## 1 Introduction  
Understanding how ductile materials accumulate damage with increasing plastic strain until reaching fracture is of great interest, as it is a key factor in engineering design to achieve the best compromises among mechanical strength, lightness, and a required level of safety. This is also crucial for establishing the deformability limits and the feasibility of cold-forming processes [1, 2, 3]. Furthermore, a better prediction of the ultimate strength of a material permits to determine whether a component can withstand overloads without reaching failure or not, increasing confidence in structural integrity, in turn. In this context, considerable efforts have been made in the last decades in estimating the conditions of plastic collapse in ductile materials, and it is now well accepted that damage accumulation is governed by the history of the stress/strain state with deformation. An early study on ductile damage accumulation and on the mechanisms influencing its evolution was conducted by [PERSON] (1968) [4] and [PERSON] (1969) [5], followed by several works from other authors [6; 7; 8], who associated failure to the evolution of the microstructure, studying the growth of cylindrical and spherical microvoids, and proved how this damage mechanism is highly influenced by the hydrostatic component of the stress tensor. Subsequently, porosity-based models were developed, guided by [PERSON]'s pioneering works (1977) [9], according to which voids volume fraction is the governing scalar variable for damage accumulation. This category of models relates fracture to the well-known mechanisms of nucleation, growth, and coalescence. [PERSON] and [PERSON] [10] provided an important contribution and enhanced the [PERSON]'s criterion, resulting in the Gurson-Tvergaard-Needleman (GTN) model, which is still well established nowadays. Another important class of damage models are those belonging to the Continuum Damage Mechanics (CDM), an approach started with the work of [PERSON] [11] and [PERSON] [12] and further developed by [PERSON] [13] and [PERSON] [14]. As opposed to other types of models that base the investigation dividing the damage evolution according to the three elementary stages (nucleation, growth, and coalescence), CDM models, using a thermodynamic framework, attempt to analyze ductile damage at a continuum scale, also including a degradation in strength and in other material properties. Concerning CDM models, several papers have been published; to mention a few: [PERSON] et al. [15], [PERSON] et al. [16], [PERSON] et al. [17], and [PERSON] et al. [18; 19; 20]. Lastly, other criteria attempt to describe the damage accumulation and final fracture at a macroscopic level through proper empirical laws based on experimental evidence. These criteria include the works of [PERSON] and [PERSON] [21], [PERSON] and [PERSON] [22], [PERSON] and [PERSON] [23]. A different general classification divides damage models into those in which damage evolution affects the plastic material behavior, called coupled models, and those in which damage and plastic flow are decoupled. The CDM and porosity-based damage models belong to the coupled models, while most empirical ones are uncoupled. Hereafter, only empirical models are considered. At the beginning, many studies [4; 5; 6; 10; 22; 24; 25] proposed damage models which are dependent on different functions of the hydrostatic pressure the material is subjected to; more specifically, damage accumulation is governed by the first invariant of the stress tensor \(I_{1}\) through the triaxiality parameters \(T\). However, these models showed a poor prediction under shear-driven failure conditions. Subsequently, in the works proposed by [PERSON] (2008) [26], [PERSON] (2005) [27], [PERSON] (2006) [28], [PERSON] (2009) [29], and [PERSON] (2015) [30], it was proved that damage depends not only on \(I_{1}\), but also on the third invariant of the deviatoric tensor \(I_{3}\) through the Lode Angle \(\theta\), introduced by [PERSON] (1926) [31].