representation of a finite element computed nonproportional, cyclic path undergone by a specific tube location in the effective strain vs. stress triaxiality space. Three different evolutions *ε* = *f*(*η*) are considered as a result of the following three approaches to account for the accumulation of damage *D* in nonproportional, cyclic paths:

(a) The envelope stress triaxiality *ηenv* based approach (Figure 12a).

$$D\_{\rm env} = \eta\_{\rm env} \overline{\varepsilon} \; \rightarrow \; \eta\_{\rm env} = \frac{1}{\overline{\varepsilon}} \int\_0^{\overline{\varepsilon}} \left( \frac{\sigma\_m}{\overline{\sigma}} \right)\_{\rm max} d\overline{\varepsilon} \tag{10}$$

where (·)*max* stands for the peak values of the stress triaxiality ratio at each cycle (circular path) of the forming tool.

(b) The average positive stress triaxiality *ηpos* based approach (Figure 12b).

$$D\_{pos} = \overline{\eta}\_{pos} \mathbb{E} \; \rightarrow \; \overline{\eta}\_{pos} = \frac{1}{\overline{\varepsilon}} \int\_0^{\overline{\varepsilon}} \langle \frac{\sigma\_m}{\overline{\sigma}} \rangle d\overline{\varepsilon} \tag{11}$$

where h·i corresponds to the Macaulay bracket to prevent accumulation of negative damage.

(c) The average stress triaxiality *η* based approach (Figure 12c), where *D* = *D* of Equation (9).

$$
\overline{D} = \overline{\eta}\overline{\varepsilon} \quad \rightarrow \ \overline{\eta} = \frac{1}{\overline{\varepsilon}} \int\_0^{\overline{\varepsilon}} \frac{\sigma\_m}{\overline{\sigma}} d\overline{\varepsilon} \tag{12}
$$

where *D* = *D* of Equation (9).

The three evolutions *ε* = *f*(*η*) resulting from these approaches are identical in case of linear, proportional strain paths because in such loading conditions, *D* = *Denv* = *Dpos* = *D*.

Figure 13 shows the finite element non-proportional, cyclic path of incremental tube forming experienced by point A of Figure 9 and the three different *ε* = *f*(*η*) evolutions that result from the integral forms of stress triaxiality *ηenv*, *ηpos* and *η* given by Equations (10)–(12). The linear piecewise *ηpiecewise* based evolution resulting from the experimental in-plane strains obtained by ARGUS® and by the linear, proportional, equal biaxial stretching *ηp* based evolution are included for comparison purposes.

̅= () ̅ ̅ **Figure 12.** Schematic representation of the non-proportional, cyclic path of incremental tube expansion experienced by an arbitrary tube location with a plot of the *ε* = *f*(*η*) evolutions based on the three different integral forms of stress triaxiality: (**a**) envelope stress triaxiality *ηenv*, (**b**) average positive stress triaxiality *ηpos* and (**c**) average stress triaxiality *η*. 

 ̅ = () **Figure 13.** Finite element computed non-proportional, cyclic path of point A (Figure 9) with several *ε* = *f*(*η*) evolutions obtained from different assumptions and integral forms of stress-triaxiality.

 ̅ ̅ As seen, the *ε* = *f*(*η*) reference evolution based on a linear, proportional, equal biaxial stress triaxiality ratio *η<sup>p</sup>* consists of a vertical line *η<sup>p</sup>* = 0.66 that extends up to an effective strain value at fracture *ε<sup>f</sup>* = 1.17 located far above the FFL. The other *ε* = *f*(*η*) reference evolution based on a linear piecewise *ηpiecewise* approximation of the experimental in-plane strains measured by ARGUS ® is not very much different from that based on *ηp*. Major differences between the two evolutions are found in the forming stages 2 to 5 due to a shift in the linear piecewise *ηpiecewise* based evolution towards plane strain.

̅ Still, the onset of fracture at *ε<sup>f</sup>* = 1.17 is nearly identical to that of the *η<sup>p</sup>* based evolution and, therefore, far above the FFL. In fact, because the linear piecewise *ηpiecewise* based evolution is built upon a direct transformation of the experimental strain paths from principal strain space to the effective strain vs. stress triaxiality space, it is understandable that the surpass of the FFL must occur in both spaces.

More important to our discussion are the *ε* = *f*(*η*) evolutions obtained for the integral forms of stress triaxiality given by *ηenv*, *ηpos* and *η* (refer to Equations (10)–(12)). As can be seen, the three evolutions reach the effective strain at fracture (*ε<sup>f</sup>* = 1.17) very far from the FFL. In particular, the evolution of *ηenv* cuts the FFL at stress triaxiality values around 0.4, suggesting that the fracture should occur much earlier than it does. The other two (*ηpos* and *η*) reach the fracture for values of stress triaxiality below 0.2 (i.e., in-between pure tension and pure shear) without crossing the FFL and in good agreement with a possible extrapolation of the FFL to the left side. The difference between the *ηpos* and *η* based evolutions is not relevant for incremental tube expansion and derives from discharging, or accounting for, the accumulation of negative damage. Although discharging negative damage is commonly executed in cold forming, there are studies recently published pointing to cut-off values of stress triaxiality up to −0.6 for the cold forming of aluminum alloys under quasi-static loading [26]. According to this and taking into account that the instantaneous stress triaxiality in the incremental tube forming oscillates between −0.6 to 0.6 (see Figure 13), the use of the average stress triaxiality *η* takes on a greater physical sense.

Taking the integral form of stress triaxiality *η* (i.e., the average stress triaxiality given by Equation (12)) into consideration, it is now important to check if the compatibility between the FFL and the above-mentioned reason for the critical in-plane strains of incremental tube expansion at fracture being far greater than those of conventional tube expansion also applies to the latter. For this purpose, we computed the *ε* = *f*(*η*) evolution for conventional tube expansion directly from the average stress triaxiality *η* and plotted the results in Figure 14. The instantaneous stress triaxiality (*η*) in the conventional tube expansion is also shown.

**Figure 14.** Finite element computed evolutions of the loading paths experienced by a point A located 1.5 mm away from the upper tube end in incremental and conventional tube expansion processes. Note: the red and blue triangular markers correspond to the experimentally determined "gauge length" strains at fracture.

̅ ≈

̅ ≈

≅ ̅

Two interesting results can be drawn. On the one hand, the level of average stress triaxiality in the conventional process at fracture (*η <sup>f</sup>* ≈ 0.47) is very well above the one obtained in the incremental process (*η <sup>f</sup>* ≈ 0.11). As suggested by Martinez-Donaire et al. [18], this difference, also observed in other incremental forming processes [27], results in a greater resistance to accumulate damage in the incremental process than in the conventional one, requiring higher levels of strain to trigger the ductile fracture. On the other hand, results also show a near coincidence of the instantaneous and the average stress triaxiality-based evolutions (*η* ∼= *η*) in the conventional process. This also makes sense and is compatible with the widely popular application of McClintock's ductile damage criterion [23] to determine the onset of cracking by tension in conventional tube forming processes [12]. These results confirm the validity of the overall approach for both nonproportional, cyclic paths of incremental tube expansion and near proportional paths of conventional tube expansion.
