## 2 Modeling
### A Condition for Warp-Free Corrugated Board  
Since the aim of corrugated board production is to eliminate warp, the case \(w\equiv 0\) is analyzed in more detail. If this board is supported by an equally flat surface, the area weight and the reaction forces of the underground compensate each other, so (16) is simplified to:  
\[\widehat{E}_{1}\cdot\frac{\partial^{3}\widehat{u}_{1}(x,y)}{\partial x^{3}}+ \widehat{G}^{*}\cdot\left(\frac{\partial^{3}\widehat{u}_{1}(x,y)}{\partial x \partial y^{2}}+\frac{\partial^{3}\widehat{v}_{1}(x,y)}{\partial x^{2} \partial y}\right)+\widehat{E}_{4}\cdot\frac{\partial^{3}\widehat{v}_{1}(x,y) }{\partial y^{3}}=0. \tag{19}\]  
Obviously, \(\widehat{u}\) only occurs as a derivative of \(x\) and \(\widehat{v}\) only as a derivative of \(y\): any constant part of the displacement function is merely a coordinate transformation, which does not cause any physical effects. Therefore, substituting \(\widehat{e}_{1,x}=\frac{\partial\widehat{v}_{1}}{\partial x}\) and \(\widehat{e}_{1,y}=\frac{\partial\widehat{v}_{1}}{\partial y}\) in accordance with (14) yields:  
\[\widehat{E}_{1}\cdot\frac{\partial^{2}\widehat{e}_{1,x}(x,y)}{\partial x^{2} }+\widehat{G}^{*}\cdot\frac{\partial^{2}\widehat{e}_{1,x}(x,y)}{\partial y^{ 2}}+\widehat{G}^{*}\cdot\frac{\partial^{2}\widehat{e}_{1,y}(x,y)}{\partial x ^{2}}+\widehat{E}_{4}\cdot\frac{\partial^{2}\widehat{e}_{1,y}(x,y)}{\partial y ^{2}}=0, \tag{20}\]  
or  
\[\left( \begin{array}{cc}\frac{\partial^{2}}{\partial x^{2}}&\frac{\partial^ {2}}{\partial y^{2}}\\ \end{array} \right)\left( \begin{array}{cc}\widehat{E}_{1}&\widehat{G}^{*}\\ \widehat{G}^{*}&\widehat{E}_{4}\\ \end{array} \right)\left( \begin{array}{c}\widehat{e}_{1,x}\\ \widehat{e}_{1,y}\\ \end{array} \right)=0. \tag{21}\]  
Using the principal-axis transformation (the parameter \(\alpha\) simply denotes the ratio between the principal values, allowing a further simplification of the equations later):  
\[\left( \begin{array}{cc}\widehat{E}_{1}&\widehat{G}^{*}\\ \widehat{G}^{*}&\widehat{E}_{4}\\ \end{array} \right)=S^{\top}\left( \begin{array}{cc}E^{*}&0\\ 0&\alpha\cdot E^{*}\\ \end{array} \right)S \tag{22}\]  
with:  
\[S=\left( \begin{array}{cc}cos(\varphi)&sin(\varphi)\\ -sin(\varphi)&cos(\varphi)\\ \end{array} \right), \tag{23}\]  
the equation is further simplified:  
\[\left( \begin{array}{cc}\frac{\partial^{2}}{\partial x^{2}}&\frac{\partial^ {2}}{\partial y^{2}}\\ \end{array} \right)S^{\top}\left( \begin{array}{cc}1&0\\ 0&\alpha\\ \end{array} \right)S\left( \begin{array}{c}\widehat{e}_{1,x}(x,y)\\ \widehat{e}_{1,y}(x,y)\\ \end{array} \right)=0. \tag{24}\]  
It should be noted that this transformation is only applied for mathematical reasons: due to the structure of the matrix in Equation (21), there must be a rotation matrix (23) which satisfies Equation (22). That means that, in a rotated coordinate system:  
\[\left( \begin{array}{c}\eta\\ \xi\\ \end{array} \right)=S\left( \begin{array}{c}x\\ y\\ \end{array} \right), \tag{25}\]the condition for warp-free corrugated board can be written as:  
\[\frac{\partial^{2}\widehat{e}_{IJ\eta}(\eta,\xi)}{\partial\eta^{2}}+\alpha\cdot \frac{\partial^{2}\widehat{e}_{IJ\xi}(\eta,\xi)}{\partial\xi^{2}}=0. \tag{26}\]  
This equation only defines the higher-order terms of the strain functions. The linear and constant terms are defined by boundary conditions. The strain components orthogonal to the edge must disappear at the boundaries. Otherwise, there would be forces orthogonal to the edges and the equilibrium of forces would be violated, as the model assumes no external forces. The condition (26) for warp-free corrugated board seems surprisingly simple. The material is effectively reduced to only two parameters \(\alpha\) and \(\varphi\). However, the equation must be fulfilled for all points of the surface. This means that the strain along the \(\eta\)- and \(\xi\)-axes must be proportional by a fixed factor \(\alpha\) to each other. The only exceptions are linear and constant terms, which are, however, completely defined by the boundary conditions.