# Thermo-Viscoelastic Characterization of 3D Printing Polymers
## 4 Results and Discussion
### Relaxation Moduli in the Time Doain  
Equations (6) and (7) may be recognized as Fourier sin and cosine transforms, respectively. The relaxation modulus \(E(t)\) can be obtained by taking the inverse Fourier transform of Equation (6) as:  
\[E(\mathbf{x},t)=\frac{2}{\pi}\int_{0}^{\infty}\frac{\tilde{E}_{R}(\mathbf{x}, \omega)}{\omega}\sin(\omega t)dt\omega \tag{9}\]  
Figure 16: Master storage modulus of Polymer C.  
Figure 17: Master storage modulus of Polymer D. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Materials** & \(\mathbf{C_{1}}\) & \(\mathbf{C_{2}}\) \\ \hline Polymer A & 13.56 & 75.21 \\ Polymer B & 15.67 & 99.75 \\ Polymer C & 57.45 & 205.3 \\ Polymer D & 42.83 & 224.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: WLF factors at reference temperature 50 \({}^{\circ}\)C.  
The dynamic complex results shown in Figures 14-17 can be readily converted into time-dependent functions. The stress relaxation modulus, \(E(t)\), can be obtained by using the inverse fast Fourier transform (IFFT) of the complex results. However, in general, IFFT is very time-consuming and often the accuracy of the classical IFFT algorithm is questionable [15]. [PERSON] and [PERSON] [16] introduced some approximation methods to obtain the relaxation (or creep) function from the real and imaginary parts of the complex modulus. In this study, the complex moduli curves that were measured in the frequency were then curve-fitted onto generalized Maxwell models by using the least squares method (see [13]). For generalized Maxwell models, the relaxation moduli for viscoelastic materials can be defined by the Prony or Dirichlet series as [13; 17]:  
\[E(\mathbf{x},t)=\sum_{i=1}^{N}E_{i}(\mathbf{x})\cdot\exp(-t/\lambda_{i}) \tag{10}\]  
where \(N\) is the number of series terms and \(\lambda\)s are relaxation times. For computational purposes, the relaxation modulus in Equation (9) can be expanded in terms of Prony series summations. Then, in the frequency domain, we have:  
\[\bar{E}_{R}(\mathbf{x},\omega)=\sum_{i=1}^{N}\frac{\omega^{2}\lambda_{i}^{2}E _{i}(\mathbf{x})}{1+\omega^{2}\lambda_{i}^{2}} \tag{11}\]  
By using the least square method that minimizes the sum of deviation, \(E_{i}\) and \(\lambda_{i}\) were determined. The experimental results for Polymers A, B, C, and D and those curve-fitted are depicted in Figures 14-17. Excellent agreement was obtained between the experimental results and those curve-fitted within less than 0.5% error. \(E_{i}\) and \(\lambda_{i}\) for Polymers A, B, C, and D are tabulated in Tables 4-7, respectively. The master curves of relaxation moduli \(E(t)\)s for Polymers A, B, C, and D at reference temperature 50 \({}^{\circ}\)C are shown in Figure 18. The time domain curves show mirror images of the frequency domain curves. As shown in Figure 18, the relaxation modulus curves show glassy, transition, and rubbery states. At low temperatures or high frequencies, polymers are hard and brittle and behave like elastic materials. In the glassy region, the modulus is not degraded with time and corresponds to the elastic one. As time or temperature increases, material behavior moves to the transition region where the material properties are very rapidly degraded with time and temperature. As time or temperature are further increased, the modulus reaches another plateau region that is called the rubbery region and where the modulus reaches the so-called relaxed value. As shown in Figure 18, after about a year, the moduli of Polymers A, B, and D decreased by 90%, 96%, and 96%, respectively, while the modulus of Polymer C was reduced by 34% only. The equilibrium modulus, \(E(\infty)\), of Polymer C was 5.63 \(\times\) 10\({}^{8}\) Pa, which was much larger than ones of Polymers A, B, and D. Polymer C was highly crosslinked. Among the four materials, Polymer C showed low modulus reduction after about a year. The experimental results show that the crosslinks of Polymer C were higher than ones of Polymers A, B, and D.