use in this model only 5 parameters instead of Mindlin’s 7 parameters. Things become even more interesting when looking at the terms in the energy which involve derivatives of the microstrain tensor $\mathbf{P}$. Indeed, our relaxed model only provides 1 extra parameter, $\alpha_{c}$, instead of Mindlin’s 11 independent parameters ($a_{1},\: a_{2},\: a_{3},\: a_{4},\: a_{5},\: a_{8},\: a_{10},\: a_{11},\: a_{13},\: a_{14},\: a_{15}$). We claim that the proposed relaxed model is able to catch the basic features of observable material behaviours of materials with microstructure.

When considering Eringen model for micromorphic continua (see [@EringenBook] p. 273 for the strain energy density), we can recover that our relaxed model can be obtained from Eringen’s one by setting $$\begin{gathered}
\mu=\mu_{e}-\mu_{c},\quad\lambda=\lambda_{e},\quad\tau=\lambda_{e}+\lambda_{h},\quad\nu=-\lambda_{e},\quad\eta=\mu_{e}+\mu_{h}-\mu_{c},\quad\sigma=\mu_{c}-\mu_{e},\quad k=2\mu_{c},\label{Identification1-1}\\
\nonumber \\
\tau_{7}=\alpha_{c},\quad\tau_{9}=-\alpha_{c},\qquad\tau_{1}=\tau_{2}=\tau_{3}=\tau_{4}=\tau_{5}=\tau_{6}=\tau_{8}=\tau_{10}=\tau_{11}=0.\label{Identification2-1}\end{gathered}$$ Also in this case, we replace Eringen’s 7 parameters with only 5 parameters and, when considering terms with derivatives of the microstrain tensor $\mathbf{P}$, we have only one additional parameter $\alpha_{c}$ instead of Eringen’s 11 parameters.

### Governing equations

Imposing the first variation of the action functional to be vanishing (i.e. $\delta\mathcal{A}=0$), integrating by parts a suitable number of times and considering arbitrary variations $\delta\boldsymbol{\chi}$ and $\delta\mathbf{P}$ of the basic kinematical fields, we obtain the strong form of the bulk equations 