## 4 Asymptotic Analysis
### Plane Strain Deformation  
Let \(\psi_{f}\) be the value of \(\psi\) at \(x_{2}=0\). For definiteness, it is assumed that \(\psi_{f}=\pi/4+\phi/2\). The case \(\psi_{f}=-(\pi/4+\phi/2)\) can be treated in a similar manner. The condition (22) requires that  
\[q=q_{0}+q_{1}x_{2}^{\alpha}+o\Big{(}x_{2}^{\alpha}\Big{)} \tag{27}\]  
as \(x_{2}\to 0\) and  
\[0<\alpha<1. \tag{28}\]Here \(q_{0}\) and \(q_{1}\) are independent of \(x_{2}\). The first term in (19) is bounded. Therefore, Equations (27) and (19) combine to give  
\[\psi=\psi_{f}+\psi_{1}x_{2}^{\alpha}+o\big{(}x_{2}^{\alpha}\big{)} \tag{29}\]  
as \(x_{2}\to 0\). Turning to Equation (20), the first and third terms of this equation are bounded. The term on its right-hand side is also bounded. Therefore, the second term must be bounded as \(x_{2}\to 0\). Expanding \(\cos 2\psi\) in the Taylor series in the vicinity of \(\psi=\psi_{f}\) one gets  
\[1+\frac{\cos 2\psi}{\sin\phi}=-2\cot\phi\big{(}\psi-\psi_{f}\big{)}+o\big{[} \big{(}\psi-\psi_{f}\big{)}\big{]} \tag{30}\]  
as \(\psi\to\psi_{f}\). Using (27), (29) and (30) the second term in (20) is represented as  
\[\bigg{(}1+\frac{\cos 2\psi}{\sin\phi}\bigg{)}\bigg{[}1+\frac{x_{2}}{R(x_{1})} \bigg{]}\frac{\partial q}{\partial x_{2}}=-2\alpha q_{1}\psi_{1}\cot\phi x_{2 }^{2\alpha-1}+o\big{(}x_{2}^{2\alpha-1}\big{)} \tag{31}\]  
as \(x_{2}\to 0\). Substituting (27), (29) and (31) into (20) shows that \(x_{2}^{2\alpha-1}=O\big{(}x_{2}^{\alpha}\big{)}\) or \(x_{2}^{2\alpha-1}=O(1)\) as \(x_{2}\to 0\). The former yields \(\alpha=1\), which contradicts (28). The latter results in  
\[\alpha=\frac{1}{2}. \tag{32}\]  
Then, (27) and (29) become  
\[q=q_{0}+q_{1}\sqrt{x_{2}}+o\big{(}\sqrt{x_{2}}\big{)}\text{ and }\psi=\psi_{f}+\psi_{1}\sqrt{x_{2}}+o\big{(}\sqrt{x_{2}}\big{)} \tag{33}\]  
as \(x_{2}\to 0\), respectively. The distribution of \(p\) neat the curve is readily found from (4) and (33) as  
\[p=\frac{q_{0}-k\cos\phi}{\sin\phi}+\frac{q_{1}}{\sin\phi}\sqrt{x_{2}}+o\big{(} \sqrt{x_{2}}\big{)} \tag{34}\]  
as \(x_{2}\to 0\).  
It is now necessary to demonstrate that (33) is compatible with the plastic flow rule. The velocity components are represented as  
\[u_{1}=U_{0}+U_{1}x_{2}^{\beta}+o\big{(}x_{2}^{\beta}\big{)}\text{ and }u_{2}=V_{0}+V_{1}x_{2}^{\lambda}+o\big{(}x_{2}^{\lambda}\big{)} \tag{35}\]  
as \(x_{2}\to 0\). Here \(U_{0}\), \(U_{1}\), \(V_{0}\) and \(V_{1}\) are independent of \(x_{2}\). Moreover, \(\beta>0\) and \(\lambda>0\). The first equation in (11) shows that \(\partial u_{2}/\partial x_{2}=O(1)\) as \(x_{2}\to 0\). Therefore \(\lambda=1\), and the second equation in (34) becomes  
\[u_{2}=V_{0}+V_{1}x_{2}+o(x_{2}) \tag{36}\]  
as \(x_{2}\to 0\). Substituting (30), (33), (35) and (36) into the second equation in (11) results in  
\[2\psi_{1}V_{1}\beta\cot\phi\sqrt{x_{2}}x_{2}^{\beta-1}=O(x_{2}) \tag{37}\]  
as \(x_{2}\to 0\). It follows from this equation that \(\beta=1/2\) and the first equation in (35) becomes  
\[u_{1}=U_{0}+U_{1}\sqrt{x_{2}}+o\big{(}\sqrt{x_{2}}\big{)} \tag{38}\]  
as \(x_{2}\to 0\).  
To the best of the authors' knowledge, two semi-analytical solutions for the double slip and rotation are available [16; 17]. The asymptotic analysis of these solutions shows that they satisfy (33), (34), (36) and (38). Several semi-analytic solutions for the double-shearing model, which is another widely used model of pressure-dependent plasticity, have been derived in [11; 12; 13] and [16]. The asymptotic analysis of these solutions also shows that they satisfy (33), (34), (36), and (38).