## 3 Elastic and Inertial Constant Identifications
To apply the procedure, it is also assumed that [51]: (i) in the shear mode SH all nodes rotate of the same and unknown angle \(\varphi\); (ii) in the flexural mode FL, the effect of micro-warping is neglected. Then, the energetic contribution of the \(j\)-fiber, behaving as a Timoshenko micro-beam, to the cell's energy, is:  
\[U_{j}=\int_{0}^{h}\left(\frac{1}{2}EA\left(\varepsilon^{(1)}\right)^{2}+\frac {1}{2}GA^{*}\gamma^{2}+\frac{1}{2}EI\kappa^{2}+\dot{N}_{j}\varepsilon^{(2)} \right)\mathrm{d}x, \tag{15}\]  
where \(\dot{N}_{j}=-P/m\) for the (prestressed) \(x\)-fibers and \(\dot{N}_{j}=0\) for the (not prestressed) \(y\)-fibers, respectively. Moreover, for the assigned modes, the energy of each fiber, Equation (15), is evaluated by describing the strain field through the interpolating functions, which exactly satisfy the linear elastic problem of the Timoshenko beam (without prestress). The cell's elastic and geometric energetic contributions are obtained by summing all the \(U_{j}\); they read:  
\[\tilde{U}^{0} :=\frac{1}{2}\,h\left[m\,EA\,\varepsilon^{2}+m\frac{12 EI}{(1+ \alpha)h^{2}}\gamma^{2}+\left(mEI+EA\sum_{j=1}^{m}y_{j}^{2}\right)\kappa^{2}\right]\] \[+\frac{1}{2}\,h\left[\frac{12 EI\left(2m-1\right)}{(1+\alpha)h^{2} }\varphi^{2}-\frac{24 EI\,m}{(1+\alpha)h^{2}}\gamma\varphi\right], \tag{16}\] \[\tilde{U}^{*} :=-P\,\frac{1}{2}\,h\left[\frac{6+10\alpha}{5(1+\alpha)^{2}} \gamma^{2}+\frac{1-5\alpha^{2}}{5(1+\alpha)^{2}}\varphi^{2}+\frac{h^{2}}{3} \kappa^{2}+h\,\gamma\,\kappa-\frac{2-10\alpha^{2}}{5(1+\alpha)^{2}}\gamma \varphi\right],\]  
where \(\alpha:=12 EI/\left(GA^{*}h^{2}\right)\) is a non-dimensional stiffness ratio. The contributions in the square bracket terms (once multiplied by \(h\)) on the right-end sides of Equation (16) are, in the order: (i) in Equation (16)-a the extensional, shear and flexural elastic energies of the \(x\)-fibers, and the flexural-shear elastic energy of the \(y\)-fibers, respectively; (ii) the flexural-shear geometric contribution of the prestressed \(x\)-fibers. By requiring \(\frac{\partial\tilde{U}}{\partial\varphi}=0\), the unknown \(\varphi\) is condensed, and \(\tilde{\phi}\) is known. Then, by imposing \(\phi=\tilde{\phi}\), \(\forall\,(\varepsilon,\gamma,\kappa)\) and enforcing the Green law, once \(\phi\) is expanded in Maclaurin series of \(P\) up to the first order, the elastic constants, Equation (5), follow:  
Figure 2: Single cell of the grid beam, and its deformation modes: (**a**) extensional (EX) mode; (**b**) shear (SH) mode; (**c**) flexural (FL) mode. \[c_{11} =m\,EA, \tag{17}\] \[c_{22} =\chi\,m\frac{12 EI}{\left(1+\alpha\right)h^{2}},\] \[c_{33} =\left(m\,EI+EA\,\sum_{j=1}^{m}y_{j}^{2}\right),\] \[c_{22}^{*} =\chi^{*}\frac{6+10\alpha}{5\left(1+\alpha\right)^{2}},\] \[c_{23}^{*} =\frac{h}{2},\] \[\tilde{U}^{0} :=\frac{1}{2}\,h\left[m\,EA\,\varepsilon^{2}+m\frac{12 EI}{(1+ \alpha)h^{2}}\gamma^{2}+\left(mEI+EA\sum_{j=1}^{m}y_{j}^{2}\right)\kappa^{2}\right]\]0  
where:  
\[\tilde{U}^{0} :=\frac{1}{2}\,h\left[m\,EA\,\varepsilon^{2}+m\frac{12 EI}{(1+ \alpha)h^{2}}\gamma^{2}+\left(mEI+EA\sum_{j=1}^{m}y_{j}^{2}\right)\kappa^{2}\right]\]1 \[\tilde{U}^{0} :=\frac{1}{2}\,h\left[m\,EA\,\varepsilon^{2}+m\frac{12 EI}{(1+ \alpha)h^{2}}\gamma^{2}+\left(mEI+EA\sum_{j=1}^{m}y_{j}^{2}\right)\kappa^{2}\right]\]2  
are non-dimensional quantities named the shear factors. However, while \(\chi\), accounting for the flexibility of the transverse fibers, is not new, since it has been already encountered in previous works [51; 53], \(\chi^{*}\) is a new corrective term, that takes into account for the flexibility of the transverse fibers also in the geometric part of the constitutive matrix \(\mathbf{C}^{*}\). Finally, it can be seen that \(\chi^{*}\) slightly varies with \(m\). Indeed, in the particular case \(\alpha=0\), i.e., when micro-beams behave as Euler-Bernoulli beams, it is \(\chi_{s}^{*}=\left(21m^{2}-22m+6\right)/\left[6(1-2m)^{2}\right]\), which ranges from \(\chi_{s}^{*}=23/27\simeq 0.85\), when \(m=2\), to \(\chi_{s}^{*}=7/8=0.875\), when \(m\to\infty\).