## 2 Modeling
### Eliminating Excess Degrees of Freedom  
Thus far, the modeling was focused on the stresses within the paper sheets. However, those stresses were initially assumed to be introduced by a deformation of the respective sheets at the time of production. This means the three-dimensional stress field \(\left(\sigma_{x},\sigma_{y},\tau_{xy}\right)^{T}\) is caused and therefore completely described by the two-dimensional displacement field (\(u\),\(v\))\({}^{T}\) (see Figure 3). Thus, it is important to note that the strain components (and therefore also the stress components) are not independent from each other, but can be derived by:  
\[\varepsilon_{x}=\frac{\partial u}{\partial x},\quad\varepsilon_{y}=\frac{ \partial u}{\partial y},\quad\gamma=\frac{1}{2}\left(\frac{\partial u}{ \partial y}+\frac{\partial v}{\partial x}\right). \tag{14}\]  
The displacement fields are again properties of the individual paper sheets. The linearity of the approach allows use of Equation (1) and to easily insert the different displacements \(\left(u_{i},v_{i}\right)^{T}\) or alternatively the displacement differences per height \(\left(\widehat{u}_{l},\widehat{\
u}_{l}\right)^{T}\), of an equivalent linear gradient, defined analogously to Equation (10):  
\[\begin{array}{c}M_{x}(x,y)=\widehat{E}_{1}\cdot\left(\frac{\partial\widehat{ u}_{l}}{\partial x}-\frac{\partial^{2}w}{\partial x^{2}}\right)+\widehat{E}_{2} \cdot\left(\frac{\partial\widehat{\
u}_{l}}{\partial y}-\frac{\partial^{2}w}{ \partial y^{2}}\right)\\ M_{y}(x,y)=\widehat{E}_{2}\cdot\left(\frac{\partial\widehat{u}_{l}}{\partial x }-\frac{\partial^{2}w}{\partial x^{2}}\right)+\widehat{E}_{4}\cdot\left(\frac {\partial\widehat{\
u}_{l}}{\partial y}-\frac{\partial^{2}w}{\partial y^{2}} \right)\\ M_{xy}(x,y)=\widehat{G}\cdot\left(\frac{1}{2}\cdot\left(\frac{\partial \widehat{u}_{l}}{\partial y}+\frac{\partial\widehat{\
u}_{l}}{\partial x} \right)-\frac{\partial^{2}w}{\partial x\partial y}\right)\end{array} \tag{15}\]  
Since the effective material parameters for the homogenized orthotropic plate already depend on the geometric details of the corrugation, the leading factors are combined with those parameters by substituting \(\widehat{E}_{*}:=E_{*}\cdot\frac{\partial}{12}\) and \(\widehat{G}:=G\cdot\frac{t^{3}}{12}\).  
Inserting this into the equilibrium of forces (5) results in:  
\[\widehat{E}_{1}\left(\frac{\partial^{3}\widehat{u}_{1}}{\partial x^{3}}-\frac {\partial^{4}w}{\partial x^{4}}\right)+\widehat{G}^{*}\left(\frac{\partial^{3} \widehat{u}_{1}}{\partial x\partial y^{2}}+\frac{\partial^{3}\widehat{v}_{1}} {\partial x^{2}\partial y}-2\cdot\frac{\partial^{4}w}{\partial x^{2}\partial y ^{2}}\right)+\widehat{E}_{4}\left(\frac{\partial^{3}\widehat{v}_{1}}{ \partial y^{3}}-\frac{\partial^{4}w}{\partial y^{4}}\right)=-p \tag{16}\]with:  
\[\widehat{G}^{*}:=\widehat{E}_{2}\cdot\widehat{G}. \tag{17}\]  
This means the corrugated board behaves identically to a board with a different shear modulus \(\widehat{G}^{*}\) but without the influence of transverse contraction.  
In this alternative material, the moments would be:  
\[\begin{split} M_{x}^{*}(x,y)&=\widehat{E}_{1}\cdot \left(\frac{\partial\widehat{u}_{1}}{\partial x}-\frac{\partial^{2}w}{ \partial x^{2}}\right),\\ M_{y}^{*}(x,y)&=\widehat{E}_{4}\cdot\left(\frac{ \partial\widehat{u}_{1}}{\partial y}-\frac{\partial^{2}w}{\partial y^{2}} \right),\\ M_{xy}^{*}(x,y)&=\widehat{G}^{*}\cdot\left(\frac{ 1}{2}\cdot\left(\frac{\partial\widehat{u}_{1}}{\partial y}+\frac{\partial \widehat{v}_{1}}{\partial x}\right)-\frac{\partial^{2}w}{\partial x\partial y }\right).\end{split} \tag{18}\]  
The Equations (16) and (18) can be used to analyze warp from measured data. In order to do this, only three effective material parameters and the effects of production parameters on three effective displacement variables need to be fitted.