## 5 Fracture Initiation Model  
It is then necessary to choose a proper fracture initiation model (ductile fracture model) that can be implemented in conjunction with the FE analysis. To date, three representative approaches incorporated with FE analysis are used for predicting the fracture of materials in the cold forming. (i) Phenomenological continuum damage model (CDM): This approach utilizes the internal damage variable (scalar if the material is isotropic and tensor if it is anisotropic) to quantify the degree of deterioration in the material stiffness and strength during elastic-plastic deformation. This approach was first proposed by [PERSON] [21] to describe the creep damage and was later extended to ductile damage in elastic-plastic deformation by [PERSON] [22] and [PERSON] [23]. CDM is coupled with constitutive equations supported by thermodynamics of irreversible processes. However, CDM generally does not explicitly account for morphology and damage mechanisms in the microstructure [24] in addition, many material parameters in this model have to be identified, making them inconvenient for applications [25]. (ii) Micromechanics-based damage model (MDM). This approach uses the growth of voids in the material as an internal variable to quantify the degree of damage. [PERSON] et al. [26] first developed a void growth model by assuming a regular array of pre-existing cylindrical voids with elliptical cross-sections in a non-hardening material. [PERSON] [27] introduced a plastic potential function sensitive to porosity (i.e., the void volume fraction) and stress triaxiality so that the influence of void growth on the yield surface could be taken into account. The Gurson-Tvergaard-Needleman [28] (GTN) model that predicts ductile fracture on the basis of the nucleation, growth, and coalescence of voids in materials  
Figure 5: (**a**) Equivalent stress–strain curves of 3.0 wt% silicon steel used for the FE analysis; (**b**) Global force–displacement response of 3.0 wt% silicon steel when Swift, Voce, and combined models were used. is most widely used. Gurson-like models are generally adequate to simulate predominate tensile fracture in moderate and high stress triaxiality conditions, but fail to predict fracture in the low and negative stress triaxiality domains [29]. However, recently, a modified GTN model [30; 31] including a void shear mechanism emerged, enabling the analysis of ductile fracture behaviors under a wide range of triaxiality. Nevertheless, it is not easy to link FE analysis and MDM to produce results. For example, the GTN model has nine material parameters that need to be calibrated. (iii) Uncoupled phenomenological ductile fracture model (DFM): This approach does not employ directly an internal variable to depict the damage. This approach introduces so-called the damage indicator \(w\), the cumulative equivalent strain up to fracture. A point in the material (or an element in FE analysis) is assumed to fracture if the value of \(w\) reaches a material dependent critical value, which can be obtained by performing fracture tests with specimens of various shapes and FE analyses of the tests. A feature of the above mentioned CDM and MDM is that the progressive deterioration of the material is implicitly embedded in the formulation of the constitutive equations (i.e., coupled approach). Meanwhile, the ductile fracture criterion has no influence on the state of constitutive equation, that is, there is no coupling between damage and the formulation of the constitutive equations. However, DFM has fewer material parameters that need to be calibrated and has proven to be able to simulate the fracture initiation of material with acceptable engineering accuracy [32; 33; 34; 35]. Hence, the uncoupled approach is adopted in the present study. DFM assumes that fracture of elastic-plastic solids begins when the cumulative damage within the material reaches a threshold. The accumulated damage is called the damage indicator \(w\) and is expressed as follows.