## 3 Results and Discussion
### Industrial Application  
The research methodology presented was later used to determine the risk of material fracture in plastic deformation processes, specifically in backward extrusion. In this process, the die cavity is initially filled through upsetting, followed by punching and extrusion of the material. As the analysis below shows, there is a risk of material fracture during the punching stage, which can result in irreversible damage to the product. Examples of fractures in the extruded part are shown in Figure 10.  
The cracks depicted in Figure 10 are considered to be caused by the local exceedance of critical process parameter values that occurs during the filling of the die cavity. The behavior of material flow leads to a tendency for fractures to appear on the product wall, and the direct cause of damage is the pressure of the upper tool, which is characterized by a complex shape. The mechanism of fracture appearance therefore corresponds to the conditions created in the laboratory test presented in this article. It is thus possible to identify the conditions that favor fracture and modify the process parameters at the technology design stage. The course of the analyzed backward extrusion process is shown in Figure 11 where the characteristic indentation pattern made by the upper tool is clearly visible in the shape of product. Figure 10: Examples of material fracture after backward extrusion process: (**a–c**) different localization of defects (marked by red circles). Numerical analysis of the industrial process of the backward extrusion of an aluminum alloy (GRADE B) has shown that the highest risk of fracture (according to the Cockcroft-Latham criterion) occurs on the internal surfaces of the product. In this area, as can be seen in Figure 12, a significant concentration of effective strain and maximum principal strain (\(\varepsilon\)1) can also be observed. The figure shows the change in the shape of the product during extrusion and the area of concentration of the fracture criterion up to the value of 1702.39. Additionally, the analysis has shown that in this area of the product, increased values of other parameters occur, such as principal strain or effective strain. By analyzing the course of the laboratory test, it can be seen that visible fractures on the outer surface of the samples are a consequence of the concentration of parameters promoting fracture on their inner surface. The direct cause of this phenomenon is the pressure of the upper tool, which in the initial stage of the test upsets the sample and then forms a recess in the material. The values of the analyzed parameters are much higher on the inner surface of the sample, which is in direct contact with the upper tool, than on the  
Figure 11: Change in product shape during backward extrusion: (**a–e**) subsequent stages of the process. Figure 12: Analysis of the distribution of parameters during backward extrusion: (**a–c**) change in the Cockcroft–Latham fracture criterion value during extrusion, (**d**) distribution of the principal strain, (**e**) distribution of the effective strain. outer surfaces, which were analyzed using the DIC system. This phenomenon is shown in Figure 13, where the values of the fracture criterion and principal strain predicted during the simulations are compared for the inner and outer surfaces. By analyzing the presented graphs (Figure 13a), it can be observed that the value of the Cockcroft-Latham fracture criterion is significantly higher in the 27 th second of the test, reaching 969.8 on the inner surface of the sample (compared to the value of 153.65 on the outer surface). This is a consequence of differences in the distribution of the principal strain. As can be seen from Figure 13b, on the inner surface of the workpiece, the principal strain reaches values of 1.93, while on the outer surface it is 0.54.