# Fractional Order Viscoelastic Model for Stress Relaxation of Polyvinyl Chloride Geomembranes
## 5 Discussion
### Stress-Relaxation Rate  
In addition to \(-\mathrm{d}\sigma/\mathrm{d}t\), \(-\mathrm{d}(\sigma/\sigma_{0})/\mathrm{d}(lnt)\) and \(-\mathrm{d}\sigma/\mathrm{d}(lnt)\) are also frequently adopted to characterize the stress relaxation rate in the dynamics on stress relaxation of polymers [39]. Figure 9 presents the stress-relaxation rate expressed by \(-\mathrm{d}(\sigma/\sigma_{0})/\mathrm{d}(lnt)\) at three tensile rates. As observed, the stress-relaxation rate was positively correlated with \(\sigma/\sigma_{0}\), the initial relaxation rate reached the maximum at 58.78% strain and the minimum at 81.09% strain at identical tensile rates, and the longitudinal specimens had higher values than the transverse specimens. The relaxation stress is related to the initial stress \(\sigma_{0}\) and the relaxation time, and \(\sigma_{0}\) depends on the tensile rate and initial strain. The comparison shows that the specimen at a larger tensile rate has a higher initial stress-relaxation rate.  
The stress-relaxation rate distributes more densely with smaller \(\sigma/\sigma_{0}\), the curves can be classified into initial and later stages, considering the denser distribution as the later stage, and vice versa as the initial stage. The boundary between the two stages is significantly affected by the tensile rate, considering the \(\sigma/\sigma_{0}\) in the range of 0.45 to 1.0 as the initial stage at \(v\) = 2.5 mm/min, 0.4 to 1.0 at \(v\) = 5.0 mm/min and 0.35 to 1.0 at \(v\) = 10.0 mm/min. This indicates that a larger tensile rate leads t  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Fitting** & \multicolumn{6}{c}{\(v\) (mm/min)} \\ \cline{2-7}
**Parameters** & **2.5** & **5.0** & **10.0** & **2.5** & **5.0** & **10.0** \\ \hline \(c\) & \(-0.020\) & \(-0.020\) & \(-0.027\) & - & - & - \\ \(d\) & 7.97 & 7.64 & 7.56 & - & - & - \\ \(\mathrm{R}^{2}\) & 0.989 & 0.982 & 0.943 & - & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fitting parameters by fitting a line between \(K\) and initial strain \(\epsilon_{0}\).  
Figure 8: Relationship between \(K\) and initial strain \(\epsilon_{0}\) at three tensile rates.  
Figure 9: Stress-relaxation rate at different tensile rates: (**a**) \(v\) = 2.5 mm/min; (**b**) \(v\) = 5.0 mm/min; (c) \(v\) = 10.0 mm/min.  
range of \(\sigma/\sigma_{0}\). This is mainly caused by the fact that the frozen polymer chain segments of rapidly stretched specimens fail to overcome the energy barriers and move in time, resulting in low energy loss, and when the stretching is completed, they gradually overcome the energy barriers and move activated by the high stored energy, resulting in the accelerated energy loss and a faster initial relaxation rate.  
The relaxation rate slowly decreases at the later stage for the reason that the polymer chain segments move slowly due to the almost stored energy in the specimen released at the initial stage, and then the relaxation stress per unit time reduces. Either a larger tensile rate or a larger initial strain leads to a lower stress relaxation rate at the later stage, indicating a longer time required for the relaxation stress to reach stability. This may be because the specimen is subjected to larger tensile deformations, more work is performed by the external force at greater strains, and more energy is stored internally.