# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
## 3 Extended Homogenization Method  
### Extension on FSDT  
The ideology of the FSDT homogenization method is based on coordinate transformation. As shown in Figure 4, the elastic constants of the flute are measured in its local coordinate system (123), rotated by an angle \(\theta\) relative to the global coordinate system (\(xyz\)). The 1-direction represents the tangential direction of the flute. Figure 4: The local and global coordinate system of the flute. Assuming the flute profile is modeled as a sinusoidal function, we have:  
\[g(x)=\frac{h_{f}}{2}\sin(\frac{2\pi x}{p}) \tag{11}\]  
\[\theta(x)=\tan^{-1}(\frac{dg(x)}{dx}) \tag{12}\]  
The transformation of stress and strain under coordinate rotation is governed by:  
\[ \begin{cases}\epsilon_{11}\\ \epsilon_{22}\\ \epsilon_{33}\\ \gamma_{23}\\ \gamma_{13}\\ \gamma_{12}\end{cases} = \begin{bmatrix}c^{2}&0&s^{2}&0&cs&0\\ 0&1&0&0&0&0\\ s^{2}&0&c^{2}&0&-cs&0\\ 0&0&0&c&0&-s\\ -2 cs&0&2 cs&0&c^{2}-s^{2}&0\\ 0&0&0&s&0&c\end{bmatrix}\begin{cases}\epsilon_{xx}\\ \epsilon_{yy}\\ \epsilon_{zz}\\ \gamma_{yz}\\ \gamma_{xx}\end{cases}  \tag{13}\]  
\[ \begin{cases}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \tau_{23}\\ \tau_{13}\\ \tau_{12}\end{cases} = \begin{bmatrix}c^{2}&0&s^{2}&0&2 cs&0\\ 0&1&0&0&0&0\\ s^{2}&0&c^{2}&0&-2 cs&0\\ 0&0&0&c&0&-s\\ -cs&0&cs&0&c^{2}-s^{2}&0\\ 0&0&0&s&0&c\end{bmatrix}\begin{cases}\sigma_{xx}\\ \sigma_{yy}\\ \sigma_{zz}\\ \tau_{yz}\\ \tau_{xz}\\ \tau_{xy}\end{cases}  \tag{14}\]  
or in compact form:  
\[ \begin{split}\epsilon_{123}&=\mathbf{T}_{\epsilon}\epsilon_{xyz}\\ \sigma_{123}&=\mathbf{T}_{\sigma}\epsilon_{xyz}\end{split}  \tag{15}\]  
where \(c=cos\theta\) and \(s=sin\theta\). The constitutive relation is described as:  
\[ \begin{split}\epsilon_{123}&=\mathbf{S}_{123}\sigma_{123}\\ \epsilon_{xyz}&=\mathbf{S}_{xyz}\sigma_{xyz}\end{split}  \tag{16}\]  
where \(\mathbf{S}\) represents the compliance matrix:  
\[ \begin{split}\mathbf{S}_{123}=\begin{bmatrix}1/E_{1}&-\
u_{21}/E_{2}&0&-\
u_{31}/E_{3}&0&0\\ -\
u_{12}/E_{2}&1/E_{2}&0&-\
u_{32}/E_{3}&0&0\\ 0&0&1/G_{12}&0&0&0\\ -\
u_{13}/E_{3}&-\
u_{23}/E_{3}&0&1/E_{3}&0&0\\ 0&0&0&0&1/G_{13}&0\\ 0&0&0&0&0&1/G_{23}\end{bmatrix}\end{split}  \tag{17}\]  
Applying Equation (15) to Equation (16), one can get:  
\[\mathbf{S}_{xyz}=\mathbf{T}_{\epsilon}^{-1}\mathbf{S}_{123}\mathbf{T}_{\sigma} \tag{18}\]  
Notice that  
\[\mathbf{T}_{\epsilon}^{-1}=\mathbf{T}_{\sigma}^{T} \tag{19}\]  
A transformation of the constitutive relation can be obtained:  
\[\mathbf{S}_{xyz}=\mathbf{T}_{\sigma}^{T}\mathbf{S}_{123}\mathbf{T}_{\sigma} \tag{20}\]  
Equations (16)-(20) are the foundation of existing homogenization approaches, which gives transformation to the compliance matrix \(\mathbf{S}\), yet the ultimate objective should be the elastic matrix \(\mathbf{Q}\). Consequently, in references [7; 10], the researchers first derived the modulus form compliance matrix using Equation (17), then substituted them into the elastic matrix of the planar form:  
\[\mathbf{Q}_{FSDT}= \begin{bmatrix}\frac{E_{1}}{1-\
u_{12}z_{21}}&\frac{\
u_{12}E_{1}}{ 1-\
u_{12}z_{21}}&\\ \frac{\
u_{12}E_{2}}{1-\
u_{12}z_{21}}&\frac{E_{2}}{1-\
u_{12}z_{21}}&\\ &G_{12}&\\ &&G_{13}&\\ &&&G_{23}\end{bmatrix}  \tag{21}\]  
Only 5 components of stress are needed in FSDT (3 for CLPT), whereas Equation (20) employs a full 3D constitutive relation containing all 6 stress components. This is an indirect approach that may have some flaws in the reasoning. However, in the case of the present work, due to a higher order expansion in the displacement field, a full 3D constitutive relation is required, therefore transformation of the elastic matrix can be derived directly.