enable one to determine $\check{\mathbf{y}}(\mathbf{x})$, $\check{\beta }(\mathbf{x})$, and $\check{l}(\mathbf{x})$. Note that equations $_1$ and $_2$ are coupled via the first Piola-Kirchhoff stress tensor and the Schmid stress containing both $\mathbf{F}$ and $\beta $, while equations $_2$ and $_3$ are coupled because both contain the gradients of $\beta $ and $l$. All equations are strongly nonlinear partial differential equations.

The above theory has been developed for the case of negligibly small resistance to dislocation motion and plastic slip. In real crystals there is however always the resistance to the dislocation motion and plastic slip causing the energy dissipation that changes the above variational principle as well as the equilibrium conditions. We assume that the dissipation function depends on the plastic slip rate $\dot{\beta }$ and on the normal velocity of the dislocation loop $v_\nu $ given by (or, equivalently, on $\dot{l}$). Thus, $$D=D(\dot{\beta },\dot{l}),$$ When the dissipation is taken into account, the above formulated variational principle must be modified. Following [@Sedov65; @Berdichevsky1967] we require that the true placement $\check{\mathbf{y}}(\mathbf{x},t)$, the true plastic slips $\check{\beta }(\mathbf{x},t)$, and the true loop function $\check{l}(\mathbf{x},t)$ obey the variational equation $$\label{eq:2.15b}
\delta I+\int_{\mathcal{V}} (\frac{\partial D}{\partial \dot{\beta }}\delta \beta +\frac{\partial D}{\partial \dot{l}}\delta l)\, dx=0$$ for all variations 