## 3 Elastic and Inertial Constant Identifications  
The identification of the elastic constants follows the lines of [53], where it has been analytically carried out by imposing an energy equivalence between the fine, i.e., the frame, and coarse models. It is worth noticing that the analytical identification of the elastic constants, can be pursued when, as in the case at hand, the fibers are arranged in a regular pattern and/or the cross-section is assumed to be rigid (see [45; 49; 51]). In the remaining cases, a numerical identification procedure must be followed (see, e.g., [52]). First a discrete map is defined that links at selected abscissas \(x_{i}:=ih\quad(i=0,1,\ldots,n)\), the displacements of the fine and coarse models. At the \(x_{i}\), \(x_{i+1}\) abscissas, the adjacent cross-sections \(\mathcal{D}_{i}\), \(\mathcal{D}_{i+1}\), respectively, bound a cell. It is assumed that the nodal points of the cross-section remain aligned, while the displacements inside the cells are unconstrained. The rigid motion of \(\mathcal{D}_{i}\) is written as:  
\[ \begin{split}\tilde{u}\left(x_{i},y\right)&=u \left(x_{i}\right)-\theta\left(x_{i}\right)y,\\ \tilde{v}\left(x_{i},y\right)&=v\left(x_{i}\right),\end{split} \qquad\text{in }\mathcal{D}_{i},\ i=0,1,\ldots,n. \tag{13}\]  
where the tilde denotes a quantity of the fine model. Then the equivalence of the energies stored by a cell and a segment of equal length of the 1D-beam is imposed under the same assigned displacements at the ends. In this step, the averaged energy density is evaluated, namely \(\tilde{\phi}:=\tilde{U}/h\), \(\tilde{U}\) being the elastic energy of a cell of the grid beam, which, by virtue of Equation (13), is a function of the configuration variables, i.e., \(\tilde{\phi}=\frac{1}{h}\tilde{U}\left(u\left(x_{i}\right),v\left(x_{i}\right),\theta\left(x_{i}\right),u\left(x_{i+1}\right),v\left(x_{i+1}\right),\theta \left(x_{i+1}\right)\right)\). Finally, the energy densities \(\tilde{\phi}\) and \(\phi\) are rendered equal. This task requires the evaluation of a suitable strain test field. Here the simplest (constant) one is assumed, and, after integration of the strain-displacement relationships (1) in the \(\left(x_{i},x_{i+1}\right)\) interval, the end displacements of the beam segment reads:  
\[ \begin{split} u\left(x_{i}\right)&=0,\qquad u\left(x _{i+1}\right)=\varepsilon\,h,\\ v\left(x_{i}\right)&=0,\qquad v\left(x_{i+1}\right) =\gamma\,h+\frac{1}{2}\kappa\,h^{2},\\ \theta\left(x_{i}\right)&=0,\qquad\theta\left(x_{i+1 }\right)=\kappa\,h.\end{split}  \tag{14}\]  
The elastic constants of matrix \(\mathbf{C}\) follow from \(\phi=\tilde{\phi}\quad\forall\left(\varepsilon,\gamma,\kappa\right)\), once use of Equation (14) is made. It is important to remark that the end displacements (14) can be interpreted as the superposition of three independent deformation modes of the cell, namely:  
* an extensional mode (EX), in which \(\mathcal{D}_{i+1}\) axially translates;
* a shear mode (SH), in which \(\mathcal{D}_{i+1}\) transversely translates;
* a flexural modes (FL), in which \(\mathcal{D}_{i+1}\) rotates around the \(z\)-axis and transversely translates. Analytical Identification of the Elastic Constants  
A cell made of \(m\)\(x\)-fibers and two \(y\)-fibers of half stiffnesses each, is considered. Only the nodal points on the \(y\)-fiber remain aligned (i.e., \(\mathcal{D}_{i}\) coincides with the set of the isolated joints), but the same fiber is free to warp around the line. In order to identify the elastic constants, the procedure previously resumed is applied (see also [53]). It calls for evaluating the elastic energy \(\tilde{U}\) of the cell, when the three deformation modes,displayed in Figure 2, are assigned. Since the cell is prestressed, it is \(\tilde{U}:=\tilde{U}^{0}+\tilde{U}^{*}\), \(\tilde{U}^{0}\) and \(\tilde{U}^{*}\) being the elastic and the geometric energetic contributions, respectively.