Appl. Sci. 2022 , 12 , 1684
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This means the moments in the equivalent orthotropic plate can be modeled as a linear combination of the stresses in the individual paper sheets. The same moments and therefore the same bending could also be realized with a linear stress gradient along the z-axis: σ I, ∗ ( x , y , z ) : = σ I, ∗ ( x , y ) · z . (10) Note that σ I , ∗ , as defined here, is given in a unit of stress per height. The respective moment would be: M I, ∗ ( x , y )= t /2 − t /2 σ I, ∗ · z 2 d z = t 3 12 · σ I, ∗ ! = n ∑ k = 1 σ I, ∗ ( x , y , z k ) · κ k . (11) Hence, the total difference in the stress per height σ I , ∗ between the top and the bottom side of the corrugated board (with an assumed linear stress gradient) can be expressed as: σ I , ∗ = n ∑ k = 1 σ I , ∗ ( x , y , z k ) · 12 · κ k t 3 . (12) Combining this result with the moments caused by bending under external forces (4) yields: M x ( x , y )= t 3 12 · σ I , x − E 1 · ∂ 2 w ∂ x 2 + E 2 · ∂ 2 w ∂ y 2 , M y ( x , y )= t 3 12 · σ I , y − E 2 · ∂ 2 w ∂ x 2 + E 4 · ∂ 2 w ∂ y 2 , M xy ( x , y )= t 3 12 · τ I , xy − G · ∂ 2 w ∂ x ∂ y . (13) Note that, while the quantities σ I , ∗ do not necessarily appear in real corrugated board, they can still be used as a measure of the effective stress as defined by (12). 2.3. 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 σ x , σ y , τ xy 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:
2
∂ x .
= 1
∂ u ∂ x , ε y
= ∂ v
∂ u ∂ y
+ ∂ v
(14)
∂ y , γ
ε x =
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 ( u i , v i ) T or alternatively the displacement differences per height ( u I , v I ) T , of an equivalent linear gradient, defined analogously to Equation (10): M x ( x , y )= E 1 · ∂ u I ∂ x − ∂ 2 w ∂ x 2 + E 2 · ∂ v I ∂ y − ∂ 2 w ∂ y 2 M y ( x , y )= E 2 · ∂ u I ∂ x − ∂ 2 w ∂ x 2 + 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 (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 E ∗ : = E ∗ · t 3 12 and G : = G · t 3 12 . Inserting this into the equilibrium of forces (5) results in: E 1 ∂ 3 u I ∂ x 3 − ∂ 4 w ∂ x 4 + G ∗ ∂ 3 u I ∂ x ∂ y 2 + ∂ 3 v I ∂ x 2 ∂ y − 2 · ∂ 4 w ∂ x 2 ∂ y 2 + E 4 ∂ 3 v I ∂ y 3 − ∂ 4 w ∂ y 4 = − p (16)
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