## 2 Basic Equations
### Plane Strain Deformation  
Let \((x_{1},\,x_{2})\) be an orthogonal coordinate system in planes of flow. The \(x_{2}-\) coordinate curves are straight (Figure 1). Then, it is possible to assume with no loss of generality that the scale factor of these coordinate curves is unity. The scale factor of the \(x_{1}-\) coordinate lines may be represented as  
\[H_{1}=1+\frac{x_{2}}{R(x_{1})}. \tag{1}\]  
Here \(R(x_{1})\) is the radius of curvature of the \(x_{1}-\)coordinate curve determined by the equation \(x_{2}=0\) (S-line in Figure 1). The equilibrium equations referred to the coordinate system chosen are [30]  
\[\frac{\partial\sigma_{11}}{\partial x_{1}}+\left[1+\frac{x_{2}}{R(x_{1})} \right]\frac{\partial\sigma_{12}}{\partial x_{2}}+\frac{2\sigma_{12}}{R(x_{1} )}=0,\,\,\,\left[1+\frac{x_{2}}{R(x_{1})}\right]\frac{\partial\sigma_{22}}{ \partial x_{2}}+\frac{\partial\sigma_{12}}{\partial x_{1}}+\frac{(\sigma_{22} -\sigma_{11})}{R(x_{1})}=0. \tag{2}\]  
Here, Equation (1) has been used. Furthermore, \(\sigma_{11}\), \(\sigma_{22}\) and \(\sigma_{12}\) are the stress components referred to the \((x_{1},\,x_{2})-\) coordinate system. As usual, it is convenient to introduce the following stress variables:  
\[p=-\frac{1}{2}(\sigma_{11}+\sigma_{22}),\ q=\frac{1}{2}\sqrt{(\sigma_{11}-\sigma_ {22})^{2}+4\sigma_{12}^{2}}>0. \tag{3}\]  
Then, the Mohr-Coulomb yield criterion reads  
\[q-p\sin\phi=k\cos\phi. \tag{4}\]  
Here \(k\) is the cohesion and \(\phi\) is the angle of internal friction. Both are material constants. Let \(\psi\) be the orientation of the major principal stress \(\sigma_{1}\) relative to the \(x_{1}-\) direction (Figure 2). Then, using (3), the stress components referred to the \((x_{1},\,x_{2})-\) coordinate system are expressed as  
\[\sigma_{11}=-p+q\cos 2\psi,\ \sigma_{22}=-p-q\cos 2\psi,\ \sigma_{12}=q\sin 2\psi. \tag{5}\]  
Substituting (5) into (2) leads to  
\[ \begin{array}{c}-\frac{\partial p}{\partial x_{1}}+\cos 2\psi\frac{ \partial q}{\partial x_{1}}-2q\sin 2\psi\frac{\partial\psi}{\partial x_{1}}+ \left[1+\frac{x_{2}}{R(x_{1})}\right]\sin 2\psi\frac{\partial q}{\partial x_{2}}+ \\ 2q\cos 2\psi\left[1+\frac{x_{2}}{R(x_{1})}\right]\frac{\partial\psi}{ \partial x_{2}}+\frac{2q\sin 2\psi}{R(x_{1})}=0,\\ -\left[1+\frac{x_{2}}{R(x_{1})}\right]\left(\frac{\partial\psi}{ \partial x_{2}}+\cos 2\psi\frac{\partial q}{\partial x_{2}}\right)+2q\left[1+ \frac{x_{2}}{R(x_{1})}\right]\sin 2\psi\frac{\partial\psi}{\partial x_{2}}+\sin 2 \psi\frac{\partial q}{\partial x_{1}}+\\ 2q\cos 2\psi\frac{\partial\psi}{\partial x_{1}}-\frac{2q\cos 2\psi}{R(x_{1})}=0. \end{array}  \tag{6}\]  
One can eliminate p in these equations using (4). As a result,  
\[ \begin{array}{c}\left(\cos 2\psi-\frac{1}{\sin \phi}\right)\frac{\partial q}{\partial x_{1}}-2q\sin 2\psi\frac{\partial\psi}{ \partial x_{1}}+\left[1+\frac{x_{2}}{R(x_{1})}\right]\sin 2\psi\frac{ \partial q}{\partial x_{2}}+\\ 2q\cos 2\psi\left[1+\frac{x_{2}}{R(x_{1})}\right]\frac{\partial\psi}{ \partial x_{2}}+\frac{2q\sin 2\psi}{R(x_{1})}=0,\\ -\left[1+\frac{x_{2}}{R(x_{1})}\right]\left(\cos 2\psi+\frac{1}{\sin \phi}\right)\frac{\partial q}{\partial x_{2}}+2q\left[1+\frac{x_{2}}{R(x_{1})} \right]\sin 2\psi\frac{\partial\psi}{\partial x_{2}}+\sin 2\psi\frac{\partial q}{ \partial x_{1}}+\\ 2q\cos 2\psi\frac{\partial\psi}{\partial x_{1}}-\frac{2q\cos 2\psi}{R(x_{1})}=0. \end{array}  \tag{7}\]  
Figure 1: Orthogonal coordinate system \((x_{1},\,x_{2})\) near a given curve (S–line) under plane strain conditions. The flow rule is [15]  
\[\xi_{11}+\xi_{22}=0,\ \xi_{12}\cos 2\psi-\frac{1}{2}(\xi_{11}-\xi_{22})\sin 2\psi+( \omega_{12}+\Omega)\sin\phi=0. \tag{8}\]  
Here \(\xi_{11}\), \(\xi_{22}\) and \(\xi_{12}\) are the strain rate components referred to the \((x_{1},\ x_{2})-\) coordinate system. Furthermore, \(\omega_{12}\) is the only non-zero spin component and \(\Omega\) is the intrinsic spin due to grain rotation. In general, the latter is an unknown variable which is governed by the equation of rotational motion.