## 2 Materials and Methods
### Mathematical Formulation
#### 2.2.1 Experimental Determination of Model Parameters
2.2 Identification of PEEK Properties according to Third-Party Sources Taking into Account the Kinetics of Crystallization  
The numerical values of the temperature dependence function of the complex modulus firstly were determined from experimental data of third-party sources [7; 16; 17; 18]. The papers describe (1) PEEK crystallization kinetics taking into account the cooling rate, (2) the PEEK viscoelastic model for predicting thermomechanical properties, and (3) the PEEK thermal deformation model. _Crystallization model formulation_. The crystallization kinetics in semi-crystalline thermoplastic composites is considered in connection with the influence of the degree of crystallinity on the mechanical properties and possible contribution of volumetric shrinkage deformation. For a pure PEEK in dynamic conditions, it is set as follows:  
\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]4  
with weight coefficients \(w_{1}\) and \(w_{2}=1-w_{1}\), where  
\[ \begin{split} F_{vc1}&=1-\exp\biggl{[}-C_{11}\int_{ 0}^{t}T\exp\biggl{\{}-n_{1}t^{(n_{1}-1)}\biggl{[}\frac{C_{11}}{T-I_{g}+51.6}+ \frac{C_{31}}{T(T_{mi}-T)^{2}}\biggr{]}\biggr{\}}dt\biggr{]},\\ F_{vc2}&=1-\exp\biggl{[}-C_{12}\int_{0}^{t}T\exp \biggl{\{}-n_{2}t^{(n_{2}-1)}\biggl{[}\frac{C_{22}}{T-I_{g}+51.6}+\frac{C_{32} }{T(T_{mi}-T)^{2}}\biggr{]}\biggr{\}}dt\biggr{]},\end{split}  \tag{24}\]  
where \(X_{vc}\) is the volume fraction of the crystal phase, \(X^{\infty}{}_{vc}\) is the equilibrium volume fraction, \(T_{g}\) is the glass transition temperature, \(T_{mi}\) is the melting point of crystals for the process \((i=1,2)\), and \(C_{1i}\), \(C_{2i}\), and \(C_{3i}\) are constants. _Viscoelastic model formulation_. The real and imaginary parts of compliance of the material \(J^{\prime}\) and \(J^{\prime\prime}\) are composed of amorphous (\(J^{\prime}_{am}\), \(J^{\prime\prime}_{am}\) ) and crystalline (\(J^{\prime}_{cr}\), \(J^{\prime\prime}_{cr}\)) components:  
\[ \begin{split} J^{\prime}&=J^{\prime}_{am}(1-X_{vc}) +J^{\prime}_{cr}X_{vc},\\ J^{\prime\prime}&=J^{\prime\prime}_{am}(1-X_{vc}) +J^{\prime\prime}_{cr}X_{vc}.\end{split}  \tag{25}\]  
The thermorelaxation transition is described by the following relations:  
\[ \begin{split} J^{\prime}_{am}&=J_{ua}+(J_{ua}-J_{ra} )+(\cos\psi)^{\alpha}\cos(\alpha\psi),\\ J^{\prime\prime}_{am}&=(J_{ra}-J_{ua})(\cos\psi)^{ \alpha}\sin(\alpha\psi).\end{split}  \tag{26}\]  
The real part of the compliance of the crystalline phase of the material is assumed to be constant, and the imaginary part is equal to zero:  
\[ \begin{split} J^{\prime}_{cr}&=J_{uc},\\ J^{\prime\prime}_{cr}&=0,\end{split}  \tag{27}\]  
\[ \begin{split}\psi&=\arccos(\omega\tau_{am}) \end{split}  \tag{28}\]where \(J_{ua}\) is the compliance of the amorphous phase in the glassy state (instantaneous), \(J_{ra}\) is the compliance of the amorphous phase in the equilibrium state (long-term), \(J_{uc}\) is the elastic compliance of the crystalline phase, \(\omega\) is the angular frequency of deformation of the sample in the DMA experiment, and \(\alpha\) is the relaxation distribution coefficient (in the range from 0 to 1), taking into account the tilt shift of the dynamic mechanical module in the glass transition domain due to changes in the degree of crystallinity. The relaxation time of the amorphous phase \(\tau_{am}\) is directly proportional to the viscosity \(\mu\) in the model of a standard viscoelastic body:  
\[\tau_{am}=\mu(J_{ra}-J_{ua}). \tag{29}\]  
The temperature dependence of the delay time is described by two different laws: Arrhenius and Williams-Landel-Ferry (WLF). The first one operates in the range below the glass transition temperature, while:  
\[\tau=\tau_{0}e^{\frac{E_{0}(1/T-1/T_{0})}{R}}\text{ for }T\leq T_{0u}. \tag{30}\]  
The base temperature \(T_{0}\) is assumed to be a linear function of the degree of crystallization in the range from \(T_{0u}\) to \(T_{0c}\). \(E_{a}\) and \(R\) are the specific activation energy and the universal gas constant, respectively.