Appl. Sci. 2022 , 12 , 1684
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with:
G ∗ : = E 2 · G .
(17) This means the corrugated board behaves identically to a board with a different shear modulus G ∗ but without the influence of transverse contraction. In this alternative material, the moments would be: M ∗ x ( x , y )= E 1 · ∂ u I ∂ x − ∂ 2 w ∂ x 2 , M ∗ y ( x , y )= E 4 · ∂ v I ∂ y − ∂ 2 w ∂ y 2 , M ∗ xy ( x , y )= G ∗ · 1 2 · ∂ u I ∂ y + ∂ v I ∂ x − ∂ 2 w ∂ x ∂ y . (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. 2.4. A Condition for Warp-Free Corrugated Board Since the aim of corrugated board production is to eliminate warp, the case w ≡ 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: E 1 · ∂ 3 u I ( x , y ) ∂ x 3 + G ∗ · ∂ 3 u I ( x , y ) ∂ x ∂ y 2 + ∂ 3 v I ( x , y ) ∂ x 2 ∂ y + E 4 · ∂ 3 v I ( x , y ) ∂ y 3 = 0. (19) Obviously, u only occurs as a derivative of x and 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 ε I, x = ∂ u I ∂ x and ε I, y = ∂ v I ∂ y in accordance with (14) yields: E 1 · ∂ 2 ε I, x ( x , y ) ∂ x 2 + G ∗ · ∂ 2 ε I, x ( x , y ) ∂ y 2 + G ∗ · ∂ 2 ε I, y ( x , y ) ∂ x 2 + E 4 · ∂ 2 ε I, y ( x , y ) ∂ y 2 = 0, (20) or ∂ 2 ∂ x 2 ∂ 2 ∂ y 2 = 0. (21) Using the principal-axis transformation (the parameter α simply denotes the ratio between the principal values, allowing a further simplification of the equations later): E 1 G ∗ G ∗ E 4 = S T E ∗ 0 0 α · E ∗ S (22) with: S = cos ( ϕ ) sin ( ϕ ) − sin ( ϕ ) cos ( ϕ ) , (23) the equation is further simplified: ∂ 2 ∂ x 2 ∂ 2 ∂ y 2 S T 1 0 0 α S = 0. (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: η ξ = S x y , (25) E 1 G ∗ G ∗ E 4 ε I, x ε I, y ε I, x ( x , y ) ε I, y ( x , y )
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