## 2 Materials and Methods
### Mathematical Formulation
#### 2.2.1 Experimental Determination of Model Parameters
The approximation of WLF is given by the expression:  
\[\tau=\tau_{0}10^{\frac{-C_{1}(T-T_{0})}{C_{2}+(T-T_{0})}}\text{ for }T>T_{0u}. \tag{31}\]  
Constants \(C_{1}\) and \(C_{2}\) are determined from experiments on deformation at variable temperatures. The relaxation experiments are carried out in the interval up to 170 \({}^{\circ}\)C to fully identify the model (30, 31). From the relations below, it is possible to calculate the real and imaginary parts of the complex modulus of the material:  
\[ \begin{split} E^{\prime}&=\frac{J^{\prime}}{J^{ \prime 2}+J^{\prime\prime\,2}},\\ E^{\prime\prime}&=\frac{J^{\prime\prime}}{J^{\prime 2 }+J^{\prime\prime\,2}}.\end{split}  \tag{32}\]  
Thermal deformation model formulationA model based on the following hypotheses was chosen to describe the temperature deformation of the PEEK material: (1) the coefficient of thermal expansion of the material is constant over the temperature ranges of the glassy and highly elastic states. At the same time, it takes the following values:  
\[ \begin{split}\alpha^{th}(T)=\alpha_{1},\ T<T_{g1}\text{glassy state}\\ \alpha^{th}(T)=\alpha_{2},\ T>T_{g2}\text{highly elastic state} \end{split}  \tag{33}\]  
The boundaries of the glass transition interval depend on the experimentally observed width of the relaxation transition:  
\[ \begin{split} T_{g1}&=T_{g}-\Delta T_{g}/2,\\ T_{g2}&=T_{g}+\Delta T_{g}/2,\end{split} \]  
where \(T_{g}\), \(\Delta T_{g}\) are the temperature and the glass transition interval, respectively. (2) In the interval of the relaxation transition, the coefficient of linear thermal expansion (CLTE) is calculated by the formula:  
\[\alpha^{th}(T)=\frac{\alpha_{2}-\alpha_{1}}{2}\sin(\pi\frac{T-T_{g}}{\Delta T _{g}})+\frac{\alpha_{2}+\alpha_{1}}{2},\ \ T_{g1}\leq T\leq T_{g2}. \tag{34}\]  
At the same time, the CTLE is interpreted as follows:  
\[\alpha^{th}(T)=\frac{d\varepsilon^{th}(T)}{dT} \tag{35}\]\[\epsilon^{th}(T)=\int_{T_{0}}^{T}\sigma^{th}(T)dT \tag{36}\]  
Total free deformation also includes the shrinkage of the material \(\epsilon^{cr}\) as a result of the partial crystallization of PEEK:  
\[\epsilon^{f}=\epsilon^{cr}+\epsilon^{th}. \tag{37}\]  
The expression for \(\epsilon^{cr}\) can be obtained by knowing the dependence of the density of the material on the volume content of the crystalline phase:  
\[\rho(X_{vc})=\rho_{am}(1-X_{vc})+\rho_{cr}X_{vc},\]  
herewith  
\[\epsilon^{cr}(X_{vc})=1/3\frac{(\rho_{am}-\rho(X_{vc}))}{\rho_{am}}. \tag{38}\]  
Thus, Equations (23)-(32) constitute the PEEK physical model (micromechanical model of crystallization kinetics taking into account the cooling rate), and Equations (33)-(38), the PEEK thermal deformation model. These expressions were used to reproduce the experimentally obtained graphs [7], which, in turn, were used to identify the constants of the Prony and WLF models. It was carried out as follows. _Crystallization model calculation._ The identification of the parameters of the crystallization model (23-24) was carried out in [16]. The verification showed that the parameter values presented in the paper (found from the condition of the minimum discrepancy between the calculation and the experiment) fail to ensure the achievement of the claimed accuracy in practice. Therefore, the search for their optimal values was implemented in the Matlab package. The following vector of the required constants varied:  
\[\mathbf{x}=\Big{\{}X_{vc\infty},n_{1},n_{2},T_{m1},T_{m2},C_{11},C_{12},C_{21},C_{22},C_{31},C_{32},T_{g},w_{1}\Big{(}\dot{T}_{i}\Big{)}\Big{\}},\;\;i= \overline{1,3}\]  
The method of a deformable polyhedron (Nelder-Mead) was used to search for the minimum specific discrepancy of the calculated and experimental data:  
\[\Phi(\mathbf{x})=\frac{\sum_{i=1}^{M_{\mathbf{c}i}}\left|X_{vci}^{e}-X_{vc}^{s }(\mathbf{x})\right|}{\sum_{i=1}^{M_{\mathbf{c}i}}\left|X_{vci}^{e}\right|} \to min. \tag{39}\]  
The experimental data of paper [7], presented in Figure 2 by red curves, were used as reference data.