## 2 Measurement of the Stress Conversion Coefficient
### Experiment Content
#### 2.2.1 Definition of the Stress Conversion Coefficient  
Under unidirectional compression, any soil mass tends to expand towards the free surface. At this time, the length of the soil mass shortens in the direction of the maximum principal stress and lengthens in the direction of other principal stress, which manifests as a \"flattened\" state at a macrolevel. We assume that there is a square cylindrical soil body with length, width b, and height h, and the first principal stress, second principal stress, and third principal stress are \(\sigma_{1}\), \(\sigma_{2}\), and \(\sigma_{3}\), respectively. In this stressed state, the soil's height gradually decreases, and the change in height is \(\delta_{1}\); thus, soil deformation in the direction of \(\sigma_{1}\) is \(\delta_{1}\). Similarly, the deformation of soil in the other two principal stress directions is \(\delta_{2}\) and \(\delta_{3}\).  
Due to the \"squishing\" state of the soil mass, under the three-way stress state, the pressure on the soil mass in one direction inevitably causes the soil mass to bulge in the other two directions. In this case, it can be considered that there is tensile stress in this direction, which is caused by compressive stress in the other directions. For example, the tensile stress caused by \(\sigma_{1}\) in the direction of \(\sigma_{2}\) is called \(\sigma_{12t}\). The force state of the soil mass of the square column is shown in Figure 3:  
[PERSON] believes that viscoelastic plastic geotechnical materials are not ideal elastic materials [2]. In the deformation and failure processes under principal stress, energy is dissipated to the outside via the closure of micro-cracks, internal friction, and the damage evolution of micro-defects and partial plastic compaction deformation, and it is stored in the material by means of elastic strain energy and partial plastic strain energy [5]. The dissipated energy is dissipated rather than converted into lateral deformation so that the remaining stress causes transverse swelling tendencies. Therefore, part of axial stress \(\sigma_{1}\) is consumed, and only the residual stress \(\sigma_{1}^{p}\) causes a transverse swelling trend.  
\[\sigma_{1}^{ep}=\sigma_{1}-\sigma_{1}^{p} \tag{1}\]  
That is,  
\[\sigma_{1}^{ep}=\xi\sigma_{1} \tag{2}\]  
Figure 3: Square column soil under compression.  
where \(\xi\) is the stress conversion coefficient: \(0<\xi<1\). This coefficient represents the ability of the soil to convert axial pressure into radial tensile stress under compression. The greater the \(\xi\), the greater the radial tension of the soil under the same pressure.  
If the triaxial test adopts a rectangular cylindrical rock sample with a high \(h\) and wide \(b\), as shown in Figure 3, axial displacement \(\delta_{1}\) is generated, resulting in the transverse tensile stress of \(\sigma_{2}\) and \(\sigma_{3}\) as -\(\sigma_{124}\) and -\(\sigma_{134}\) (\"-\" indicates the tensile stress) and the transverse displacement of \(\sigma_{2}\) and \(\sigma_{3}\) as \(\delta_{12}\) and \(\delta_{13}\), respectively. According to [PERSON]'s swap theorem in elasticity, the following is the case.  
\[\sigma_{124}\cdot bh\cdot\delta_{12}+\sigma_{134}\cdot bh\cdot\delta_{13}= \sigma_{1}^{ep}\cdot b^{2}\cdot\delta_{1} \tag{3}\]  
If soil comprises isotropic materials, Equation (2) is substituted into Equation (3), and we can observe the following:  
\[\xi=\frac{W_{\sigma_{1}}}{W_{\sigma_{1}^{ep}}}=\frac{W_{\sigma_{1}}}{W_{\sigma _{124}+\sigma_{134}}} \tag{4}\]  
where \(W_{\sigma_{1}}\) is the operation carried out by the first principal stress, and \(W_{\sigma_{124}+\sigma_{134}}\) is the operation carried out by the tension stress generated by it.