## 4 Discussion on the Effective Stress Principle for Saturated Claystone  
### The Value of Effective Stress Coefficient for Claystone  
The incremental form of Equation (2) can be written as follow:  
\[\Delta\sigma_{e}=\Delta\sigma_{c}-\eta\Delta\sigma_{p} \tag{3}\]  
where \(\Delta\sigma_{e}\) is the increment of effective stress; \(\Delta\sigma_{c}\) is the increment of confining pressure; \(\Delta\sigma_{p}\) is the increment of pore pressure.  
Figure 7: The pressures and strains curve during Stage two: (**a**) sample one; (**b**) sample two.  
The largest linear elastic deformation of the samples during Stage one can be obtained from Figure 6 (Table 5).  
When \(\Delta\sigma_{p}=0.2\) MPa and \(\Delta\sigma_{c}=0.18\) MPa, the samples are dilatant, meaning \(\Delta\sigma_{e}<0\) under this stress condition.  
When \(\Delta\sigma_{p}=0.2\) MPa and \(\Delta\sigma_{c}=0.22\) MPa, the samples are compressed, meaning \(\Delta\sigma_{e}>0\) under this stress condition.  
Substituting the above results into Equation (3) gives the following:  
\[\left.\begin{array}{l}0.18-0.2\eta<0\\ 0.22-0.2\eta>0\end{array}\right\}\Rightarrow 0.9<\eta<1.1 \tag{4}\]  
[PERSON] [11] presented that the effective stress coefficient of homogeneous rock was not expected to exceed one. Thus, Equation (4) evolves into \(0.9<\eta<1.0\).  
When \(\Delta\sigma_{p}=0.2\) MPa and \(\Delta\sigma_{c}=0.2\) MPa, the deformation of samples are not obvious, meaning \(\Delta\sigma_{e}\approx 0\) under this condition.  
Then the following equation is deduced:  
\[0.2-0.2\eta\approx 0\Rightarrow\eta\approx 1 \tag{5}\]  
The above conclusions indicate that \(\eta\) is a value greater than 0.9 and less than but close to 1.  
The effective stress coefficient will be further analyzed according to the above conclusions. The relationship between the stress increments and axial strain increments can be expressed by a generalized [PERSON]'s law:  
\[\Delta\varepsilon_{z}=\frac{1}{E_{v}}[\Delta\sigma_{z}^{e}-v_{vh}(\Delta \sigma_{x}^{e}+\Delta\sigma_{y}^{e})] \tag{6}\]  
where \(\Delta\varepsilon_{z}\) in the axial strain increment; \(\Delta\sigma_{x}\), \(\Delta\sigma_{y}\), and \(\Delta\sigma_{z}\) are the effective stress increments in x, y, and z direction; \(E_{v}\) is the vertical elastic modulus; \(v_{vh}\) is the Poison ratio in the horizontal and vertical plane.  
According to the results of the stress (Figure 6) and strain (Table 5) variations, the back analysis was performed to determine the effective stress coefficient. The FMINSEARCH function of MATLAB (version number: 8.3) was selected for the optimization calculation. The optimization calculation process is shown in Figure 8, and the objective function of the back analysis can be expressed as the following:  
\[\sum_{i=1}^{n}\biggl{\{}\Delta\varepsilon_{zi}-\frac{1}{E_{v}}[\Delta\sigma_{zi }^{e}-v_{vh}(\Delta\sigma_{xi}^{e}+\Delta\sigma_{yi}^{e})]\biggr{\}}^{2} \rightarrow\min \tag{7}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Deformation of Sample One**} & \multicolumn{3}{c}{**Deformation of Sample Two**} \\ \cline{2-7}
**The Number of Cycles** & **Axial Strain** & **Radial Strain** & **Volumetric Strain** & **Axial Strain** & **Radial Strain** & **Volumetric Strain** \\ \hline The first & \(-4.95\) & \(-0.81\) & \(-6.57\) & \(-5.24\) & \(-0.73\) & \(-6.70\) \\ The second & \(-2.36\) & \(-0.23\) & \(-2.82\) & \(-2.46\) & \(-0.30\) & \(-3.06\) \\ The third & \(-1.36\) & \(-0.07\)  
To ensure the accuracy of this back analysis, the range of physical parameters is shown in Table 6.  
The analysis results showed that the effective stress coefficient of claystone was 0.991 for sample one and 0.995 for sample two.