**7. Effect of Selected Loading Conditions on Void Development**

The effect of stress triaxiality on void development and ductile failure has been very extensively documented in the literature [63,116]. Moreover, Bao and Wierzbicki [7], examining the 2024-T351 aluminum alloy, showed that the strain at failure is not a monotonic function of triaxiality, but the domain of low and high triaxiality should be distinguished (Figure 8a). In the latter case, an increase in triaxiality corresponds to a decrease in the critical strain.

**Figure 8.** (**a**) Effect of stress triaxiality on strain at failure, from [7]; (**b**) dependence of the mean void nucleation strain on stress triaxiality [47].

A similar relationship for medium and high triaxialities was obtained in the already mentioned studies [47–49], but in this case, the analysis concerned the void nucleation strain in the S355 structural steel. In the range of triaxialities from 0.516 to 1.345, along with the increase in *η*, a decrease in *ε<sup>N</sup>* was observed (Figure 8b).

High values of triaxiality favor an increase in the volume of voids. Figure 9 presents the experimentally determined (by examining the fracture surfaces) relationship between triaxiality *η* and the volume fraction of voids at failure in S355J2G3 steel.

Depending on triaxiality, the volume fraction of voids ranged between 59.7 and 71.2%. For comparison, the fraction of voids in the unstrained material was 0.09%.

The stress triaxiality affects not only the intensity of the void growth, but also their shape (Figure 10). In areas with high triaxiality, the voids are spherical in shape (a large share of the stress hydrostatic component forces the voids to grow steadily in all directions— Figure 10a).

As triaxiality decreases (decreasing influence of the spherical component of the stress tensor), the voids become more elongated (Figure 10b,c), because the process of microdamage development in these cases is mainly controlled by shear stresses [117].

As shown by Lin et al. in [118], the reduction of the triaxiality of stresses increases the level of strains at which cracking occurs. For greater triaxial stresses (axis of the specimen), the microvoids grow intensively in the plane perpendicular to the tensile, joining with each other.

Morgeneyer and Besson in [119] presented an example of the successive occurrence of both these mechanisms (regime of high and low triaxiality) in the test of a plate tearing (Kahn test). The observations made by the X-ray microtomography showed that the specimen failure was initiated inside the plate, under conditions of high triaxiality. As the microdamage propagated towards the plate faces, an increasing number of elongated voids was observed, which results from the decrease in triaxiality and the increasing role of shear in the process of void development. It is worth noting that while the shape of the voids undergoes large changes in this case, their volume fraction does not change significantly.

The shear induced failure propagation is also visible on the macroscopic level by the inclination of the fracture surface near the specimen edges (Figure 11).

**Figure 9.** Effect of stress triaxiality on the void volume fraction at failure [47].

**Figure 10.** Fracture surfaces of notched tensile specimens made of S355J2G3 steel, subjected to various stress triaxialities: (**a**) 1.345; (**b**) 0.739; (**c**) 0.516.

The development of voids under low triaxiality conditions has been intensively researched in recent years. The small share of the hydrostatic component results in a relatively low increase in the volume of voids. However, in such cases, the action of shear stresses and the associated deformation (change of shape) of the void becomes of primary importance. As it has been shown, in the face of a small value of triaxiality *η*, the development of voids is in this case are controlled by the value of the Lode parameter *ξ* (Section 3), although the exact relationships have not yet been defined.

Under shear dominant conditions, voids can take the form of penny-shaped cracks. In addition, the presence of shear stress may cause the voids to rotate (Figure 12), which additionally affects the location of their coalescence area and implies a failure mechanism [11].

**Figure 11.** Macroscopic photograph of a fracture surface of a tensile notched specimen. Transition from the flat failure mechanism in the specimen center to the slant fracture at the edges is clearly visible. The first mechanism involves normal stress, while in the latter, shear stress plays a dominant role.

**Figure 12.** Void deformation and rotation under low triaxiality.

In fact, second phase particles still remain inside the void, limiting the possibility of its deformation, especially in the situation of incomplete decohesion at the particle– matrix interface.

Current models of void development primarily take into account the prevailing state of stress, but mostly do not take into account other factors such as deformation rate, temperature, and others, which undoubtedly is of great importance in modeling engineering structures.

Some of the few works taking into account the above factors are [120,121]. It was shown that good results of material description at high temperature are achieved by the use of an Arrhenius phenomenological model [120]. The model takes into account the strain rate for an elevated tensile temperature. The tests were carried out on steel with a bainite structure.

Similarly, the analysis of the influence of very high triaxial strain rates and temperatures in the range of 300–2000 K on the process of nucleation and void development is discussed in [122]. Using the void nucleation and growth model (NAG procedure) as well as molecular dynamics (MD) simulation, the increase in void volume fraction over time was determined, taking into account the process of nucleation, growth, and coalescence. It was found that, while the increase in temperature led to a significant acceleration of the nucleation of the voids, the influence of temperature in the case of the growth of voids was slight.

Another very interesting example of the analysis of the development of voids under dynamic load conditions (spall failure) is [123], in which the phenomenon of stress wave interaction and the associated negative pressure formation, which results in rapid nucleation and void growth, was investigated. Velocimetry on the specimen free surface was used to estimate changes in stress distribution over time. A comparative analysis of the obtained stress distributions and microtomographic photographs of the voids formed made it possible to determine the critical negative pressure necessary to initiate the void, at the level of about 1–2 GPa.

The problem of dynamic development of voids has also been thoroughly discussed in the article [124]. It was found that in the first phase of the failure initiation, the inertia of the material surrounding the void slows down its development, but at a higher value of strain it promotes the void growth. Moreover, in the range of void diameters from 0.1 to 1 μm, the strain gradients around the void contributed to a significant, local increase in the yield point. On the other hand, however, the increase in thermal energy accompanying rapid deformation caused a local increase in temperature even above the melting point, thereby lowering the yield stress, and the stress values necessary to initiate a void, and thus also increasing the intensity of void development.
