These theories are based on understanding that no material is perfectly continuous. According to Griffith, the deformation of voids inside the materials increases the inner tension inside the crystalline matrix. This increase in the tension can also be influenced by nucleation, coalescence, and void growth.

According to Gross and Seeling [11]^[1], damage can be initiated by the nucleation, growth, and coalescence of micro-voids. Grain boundaries and different phases or inclusions can be a barrier to the dislocation movement, resulting in the nucleation of a crack. Thus, microscopic plastic flow or micro-void density generated by punching is not the dominant factor in the stretch–flange –formability of UHSS sheets.

The damage theory assumes that the voids are far from each other and do not interact. The elastic part is dominated by Hooke’s law, and the plastic part is dominated by the yield condition. Anderson [12]^[2], based on the observations, says that void nucleation is faster under high triaxiality stresses. Void nucleation in large particles is facilitated by increasing the number of defects that cause crack initiations.

Orowan [13]^[3] explained that Griffith’s theory for brittle-behavior solids could not be applied to ductile-behavior materials. The same author observed that laboratory tests showed that cleavage in a steel fracture can increase at slow deformation rates, different from brittle materials that grow fast. There is significant plastic deformation due to the plastic constraints that result in fractures by cleavage. Steel fractures may begin with ductile cracking followed by considerable plastic deformation. Orowan could not adjust Griffith’s theory to include plasticity. Years later, based on Orowan’s theory, Irwin [14]^[4] improved existing theories. Irwin observed extensive plastic deformations in X-ray tests, highlighting the flaw in Griffith’s theory.

Irwin (1961) [14]^[4] structured the analysis using stored strain energy, surface energy, and the work for plastic deformation. The analysis was based on the energy balance defined by Griffith.

Based on that principle, Cockcroft–Latham [15]^[5] formulated the theory of maximum principal stress, using multiaxial fractures related to crack flow. Cockcroft–Latham (1968) [15]^[5] proposed the energy fracture criterion, which states that fractures depend on the integral of the principal tensile stress. Thus, for a given material, this criterion suggests that fractures occur when the integral of the tensile stress reaches the critical value.

Lee (2005) [17]^[6] compares seven fracture criteria:

• Bao–Wierzbicki model: best adapted to most mechanical tests due to Lee’s calibration in his study;
• Maximum shear stress: similar to Tresca’s yield condition, but using the critical shear stress, not the shear flow stress;
• Cockcroft–Lathan: as explained before, best captured the process of crack formation;
• Wilkins’s model: also based on the critical value and on the critical dimension, which should be calibrated for each designated test;
• Johnson–Cook fracture criterion: postulates that fractures are a function of stress triaxiality, temperature, and strain rate, but is good to express the nonconstant triaxiality loading process;
• Constant equivalent strain fracture criterion: to reach the plastic strain reaches a critical value for fractures;
• The fracture forming limit diagram: based on the relation between the maximum and the minimum stress, and it is highly used in stamping processes to understand critical zones for tooling processes, but it is however inappropriate to give information on the crack beginning [17]^[6].

2. Mechanical Test

There are different mechanical tests, and the most used test to define steel properties is the tensile test.  This test could provide a basis for FEA analysis, but it is not enough; there are two tests derived from the tensile test with equal relevance: the plastic strain ratio (r) test [19]^[7] and the tensile strain-hardening exponents [20]^[8] test. These results quantify the capacity of the material to reduce its thickness during the tensile test and the hardening capacity of the material during the tensile test for homogeneous elongation, respectively. Another possibility is a forming limit diagram (FLD) [21]^[9]. An FLD is a graphical representation of the limits to forming, i.e., the major and minor stresses where local necking occurs. Different specimen geometries generate different deformation patterns, as expressed by the Nakajima test [22]^[10]. Another example is the cup test by Erichsen. Although very simple, it is sometimes still used to understand the maximum capacity of the equiaxial deformation up to the rupture and to compare the ductility of different materials [24]^[11]. Finally, there is the hole expansion ratio test (HERT). This test is important to compare the flangeability of sheet materials. Its characteristics are discussed in Section 5.

3. Trimming

The quality of the trimming process in stamping processes must be constantly evaluated regarding the sheared edge morphology since it represents one of the greatest influencing factors on edge flangeability during the drawing process. The goal of stamping processes is to provide a part with the correct geometry, required surface quality (variables depending on application), and desired features (e.g., holes of the correct size and location, properly oriented with no splits or objectionable wrinkles). The latter two, tearing (splits) and wrinkles, are strongly influenced by the selection of material. The trimming process occurs before hole expansion is standardized; there are thus no considerable differences between two materials of a similar resistance and elongation. In this part, the texture does not prevail over the resistance [25]^[12]. The quality of trimming is a function of the clearance and the resistance; when clearance decreases and resistance increases, burr and rollover are reduced, but the wear of the tool increases. Four zones can be observed: rollover, burnish, fracture, and burr. When the resistance increases, it is normal to see the alternation of burnish, caused by shear stress and fractures. Different hole finishing preparation methods (e.g., punching, milling, and laser cutting) can increase the damage levels causing crack initiation at the hole edge. Maximum damage is normally caused by conventional punching, followed by milling, and later by lasers. All these processes can cause small cracks and initiate a fracture [25]^[12]. The stretch flangeability of an ultra-high-strength steel sheet with a small ductility can be increased by improving the quality of the sheared edge [25]^[12]. Warm and hot punching using resistance heating was developed to improve the quality of sheared edges of an ultra-high-strength steel sheet. As the heating temperature increased, the depth of the shiny burnished surface on the sheared edge increased and that of the rough fracture surface decreased. The rollover depth and burr height of the sheared edge exceeded 800 °C. Although the roughness of the burnished surface was almost constant, the roughness of the fracture surface increased by 650 °C. The local resistance heating of the shearing region was efficient for warm and hot shearing. Warm and hot shearing of ultra-high-strength steel sheets has been found to be effective for improving the quality of the sheared edge and in reducing the shearing load. Since deep drawing has problems such as fractures, seizures, and tool wear due to large deformations, bending is a preferable process for ultra-high-strength steel [25]^[12].

4. Microstructure

Raabe (2020) [5]^[13] conducted a very detailed study on advanced high-strength steels and exposed the relation between the microstructure and hole expansion. He presented a graph containing the relation of multiphase steel elongation, oppositely proportional to the hole expansion performance. This aids in the understanding of why the present sentudry focuses on the crack area to identify void growth [5]^[13]. The void continued growing until a barrier was found; this barrier could be another grain of martensite. To understand this phenomenon, Zhang (2019) tested the relationship between the texture of different reduction rates and how this could influence the material resistance [31]^[14]. The relation between premature cracking and flangeability was discussed by Akela, Kumar, and Bakachndran (2021), comparing two equivalent tensile strengths, DP 600 and HS 700 (high-strength low alloy). In their observations, the materials with a more uniform distribution and a finer-grained ferritic matrix had a greater hole expansion ratio compared to HS 700 [32]^[15].

5. Correlation between Experimental Mechanical Testing and Simulations

To replicate the test in a digital twin, it was necessary to recreate the intrinsic and extrinsic conditions for the material and the tool and the mechanical properties, such as flow stress, Young’s module, Poisson’s ratio, Hill’s quadratic median anisotropy for the three directions [19]^[7], and density [33]^[16]. The hole expansion test was used before to calibrate the methodology of Bao–Wierzbicki, stress-triaxiality dependent. This process allowed for the determination of the ductile damage and the progressive damage of aluminum 1050–H14 (monophasic). This was a good example of using a hybrid method to calibrate the parameters to introduce in the damage model. This exposed the need to identify a damage model that can incorporate into FEA to predict the metal-forming process cracks [34]^[17]. There are different methods to calculate the damage criteria and evaluate damage; the method chosen for the present paperntry was the damage model of Cockcroft–Latham. According to the literature, this method is very accurate and is easily implemented to evaluate the workability of the material. The Cockcroft–Latham criteria relates plastic deformation and tensile stress, observing that voids grow by deformation [35]^[18]. There are many other ways to determine the damage module such as the principle of equivalence in deformation; this principle is based on thermodynamics and expresses the possibility of understanding the microstructure applied to rheological dislocations [36]^[19]. This is important to improve tool geometry, calculate the thickness distribution, and optimize the material flow to avoid premature failures. Some studies show that using a combination of practical and theoretical models to improve quality reduces development time and costs [37,38]^[20][21].