Appl. Sci. 2022 , 12 , 1684
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so that a lever arm exists within the deformed plate. The internal forces discussed here, however, will always be aligned with the plate. The in-plane deformation of the board is neglected since it is, contrary to warpage, not relevant in industrial practice. For the rest of this section, the relevant equations for modeling orthotropic Kirchhoff plates under orthogonal surface load are summarized based on standard textbook knowl- edge (e.g., [22] (pp. 193–208)). It is important to note here that the subsequent equations refer to an orthotropic plate, which would show the same reaction as the corrugated board to a pure bending load; the equivalent material parameters used here cannot be used to describe in-plane deformation. The stress–strain relation of an orthotropic material within aplane is:
⎛ ⎝
⎞ ⎠ =
⎛ ⎝
⎛ ⎝
⎞ ⎠ .
⎞ ⎠ ·
E 1 E 2 0 E 2 E 4 0 0 0 G
σ x σ y τ xy
ε x ε y γ
(1)
As usual, σ ∗ , and τ xy denote the normal and shear stress, respectively (with the orientation given in the index), ε ∗ and γ normal and shear strain, and the matrix includes the elastic parameters of the material. Under the assumptions of Kirchhoff theory, the strain is approximated from the deflection w by: ε x = − z · ∂ 2 w ∂ x 2 , ε y = − z · ∂ 2 w ∂ y 2 , γ = − z · ∂ 2 w ∂ x ∂ y . (2) The moments in a plate of thickness t (aligned with the z -coordinate as shown in Figure 2b) are given by integrals over the stresses: M x ( x , y )= t /2 − t /2 σ x ( x , y , z ) · zdz , M y ( x , y )= t 2 − t 2 σ y ( x , y , z ) · zdz , M xy ( x , y )= t 2 − t 2 τ xy ( x , y , z ) · zdz , (3) Note that Equation (3) is derived by analyzing an infinitesimal section of the plate. The moments are therefore given per length of the section plane and appear to have the unit of forces. One would obtain the actual moments by integrating the terms above over the length of the respective edge Inserting the relations (2) yields:
12 · E 1 · 12 · E 2 ·
∂ 2 w ∂ y 2 ∂ 2 w ∂ y 2
M x ( x , y )= − t 3 M y ( x , y )= − t 3
∂ 2 w ∂ x 2 ∂ 2 w ∂ x 2
+ E 2 · + E 4 ·
(4)
M xy ( x , y )= − t 3
∂ 2 w ∂ x ∂ y
12 · G ·
From the equilibrium of forces over an infinitesimal element, one obtains:
∂ 2 M
∂ 2 M
∂ 2 M
xy
y
x
+ 2 ·
= − p ( x , y )
(5)
+
∂ x 2
∂ y 2
∂ x ∂ y
Here, p denotes the surface load of the plate, which consists of the weight of the board and the reaction forces of the underground, as stated above. Combining these equations leads to the behavior of plates under external forces without any internal stress and therefore without warp.
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