# Fractional Order Viscoelastic Model for Stress Relaxation of Polyvinyl Chloride Geomembranes
## 4 Results  
### Wide-Width Strip Tensile Test Results  
Engineering strain is expressed by a ratio of the increment of gauge length to the initial gauge length (Equation (13)) in engineering practice. It is proved that the engineering strain is merely appropriate at small strains. However, it varies remarkably with the true strain at larger strains. [PERSON] and [PERSON] deduced the formula for the theoretical strain in the gauge by calculus (Equation 14), already included in ASTM D5323-92 [38]. \[\varepsilon_{aE}=\frac{\Delta L}{L_{0}}=\frac{L_{f}-L_{0}}{L_{0}} \tag{13}\]  
\[\varepsilon_{a}=\int_{L_{0}}^{L_{f}}\frac{dL}{L}=ln\bigg{(}\frac{L_{f}}{L_{0} }\bigg{)}=ln\bigg{(}\frac{L_{0}+\Delta L}{L_{0}}\bigg{)}=ln(1+\varepsilon_{aE}) \tag{14}\]  
where \(\varepsilon_{aE}\) is the engineering strain, \(\Delta L\) is the increment of the gauge length, \(L_{0}\) is the initial gauge length of the specimen, \(L_{f}\) is the distance between gauge marks at any time, and \(\varepsilon_{a}\) is the theoretical strain. Both the width and thickness of the specimen decrease during the uniaxial tensile test. [PERSON] [35] measured the Poisson's ratios in the transverse and longitudinal directions of the material using the digital image correlation (DIC) non-contact measurement method, and concluded that the Poisson's ratio \(\mu_{y}\) and \(\mu_{z}\) of PVC GMB were 0.51, 0.49, respectively. The relaxation stress is calculated by tension divided by the deformed width and thickness (Equation (15)). \[\sigma=\frac{F}{A_{u}}=\frac{F}{W_{\varepsilon_{a}=0}T_{\varepsilon_{a}=0} \big{(}1-\mu_{y}\varepsilon_{a}\big{)}(1-\mu_{z}\varepsilon_{a})} \tag{15}\]  
where \(\sigma\) is the relaxation stress, \(F\) is the measured force acting on the specimen during the uniaxial tension test, \(A_{u}\) is the cross-sectional area of the specimen during the uniaxial tension test, \(W_{\alpha=0}\) is the initial width of the specimen, \(T_{\alpha=0}\) is the initial thickness of the specimen, \(\mu_{y}\) is the transverse [PERSON]'s ratio, and \(\mu_{z}\) is the longitudinal Poisson's ratio. Figure 4 reports the stress-relaxation curves of transverse and longitudinal specimens at distinct tensile rates and initial strains. The variation of relaxation stress with time follows similar patterns among the specimens. As expected, the relaxation stress declined sharply at the initial stage, followed by a slow degradation until it reached stability. According to the relaxation rate, the stress relaxation curve was classified into two typical stages, characterized by a faster stress-relaxation rate at the initial stage and a relatively low value subsequently. Moreover, the dividing point between the two stages is approximately located at a relaxation time of 2 h.  
The longitudinal specimens shared similar patterns with the transverse specimens at the same tensile rate and initial strain, and the relaxation stress in the former was slightly higher than that in the latter. This may be attributed to the stress state of the material during the production process. At the same tensile rate, a greater initial strain resulted in a higher relaxation stress, and hence the time for relaxation stress to stabilize was longer. For example, at a tensile rate of 2.5 mm/min, the average relaxation stress and the time for relaxation stress to stabilize at the initial strain of 81.09%, 58.78%, and 40.55% were (4.75 MPa, 39.5 h), (3.21 MPa, 29.9 h), and (2.10 MPa, 25.8 h), respectively. One possible explanation for this discrepancy is that the specimen was subjected to a greater tension at a larger tensile strain, while the width and thickness decreased. Consequently, the initial strain affects the relaxation stress and the time for relaxation stress to stabilize. The comparison of the stress-time curves for distinct tensile rates suggests that at the same initial strain, a slower tensile rate resulted in a lower relaxation stress and shorter time for relaxation stress to stabilize, namely, viscous response of the material.