## 3 Finite Element Model
### Model Description  
The model was built in Abaqus/CAE 2017 by resorting to brick-type elements for each layer of the beam. The finite element type adopted is C3D8 RH, usually used for purely mechanical analysis, with a total of 16,269 nodes and 10,752 elements. The elements comprising the model are defined as deformable elements with associated non-linearity and cracking characteristics. Material-related constitutive non-linearities were simulated by defining appropriate constitutive bonds: anisotropic elasto-plastic for wood and Yeoh-like hyperelastic for polyurethane. An elasto-plastic constitutive bond was adopted for wood boards, in which the anisotropy of the material in the three main directions was considered. Linear elasticity in an orthotropic material is most easily defined by giving the \"engineering constants\":the three moduli E\({}_{1}\), E\({}_{2}\), E\({}_{3}\); Poisson's ratio \(\
u_{12}\), \(\
u_{13}\), \(\
u_{23}\); and the shear moduli G\({}_{12}\), G\({}_{13}\), and G\({}_{23}\) associated with the principal directions of the material. These moduli define the elastic yielding according to:  
\[\left\{ \begin{matrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{13}\\ \gamma_{23}\end{matrix} \right\}=\left[ \begin{matrix}1/E_{1}&-\
u_{21}/E_{2}&- \
u_{31}/E_{3}&0&0&0\\ -\
u_{12}/E_{1}&1/E_{2}&-\
u_{32}/E_{3}&0&0&0\\ -\
u_{13}/E_{1}&-\
u_{23}/E_{2}&1/E_{3}&0&0&0\\ 0&0&0&1/G_{12}&0&0\\ 0&0&0&0&1/G_{13}&0\\ 0&0&0&0&0&1/G_{23}\end{matrix} \right]\left\{ \begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\end{matrix} \right\} \tag{1}\]  
The quantity \(\
u_{ij}\) has the physical interpretation of Poisson's ratio that characterizes the transverse deformation in the j direction when the material is stressed in the i direction. In general, \(\
u_{ij}\) if it is not equal to \(\
u_{ij}\): is related by \(\
u_{ij}/E_{i}=\
u_{ji}/E_{j}\). Engineering constants can also be given as functions of temperature and other predefined fields, if necessary. The elastic solution must meet the following stability conditions:  
\[ \begin{array}{l}E_{1},E_{2},E_{3},G_{12},G_{13},G_{23},>0,\\ |\
u_{12}|<(E_{1}/E_{2})^{1/2},\\ |\
u_{13}|<(E_{1}/E_{3})^{1/2},\\ |\
u_{23}|<(E_{2}/E_{3})^{1/2},\\ 1-\
u_{12}\
u_{21}-\
u_{23}\
u_{32}-\
u_{31}\
u_{13}-2\
u_{21}\
u_{32}\
u_{13}> 0\end{array}  \tag{2}\]  
When the left side of the inequality approaches zero, the material exhibits incompressible behavior. Using the relations \(\
u_{ij}/E_{i}=\
u_{ji}/E_{j}\), the second, third and fourth restrictions of the previous series can also be expressed as:  
\[ \begin{array}{l}|\
u_{21}|<(E_{2}/E_{1})^{1/2},\\ |\
u_{31}|<(E_{3}/E_{1})^{1/2},\\ |\
u_{32}|<(E_{3}/E_{2})^{1/2}.\end{array}  \tag{3}\]  
The following Tables 3 and 4 show the parameters defined in the FE model for the representation of wooden planks. [PERSON]'s model, in which the material is defined as hyperelastic isotropic, was adopted for the polyurethane layer. According to this phenomenological model, the elastic characteristics of rubber can be represented by a strain energy density func \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Plastic Behavior of Wood**} \\ \hline \(\sigma_{y}\) (MPa) & 104 & parametric \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Post-elastic characteristic of wood. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{5}{c}{**Elastic Behavior of Wood**} \\ \hline E\({}_{1}\) (MPa) & 390 & \\ \hline E\({}_{2}\) (MPa) & 690 & \\ E\({}_{3}\) (MPa) & 15,900 & \\ \(\
u_{12}\) & 0.25 & \\ \(\
u_{13}\) & 0.01 & [PERSON] et al., 1935 [43; 44] \\ \(\
u_{23}\) & 0.03 & \\ G\({}_{12}\) (MPa) & 36 & \\ G\({}_{13}\) (MPa) & 770 & \\ G\({}_{23}\) (MPa) & 620 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mechanical parameters of wood. of powers of the strain invariants I\({}_{1}\), I\({}_{2}\) and I\({}_{3}\) of the Cauchy-Green strain tensors. This consideration was the basis of [PERSON]'s theory, from which [PERSON]'s model starts.