+w_{\nabla \beta }\cdot \mathbf{n}\, \delta \beta]\, da + \int_{\partial \mathcal{V}}w_{\nabla l }\cdot \mathbf{n}\, \delta l \, da=0.\end{gathered}$$ Equation implies that the minimizer must satisfy in $\mathcal{V}$ the equilibrium equations $$\label{eq:2.10}
\mathbf{P}\cdot \nabla =0,
\quad
-w_\beta +\nabla \cdot w_{\nabla \beta }=0, \quad \nabla \cdot w_{\nabla l} =0,$$ subjected to the kinematic boundary conditions and at $\partial _k$, and the following natural boundary conditions $$\label{eq:2.11}
\mathbf{P}\cdot \mathbf{n}=\mathbf{t},
\quad
w_{\nabla \beta }\cdot \mathbf{n}=0 , \quad w_{\nabla l }\cdot \mathbf{n}=0 \quad \text{at $\partial _s$}.$$ We call $\mathbf{P}$ the first Piola-Kirchhoff stress tensor, $\tau _r=-w_\beta $ the resolved shear stress (or Schmid stress), and $\varsigma =-\nabla \cdot w_{\nabla \beta }$ the back stress. The first equation of is nothing else but the equilibrium of macro-forces acting on the crystal, the second equation represents the equilibrium of micro-forces acting on dislocations, while the last one expresses the equilibrium condition for the curved dislocation lines.

The constitutive equations for $\mathbf{P}=w_{\mathbf{F}}$, $-w_\beta $, $w_{\nabla \beta }$, and $w_{\nabla l}$ can easily be obtained from the free energy density . First, we express $\mathbf{F}^e$ in terms of $\mathbf{F}$ and $\beta $ with the use of and $$\mathbf{F}^e=\mathbf{F}\cdot \mathbf{F}^{p-1}=\mathbf{F}\cdot (\mathbf{I}-\beta \mathbf{s}\otimes \mathbf{m}).$$ Now, the standard differentiation using the chain rule 