## 1 Introduction  
A vast variety of numerical models are used to investigate the evolution of thte microstructure in a deformed material or to assess the distribution of strain/stress in a particular direction. The most widely used finite element approaches are based on the principles of continuum mechanics. However, these numerical methods demand information on the behavior of the material during elastic/plastic deformation. In view of this, a whole spectrum of constitutive models was developed. These material models [1; 2; 3; 4; 5; 6] describe analytically the relationship between the strain, strain rate, temperature, and other technological parameters. Depending on both the complexity of the algorithm and the number of examined parameters, the accuracy of the constitutive laws can differ from each other. For an effective numerical calculation, a relatively simple but accurate and physically sound approach is of particular importance. Various analytical equations, which relate strain and stress, are incorporated in the frame of finite element modeling. Typical examples are Ramberg-Osgood [3; 4], bilinear [5; 6], cubic [5; 6], and linear hardening [7]material models. The plasticity approximations, such as Johnson-Cook-type models [8; 9] (see Equation (1)), are capable of capturing the effect of temperature \(T\) and the strain rate on hardening. However, the major drawback of these type of descriptive models is that the determination of the fitting parameters requires numerous experiments as given below:  
\[\sigma=\left(\sigma_{00}+Be_{pl}^{n}\right)\left(1+C\ln\left(\frac{\dot{\epsilon }}{\dot{\epsilon}_{ref}}\right)\right)\left(1-\left(\frac{T-T_{ref}}{T_{m}-T_{ ref}}\right)^{m*}\right), \tag{1}\]  
where \(\sigma\) is equivalent stress, \(\epsilon_{pl}\) is the plastic strain, \(\dot{\epsilon}\) is a strain rate, \(T\) is the temperature, \(\sigma_{00}\) is the yield stress, \(B\) is the hardening variable, \(n\) is the hardening exponent, \(\dot{\epsilon}_{ref}\) is the reference strain rate, \(C\) is the strain-rate sensitivity factor, \(T_{ref}\) is the reference temperature, \(T_{m}\) is the melting point of the material, and \(m^{*}\) is the thermal softening factor. In the case of cold-forming processes, a relatively simple elastoplastic material model is used (Ramberg-Osgood approximation) [3; 4]:  
\[\varepsilon_{t}=\epsilon_{el}+\epsilon_{pl}=\frac{\sigma}{E}+\left(\frac{ \sigma}{K}\right)^{1/n}, \tag{2}\]  
where \(\epsilon_{t}\) is the total strain, \(\epsilon_{el}\) is the elastic strain, \(\epsilon_{pl}\) is the plastic strain, \(\sigma\) is the stress, \(E\) is the linear elastic modulus, and \(K\) is the hardening coefficient in the Hollomon equation [10] (second term of the equation), which takes into account the effect of the strain rate \(\dot{\epsilon}\) on \(\sigma\):  
\[\sigma=Ke^{n}\dot{\epsilon}^{m}+\sigma_{0}, \tag{3}\]  
where the exponent \(m\) expresses the dependence of stress on the strain rate, and \(\sigma_{0}\) is the stress required to initiate the plastic deformation. It should be noted that, in the case of cold deformation, the effect of the strain rate on stress is negligible, and therefore \(m\) is generally assumed to be 0 [11]. All three material constants (\(E\), \(K\), and \(n\)) of Equation (2) can be determined from the tensile test. The necking point on the tensile curve can be determined by employing the Considere criterion [12] in the form of Equation (4) [13]. The necking process in metals starts at the advanced stages of tensile testing, leading to a reduced local cross-section:  
\[\sigma(\epsilon=\epsilon_{n})=\frac{d\sigma(\epsilon=\epsilon_{n})}{d\epsilon}, \tag{4}\]  
where \(\epsilon_{n}\) is the strain at the necking point. The strain rate dependency can be accounted for by employing the Hart criterion [12].