evolve and the dislocation lines cannot move: $\dot{\beta}=0$ and $\dot{l}=0$. Thus, they are frozen in the crystal.

Extension to multiple slip
==========================

The extension to the case of single crystals having $n$ active slip systems can be done straightforwardly under the assumption[^2] $$\label{eq:2.16}
\mathbf{F}^p=\mathbf{I}+\sum_{\mathfrak{a}=1}^n \beta ^\mathfrak{a}(\mathbf{x}) \mathbf{s}^\mathfrak{a}\otimes \mathbf{m}^\mathfrak{a} ,$$ with $\beta ^\mathfrak{a}$ being the plastic slip, where the pair of constant and mutually orthogonal unit vectors $\mathbf{s}^\mathfrak{a}$ and $\mathbf{m}^\mathfrak{a}$ is used to denote the slip direction and the normal to the slip planes of the corresponding $\mathfrak{a}$-th slip system, respectively. Here and later, the Gothic upper index $\mathfrak{a}$ running from 1 to $n$ numerates the slip systems, so one could clearly distinguish $\beta ^{\mathfrak{a}}$ from the power function. We denote by $\mathbf{p}^{\mathfrak{a}}$ the unit vector lying in the slip plane such that $\mathbf{s}^{\mathfrak{a}}$, $\mathbf{p}^{\mathfrak{a}}$, and $\mathbf{m}^{\mathfrak{a}}$ form a right-handed basis vectors. For each slip system we can introduce the coordinates associated with these basis vectors $$\label{eq:2.17}
\xi ^{\mathfrak{a}}_1=\mathbf{s}^{\mathfrak{a}}\cdot \mathbf{x}, \quad \xi ^{\mathfrak{a}}_2=\mathbf{p}^{\mathfrak{a}}\cdot \mathbf{x},\quad \xi ^{\mathfrak{a}}_3=\mathbf{m}^{\mathfrak{a}}\cdot \mathbf{x}.$$ Equations can be regarded as the one-to-one linear transformation relating ${\boldsymbol{\xi}}^{\mathfrak{a}}$ and $\mathbf{x}$ according to $${\boldsymbol{\xi}}^{\mathfrak{a}}=\mathbf{M}^{\mathfrak{a}}\cdot \mathbf{x}, \quad \mathbf{x}=\mathbf{M}^{{\mathfrak{a}}-1}{\boldsymbol{\xi}}^{\mathfrak{a}},$$ where $\mathbf{M}^{\mathfrak{a}}$ is the $3\times 3$ matrix whose rows are basis vectors $\mathbf{s}^{\mathfrak{a}}$, $\mathbf{p}^{\mathfrak{a}}$, and 