## 1 Introduction  
The prediction of the moment of fractures appearing in engineering materials (such as wood [1], rock [2], composite materials [3] or metals and their alloys, e.g., steel [4]) which are subjected to loads causing plastic deformation is still a current topic for many scientific studies and is particularly important for several critical components, such as pipelines [5]. The aim of the tests is usually to determine the deformation limit and the process parameters present at the moment of crack initiation. Such knowledge makes it possible, for example, to determine the scope of work of structural elements in real operational conditions [2]. Thanks to tests in laboratory conditions, it is possible to create a specialized stand that will enable the moment of the occurrence of a crack in the material under the influence of a specific state of stress that is conducive to the separation of the material to be forecast. For that purpose, specific sets of tools [1,4] and precisely selected shapes for the tested samples were designed [1,3]. An important support in the case of such tests is the use of the DIC (digital image correlation) system, which allows the analysis of the deformation distribution in specific areas of the sample and the general mechanical condition at the point of crack initiation [6]. Additionally, supporting the analysis with the digital image correlation system makes it possible to verify the models of the fracture process adopted in the numerical simulation [1, 2, 3, 4]. Crack propagation can also be detected by combined studies using the DIC technique along with microscopy techniques such as SEM for enhanced resolution [7, 8, 9, 10] and other image-based techniques can also be applied in order to realistically take into account the workpiece's microstructure for the purpose of crack detection [11]. In forging processes, the phenomenon of material cracking causes rejection of the final product in most cases. Actually, cracks may occur in different parts of the workpiece such as the surface or the flash region [12]. The reasons for fracture may result from many factors, such as incorrect selection of technological parameters, e.g., process temperature or tool velocity or reduced quality of the workpiece [12,13]. Especially in the case of aluminum [14], magnesium [15] or titanium [16], this phenomenon is often discussed in the literature. Fortunately, an appropriate procedure enables the proper selection of fracture models and predicts the risk of element damage already at the technology design stage, which significantly reduces production costs [17]. Comprehensive experimental work can also lead to detailed mapping of forming conditions with respect to process parameters such as height reduction and temperature in order to avoid unstable conditions, oxidation or cracking [18]. Moreover, the determination of forming limit diagrams using appropriate experimental tests is also important for establishing the formability limits for different materials [19]. Numerical simulations supported by material formability tests play an important role for achieving optimal forging conditions, as they can assist in the procedure of crack detection and product quality improvement [20]. Usually, during forging operations, formability is limited by the ductile fracture [21]. The main division of ductile fracture models distinguishes this type of material separation into coupled (with different material properties) and uncoupled criteria [22,23]. However coupled models can be further distinguished into phenomenological- and micromechanics-based models [24]. In coupled models, the quantitative amount of damage is closely related to the mechanical properties of the material that change during deformation and especially damage arising during the deformation process affects model plasticity [23]. Furthermore, in coupled models, fracture initiation is caused by a constitutive variable which reaches a limit value [23].