# Fractional Order Viscoelastic Model for Stress Relaxation of Polyvinyl Chloride Geomembranes
## 3 Formulation of the FOVM  
As the classical viscoelastic model is constructed based on springs and dashpots, and these equations take the form of integer order differential or integral, simple combinations of elements fail to accurately describe the long-term stress relaxation. Increasing the number of elements and changing the form of the combination are generally employed to improve the fitting accuracy; however, the increased parameters lead to the instability of the model [36]. The spring-pot is used instead of springs and dashpots, and the constitutive equation based on fractional-order differential can describe the stress relaxation of polymers with good stability [30; 37]. A fractional-order Maxwell model (illustrated in Figure 3b) is constructed by replacing the spring and dashpot in the classical Maxwell model (see Figure 3a) with the spring-pot. By [PERSON]'s law and [PERSON]'s law, we can write out the following equations. \[\sigma(t)=E\epsilon(t)=E_{a}^{RL}D_{t}^{0}\epsilon(t) \tag{1}\]  
Figure 3: Viscoelastic model: (**a**) classical Maxwell model; (**b**) fractional order Maxwell model. \[\sigma(t)=\eta\frac{d\epsilon(t)}{dt}=\eta_{a}^{RL}D_{t}^{1}\epsilon(t) \tag{2}\]  
where \(\sigma\) is the relaxation stress, \(E\) is the elastic modulus, \(\eta\) is the viscosity coefficient, \(\frac{\kappa L}{a}D_{t}^{0}\epsilon(t)\) and \(\frac{\kappa L}{a}D_{t}^{1}\epsilon(t)\) represent the zero-order and first-order derivative of \(\epsilon(t)\) with respect to \(t\), respectively, \(\epsilon(t)\) is the strain as a function of time, and \(t\) is the relaxation time. Based on the Bollzman superposition principle, the relaxation stress can be formulated as  
\[\sigma(t)=\int_{-\infty}^{t}dt^{\prime}G\big{(}t-t^{\prime}\big{)}\frac{d \epsilon(t^{\prime})}{dt^{\prime}} \tag{3}\]  
\(G(t)\) is the relaxation modulus, and its self-similar power law form is:  
\[G(t)=\frac{E}{\Gamma(1-\beta)}(t/\tau)^{-\gamma} \tag{4}\]  
\(\Gamma(z)\) is the Gamma function, defined as:  
\[z=\int_{0}^{+\infty}x^{z-1}e^{-x}dx \tag{5}\]  
As the PVC GMB is viscoelastic, the \(\beta\)-order Riemann-Liouville derivative is introduced here as follows:  
\[_{a}^{RL}D_{x}^{\beta}f(x)=\frac{1}{\Gamma(m-\beta)}\cdot\frac{d^{m}}{dx^{m}} \int_{a}^{x}(x-t)^{m-\beta-1}f(t)dt \tag{6}\]  
where \(\beta\) is the fractional derivative order. \(0\leq a\) and m \(-1\)\(<\beta\ <\ m\), \(m\) is the minimal integer greater than \(\beta\), \(m=1\), \((x-t)^{m-\beta-1}f(t)\) is integrable on \(a\), \(+\infty\) and \(\int_{a}^{x}(x-t)^{m-\beta-1}f(t)dt\) is differentiable on \(a\), \(+\infty\). Combining Equations (3), (4) and (6) yields  
\[\sigma(t)=E\tau^{\beta}\frac{d^{\beta}\epsilon(t)}{dt^{\beta}}=E\tau^{\beta RL }_{0}D_{t}^{\beta}\epsilon(t)=\frac{E\tau^{\beta}\epsilon(t)}{\Gamma(1-\beta )}t^{-\beta} \tag{7}\]  
Let \(K=E\tau^{\beta}\)  
\[\sigma(t)=K\frac{\epsilon(t)}{\Gamma(1-\beta)}t^{-\beta} \tag{8}\]  
when \(\beta=0\), Equation (8) is the same as Equation (1), and the spring-pot exhibits the properties of a spring, and \(K\) is equivalent to the modulus of elasticity. When \(\beta=1\), Equation (8) is the same as Equation (2), and the spring-pot exhibits the properties of a dashpot, and K is equivalent to the coefficient of viscosity. Therefore, the spring-pot is a viscoelastic element with the properties of between a spring and a dashpot. The initial strain of the specimen in the relaxation stress state is a constant \(\epsilon_{0}\), and Equation (8) becomes  
\[\sigma(t)=\frac{K\epsilon_{0}}{\Gamma(1-\beta)}t^{-\beta} \tag{9}\]  
Equation (9) can be simplified by taking the logarithm of both sides, namely  
Figure 3:0  
Let  
Figure 3:1  
Equation (10) simplifies to  
Figure 3:2where \(\alpha\) is the logarithm of relaxation stress at the relaxation time of 1 h.  
Thus, the FOVM expressed by Equation (12) was established to evaluate the stress relaxation behavior of PVC GMBs.