## 2 Basic Equations
### Axisymmetric Deformation  
Let \((x_{1},\ x_{2})\) be an orthogonal coordinate system in the generic plane of a cylindrical coordinate system \((r,\ \theta,z)\) and \(\gamma\) is the angle between the r-axis and the \(x_{1}-\) direction measured from the r-axis anticlockwise (Figure 3). As in the case of plane strain deformation, the \(x_{2}-\) lines are straight. Then, the scale factor of these coordinate lines are unity and the scale factor of the \(x_{1}-\) coordinate lines is given by (1). The solution is independent of \(\theta\). Moreover, the circumferential velocity and the shear stresses \(\sigma_{1\theta}\) and \(\sigma_{2\theta}\) vanish everywhere. The non-trivial equilibrium equations are [30]  
Figure 2: Orientation of the major principal stress. \[ \begin{split}\frac{\partial\sigma_{11}}{\partial x_{1}}+\left[1+\frac{x_ {2}}{R(x_{1})}\right]\frac{\partial\sigma_{12}}{\partial x_{2}}+\frac{( \sigma_{11}-\sigma_{00})}{r}\frac{\partial r}{\partial x_{1}}+\sigma_{12} \left[\left(1+\frac{x_{2}}{R(x_{1})}\right)\frac{\partial r}{\partial r\partial x _{2}}+\frac{2}{R(x_{1})}\right]=0,\\ \frac{\partial\sigma_{22}}{\partial x_{2}}+\left[1+\frac{x_{2}}{R( x_{1})}\right]^{-1}\frac{\partial\sigma_{12}}{\partial x_{1}}+\frac{( \sigma_{22}-\sigma_{00})}{r}\frac{\partial r}{\partial x_{2}}+\frac{(\sigma_{ 22}-\sigma_{11})}{R(x_{1})+x_{2}}+\left(1+\frac{x_{2}}{R(x_{1})}\right)^{-1} \frac{\sigma_{12}}{r}\frac{\partial r}{\partial x_{1}}=0.\end{split}  \tag{12}\]  
It follows from the geometry of Figure 3 that \(\partial r/\partial x_{1}=(1+x_{2}/R(x_{1}))\cos\gamma\) and \(\partial r/\partial x_{2}=-\sin\gamma\). Then, the equations in (12) become  
\[ \begin{split}\frac{\partial\sigma_{11}}{\partial x_{1}}+\left[1+ \frac{x_{2}}{R(x_{1})}\right]\frac{\partial\sigma_{12}}{\partial x_{2}}+\frac{ (\sigma_{11}-\sigma_{00})}{r}\left[1+\frac{x_{2}}{R(x_{1})}\right]\cos\gamma+ \sigma_{12}\left[\frac{2}{R(x_{1})}-\left(1+\frac{x_{2}}{R(x_{1})}\right) \frac{\sin\gamma}{r}\right]=0,\\ \frac{\partial\sigma_{22}}{\partial x_{2}}+\left[1+\frac{x_{2}}{R( x_{1})}\right]^{-1}\frac{\partial\sigma_{12}}{\partial x_{1}}-\frac{( \sigma_{22}-\sigma_{00})\sin\gamma}{r}+\frac{(\sigma_{22}-\sigma_{11})}{R(x_{ 1})+x_{2}}+\frac{\sigma_{12}\cos\gamma}{r}=0.\end{split}  \tag{13}\]  
There are several flow regimes associated with difference faces and edges of the Mohr-Coulomb yield criterion (Figure 4). One of the principal stresses is the circumferential stress \(\sigma_{\theta\theta}\). The other principal stresses are denoted as \(\sigma_{1}\) and \(\sigma_{2}\). It is assumed that \(\sigma_{1}>\sigma_{2}\). In this case, there are two regimes in which the hypothesis of [PERSON] and [PERSON], that \(\sigma_{1}=\sigma_{\theta\theta}\) or \(\sigma_{2}=\sigma_{\theta\theta}\), is satisfied. These regimes correspond to point A and F in Figure 4. Using (3) the yield criterion can be represented as  
\[\sigma_{\theta\theta}=-p-mq,\;q=p\sin\phi+k\cos\phi \tag{14}\]  
where \(m=+1\) for regime A, and \(m=-1\) for regime F. The equations in (5) are valid. Substituting (5) and the first equation in (14) into (13) yields  
\[ \begin{split}-\frac{\partial p}{\partial x_{1}}+\cos 2\psi\frac{ \partial q}{\partial x_{1}}-2q\sin 2\psi\frac{\partial\psi}{\partial x_{1}}+\left[1+ \frac{x_{2}}{R(x_{1})}\right]\left(\sin 2\psi\frac{\partial q}{\partial x_{2}}+2q \cos 2\psi\frac{\partial\psi}{\partial x_{2}}\right)+\\ \left[1+\frac{x_{2}}{R(x_{1})}\right]\frac{(\cos 2\psi+m)q\cos \gamma}{r}+\left[\frac{2}{R(x_{1})}-\left(1+\frac{x_{2}}{R(x_{1})}\right) \frac{\sin\gamma}{r}\right]q\sin 2\psi=0,\\ -\frac{\partial p}{\partial x_{2}}-\cos 2\psi\frac{\partial q}{ \partial x_{2}}+2q\sin 2\psi\frac{\partial\psi}{\partial x_{2}}+\left[1+\frac{x_{2}}{R(x_{1} )}\right]^{-1}\left(\sin 2\psi\frac{\partial q}{\partial x_{1}}+2q\cos 2\psi\frac{ \partial\psi}{\partial x_{1}}\right)+\\ \frac{(\cos 2\psi-m)q\sin\gamma}{r}-\frac{2q\cos 2\psi}{R(x_{1})+x_{2 }}+\frac{q\sin 2\psi\cos\gamma}{r}=0.\end{split}