will be used in the model with dissipation.

From the above discussion of kinematics we see that a single crystal with $n$ active slip systems is a generalized continuum with $3+2n$ degrees of freedom at each point: $\mathbf{y}(\mathbf{x})$, $\beta ^{\mathfrak{a}}(\mathbf{x})$, and $l^{\mathfrak{a}}({\boldsymbol{\xi}}^{\mathfrak{a}}(\mathbf{x}))$, ${\mathfrak{a}}=1,\ldots ,n$. We require that the free energy per unit volume of crystal (assumed as macroscopically homogeneous) must be a function of $\mathbf{C}^e=\mathbf{F}^{eT}\cdot \mathbf{F}^e$ (where $\mathbf{F}^e=\mathbf{F}\cdot \mathbf{F}^{p-1}$), $\rho^{\mathfrak{a}}_\perp $, and $\rho^{\mathfrak{a}}_\parallel $ $$\psi = \psi (\mathbf{C}^e,\rho^{\mathfrak{a}}_\perp ,\rho^{\mathfrak{a}}_\parallel ).$$ Under the same loading condition as for the crystal with single slip we write down the energy functional $$\label{eq:2.22}
I[\mathbf{y}(\mathbf{x}),\beta ^{\mathfrak{a}}(\mathbf{x}),l^{\mathfrak{a}}({\boldsymbol{\xi}}^{\mathfrak{a}}(\mathbf{x}))]=\int_{\mathcal{V}}w(\mathbf{F},\beta ^{\mathfrak{a}},\nabla \beta ^{\mathfrak{a}},\nabla l^{\mathfrak{a}})\, dx-\int_{\partial _s} \mathbf{t} \cdot \mathbf{y}\, da,$$ where $$w(\mathbf{F},\beta ^{\mathfrak{a}},\nabla \beta ^{\mathfrak{a}},\nabla l^{\mathfrak{a}})=\psi (\mathbf{C}^e,\rho^{\mathfrak{a}}_\perp ,\rho^{\mathfrak{a}}_\parallel ).$$ Provided the resistance to the dislocation motion is negligibly small, we formulate the following variational principle for single crystals with $n$ active slip systems: the true placement vector $\check{\mathbf{y}}(\mathbf{x})$, the true plastic slips $\check{\beta }^{\mathfrak{a}}(\mathbf{x})$, and the true loop functions $\check{l}^{\mathfrak{a}}({\boldsymbol{\xi}}^{\mathfrak{a}}(\mathbf{x}))$ in the [*final*]{} equilibrium state of deformation minimize energy functional among all continuously differentiable fields $\mathbf{y}(\mathbf{x})$, $\beta ^{\mathfrak{a}}(\mathbf{x})$, and $l^{\mathfrak{a}}({\boldsymbol{\xi}}^{\mathfrak{a}}(\mathbf{x}))$ satisfying the constraints $$\label{eq:2.23}
\mathbf{y}(\mathbf{x})=\mathbf{x}+\mathbf{u}_0(\mathbf{x}),\quad \beta ^{\mathfrak{a}}(\mathbf{x})=0, \quad l^{\mathfrak{a}}(\mathbf{x})=0 \quad \text{at $\partial _k$},$$

Applying the same calculus of variation 