# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
## Appendix A  
This appendix supplements details of the theoretical derivation. We can rewrite Equation (1) in matrix form:  
\[\mathbf{d}\mathbf{=}\mathbf{X}\mathbf{u} \tag{10}\]  
where \(\mathbf{d}= \begin{bmatrix}u&v&w\end{bmatrix} ^{T}\) and \(\mathbf{u}\) is defined in Equation (5). Under the displacement field, the strain can be derived as follows:  
\[ \begin{array}{l}\varepsilon_{x}=\frac{\partial u}{\partial x}=\varepsilon_{x }^{0}+z\kappa_{x};\varepsilon_{y}=\frac{\partial v}{\partial y}=\varepsilon_{y }^{0}+z\kappa_{y}\\ \gamma_{xy}=\frac{\partial v}{\partial x}+\frac{\partial u}{\partial y}= \gamma_{xy}^{0}+z\kappa_{xy};\varepsilon_{z}=\frac{\partial v}{\partial z}= \varepsilon_{z}^{0}+z\kappa_{z}\\ \gamma_{xz}=\frac{\partial u}{\partial x}+\frac{\partial u}{\partial z}= \gamma_{xz}^{0}+z\kappa_{xz}+z^{2}\theta_{xz}\\ \gamma_{yz}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial z}= \gamma_{yz}^{0}+z\kappa_{yz}+z^{2}\theta_{yz}\end{array}  \tag{11}\]  
with  
\[ \begin{array}{l}\varepsilon_{x}^{0}=\frac{\partial u_{0}}{\partial x}; \varepsilon_{y}^{0}=\frac{\partial v_{0}}{\partial y};\gamma_{xy}^{0}=\frac{ \partial v_{0}}{\partial x}+\frac{\partial u_{0}}{\partial y};\\ \varepsilon_{z}^{0}=\psi;\gamma_{xz}^{0}=\varphi_{y}+\frac{\partial u_{0}}{ \partial x};\gamma_{yz}^{0}=-\varphi_{x}+\frac{\partial u_{0}}{\partial y};\\ \kappa_{x}=\frac{\partial\varphi_{y}}{\partial x};\kappa_{y}=-\frac{\partial \varphi_{x}}{\partial y};\kappa_{xy}=\frac{\partial\varphi_{y}}{\partial y}- \frac{\partial\varphi_{x}}{\partial x};\\ \kappa_{z}=2\zeta;\kappa_{xz}=\frac{\partial\psi}{\partial x};\kappa_{yz}= \frac{\partial\psi}{\partial y};\theta_{xz}=\frac{\partial^{2}_{z}}{\partial x };\theta_{yz}=\frac{\partial^{2}_{z}}{\partial y}\end{array}  \tag{12}\]  
In matrix form:  
\[\mathbf{\varepsilon}\mathbf{=}\mathbf{Z}\mathbf{\varepsilon}_{0} \tag{13}\]  
\[\mathbf{\varepsilon}_{0}=\mathbf{L}\mathbf{u} \tag{14}\]  
where \(\mathbf{\varepsilon}= \begin{bmatrix}\varepsilon_{x}&\varepsilon_{y}&\gamma_{xy}& \varepsilon_{z}&\gamma_{xz}&\gamma_{yz}\end{bmatrix} ^{T}\)  
\[\mathbf{\varepsilon}_{0}= \begin{bmatrix}\varepsilon_{x}^{0}&\varepsilon_{y}^{0}& \gamma_{xy}^{0}&\varepsilon_{z}^{0}&\gamma_{xz}^{0}&\gamma_{yz}^{0}&\kappa_{x }&\kappa_{y}&\kappa_{xy}&\kappa_{zy}&\kappa_{xz}&\kappa_{yz}&\theta_{xz}& \theta_{yz}\end{bmatrix} ^{T} \tag{15}\]  
which is called generalized strain, and  
\[\mathbf{Z}= \begin{bmatrix}1&&&&z&&&&\\ &1&&&&z&&&&\\ &&1&&&&z&&&&\\ &&1&&&&z&&&\\ &&&1&&&&z&&\\ &&&&1&&&&z&&z^{2}\end{bmatrix}  \tag{16}\]  
\[\mathbf{L}= \begin{bmatrix}\frac{\partial}{\partial x}&&\frac{\partial}{\partial y}&&&& &&&&\\ &\frac{\partial}{\partial y}&\frac{\partial}{\partial x}&&&&&&&&\\ &&\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&&&&&&&\\ &&\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&&&&&&&\\ &&&-1&&-\frac{\partial}{\partial y}&-\frac{\partial}{\partial x}&&&&\\ &&1&&\frac{\partial}{\partial x}&&\frac{\partial}{\partial y}&&&&\\ &&1&&&&\frac{\partial}{\partial x}&\frac{\partial}{\partial y}&&\\ &&&&2&&\frac{\partial}{\partial x}&\frac{\partial}{\partial y}\end{bmatrix} ^{T} \tag{17}\]  
A complete constitutive relation containing all 6 strain components should be used in matrix form:  
\[\sigma=\mathbf{Q}\epsilon\] (A9)  
with \(\sigma=\left[\sigma_{x}\quad\sigma_{y}\quad\tau_{xy}\quad\sigma_{z}\quad\tau_{ xz}\quad\tau_{yz}\right]^{T}\)  
The stress resultant and stress couples are defined as:  
\[{\sigma_{0}}^{T}=\int_{-h/2}^{h/2}\sigma^{T}\mathbf{Z}dz\] (A10)  
with  
\[\sigma_{0}=\left[N_{x}\quad N_{y}\quad N_{xy}\quad N_{z}\quad Q_{xz}\quad Q_{ yz}\quad M_{x}\quad M_{y}\quad M_{xy}\quad M_{z}\quad R_{xz}\quad R_{yz} \quad S_{xz}\quad S_{yz}\right]^{T}\] (A11)  
Substituting Equations (A1), (A4), (A5), (A9), and (A10), the overall constitutive matrix of the plate can be given:  
\[\sigma_{0}=\mathbf{Q}_{0}\epsilon_{0}\] (A12)  
with  
\[\mathbf{Q}_{0}=\int_{-h/2}^{h/2}\mathbf{X}^{T}\mathbf{Q}\mathbf{X}dz\] (A13)  
which corresponds to Equation (2).