## 2 Materials and Methods
### Mathematical Formulation
#### 2.2.1 Experimental Determination of Model Parameters
The experimental curves of the degree of crystallization for cooling rates \(T_{i}=1\), \(10\) and \(35\) K\(\cdot\)s\({}^{-1}\) are given in the work. The result of (39) is presented in Tables 1 and 2. \begin{table}
\begin{tabular}{c c c c c c} \hline \(X_{vc\infty}\) & \(n_{1}\) & \(n_{2}\) & \(T_{m1}\), \(\mathbf{K}\) & \(T_{m2}\), \(\mathbf{K}\) & \(C_{11}\), \(\mathbf{c}^{-n}\mathbf{K}^{-1}\) \\ \hline
0.2721 & 2.7986 & 1.4963 & \(5.9246\times 10^{2}\) & \(6.1696\times 10^{2}\) & \(3.4052\times 10^{10}\) \\ \hline \(C_{12}\), \(\mathbf{c}^{-n}\mathbf{K}^{-1}\) & \(C_{21}\), \(\mathbf{K}\) & \(C_{22}\), \(\mathbf{K}\) & \(C_{31}\), \(\mathbf{K}^{3}\) & \(C_{32}\), \(\mathbf{K}^{3}\) & \(T_{g}\), \(\mathbf{K}\) \\ \hline \(3.4052\times 10^{10}\) & \(5.0698\times 10^{3}\) & \(4.2589\times 10^{3}\) & \(3.6516\times 10^{7}\) & \(4.6533\times 10^{7}\) & \(4.1700\times 10^{2}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Values of parameters of the PEEK material classification kinetics model. \begin{table}
\begin{tabular}{c c c c c} \hline \(\dot{T}_{\mu}\) \(\mathbf{K}\)\(\cdot\)s\({}^{-1}\) & \(\mathbf{1}\) & \(\mathbf{10}\) & \(\mathbf{35}\) \\ \hline \(w_{1}(\dot{T}_{i})\) & \(8.1209\times 10^{-2}\) & \(7.1103\times 10^{-1}\) & \(9.9980\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Weight coefficients of the mechanisms of crystallization of the PEEK material. _Viscoelastic model calculation._ The results from calculating the temperature dependence of the complex modulus according to (32) from paper [7] were used as a reference to identify the parameters of (9)-(20) available in ANSYS. The values of the material constants accepted in the calculation are presented in Table 3.  
The calculation result is shown in Figure 3a,b shows experimental graphs of the complex modulus amplitude from [7]. _Thermal deformation model calculation._ Figure 4 shows the results of calculating the free deformation \(\varepsilon/=\varepsilon^{\sigma\tau}+\varepsilon^{th}\) and its component obtained from (33)-(38). The thermomechanical constant values of the model are presented in Table 4. \begin{table}
\begin{tabular}{c c c c c c} \hline \(\rho_{am}\), \(\mathbf{kg}\cdot\mathbf{m}^{-3}\) & \(\rho_{cr}\), \(\mathbf{kg}\cdot\mathbf{m}^{-3}\) & \(\alpha_{2}\), \(\mathbf{K}^{-1}\) & \(\alpha_{2}\), \(\mathbf{K}^{-1}\) & \(\Delta T_{g}\), \(\mathbf{K}\) \\ \hline
1262.6 & 1400.6 & \(50\times 10^{-6}\) & \(200\times 10^{-6}\) & 10 \\ \hline \end{tabular}
\end{table} 
Table 4: Thermomechanical values of constants. \begin{table}
\begin{tabular}{c c c c c c} \hline \(J_{am}\), \(\mathbf{Pa}^{-1}\) & \(J_{nr}\), \(\mathbf{Pa}^{-1}\) & \(J_{uc}\), \(\mathbf{Pa}^{-1}\) & \(\omega\), \(\mathbf{rad}\)\(\mathbf{s}^{-1}\) & \(\tau_{0}\), \(\mathbf{s}\) & \(E_{av}\), \(\mathbf{j}\) mol\({}^{-1}\) & \(R\), \(\mathbf{j}\) mol\({}^{-1}\)\(\mathbf{K}^{-1}\) \\ \hline
380 \(\times 10^{-12}\) & 15000 \(\times 10^{-12}\) & 120 \(\times 10^{-12}\) & 1 & 1 & \(600\times 10^{3}\) & 8.314462618 \\ \hline \(T_{0 av}\), \(\mathbf{K}\) & \(T_{0c}\), \(\mathbf{K}\) & \(C_{1}\) & \(C_{2}\) & \(\alpha\) & \(\alpha_{am}\) & \(\alpha_{cr}\) \\ \hline
414 & 533 & 25 & 100 & 0.5 & 1 & 0.4 \\ \hline \end{tabular}
\end{table} 
Table 3: Values of constants for calculating the dependence of a complex modulus. Figure 2: Temperature dependence of the degree of crystallization of PEEK during cooling at different speeds: (**a**) without detailing the mechanisms of crystal nucleation; (**b**) with detailing. Black, green, and blue lines show calculations; red ones show experiments [7]. _Identification of the Prony model constants from graph data._ The temperature dependence of the complex modulus at the cooling rate of \(\dot{T}\)= 1 K\(\cdot\)s\({}^{-1}\), shown in Figure 3a with thin lines, was used to determine the parameters of the Prony model with the temperature-time analogy WLF (9)-(20). The formulation of the problem of finding constants of the model is similar to (39). The obtained values are shown in Figures 5 and 6.