**4. Results and Discussion**

In this section, the 60 × 110 × 4 elements mesh distribution is used to present the results. The effect of *kw* on the force as a function of tensile strain under ambient pressure (*α* = 0.0) is shown in Figure 5. It is clearly observed that the fracture delays with a decrease

in *kw*. Shear damage growth increases with an increase in *k<sup>w</sup>* according to Equation (4). Therefore, the total void volume fraction increases with *k<sup>w</sup>* as shown in Equation (5). It should be noted that the result for the case with *k<sup>w</sup>* = 0.0 corresponds to using the conventional GTN model with the effect of shear damage mechanism not being considered. Additionally, the deformed shape of the fractured specimen is shown in Figure 6. Necking and localized deformation is clearly observed in the specimen. Damage is very low, close to zero, before necking and it starts to grow when localized deformation happens. = 0.0 <sup>௪</sup> <sup>௪</sup> <sup>௪</sup> <sup>௪</sup> = 0.0 = 0.0 <sup>௪</sup> <sup>௪</sup> <sup>௪</sup> <sup>௪</sup> = 0.0

∗

 **Figure 5.** Effect of *kw* on the normalized force–tensile strain curve. 

**Figure 6.** Deformed shape of the specimen after fracture.

As mentioned previously, the effect of pressure on ductility and bendability has been determined in [1,4] using the conventional GTN model when the effect of the shear damage mechanism is not considered. It is explained in [1,4] that the superimposed hydrostatic pressure lowers the stress triaxiality, which retards void growth and increases the fracture strain. In the present study, the influence of superimposed hydrostatic pressure *p* = −*ασ<sup>y</sup>* on fracture under tension is considered while accounting for the shear damage mechanism by using the modified GTN model. Figure 7 shows the effect of *α* on the force–tensile strain ( = −௬)

curve. It was found that as *α* increases, the tensile strength is unaffected and the fracture strain of the material increases. 

**Figure 7.** Effect of superimposed hydrostatic pressure on normalized force–tensile strain curves.

(ଵଵ + ଶଶ + ଷଷ) Figure 8 shows the volumetric strain (*ε*<sup>11</sup> + *ε*<sup>22</sup> + *ε*33) at the center of the specimen for sheets under a range of superimposed hydrostatic pressures. It is found that the volumetric strain decreases with increasing *α* as shown in Figure 8. According to Equation (1), the decrease in volumetric strain renders void growth less favorable and leads to higher ductility.

**Figure 8.** Effect of superimposed hydrostatic pressure on volumetric strain at the center of the specimen.

ு = (1 3⁄ )൫<sup>௫௫</sup> + <sup>௬௬</sup> + ௭௭൯ ఙಹ ఙഥ The delay of void growth and the concomitant increase in fracture strain caused by the increase in applied pressure can be explained in terms of how this pressure influences the hydrostatic pressure and stress triaxiality at the center of the specimen. Figure 9 presents the hydrostatic pressure *σ<sup>H</sup>* = (1/3) *σxx* + *σyy* + *σzz* and stress triaxiality *<sup>σ</sup><sup>H</sup> σ* at the center of the specimen, where fracture initiates as a function of tensile strain under

=0

various superimposed hydrostatic pressures. At room pressure *p* = 0, both hydrostatic pressure and stress triaxiality develop in a way to assist void growth. However, under a superimposed hydrostatic pressure *p* = −*ασy*, both values are initially compressive. This result implies that void growth is delayed until a sufficiently large component of tensile stress is introduced.

**Figure 9.** Effect of superimposed hydrostatic pressure on (**a**) pressure and (**b**) stress triaxiality at the center of the specimen.

The effect of superimposed double-sided pressure on the formability of a biaxially stretched age-hardenable aluminum sheet metal (AA6111-T4) was studied in [33], where the researchers numerically employed the GTN model. It was found that double-sided pressure increased formability while void nucleation remained invariable. Furthermore, only the extent of void growth was observed to change, decreasing with an increase in pressure. Figure 10a shows the effect of superimposed hydrostatic pressure on void nucleation. It is demonstrated that the final value of nucleated void volume fraction is not a function of superimposed hydrostatic pressure as the GTN model used in this study assumes that the nucleation is strain controlled (Equations (2) and (3)). A previous study [1] investigated the effect of superimposed hydrostatic pressure on bendability by using both the strain- and stress-controlled void nucleating GTN model. It was found that the final value of nucleating void volume fraction is constant when the strain void nucleating GTN model is used [1]. On the contrary, the final value of nucleating void volume fraction decreases with increasing pressure when the stress-controlled void nucleating GTN model is used. However, the effect of superimposed hydrostatic pressure on void growth is shown in Figure 10b and it is clearly seen that hydrostatic pressure delays void growth. The reduction in void growth due to an increase in the hydrostatic pressure has been reported in other studies for sheets under tension [4] and bending [28]. Figure 10c shows the effect of hydrostatic pressure on the prevalence of the shear damage mechanism and it is clearly observed that it dominates at higher values of hydrostatic pressure. As mentioned previously, the void sheet mechanism is excluded under external applied pressure, leaving shear decohesion as the dominant failure mechanism [7]. It is interesting to note that while the shear damage mechanism becomes more dominant as pressure is increased, both it and the void growth mechanism calculated using the modified GTN model become more delayed as the superimposed hydrostatic pressure is increased. Finally, Figure 10d shows the total void volume fraction under various superimposed hydrostatic pressures. It is found that the total void volume fraction delays and it will be shown that it causes the fracture strain to increase.

(**a**) Void nucleation volume fraction

**Figure 10.** *Cont*.

(**b**) Void growth volume fraction

(**c**) Shear void growth volume fraction

**Figure 10.** *Cont*.

ln 

(**d**) Total void volume fraction

 = ௧ ௧

**Figure 10.** Effect of superimposed hydrostatic pressure on (**a**) void nucleation volume fraction, (**b**) void growth volume fraction, (**c**) shear void growth volume fraction and (**d**) total void volume fraction at the center of the specimen.

= =

ቀ ቁ ൫൯ = Figures 11 and 12 plot the influence of superimposed hydrostatic pressure on the normalized minimum cross-section area *Amin Ao* and the fracture strain *εf* in the middle section of the specimen, respectively. Here, the fracture strain *ε<sup>f</sup>* is defined as *ε<sup>f</sup>* = ln *<sup>A</sup><sup>o</sup> Amin* , where *Amin* is the minimum cross-sectional area of the sheet when fracture is complete. It is to be noted that *Amin* = *tmin* and *A<sup>o</sup>* = *t<sup>o</sup>* considering the plane strain condition and in this way, *<sup>A</sup><sup>o</sup> Amin* = *to tmin* . The following equation will be obtained to calculate *ε<sup>f</sup>* :

$$
\varepsilon\_f = \ln \frac{t\_o}{t\_{min}} \tag{9}
$$

 = ln It is found that the minimum cross-sectional area follows an inverse relationship with the level of hydrostatic pressure. Therefore, as the pressure increases, the minimum cross-sectional area at fracture decreases and the specimen can deform more before failure, which is manifested as an increase in fracture strain.

**Figure 11.** Effect of superimposed hydrostatic pressure on normalized minimum cross-sectional area.

**Figure 12.** Effect of pressure on fracture strain.
