Appl. Sci. 2022 , 12 , 1684
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the condition for warp-free corrugated board can be written as: ∂ 2 ε I, η ( η , ξ ) ∂η 2 + α · ∂ 2 ε I, ξ ( η , ξ ) ∂ξ 2 = 0.
(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 α and ϕ . However, the equation must be fulfilled for all points of the surface. This means that the strain along the η - and ξ -axes must be proportional by a fixed factor α to each other. The only exceptions are linear and constant terms, which are, however, completely defined by the boundary conditions. Since corrugated board is produced in a continuous process, it has two clearly distinct directions. In machine direction (aligned with the fluting, coordinate axis y in Figure 2), the individual sheet is cut from a quasi-infinite band, while the width in cross direction (orthogonal to machine direction, coordinate axis x in Figure 2) is limited by the width of the machines. Therefore, it is difficult to imagine a way of matching the strains in two orthogonal axes in practice by influencing production parameters. The only practical way to satisfy (26) seems to be a completely strain-free board, i.e., ε I , η ≡ 0and ε I , ξ ≡ 0. 3. Applying the Model on Measured Surface Data The primary intended use for the proposed model is the post-processing of mea- sured surface data. This allows both the interpolation of the surface and the separation of warp and other surface defects which are not caused by warp. The basic idea, as shown in Figure 4, is to fit the parameters E 1/4 , G ∗ , the surface load p , and the effective dis- placement field ( u , v ) T ( x , y ) —together called “displacement and stiffness parameters” in Figure 4—so that the surface geometry predicted by the model matches the surface height, measured at defined sample points and stored in order in a measurement vector → m (“Mea- sured data at sample points”). While the location of those sample points in the xy -plane must be known, the method proposed here does not depend on a specific sample pattern (e.g., rectangular grid), although the location will influence the numerical precision. The outer loop (consisting of everything in Figure 4 outside of the grey area) follows the general principle of regression by quadratic error minimization, where an optimizer tries to find an input to an abstract function block (the grey area in Figure 4) in order to minimize the quadratic difference between the output of this function block and some external value (in this case, the measured data).
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