## 2 Governing Equations
### Evolution of the Fractional-Order  
The fractional-order \(\alpha\) in the fractional elastoplastic models is typically assumed to be constant for simplicity [16; 17; 18]. However, this assumption may be against the basic definition of the critical state concept, which requires the soil sample to perform fluid-like deformation without volumetric change. In this paper, the fractional derivatives of \(f\) with respect to \(p\) and \(q\) are given in Equations (13) and (14).  
\[\frac{\partial^{\alpha}f}{\partial p^{\alpha}}=\frac{\alpha M_{cs}^{2}p^{2}-( 2-\alpha)q^{2}}{\Gamma(3-\alpha)p^{\alpha}} \tag{13}\]  
\[\frac{\partial^{\alpha}f}{\partial q^{\alpha}}=\frac{2q^{2-\alpha}}{\Gamma(3 -\alpha)} \tag{14}\]  
A new stress-dilatancy relation according to the fractional-order flow rule is therefore derived:  
\[d=\frac{\alpha M_{cs}^{2}-(2-\alpha)\eta^{2}}{2\eta^{2-\alpha}} \tag{15}\]  
Equation (15) will return to the stress-dilatancy relation of the modified Cam-Clay model in the case of \(\alpha\) equals 1.0. The dilatancy ratio \(d\) (defined as \(d=de_{v}^{p}/de_{g}^{p}\)) is equal to zero when \(\eta=M_{cs}\) at the critical state with \(de_{v}^{p}\) being zero, and the basic definition of the critical state concept is satisfied. However, if \(\alpha\) is a constant that does not equal 1.0, excessive plastic volumetric deformation will be obtained at the critical state. After that, a better assumption is that \(\alpha\) is changeable during shearing and equals 1.0 at the critical state. As shown in Figure 4, changing the value of \(\alpha\) at a specific stress level significantly affects the value of \(d\). [PERSON] and [PERSON] [16] suggested that \(\alpha\) depends on the state parameter [22] for granular matters, obeying an empirical expression. With the increase in the value of \(\alpha\), the stress-dilatancy curve rotates clockwise.  
In fact, there is another phase transformation state ([PERSON] et al. [23]) where the value of \(d\) is zero before rock samples reach to the critical state. The slope of the phase transformation line in the \(p-q\) plane is denoted as \(M_{c}\). At the phase transformation state (e.g., \(\eta\) equals to \(M_{c}\)), a transition from compaction to dilation in the volume of rock samples is observed, and the fractional stress-dilatancy rule is rewritten as:  
\[d_{\eta=M_{c}}=\frac{\alpha M_{cs}^{2}-(2-\alpha)M_{c}^{2}}{2M_{c}^{2-\alpha} }=0 \tag{16}\]  
Rearranging Equation (16) leads to:  
\[\alpha=\frac{2M_{c}^{2}}{M_{cs}^{2}+M_{c}^{2}} \tag{17}\]  
To consider the effect of overconsolidation on the dilatancy of soft rocks, \(M_{c}\) is assumed to be a function of the similarity ratio \(R\) with \(M_{c}=M_{cs}R^{n}\), where \(n\) is a material constant. Once the critical state is reached due to shearing, we have \(R=1\) and \(M_{c}=M_{cs}\), then a unit value of \(\alpha\) is obtained. It can be found that the expression of \(\alpha\) is a function of \(M_{cs}\) and \(R\), but it is challenging to calibrate the additional parameter \(n\) using the laboratory tests, which is the main limitation of the proposed model.  
Figure 5 shows the fitting results of the fractional stress-dilatancy relation on the experimental data of Bourke clay ([PERSON] and [PERSON] [24]) at different temperatures. It can be found that the fractional stress-dilatancy relation has a better performance than the stress-dilatancy relation of the modified Cam-clay model, especially at a low-stress level.  
Figure 4: Evolution of the dilatancy ratio \(d\) with respect to the fractional\(-\)order \(\alpha\) and the stress ratio \(\eta\).  
Figure 5: Stress–dilatancy relation on the Bourke clay at different temperatures (data from ([PERSON] and [PERSON] [24])).