strain defined by $$\mathbf{C}^e=\mathbf{F}^{eT}\cdot \mathbf{F}^e.$$

Let the undeformed single crystal occupy some region $\mathcal{V}$ of the three-dimensional space. The boundary of this region, $\partial \mathcal{V}$, is assumed to be the closure of union of two non-intersecting surfaces, $\partial _k$ and $\partial _s$. Let the placement be a given smooth function of coordinates $$\label{eq:2.6}
\mathbf{y}(\mathbf{x})=\mathbf{x}+\mathbf{u}_0(\mathbf{x}) \quad \text{at $\partial _k$},$$ where $\mathbf{u}_0(\mathbf{x})=\mathbf{y}(\mathbf{x})-\mathbf{x}$ is the given displacement vector. Such condition does not admit dislocations to reach this part $\partial _k$ of the boundary, so we set $$\label{eq:2.6a}
\beta (\mathbf{x})=0, \quad l(\mathbf{x})=0 \quad \text{at $\partial _k$}.$$ At the remaining part $\partial _s$ the “dead” load (traction) $\mathbf{t}$ is specified. Note that, in case the whole boundary is free, we do not have any kinematic constraint at $\partial \mathcal{V}$. However, as the energy density is invariant with respect to the shift of the loop function $l(\mathbf{x})$ on an arbitrary constant which does not change the dislocation densities, we can impose on this scalar function the following constraint $$\int_{\mathcal{V}}l(\mathbf{x})\, dx=0,$$ where $dx=dx_1dx_2dx_3$ denotes the volume element. If no body force acts on this crystal, then its energy functional is defined as $$\label{eq:2.7}
I[\mathbf{y}(\mathbf{x}),\beta (\mathbf{x}),l(\mathbf{x})]=\int_{\mathcal{V}}w(\mathbf{F},\beta ,\nabla \beta ,\nabla l)\, dx-\int_{\partial _s} \mathbf{t} \cdot \mathbf{y}\, da,$$ where $$\label{eq:2.8}
