# Incorporating Transverse Normal Strain in the Homogenization of Corrugated Cardboards
## Appendix A
The strain energy is:  
\[U=\frac{1}{2}\int_{V}\sigma^{T}\epsilon dV\] (A14)  
Applying Equations (A5) and (A10), one can obtain:  
\[U=\frac{1}{2}\int_{A}\sigma_{0}^{T}\epsilon_{0}dA=\frac{1}{2}\int_{A}\epsilon _{0}^{T}\mathbf{Q}_{0}\epsilon_{0}dA\] (A15)  
The external work is:  
\[W=\int_{V}\mathbf{b}^{T}\mathbf{u}dV+\int_{S}\overline{p}^{T}\mathbf{u}dS\] (A16)  
where \(b^{T}=\left(b_{x}\quad b_{y}\quad b_{z}\right)\) stands for body force and \(\overline{p}^{T}=\left(\overline{p_{x}}\quad\overline{p_{y}}\quad\overline{p_ {z}}\right)\) stands for surface traction. Break the whole surface \(S\) into the top and bottom surface and a ring around the plate, integrating along the thickness, one can get:  
\[ \begin{array}{l}W=\int_{A}\Big{(}\begin{array}{cccccc}n_{x}&n_{y}&n_{z}&m _{y}&m_{x}&m_{z}\end{array}\Big{)}\mathbf{u}dA\\ +\int_{l}\Big{(}\begin{array}{cccccc}\overline{N_{x}}&\overline{N_{y}}& \overline{N_{z}}&\overline{M_{y}}&\overline{M_{x}}&\overline{M_{z}}\end{array} \Big{)}\mathbf{u}dl\end{array} \] (A17)with  
\[ \begin{array}{l}n_{x}=\int_{-h/2}^{h/2}b_{x}dz+\left[\tau_{xz}(h/2)-\tau_{xz}(-h/ 2)\right]\\ n_{y}=\int_{-h/2}^{h/2}b_{y}dz+\left[\tau_{yz}(h/2)-\tau_{yz}(-h/2)\right]\\ n_{z}=\int_{-h/2}^{h/2}b_{z}dz+\left[\sigma_{z}(h/2)-\sigma_{z}(-h/2)\right]\\ m_{y}=-\int_{-h/2}^{h/2}b_{y}dzdz-\left[\tau_{yz}(h/2)+\tau_{yz}(-h/2)\right] \frac{h}{2}\\ m_{x}=\int_{-h/2}^{h/2}b_{x}dz+\left[\tau_{xz}(h/2)+\tau_{xz}(-h/2)\right] \frac{h}{2}\\ m_{z}=\int_{-h/2}^{h/2}b_{z}dz+\left[\sigma_{z}(h/2)+\sigma_{z}(-h/2)\right] \frac{h}{2}\\ s_{z}=\int_{-h/2}^{h/2}b_{z}z^{2}dz+\left[\sigma_{z}(h/2)-\sigma_{z}(-h/2) \right]\frac{h^{2}}{4}\\ \left[\overline{N_{x}}\quad\overline{N_{y}}\quad\overline{Q_{z}}\right]=\int _{-h/2}^{h/2}\left(\overline{p^{s}_{x}}\quad\overline{p^{s}_{y}}\quad\overline {p^{s}_{z}}\right)dz\\ \left[\overline{M_{x}}\quad\overline{M_{y}}\quad\overline{R_{z}}\right]=\int _{-h/2}^{h/2}z\left(-\overline{p^{s}_{y}}\quad\overline{p^{s}_{x}}\quad \overline{p^{s}_{z}}\right)dz\\ \overline{S_{z}}=\int_{-h/2}^{h/2}z^{2}\overline{p^{s}_{z}}dz\end{array} \]  
where the loop integral is along the edge of the middle surface and superscript \(s\) means traction on section. \(\sigma_{z}(h/2)\) means the traction force in the \(z\) direction on the top surface, and similarly for \(\tau_{xz}\) and \(\tau_{yz}\). Body force is usually ignored in most cases for corrugated cardboard, but the \(n_{z}\), \(m_{z}\), and \(s_{z}\) applying on top and bottom surface are not zero. According to the principle of minimum potential:  
\[\delta\Pi=\delta(U-W)=0\] (A18)  
After variational derivations, equilibrium equations can be obtained:  
\[ \begin{array}{l}N_{x,x}+N_{xy,y}+n_{x}=0\\ N_{xy,x}+N_{y,y}+n_{y}=0\\ Q_{x,x}+Q_{y,y}+n_{z}=0\\ M_{xy,x}+M_{y,y}-Q_{y}-m_{y}=0\\ M_{x,x}+M_{xy,y}-Q_{x}+m_{x}=0\\ R_{x,x}+R_{y,y}-N_{z}+m_{z}=0\\ S_{x,x}+S_{y,y}-2M_{z}+s_{z}=0\end{array} \] (A19)  
and boundary conditions:  
\[ \begin{array}{l}N_{x}n_{1}+N_{xy}n_{2}=\overline{N_{x}}\qquad u_{0}= \overline{u_{0}}\\ N_{xy}n_{1}+N_{y}n_{2}=\overline{N_{y}}\qquad v_{0}=\overline{v_{0}}\\ Q_{x}n_{1}+Q_{y}n_{2}=\overline{Q_{z}}\qquad w_{0}=\overline{w_{0}}\\ M_{xy}n_{1}+M_{y}n_{2}=\overline{M_{y}}\quad\mbox{or}\quad\varphi_{y}= \overline{\varphi_{y}}\\ M_{x}n_{1}+M_{xy}n_{2}=\overline{M_{x}}\qquad\varphi_{x}=\overline{\varphi_{x}} \\ R_{x}n_{1}+R_{y}n_{2}=\overline{R_{z}}\qquad\psi=\overline{\psi}\\ S_{x}n_{1}+S_{y}n_{2}=\overline{S_{z}}\qquad\qquad\zeta=\overline{\zeta}\end{array} \] (A20)  
where \(n_{1}\) and \(n_{2}\) mean the tangential and normal direction of the edge, respectively.