# Fractional Order Viscoelastic Model for Stress Relaxation of Polyvinyl Chloride Geomembranes
## 1 Introduction  
Polyvinyl chloride (PVC) geomembranes (GMBs) have been applied in membrane faced rockfill dams (MFRDs) for their outstanding flexibility, strong adaptability to deformation, and low permeability, thereby making them candidates as anti-seepage materials [1, 2]. As known, they have a nonlinear stress-strain behavior that depends on strain rate and time [3, 4, 5, 6, 7], and undergo time-dependent decreases in stress under constant strain (i.e., stress relaxation). The stress relaxation behavior plays a critical role in the mechanical properties of PVC GMBs, and tests can be performed by applying a constant strain and measuring the corresponding stress required to maintain the deformation as a function of time.  
At the peripheral joints of the MFRD, the GMB is placed with one end on the dam surface, relying on the interface friction to maintain stability when subjected to water load [8, 9, 10], and the other end anchored on the plinth of the dam abument or the top of the concrete antiseepage wall (See Figure 1). Generally, the GMB on the dam surface can accommodate the local small deformation at the bottom of the gravel cushion, while that at the peripheral joints is subjected to large tensile deformation due to differential displacement between the dam body and abutment. The tensile deformation mainly in the initial impoundment of the reservoir and operation results in decreasing stress, and then the anti-seepage structure is susceptible to failure.  
There has been extensive research on the establishment of a constitutive model to theoretically describe stress relaxation behavior. The simplest was the classical Maxwell model (one spring and one dashpot in series), and then several generalized Maxwell modelswere established [11; 12; 13], combining springs and dashpots in diverse ways. Thereafter, alternative models were developed, including linear and nonlinear viscoelastic models [14], viscoplastic models [15; 16], three-component models [17; 18; 19; 20], Zener models [21; 22], Poynting Thomson models [23], and viscoelastic-viscoplastic models [24; 25]. All models can evaluate the stress relaxation behavior of polymers. However, they are based on constitutive equations with differential operators of integer order. Pronounced relaxation behavior that spans over several decades requires many Kelvin-Voigt or Maxwell elements, respectively, leading to numerous parameters to be identified from experimental data. For this reason, the models fail to predict accurately the relaxation stress at specified relaxation time.  
In contrast, the FOVM based on fractional-order calculus was found to describe complex stress relaxation of polymers with fewer parameters [26; 27]. The use of fractional operators in viscoelastic material models was justified, consistent with molecular theories by [PERSON] and [PERSON] [28]. Therefore, the FOVM was successfully applied in describing the relaxation behavior of other viscoelastic materials [29; 30]. However, rather less attention has been paid to its application to the GMBs.  
The purpose of this paper is to provide and validate a viscoelastic constitutive model, i.e., FOVM, for modeling the stress relaxation in a 2.0 mm thick, PVC GMB at 20 \(\pm\) 1 \({}^{\circ}\)C: (1) Formulate the FOVM based on fractional-order calculus; (2) Identify the model parameters by best fit to results from wide-width strip tensile tests conducted over a range of tensile rates; (3) Verify the feasibility of FOVM in simulating the stress relaxation response of PVC GMBs; (4) Investigate the effects of tensile rate and initial strain on the model parameters, stress relaxation rate, and relaxation modulus, and analyse the residual stress during the relaxation process.