^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;2}}+\psi _{\rho _\parallel ^{\mathfrak{a}}}\frac{\partial \rho _\parallel ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;2}}\right)\mathbf{p}^{\mathfrak{a}}.$$ Differentiating formulas for the dislocation densities $\rho^{\mathfrak{a}}_\perp $ and $\rho^{\mathfrak{a}}_\parallel $ with respect to $l^{\mathfrak{a}}_{;1}$ and $l^{\mathfrak{a}}_{;2}$, we get $$\begin{aligned}
\frac{\partial \rho _\perp ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;1}}&=\frac{1}{b} {\text{sign}}(s^{\mathfrak{a}}_\nu \partial _{\nu ^{\mathfrak{a}}}\beta ^{\mathfrak{a}})\left( -\frac{2(l^{\mathfrak{a}}_{;1})^2(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{((l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2)^2}+\frac{2\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2}}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2} \right) ,
\\
\frac{\partial \rho _\parallel ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;1}}&=\frac{1}{b}{\text{sign}}(s^{\mathfrak{a}}_\tau \partial _{\nu ^{\mathfrak{a}}}\beta ^{\mathfrak{a}})\left( -\frac{2l^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;2}(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{((l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2)^2}+\frac{\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;2}}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2} \right) ,
\\
\frac{\partial \rho _\perp ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;2}}&=\frac{1}{b}{\text{sign}}(s^{\mathfrak{a}}_\nu \partial _{\nu ^{\mathfrak{a}}}\beta ^{\mathfrak{a}})\left( -\frac{2l^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;2}(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{((l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2)^2}+\frac{\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;1}}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2} \right) ,
\\
\frac{\partial \rho _\parallel ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;1}}&=\frac{1}{b} {\text{sign}}(s^{\mathfrak{a}}_\tau \partial _{\nu ^{\mathfrak{a}}}\beta ^{\mathfrak{a}})\left( -\frac{2(l^{\mathfrak{a}}_{;2})^2(\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2})}{((l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2)^2}+\frac{\beta ^{\mathfrak{a}}_{;1}l^{\mathfrak{a}}_{;1}+2\beta ^{\mathfrak{a}}_{;2}l^{\mathfrak{a}}_{;2}}{(l^{\mathfrak{a}}_{;1})^2+(l^{\mathfrak{a}}_{;2})^2} \right) .\end{aligned}$$ Substituting the constitutive equations - into - we get the completely new system of equations and boundary conditions which, together with , enables one to determine $\check{\mathbf{y}}(\mathbf{x})$, $\check{\beta }^{\mathfrak{a}}(\mathbf{x})$, and $\check{l}^{\mathfrak{a}}(\mathbf{x})$.

For the case of non-zero resistance to dislocation motion leading to the energy dissipation we take the dissipation function in the form $$D=D(\dot{\beta }^{\mathfrak{a}},\dot{l}^{\mathfrak{a}}),$$ We require that the true placement $\check{\mathbf{y}}(\mathbf{x},t)$, the true plastic slips $\check{\beta }^{\mathfrak{a}}(\mathbf{x},t)$, and the true loop functions $\check{l}^{\mathfrak{a}}(\mathbf{x},t)$ obey the variational equation $$\label{eq:2.32}
\delta I+\int_{\mathcal{V}} \sum_{{\mathfrak{a}}=1}^n (\frac{\partial D}{\partial \dot{\beta }^{\mathfrak{a}}}\delta \beta ^{\mathfrak{a}}+\frac{\partial D}{\partial \dot{l}^{\mathfrak{a}}}\delta l^{\mathfrak{a}})\, dx=0$$ for all variations of admissible fields $\mathbf{y}(\mathbf{x},t)$, $\beta ^{\mathfrak{a}}(\mathbf{x},t)$, and $l^{\mathfrak{a}}(\mathbf{x},t)$ satisfying 