w(\mathbf{F},\beta ,\nabla \beta ,\nabla l)=\psi (\mathbf{C}^e,\rho_\perp ,\rho_\parallel ),$$ Provided the resistance to the dislocation motion is negligibly small and no surfaces of discontinuity occur inside crystals, then the following variational principle is valid for single crystals with one active slip system: the true placement vector $\check{\mathbf{y}}(\mathbf{x})$, the true plastic slip $\check{\beta }(\mathbf{x})$, and the true loop function $\check{l}(\mathbf{x})$ in the [*final*]{} equilibrium state of deformation minimize energy functional among all continuously differentiable fields $\mathbf{y}(\mathbf{x})$, $\beta (\mathbf{x})$, and $l(\mathbf{x})$ satisfying constraints and .

Let us derive the equilibrium equations from this variational principle. We compute the first variation of functional $$\delta I= \int_{\mathcal{V}}
\left( \mathbf{P}\mathbf{:}\delta \mathbf{y}\nabla +\frac{\partial w}{\partial \beta }\delta \beta +\frac{\partial w}{\partial \nabla \beta }\cdot \nabla \delta \beta +\frac{\partial w}{\nabla l}\cdot \nabla \delta l \right) dx
- \int_{\partial _s} \mathbf{t} \cdot \delta \mathbf{y}\, da ,$$ where $\mathbf{P}=\partial w /\partial \mathbf{F}$. Integrating the first, third, and fourth term by parts with the help of Gauss’ theorem and taking the conditions and into account, we obtain $$\begin{gathered}
\label{eq:2.9}
\delta I = \int_{\mathcal{V}}[ -\delta \mathbf{y}\cdot (\mathbf{P}\cdot \nabla ) +(w_\beta -\nabla \cdot w_{\nabla \beta })\delta \beta - (\nabla \cdot w_{\nabla l})\delta l ] \, dx
\\
+ \int_{\partial _s} [(\mathbf{P}\cdot \mathbf{n}-\mathbf{t} )\cdot \delta \mathbf{y} 