and taking into account the arbitrariness of the variations of $\mathbf{y}(\mathbf{x})$, $\beta (\mathbf{x})$, and $l({\boldsymbol{\xi}}^{\mathfrak{a}}(\mathbf{x}))$ in $\mathcal{V}$ as well as at $\partial _s$, one can show that the minimizer must satisfy in $\mathcal{V}$ the equilibrium equations $$\label{eq:2.25}
\mathbf{P}\cdot \nabla =0,
\quad
-w_\beta ^{\mathfrak{a}}+\nabla \cdot w_{\nabla \beta ^{\mathfrak{a}}}=0, \quad \nabla \cdot w_{\nabla l^{\mathfrak{a}}} =0,$$ subjected to the kinematic boundary conditions at $\partial _k$ and the following natural boundary conditions $$\label{eq:2.26}
\mathbf{P}\cdot \mathbf{n}=\mathbf{t},
\quad
w_{\nabla \beta ^{\mathfrak{a}}}\cdot \mathbf{n}=0, \quad w_{\nabla l^{\mathfrak{a}}}\cdot \mathbf{n}=0 \quad \text{at $\partial _s$}.$$ The constitutive equations for $\mathbf{P}=w_{\mathbf{F}}$, $-w_{\beta ^{\mathfrak{a}}}$, $w_{\nabla \beta ^{\mathfrak{a}}}$, and $w_{\nabla l^{\mathfrak{a}}}$ can easily be obtained from the above free energy density by standard differentiation. For the first Piola-Kirchhoff stress tensor and the Schmid stresses we have $$\label{eq:2.27}
\mathbf{P}=w_{\mathbf{F}}=2\mathbf{F}^e\cdot \psi _{\mathbf{C}^e} \cdot \mathbf{F}^{p-T}.$$ $$\label{eq:2.28}
\tau ^{\mathfrak{a}}_r=-w_{\beta ^{\mathfrak{a}}}=2\mathbf{s}^{\mathfrak{a}}\cdot \mathbf{F}^{p-T}\cdot \mathbf{C}^e \cdot \psi _{\mathbf{C}^e}\cdot \mathbf{m}^{\mathfrak{a}}.$$ Likewise, from follows $$\label{eq:2.29}
w_{\nabla \beta ^{\mathfrak{a}}}=\frac{1}{b}[\psi _{\rho^{\mathfrak{a}}_\perp }{\text{sign}}(s^{\mathfrak{a}}_\nu \partial _{\nu ^{\mathfrak{a}}} \beta ^{\mathfrak{a}})s^{\mathfrak{a}}_\nu +\psi _{\rho^{\mathfrak{a}}_\parallel }{\text{sign}}(s^{\mathfrak{a}}_\tau \partial _{\nu ^{\mathfrak{a}}} \beta ^{\mathfrak{a}})s^{\mathfrak{a}}_\tau ] {\boldsymbol{\nu}}^{\mathfrak{a}}.$$ Thus, the vectors $w_{\nabla \beta ^{\mathfrak{a}}}$ are two-dimensional. Finally, for $w_{\nabla l^{\mathfrak{a}}}$ we have $$\label{eq:2.30}
w_{\nabla l^{\mathfrak{a}}}=\left( \psi _{\rho _\perp ^{\mathfrak{a}}}\frac{\partial \rho _\perp ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;1}}+\psi _{\rho _\parallel ^{\mathfrak{a}}}\frac{\partial \rho _\parallel ^{\mathfrak{a}}}{\partial l^{\mathfrak{a}}_{;1}}\right)\mathbf{s}^{\mathfrak{a}}+\left( \psi _{\rho _\perp ^{\mathfrak{a}}}\frac{\partial \rho _\perp 