## 2 Modeling
### Effects of Internal Stress  
Equation (4) is usually used to describe the moments and, in combination with (3), the stresses in a plate deformed by external forces. The linearity of [PERSON] theory allows use of the (negative) internal stresses of the flat plate instead: the deformation by external forces that would cause exactly the negative stress values in a plate without internal stress leads to a compensation and therefore a relaxed state in a plate with internal stresses. The proposed model approximates the corrugated board as a homogenous orthotropic plate; however, internal stress must ultimately be introduced by the paper sheets, which are of negligible thickness compared to the board. Therefore, the relationship between the corrugated board and the idealized plate model needs to be analyzed. For the non-corrugated layers of paper (liners), the stress value at their respective \(z\)-positions in the plate is simply the stress introduced in the paper during production. In the case of fluting, the situation seems to be more complex, as their \(z\)-position is a function of the location in the \(xy\)-plane. However, the warp which is to be modeled here is by its definition the average effect on larger scales. It is therefore reasonable to assume that there is an average \(z\)-position for each corrugated layer which sees the stress of this layer. This average position does not need to be the arithmetic mean; the important point is that, for all layers of paper, there exists a \(z\)-position within the equivalent orthotropic plate at which the internal stress of the paper is equal to the stress in the orthotropic plate. At this point, the internal stress \(\sigma_{I,*}(x,y,\widehat{z}_{i})\) at \(n\) different \(z\)-positions \(\widehat{z}_{i}\) is known, where \(n\) denotes the number of paper layers in the corrugated board. Since the equations for the three components are identical in principle, \(\sigma_{I,*}\) is used to represent \(\sigma_{I,x}\), \(\sigma_{I,y}\), or \(\tau_{I,xy}\). The additional index I denotes quantities resulting from internal stress. Between the positions \(\widehat{z}_{i}\), some kind of interpolation must exist. For this model, we will assume that the interpolation can be written as:  
\[\sigma_{I,*}(x,y,z)=-\sum_{k=1}^{n}f_{k}(z)\cdot\sigma_{I,*}(x,y,\widehat{z}_{ k}) \tag{6}\]  
where \(f_{k}(z)\) can be chosen from all functions which satisfy the condition \(f_{k}(\widehat{z}_{l})=\delta_{kl}\) with the Kronecker symbol \(\delta_{kl}\). While this condition may seem to be trivially satisfied, it is important to note that \(f_{k}(\widehat{z})\) is required to be independent of the coordinate in the \(xy\)-plane. This would be natural to assume for an infinite uniform plane, where there is no physical difference between two different points. Therefore, it can be seen as a way of neglecting boundary effects. In other words, this means that the stress in any height can be expressed as a (height-dependent) linear combination of the stress values in the paper planes. This assumption is reasonable, as the entire theory is based on linear material properties. Inserting this interpolation approach into the (internal) moment equations yields:  
\[M_{I,*}=\int_{-t/2}^{t/2}\sum_{k=1}^{n}f_{k}(z)\cdot\sigma_{I,*}(x,y,\widehat{ z}_{k})\cdot z\;\mathrm{d}z=\sum_{k=1}^{n}\sigma_{I,*}(x,y,\widehat{z}_{k}) \cdot\int_{-t/2}^{t/2}f_{k}(z)\cdot z\;\mathrm{d}z. \tag{7}\]  
As seen here, the expression for the moment can be separated into a position-dependent and a position-independent part. By substituting:  
\[\kappa_{k}:=\int_{-t/2}^{t/2}f_{k}(z)\cdot z\;\mathrm{d}z, \tag{8}\]  
one can rewrite the equation as:  
\[M_{I,*}(x,y)=\sum_{k=1}^{n}\sigma_{I,*}(x,y,\widehat{z}_{k})\cdot\kappa_{k}. \tag{9}\]This means the moments in the equivalent orthotropic plate can be modeled as a linear combination of the stresses in the individual paper sheets.