and the above relation yields the first Piola-Kirchhoff stress tensor $$\label{eq:2.12}
\mathbf{P}=w_{\mathbf{F}}=2\mathbf{F}^e\cdot \psi _{\mathbf{C}^e} \cdot \mathbf{F}^{p-T}.$$ For the resolved shear stress (Schmid stress) we get $$\label{eq:2.13}
\tau _r=-w_\beta =2\mathbf{s}\cdot \mathbf{F}^{p-T}\cdot \mathbf{C}^e \cdot \psi _{\mathbf{C}^e}\cdot \mathbf{m}.$$ Likewise, from follows $$\label{eq:2.14}
w_{\nabla \beta }=\frac{1}{b}[\psi _{\rho_\perp }{\text{sign}}(s_\nu \partial _\nu \beta )s_\nu +\psi _{\rho_\parallel }{\text{sign}}(s_\tau \partial _\nu \beta )s_\tau ] {\boldsymbol{\nu}}.$$ Thus, the vector $w_{\nabla \beta }$ is two-dimensional. Finally, we compute $w_{\nabla l }$ directly in components using formulas . Since $\rho _\perp $ and $\rho _\parallel $ do not depend on $l_{,3}$, so $w_{l_{,3}}=0$, and the vector $w_{\nabla l}$ is also two-dimensional. For its first two components we have $$\label{eq:2.15}
\begin{split}
w_{l_{,1}}=\frac{1}{b}\left[ \psi _{\rho _\perp }{\text{sign}}(s_\nu \partial _\nu \beta )\left( -\frac{2l_{,1}^2(\beta _{,1}l_{,1}+\beta _{,2}l_{,2})}{(l_{,1}^2+l_{,2}^2)^2}+\frac{2\beta _{,1}l_{,1}+\beta _{,2}l_{,2}}{l_{,1}^2+l_{,2}^2} \right) \right.
\\
+\left. \psi _{\rho _\parallel }{\text{sign}}(s_\tau \partial _\nu \beta )\left( -\frac{2l_{,1}l_{,2}(\beta _{,1}l_{,1}+\beta _{,2}l_{,2})}{(l_{,1}^2+l_{,2}^2)^2}+\frac{\beta _{,1}l_{,2}}{l_{,1}^2+l_{,2}^2} \right) \right] ,
\\
w_{l_{,2}}=\frac{1}{b}\left[ \psi _{\rho _\perp }{\text{sign}}(s_\nu \partial _\nu \beta )\left( -\frac{2l_{,1}l_{,2}(\beta _{,1}l_{,1}+\beta _{,2}l_{,2})}{(l_{,1}^2+l_{,2}^2)^2}+\frac{\beta _{,2}l_{,1}}{l_{,1}^2+l_{,2}^2} \right) \right.
\\
+\left. \psi _{\rho _\parallel }{\text{sign}}(s_\tau \partial _\nu \beta )\left( -\frac{2l_{,2}^2(\beta _{,1}l_{,1}+\beta _{,2}l_{,2})}{(l_{,1}^2+l_{,2}^2)^2}+\frac{\beta _{,1}l_{,1}+2\beta _{,2}l_{,2}}{l_{,1}^2+l_{,2}^2} \right) \right] .
\end{split}$$ Substituting the constitutive equations - into - we get the completely new system of equations and boundary conditions which, together with and , 