# A Variable-Order Dynamic Constitutive Model for Clay Based on the Fractional Calculus
## 4 Establishment of Dynamic Constitutive Model
### The Strain Response of Clay under Long-Term Alternating Load  
#### 4.2.1 Dynamic Response Theory of Alternating Strain \(\varepsilon_{a}(t)\)  
Viscoelastic materials, such as clay, show a viscous effect under subjection to alternating stress, Energy storage and dissipation are continuously accompanied by the process of soil deformation under long-term alternating loading [50]. The phase differences between dynamic strain and dynamic stress for elastic and ideal viscous materials are 0 and \(\pi/2\), respectively. Clay is regarded as a kind of viscoelastic material, so that the alternating strain \(\varepsilon_{a}(t)\) lags behind alternating stress \(\sigma_{a}(t)\), assuming that the phase difference between \(\varepsilon_{a}(t)\) and \(\sigma_{d}(t)\) is \(\theta\); thus, \(0<\theta<\pi/2\), retardation time is \(\theta/\omega\), and \(\theta\) is the complementary angle of energy dissipation angle in viscoelastic mechanics. On the basis of the creep constitutive model portrayed in Figure 11, the alternating stress on clay can be therefore be expressed as  
\[\sigma_{a}(t)=\sigma_{1}\sin(wt)=\sigma_{1}e^{iwt} \tag{27}\]  
where \(\sigma_{1}\) is the amplitude of alternating stress, and \(\sigma_{1}=(\sigma_{\text{d.max}}-\sigma_{\text{d.min}})/2\), while \(i\) is the imaginary unit. According to viscoelasticity theory, the strain response can be calculated thus:  
\[\varepsilon_{a}(t)=\varepsilon_{1}\sin(wt+\theta)=\varepsilon_{1}e^{i(wt+ \theta)}=\varepsilon^{*}e^{iwt} \tag{28}\]  
Figure 11: Improved variable-order fractional creep constitutive model.  
Figure 10: Viscoplastic body.  
where \(\varepsilon_{1}\) is the amplitude of alternating strain, \(\varepsilon^{*}\) is the amplitude of complex strain, and \(\varepsilon_{1}\) and \(\theta\) are given by  
\[\varepsilon_{1}=\sigma_{1}\sqrt{\frac{1}{2}+J_{2}^{2}} \tag{29}\]  
\[\theta=\arctan(J_{2}/J_{1}) \tag{30}\]  
where \(J_{1}\) and \(J_{2}\) are represent storage and loss compliance, respectively.  
Following the analysis above, the strain response equation of the hook body under alternating stress \(\sigma_{1}e^{iwt}\) is expressed as  
\[\sigma_{1}e^{iwt}=E_{0}\cdot\varepsilon^{*}e^{iwt} \tag{31}\]  
By rewriting Equation (31) into the form of complex compliance, then  
\[J^{*}(iw)=\frac{\varepsilon^{*}}{\sigma_{1}}=\frac{1}{E_{0}}=J_{1}-iJ_{2} \tag{32}\]  
Where storage compliance \(J_{1}=1/E_{0}\), and loss compliance \(J_{2}=0\).  
Variable-Order Fractional Abel Dashpot  
For the variable-order fractional Abel dashpot, the strain response equation can be derived in accordance with Equation (13) and Fourier transform thus:  
\[\sigma_{1}e^{iwt}=\eta_{a(t)}(iw)^{a(t)}\varepsilon^{*}e^{iwt} \tag{33}\]  
Equation (33) can be rewritten as  
\[J^{*}(iw)=\frac{\varepsilon^{*}}{\sigma_{1}}=\frac{1}{\eta_{a(t)}(iw)^{a(t)}} \tag{34}\]  
Due to \((i)^{a(t)}=\cos(\frac{a(t)\pi}{2})+i\cdot\sin(\frac{a(t)\pi}{2})\), Equation (34) can now be rewritten as  
\[\frac{\varepsilon^{*}}{\sigma_{1}}=\frac{\cos(\frac{a(t)\pi}{2})}{\eta_{a(t)} w^{a(t)}}-i\frac{\sin(\frac{a(t)\pi}{2})}{\eta_{a(t)}w^{a(t)}}=J_{1}-iJ_{2} \tag{35}\]  
Accordingly, storage compliance \(J_{1}=\frac{\cos(\frac{a(t)\pi}{2})}{\eta_{a(t)}w^{a(t)}}\), loss compliance \(J_{2}=\frac{\sin(\frac{a(t)\pi}{2})}{\eta_{a(t)}w^{a(t)}}\).  
Viscoplastic Body  
For the viscoplastic body, the strain response equation can be derived in accordance with Equation (25) and Fourier transform, thus:  
\[\frac{\varepsilon^{*}}{\sigma_{1}}=\frac{e^{\beta t}}{\eta}=J_{1}-iJ_{2} \tag{36}\]  
Consequently, storage compliance \(J_{1}=\frac{e^{\beta t}}{\eta}\), loss compliance \(J_{2}=0\).