## 2 Unified Strength Theory Formula under Plane Strain State  
The intermediate principal stress has an important influence on the failure of different materials. The twin-shear unified strength theory was proposed, containing all the stress components acting on the orthogonal octahedron element and different effects on material failure [30]. In addition, it can adapt to geotechnical materials with different tensile and compressive strengths and related to hydrostatic stress, and makes up for the deficiencies of the Mohr-Coulomb theory. This theory collects a series of failure criteria, covering all areas between the upper and lower limits, and has good applicability to different materials. The principal stress form of the twin-shear unified strength theory is expressed as:  
\[\begin{cases}f=\sigma_{1}-\frac{\alpha(\log_{2}+\sigma_{3})}{1+b}=\sigma_{t} \left(\sigma_{2}\leq\frac{\sigma_{1}+\alpha\sigma_{1}}{1+a}\right)\\ f^{\prime}=\frac{(b\
u_{2}+\sigma_{1})}{1+b}-\alpha\sigma_{3}=\sigma_{t}\left( \sigma_{2}\geq\frac{\sigma_{1}+\alpha\sigma_{3}}{1+a}\right)\end{cases}, \tag{1}\]  
\[\alpha=\frac{\sigma_{t}}{\sigma_{c}}=\frac{1-\sin\varphi_{0}}{1+\sin\varphi_{ 0}},\sigma_{t}=\frac{2c_{0}\cos\varphi_{0}}{1+\sin\varphi_{0}}, \tag{2}\]  
where \(b\) is the intermediate principal stress weight coefficient, which can express the influence of the intermediate principal stress on the yield of the material, and \(0\leq b\leq 1\)[29]; \(\sigma_{1}\), \(\sigma_{2}\), \(\sigma_{3}\) are three principal stresses; \(\sigma_{t}\) and \(\sigma_{c}\) are the material tensile and compressive stress, respectively; \(a\) is the material tension-compression ratio parameter; and \(c_{0}\) and \(\varphi_{0}\) are the soil cohesion and internal friction angle obtained from the triaxial test, respectively.  
The \(\pi\) plane limit lines of the unified strength theory are shown in Figure 1. The unified strength theory can degenerate to more calculation criteria under different \(b\) values. The limit lines are convex when \(0\leq b\leq 1\). When \(b<0\) or \(b>1\), the limit lines are non-convex. Therefore, the unified strength theory is not limited by the traditional convexity theory. In addition to forming a series of convex limit surfaces, it can also form a series of non-convex limit surfaces, which can be flexibly adapted to various materials. The innermost is the single-shear strength theory--that is, the Mohr-Coulomb strength theory (\(b=0\)). The outermost is the twin-shear strength theory (\(b=1\)).  
Because the intermediate principal stress \(\sigma_{2}\) is between the maximum principal stress \(\sigma_{1}\) and the minimum principal stress \(\sigma_{3}\), the following relationship is given:  
\[\sigma_{2}=\frac{m(\sigma_{1}+\sigma_{3})}{2}, \tag{3}\]  
where \(m\) is the intermediate principal stress coefficient. The elastic solution of the plane strain problem can be known: \(\varepsilon_{y}=0\). Therefore, \(m\) can be obtained by the generalized [PERSON]'s law: \(0<m\leq 1\). In plane strain, volume deformation in the plastic state is ignored, that is, \(m=1\). Moreover, when \(m=1\), \(b=0,\ 0.5,\ 1\), the twin-shear unified strength theory  
Figure 1: Yield loci of the unified strength theory.  
is a special case or linear approximation of Tresca, Mises, and twin-shear stress yield criteria. Different \(b\) values can reflect different intermediate principal stress effects. Therefore, \(b=0,\ 0.5,\ 1\) is taken as the representative value in subsequent analysis, combined with the influence of the intermediate principal stress. By combining Equations (1) and (2), the modified soil friction angle \(\varphi_{t}\) and cohesion \(c_{t}\) are determined as:  
\[\frac{\sigma_{1}-\sigma_{3}}{2}=\frac{\sigma_{1}+\sigma_{3}}{2}\sin\varphi_{t }+c_{t}\cos\varphi_{t}, \tag{4}\]  
\[\begin{cases}\varphi_{t}=\arcsin\left[\frac{2(1+b)\sin\varphi_{0}}{2(1+b+\sin \varphi_{0})}\right]\\ c_{t}=\frac{2(1+b)c_{0}\cos\varphi_{0}}{2+b(1+\sin\varphi_{0})}.\end{cases}. \tag{5}\]