## 2 Materials and Methods
### Mathematical Formulation
#### 2.2.1 Experimental Determination of Model Parameters  
The use of the viscoelastic Prony model implies a preliminary determination of the material constants of the relaxation function. It is assumed that the material experiences only shear relaxation, and the volume compression modulus is constant in the model presented above. Traditionally, data from uniaxial tension-compression tests, rather than tests for shear, are used to obtain experimental parameters. The relaxation module for uniaxial stretching-compression has the form similar to (10):  
\[E(t)=E_{0}\Bigg{[}c_{\infty}+\sum_{i=1}^{N_{e}}c_{i}\exp\left(-\frac{t}{\beta _{i}}\right)\Bigg{]},\]  
where \(c_{i}\) are the relative modules of tension-compression for relaxation times \(\beta_{i}\) and \(N_{e}\) is the number of relaxation times of tension-compression. From the conditions \(E_{0}=\left.E\right|_{t=0}\), \(E_{\infty}=\left.E\right|_{t=0\sigma}\), it follows that:\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]  
Assuming that the relaxation times for shear (\(\tau_{i}^{G}\)) and stretch-compression (\(\beta_{i}\)) coincide and their number is equal to (\(N_{e}=n_{G}=n\)). Then, from (11) and (12), we can obtain a relation of the form:  
\[\sum_{i=1}^{n}\alpha_{i}^{G}=\sum_{i=1}^{n}c_{i}\left[\frac{\mathcal{G}_{0}-G_ {\infty}}{G_{0}}\cdot\frac{E_{0}}{E_{0}-E_{\infty}}\right].\]  
Denoting \(B=\left[\frac{\mathcal{G}_{0}-G_{\infty}}{G_{0}}\cdot\frac{E_{0}}{E_{0}-E_{ \infty}}\right]\), we receive  
\[\alpha_{i}^{G}=c_{i}\cdot B, \tag{13}\]  
It is known that  
\[G_{0}=\frac{E_{0}}{2(1+v_{0})}, \tag{14}\]  
\[G_{\infty}=\frac{E_{\infty}}{2(1+v_{\infty})}, \tag{15}\]  
where \(v_{0}\) and \(v_{\infty}\) are the values of the [PERSON]'s ratio at the times \(t=0\) and \(t=\infty\), respectively. We find \(v_{\infty}\) from the condition of constancy of the volume compression modulus \(K^{*}\):  
\[K^{*}=\frac{E_{0}}{3(1-2v_{0})}=const, \tag{16}\]  
\[K_{\infty}=K^{*}=\frac{E_{\infty}}{3(1-2v_{\infty})}\Rightarrow v_{\infty}=0. 5\bigg{(}1-\frac{E_{\infty}}{3K^{*}}\bigg{)}. \tag{17}\]  
Substituting (17) into (15), we receive:  
\[G_{\infty}=\frac{E_{\infty}}{2\Big{(}1+0.5\Big{(}1-\frac{E_{\infty}}{3K^{*}} \Big{)}\Big{)}}=E_{\infty}\frac{3}{9-\frac{E_{\infty}}{K^{*}}}. \tag{18}\]  
Consequently, from (13), (16), (14), and (18) it is possible to calculate \(\alpha_{i}^{G}\), \(K^{*}\), \(G_{0}\), and \(G_{\infty}\), having previously found \(c_{i}\), \(E_{0}\), \(E_{\infty}\), and \(v_{0}\) from the tension-compression experiment. For this, we used the constitutive relations for the uniaxial case in the form:  
\[\sigma(t)=\int_{0}^{t}\bigg{[}E_{\infty}+E_{0}\sum_{i=1}^{N_{c}}c_{i}\exp \left(-\frac{t-\tau}{\beta_{i}^{\prime}}\right)\bigg{]}d\varepsilon(\tau), \tag{19}\]  
where \(\beta_{i}^{\prime}=\frac{\beta_{i}}{A(T)}\) is the reduced time, \(A(T)\) is the shift function. The material is assumed to be thermorheologically simple, so the Williams-Landel-Ferry shift function is used:  
\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]0  
where \(T\) is the current temperature, \(T_{r}\) is the constant base temperature, and \(C_{1}\) and \(C_{2}\) are the empirical constants for the material. When representing the deformation as a harmonic function:  
\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]1  
expression (19) is converted to the form:  
\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]2  
\[c_{\infty}=\frac{E_{\infty}}{E_{0}},\ \sum_{i=1}^{N_{c}}c_{i}=\frac{E_{0}-E_{ \infty}}{E_{0}}. \tag{12}\]3  
Thus, Equations (9)-(20) constitute the PEEK physical model (Prony model and WLF), the parameters (material constants of the relaxation function and WLF constants) of which can be calculated from (21) and (22), knowing the temperature dependence of the real and imaginary parts of the PEEK complex modulus.