of motion of considered system which read $$\begin{aligned}
\rho\,\mathbf{u}_{tt} & =\mathrm{Div}\left[2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1+2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right],\nonumber \\
\label{eq:bulk-mod-3}\\
\eta\,\mathbf{P}_{tt} & =2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1-2\,\mu_{h}\,\mathrm{sym}\,\mathbf{P}-\lambda_{h}\mathrm{tr}\,\mathbf{P}\:\mathds1\nonumber \\
 & +2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)-\alpha_{c}\,\mathrm{\mathrm{Cur}l}\left(\mathrm{\mathrm{Cur}l}\,\mathbf{P}\right)\nonumber\end{aligned}$$ or equivalently, in index notation[^5] $$\begin{aligned}
\rho\,\ddot{u}_{i} & =\mu_{e}\left(u_{i,jj}-P_{ij,j}+u_{j,ij}-P_{ji,j}\right)+\lambda_{e}\left(u_{j,ji}-P_{jj,i}\right)\delta_{ij}+\mu_{c}\left(u_{i,jj}-P_{ij,j}-u_{j,ij}+P_{ji,j}\right),\nonumber \\
\label{eq:bulk-mod-3-1}\\
\eta\,\ddot{P}_{ij} & =\mu_{e}\left(u_{i,j}-P_{ij}+u_{j,i}-P_{ji}\right)+\lambda_{e}\left(u_{k,k}-P_{kk}\right)\delta_{ij}-\mu_{h}\left(P_{ij}+P_{ji}\right)-\lambda_{h}P_{kk}\delta_{ij}\nonumber \\
 & +\mu_{c}\left(u_{i,j}-P_{ij}-u_{j,i}+P_{ji}\right)+\alpha_{c}\left(P_{ij,kk}-P_{ik,jk}\right).\nonumber\end{aligned}$$

The classical micromorphic continuum
------------------------------------

From here on, we call classical micromorphic continuum a medium the energy of which is given by $$\begin{aligned}
W & =\mu_{e}\left\Vert \,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\lambda_{e}}{2}\left(\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right)^{2}+\mu_{h}\left\Vert \,\mathrm{sym}\,\mathbf{P}\,\right\Vert ^{2}+\frac{\lambda_{h}}{2}\left(\mathrm{tr}\,\mathbf{P}\right)^{2}\label{KinPot-1-1}\\
 & +\mu_{c}\left\Vert \,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\alpha_{g}}{2}\left\Vert \nabla\,\mathbf{P}\right\Vert ^{2}.\nonumber\end{aligned}$$

### Comparison with the Mindlin and Eringen models

Analogously to what done for the relaxed case, we can recover that the classical micromorphic continuum can be seen as a particular case of Mindlin’s model by means of the parameter identification (\[Identification1\]) to which one must add: $$a_{10}=\alpha_{g},\qquad a_{1}=a_{2}=a_{3}=a_{4}=a_{5}=a_{6}=a_{7}=a_{8}=a_{9}=a_{11}=a_{12}=a_{13}=a_{14}=a_{15}=0.$$ Analogously, the classical micromorphic continuum can be obtained from Eringen’s model by means of the parameter identification (\[Identification1-1\]) to which one must add the conditions $$\tau_{7}=\alpha_{g},\qquad\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau_{5}=\tau_{6}=\tau_{8}=\tau_{9}=\tau_{10}=\tau_{11}=0.$$

### Governing equations

The equations of motion are the same as Eqs. (\[eq:bulk-mod-3\]), except for the gradient term in the second equation: $$\begin{aligned}
\rho\,\mathbf{u}_{tt} & =\mathrm{Div}\left[2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1+2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right],\nonumber \\
\label{eq:bulk-mod-3-2-1}\\
\eta\,\mathbf{P}_{tt} & =2\,\mu_{e}\,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\lambda_{e}\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\mathds1-2\,\mu_{h}\,\mathrm{sym}\,\mathbf{P}-\lambda_{h}\mathrm{tr}\,\mathbf{P}\:\mathds1\nonumber \\
 & +2\,\mu_{c}\,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)+\alpha_{g}\:\mathrm{Div}\left(\nabla\mathbf{P}\right).\nonumber\end{aligned}$$ The second equation can hence be rewritten in index notation as $$\begin{aligned}
\eta\,\ddot{P}_{ij} & =\mu_{e}\left(u_{i,j}-P_{ij}+u_{j,i}-P_{ji}\right)+\lambda_{e}\left(u_{k,k}-P_{kk}\right)\delta_{ij}-\mu_{h}\left(P_{ij}+P_{ji}\right)-\lambda_{h}P_{kk}\delta_{ij}\label{EqBulkClass}\\
 & +\mu_{c}\left(u_{i,j}-P_{ij}-u_{j,i}+P_{ji}\right)+\alpha_{g}P_{ij,kk}.\nonumber\end{aligned}$$

Plane wave propagation 