$\boldsymbol{\epsilon}$ is the standard Levi-Civita tensor.

For any second order tensor $\mathbf{X}\in\mathbb{R}^{3\times3}$ we introduce its symmetric, skew-symmetric, spheric and deviatoric part respectively as $$\mathrm{sym}\:\mathbf{X}=\frac{1}{2}\left(\mathbf{X}+\mathbf{X}^{T}\right),\quad\mathrm{skew}\:\mathbf{X}=\frac{1}{2}\left(\mathbf{X}-\mathbf{X}^{T}\right),\quad\mathrm{sph}\:\mathbf{X}=\frac{1}{3}\,\mathrm{tr}\,\mathbf{X}\:\mathds1,\quad\mathrm{dev}\,\mathbf{X}=\mathbf{X}-\mathrm{sph}\:\mathbf{X},$$ or equivalently, in index notation $$\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\negthickspace\left(\mathrm{sym}\:\mathbf{X}\right)_{ij}=\frac{1}{2}\left(X_{ij}+X_{ji}\right),\ \ \left(\mathrm{skew}\:\mathbf{X}\right)_{ij}=\frac{1}{2}\left(X_{ij}-X_{ji}\right),\ \ \left(\mathrm{sph}\:\mathbf{X}\right)_{ij}=\frac{1}{3}\, X_{kk}\:\delta_{ij},\ \ \left(\mathrm{dev}\,\mathbf{X}\right)_{ij}=X_{ij}-\left(\mathrm{sph}\:\mathbf{X}\right)_{ij},$$ where $\delta_{ij}$ is the Kronecher delta tensor which is equal to $1$ when $i=j$ and equal to $0$ if $i\neq j$. The *Cartan-Lie-algebra decomposition* for the tensor $\mathbf{X}$ is introduced as $$\mathbf{X}=\mathrm{dev\: sym}\,\mathbf{X}+\mathrm{skew}\,\mathbf{X}+\mathrm{sph}\,\mathbf{X}.\label{CartanLie}$$

Equations of motion in strong form
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We describe the deformation of the considered continuum by introducing a Lagrangian configuration $B_{L}\subset\mathbb{R}^{3}$ and a suitably regular kinematical field $\boldsymbol{\chi}(\mathbf{X},t)$ which associates to any material point $\mathbf{X}\in B_{L}$ its current position $\mathbf{x}$ at time $t$. The image of the function $\boldsymbol{\chi}$ gives, at any instant $t$ the current shape of the body $B_{E}(t)$ which is often referred to as Eulerian configuration of the system. Since we will use it in the following, we also introduce the displacement field $\mathbf{u}(\mathbf{X},t)=\boldsymbol{\chi}(\mathbf{X},t)-\mathbf{X}$. The kinematics of the continuum is then enriched by adding a second order tensor field $\mathbf{P}(\mathbf{X},t)$ which accounts for deformations associated to the microstructure of the continuum itself. Hence, the current state of the considered continuum is identified by 12 independent kinematical fields: 