3 components of the displacement field and 9 components of the micro-deformation field. Such a theory of a continuum with microstructure has been derived in [@Mindlin] for the linear-elastic case and re-proposed e.g. in [@EringenI; @Eringen; @II; @Forest0; @Forest] for the case of non-linear elasticity.

Once the used kinematics has been made clear, we can introduce the action functional of the considered micromorphic system as $$\mathcal{A}=\int_{0}^{T}\int_{B_{L}}\left(T-W\right)d\mathbf{X}\, dt$$ where $T$ and $W$ are the kinetic and potential energies of the considered system respectively. Denoting by $\rho$ and $\text{\ensuremath{\eta}}$ the macroscopic and microscopic mass densities respectively, we choose the kinetic energy of the system to be $$\begin{gathered}
T=\frac{1}{2}\,\rho\left\Vert \,\mathbf{u}_{t}\,\right\Vert ^{2}+\frac{1}{2}\,\eta\left\Vert \,\mathbf{P}_{t}\,\right\Vert ^{2},\label{Kinetic}\end{gathered}$$ where we denote by a subscript $t$ partial derivative with respect to time. We remark that, since the micro-deformation tensor $\mathbf{P}$ is dimensionless, the micro-density $\eta$ has the dimensions of a bulk density ($Kg/m^{3}$) times the square of a length. This means that, if $\rho'$ is the true density (per unit of macro volume) of the material constituting the underlying microstructure of the medium one can think to write the homogenized density $\eta$ as (see also [@Mindlin]) $$\eta=d^{2}\:\rho',$$ where we denoted by $d$ the characteristic length of the microscopic inclusions[^4].

