## 3 Modelling of Wave Propagation in Wooden Poles
### Modelling of Wood Material  
Wave propagation in wood is a complicated dynamic procedure controlled by the properties, orientation and microstructure of the fiber, and perhaps more importantly, by the geometry of the material [28]. As a result, accurate modelling of wood materials is the priority issue to get insight into wave propagation behaviour in wood poles.  
To investigate the influence of the material characteristics on the wave behaviour, FE modelling of material characteristics should be divided into three categories: isotropic, transversely isotropic and orthotropic. Isotropic material has the uniform values of a property in all directions, none of the properties depend on the orientation and the material is perfectly rotationally symmetrical. Transversely isotropic material has the uniform properties in one plane and different properties in the direction normal to this plane. An orthotropic material has at least two symmetric orthogonal planes, where material property is unique and independent in the direction within each plane. A material without any symmetric planes in defined as orthotropic material [29; 30].  
According to the general Hooke's law, the orthotropic material is able to be mathematically expressed using the following matrix:  
\[\left[\begin{array}{c}\varepsilon_{LL}\\ \varepsilon_{RR}\\ \varepsilon_{TT}\\ \gamma_{LR}\\ \gamma_{LT}\\ \gamma_{RT}\end{array}\right]=\left[\begin{array}{cccccc}\frac{1}{E_{L}}&- \frac{\gamma_{RL}}{E_{R}}&-\frac{\gamma_{TL}}{E_{T}}&0&0&0\\ -\frac{\gamma_{LR}}{E_{L}}&\frac{1}{E_{R}}&-\frac{\gamma_{TR}}{E_{T}}&0&0&0\\ -\frac{\gamma_{LT}}{E_{L}}&-\frac{\gamma_{LT}}{E_{R}}&\frac{1}{E_{T}}&0&0&0\\ 0&0&0&\frac{1}{G_{LR}}&0&0\\ 0&0&0&0&\frac{1}{G_{LT}}&0\\ 0&0&0&0&0&\frac{1}{G_{RT}}\end{array}\right]\times\left[\begin{array}{c} \sigma_{LL}\\ \sigma_{RR}\\ \sigma_{TT}\\ \tau_{LR}\\ \tau_{LT}\\ \tau_{RT}\end{array}\right] \tag{1}\]  
where \(\varepsilon\) denotes the elastic strain vector; \(\gamma\) denotes the shear strain vector; \(\sigma\) denotes the stress vector; \(E\) denotes elastomer modulus; \(G\) denotes the shear modulus; \(v\) denotes the Poisson's ratio. The subscripts \(L\), \(R\) and \(T\) denote the grain directions, with \(L\) representing the longitudinal direction, \(R\) representing the radius direction and \(T\) representing the tangential direction, shown in Figure 1.  
The matrix can be simplified to the following equation:  
\[\{\varepsilon\}=[\mathbf{C}]_{orth}\{\sigma\} \tag{2}\]  
where \(\{\varepsilon\}\) and \(\{\sigma\}\) denotes the strain and stress tensors, respectively; \([\mathbf{C}]_{orth}\) denotes a 6 \(\times\) 6 orthotropic elastic matrix. For the orthotropic material, the following expression is valid:  
\[\frac{v_{ij}}{E_{i}}=\frac{v_{ji}}{E_{j}},\ i\
eq j,\ i,j=L,R,T \tag{3}\]  
Accordingly, the orthotropic elastic matrix \([\mathbf{C}]_{orth}\) is able to be expressed as a symmetric matrix, with nine parameters required to be defined as the wood material properties.  
Furthermore, the material properties for the isotropic material model are defined as 23,000 MPa of elastic modulus and 0.3 of Poisson's ratio. For orthotropic and transversely isotropic material models, the corresponding parameters are given in Table 1[31].  
Figure 1: Three dimensional structure of wood.