variational principles of the three-dimensional CDT and derives its governing equations. Section 4 extends this nonlinear theory to the case of single crystals with $n$ active slip system. Section 5 studies the three-dimensional small strain CDT. Section 6 is devoted to the analytical and asymptotic solutions of the two-dimensional energy minimization problem of a single crystal beam deforming in simple shear. Finally, Section 7 concludes the paper.

3-D kinematics for single crystals deforming in single slip {#sec:ncdt}
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Nonlinear CDT starts from the basic kinematic resolution of the deformation gradient $\mathbf{F}=\partial \mathbf{y}/\partial \mathbf{x}$ into elastic and plastic parts [@Bilby57] $$\label{eq:2.1}
    \mathbf{F}=\mathbf{F}^e\cdot \mathbf{F}^p.$$ We attribute an active role to the plastic deformation: $\mathbf{F}^p$ is the deformation [*creating*]{} dislocations (either inside or at the boundary of the volume element) or [*changing*]{} their positions in the crystal without distorting the lattice parallelism (see Fig. \[fig:Nhslip\]). On the contrary, the elastic deformation $\mathbf{F}^e$ deforms the crystal lattice having [*frozen*]{} dislocations [@Le2014nonlinear]. Note that the lattice vectors remain unchanged when the plastic deformation is applied, while they change together with the shape vectors by the elastic deformation.

![Multiplicative decomposition[]{data-label="fig:Nhslip"}](Fp){height="6.5cm"}

We consider first a single crystal deforming in single slip. In this case let us denote 