On the other hand, we will specify the choice of the strain energy density $W$ in the next subsections by discussing the cases of the relaxed micromorphic continuum introduced in [@NeffRelaxed] and of the classical micromorphic continuum.

The relaxed micromorphic continuum
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The strain energy density for the relaxed micromorphic continuum can be written as

$$\begin{aligned}
W & =\mu_{e}\left\Vert \,\mathrm{sym}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\lambda_{e}}{2}\left(\mathrm{tr}\left(\nabla\mathbf{u}-\mathbf{P}\right)\right)^{2}+\mu_{h}\left\Vert \,\mathrm{sym}\,\mathbf{P}\,\right\Vert ^{2}+\frac{\lambda_{h}}{2}\left(\mathrm{tr}\,\mathbf{P}\right)^{2}\label{KinPot}\\
 & +\mu_{c}\left\Vert \,\mathrm{skew}\left(\nabla\mathbf{u}-\mathbf{P}\right)\,\right\Vert ^{2}+\frac{\alpha_{c}}{2}\left\Vert \,\mathrm{\mathrm{Cur}l}\,\mathbf{P}\,\right\Vert ^{2},\nonumber\end{aligned}$$

where all the introduced elastic coefficients are assumed to be constant. This decomposition of the strain energy density, valid in the isotropic, linear-elastic case, has been proposed in [@NeffRelaxed; @Ghiba] where well-posedness theorems have also been proved. It is clear that this decomposition introduces a limited number of elastic parameters and we will show how this may help in the physical interpretation of these latter. Positive definiteness of the potential energy implies the following simple relations on the introduced parameters $$\mu_{e}>0,\qquad\mu_{c}>0,\qquad3\lambda_{e}+2\mu_{e}>0,\qquad\mu_{h}>0,\qquad3\lambda_{h}+2\mu_{h}>0,\qquad\alpha_{c}>0.\label{DefPos}$$ One of the most interesting features of the proposed strain energy density is the reduced number of elastic parameters which are needed to fully describe the mechanical behavior of a micromorphic continuum. Indeed, each parameter can be easily related to specific micro and macro deformation modes. In the following, we 