# Fractional Order Viscoelastic Model for Stress Relaxation of Polyvinyl Chloride Geomembranes
## 6 Conclusions  
This paper presents a viscoelastic constitutive model, namely the FOVM, for simulating the stress relaxation behavior of PVC GMBs in a single direction at the peripheral joints of MFRD, which cannot represent actual deformation. Model parameters were obtained by best fit to results from wide-width strip tensile tests on longitudinal and transverse specimens conducted at the initial strain of 40.55%, 58.78%, and 81.09%. The accuracy of the FOVM in predicting the relaxation stress at any time was then verified by performing a long-term stress relaxation test for 90 d, and the FOVM proved to be an efficient approach to estimating the stress relaxation properties of PVC GMBs. Based on the results of the present investigation, the following conclusions can be drawn. 1. The stress relaxation behavior of PVC GMB was strongly associated with tensile rate, initial strain, and longitudinal and transverse directions, and the higher the initial strain and tensile rate, the longer time for the relaxation stress to stabilize. The relaxation stress in longitudinal direction was higher than that in transverse direction under the same initial strain and tensile rate. 2. The fractional-order Maxwell model constructed from a single spring-pot contained two parameters, \(\beta\) and K. The relaxation stress of PVC GMB deduced by the model was in good agreement with the measured value. If the criteria for relaxation stress at failure are provided, the FOVM could predict the corresponding relaxation time, and then examine whether the project satisfies the service cycle. 3. The fractional derivative order \(\beta\) was found to be independent of initial strain and tensile rate. Whereas the viscoelastic parameter \(K\) was associated with initial strain and tensile rate, and decreased linearly with the tensile rate for both transverse and longitudinal specimens. A linear decrease in the \(K\) of transverse specimens with the initial strain was observed, while a nonlinear relation was detected in the longitudinal specimens. 4. Due to the orientation of polymer chain segments, the residual stress appeared in longitudinal specimens during the production process, whereas no residual stress existed in transverse specimens, which primarily account for the difference in relaxation stress between the longitudinal and transverse specimens. Figure 11: Relationship between \((-\mathrm{d}\sigma/\mathrm{d}lnt)\mathrm{max}\) and \(\sigma_{0}\): (**a**) longitudinal specimens; (**b**) transverse specimens. The FOVM provides some insight into the stress relaxation behavior of PVC GMBs and predicts the relaxation stress at any time with great accuracy. Moreover, it has great potential for applications in predicting the long-term deformation of PVC GMBs at the peripheral joints of MFRD to prolong the relaxation time at failure and the service life of the project. For example, the PVC GMBs can be subjected to lower initial strain and tensile rate by taking measures to reduce their differential displacement and settlement rate at the peripheral joints of MFRD. The PVC GMBs at the peripheral joints in practice are subjected to horizontal opening displacement, vertical settlement displacement, and shear displacement. Although the tensile deformation produced by the former two can be synthesized in a single direction, that induced by the shear displacement still exists. This paper focuses on the stress relaxation behavior of PVC GMBs in a single direction, which cannot represent their true deformations. Future work should take into account the shear strain in the stress relaxation properties of PVC GMBs. Conceptualization, Y.W. ; methodology, C.Y. and Y.W. ; validation, X.G. ; formal analysis, Y.W. ; data curation, Y.W. ; writing--original draft preparation X.Z. ; writing--review and editing, Y.W. and X.Z. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (grantnumbers 51079047, 51379069).