## 4 Numerical Simulations
### Initial Values of Material Parameters for Three Material Models (SAC305)  
The initial parameter values were determined separately for the Perzyna, Chaboche and Norton models. The relationships usually used for the given material models were used. The initial values of the parameters for the Norton model were determined according to Figure 10, where the rates of the secondary part of the creep are given as a function of the nominal stress. The parameter values for the Norton model \(C_{4}\) and \(C_{5}\) can be determined directly by comparing Equation (3) with the power approximation function of the equation shown in Figure 10 for the hourglass specimen (it is necessary to consider the stress units used in the Ansys 2022 R2 software). Then, the initial values of the parameters for the Norton model are \(C_{5}\) = 9.62728 and \(C_{4}\) = 7.97016 \(\times\) 10\({}^{-19}\) s\({}^{-1}\). The parameters of the Perzyna model can be estimated from at least two tensile tests performed at different strain rates. The procedure is shown for two tensile diagrams at position rates of 5 mm/min and 10 mm/min performed at a temperature of 22 \({}^{\circ}\)C. Equation (1) is adjusted to the following form:  
\[\sigma=\alpha+\sigma_{y}+\sigma_{vp}, \tag{4}\]  
where \(\sigma_{vp}\) is a viscoplastic stress, which is calculated from Equation (1) as follows:  
\[\sigma_{vp}=\sigma_{y}\left(\frac{\dot{\varepsilon}_{p}}{\
u}\right)^{m}, \tag{5}\]  
where the plastic strain rate \(\dot{\varepsilon}_{p}\) corresponds directly to the accumulated plastic strain rate \(\dot{p}\). The value of viscoplastic stress is not known for the case of the tensile test with the lowest  
Figure 10: Determination of initial parameters of the Norton model. Figure 9: Displacement boundary condition for (**a**) tensile test at rate 1 mm/min and (**b**) tensile tests at rate 5 and 10 mm/min (SAC305 solder alloy). strain rate and must be estimated. We assume a value of \(\sigma_{vp}\) = 3.4 MPa. The relation (5) is converted into the following form:  
\[ln\left(\frac{\sigma_{vp}}{\sigma_{y}}\right)=mln\dot{\epsilon}_{p}+lnv^{-m}, \tag{6}\]  
Equation (6) can be interpreted as the equation of a straight line \(y=mx+q\), where \(y=ln\left(\frac{\sigma_{vp}}{\sigma_{y}}\right)\), \(x=ln\dot{\epsilon}_{p}\), and \(q=lnv^{-m}\) (logarithmic coordinates). For the selected value of plastic strain \(\varepsilon_{p}\) (the highest value common to all tensile tests is used most often), the unknown component of viscoplastic stress \(\left(\sigma-\sigma_{y}-\alpha\right)\) and the value of plastic strain rate \(\dot{\epsilon}_{p}\) are added. The courses of the plastic strain rate for the 2 nd and 3 rd tensile tests are shown in Figures 11a and 11b, respectively. The plastic strain rate is related to the plastic strain value \(\varepsilon_{p}=0.06\), and the determined values for both tests are shown in Table 5. Viscoplastic stress values are also listed in the table mentioned. The yield stress value is assumed to be the same for both experiments. The last two columns in Table 5 represent the points from which the parameters of the straight line \(m\) and \(q\) can be determined, and therefore the parameters \(m\) and \(\
u\) of the Perzyna model. The situation is shown in Figure 12. The resulting initial values of the parameters of the Perzyna model are \(m\) = 9.69242 \(\times\) 10\({}^{-1}\) and \(\
u\) = 5.30916 \(\times\) 10\({}^{-3}\) s\({}^{-1}\). Figure 11: The courses of the plastic strain rate for Perzyna initial parameter determination (**a**) for the tensile test at rate of 5 mm/min and (**b**) 10 mm/min. Figure 12: The coefficients of the straight-line equation were used to determine the parameters of the Perzyna model.