# A Creep Model of Asphal Mixture Based on Variable Order Fractional Derivative
## 4 Variable Order Fractional Viscoelastoplastic Creep Model of Asphal Mixture  
As shown in Figure 3, the creep behavior of asphalt mixture can be divided into two types according to the stress level. From the curve (a) in Figure 3, the asphalt mixture mainly exhibits linear viscoelasticity at low stress level, and the creep process only includes two stages: primary stage (\(0<t\leq t_{1}\)) and steady stage (\(t>t_{1}\)). At higher stress levels, damage will occur in the creep process after a certain time (\(t_{2}\)), and, in turn, damage evolution will accelerate the creep. The time (\(t_{2}\)) is called the damage incubation time, which is determined by the stress level. It is well accepted that different constant applied stresses correspond to different damage incubation times, and the higher the applied stress, the shorter the damage incubation time required. It is expected, therefore, that there exist two different critical stresses at a given temperature: one being the so-called critical damage initiation stress, \(\sigma_{\mathrm{d}}\), which is the minimum value of stress below which no damage occurs for any elapsed loading time, and the other, the critical fracture stress, \(\sigma_{\mathrm{f}}\), at which the material fractures instantaneously, leading to a zero damage incubation time. Thus, the damage incubation time can be expressed by the model as (21):  
\[t_{2}=A\left(\frac{\sigma_{\mathrm{f}}-\sigma}{\sigma-\sigma_{\mathrm{d}}} \right)^{n},\ (\sigma_{\mathrm{d}}<\sigma\leq\sigma_{\mathrm{f}}) \tag{16}\]  
where \(A\) and \(n\) are material constants determined from the test data fitting. As seen from curve (b) in Figure 3, when the creep stress is higher than the damage initiation stress \(\sigma_{\mathrm{d}}\), besides the primary stage (\(0<t\leq t_{1}\)) and steady stage (\(t_{1}<t\leq t_{2}\)), the tertiary stage will occur for \(t>t_{2}\) During the primary stage, the strain rate gradually decreases with time; during the steady stage, the strain rate basically keeps constant, and the strain increases with a constant speed or tends to be a stable value; during the tertiary stage, because of the damage evolution, the viscosity of the material decreases with time and the strain rate increases with time, which results in a rapid increase of deformation and ultimate failure. According to the above two types of creep behavior and their characteristics, a fractional viscoelastic creep model consisting of variable order fractional viscoelastic element and time-varying viscosity dashpot element will be constructed to analyze and describe the creep behavior of asphalt mixture. Figure 2: Dashpot element with time-varying viscosity. The creep process was divided into primary stage, steady stage and tertiary stage according to the time history, and each stage was described by different viscoelastic elements or viscoplastic elements. Considering the instantaneous elasticity, viscoelasticity and viscoplasticity of the asphalt mixture, a variable order fractional viscoelasticity creep model was proposed. Figure 4 shows the schematic diagram of the model. If the creep process only consists of the primary stage and steady stage, the time-varying viscoplastic element will be removed from the model, which is then reduced to the variable order fractional viscoelastic creep model. Compared with the classic viscoelastic and or viscoplastic model, the viscoelastic element and/or viscoplastic element in the present model only play their roles in the corresponding creep stages. At low stress levels (\(\sigma\leq\sigma_{\text{d}}\)), the asphalt mixture only exhibits the primary creep and steady creep. As seen in Figure 4, at the moment of loading, the asphalt mixture produces an instantaneous elastic deformation, which can be described by the spring element in the model:  
\[\varepsilon_{0}=\frac{\sigma}{E_{0}} \tag{17}\]  
In the primary stage, due to the low elastic modulus of the asphalt component in the mixture, viscous flow will occur, which will reduce the aggregate gap.