Materials 2022 , 15 , 663
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If, instead of bending, the corrugated board cross-section is compressed or stretched in MD, the equation for compression/tensile stiffness can also be derived:
N ∑
N ∑ i = 1
L i
P i δ i
P i L i δ i
i
.
(14)
EA =
dx =
0
The theoretical bending stiffness (valid for a perfect model without imperfections) as the product of the stiffness modulus in MD and the moment of inertia of liners only is:
t 3 i 12
N i = 1 E i b
2 i .
EI = ∑
(15)
+ t i z
In order to normalize the theoretical values and the results of four-point bending tests, both values are divided by the width of the sample b . Hence, ultimately, the bending stiffness is: BS = EI b . (16) The presented derivation allows to explain the differences between the bending stiff- ness obtained from testing of the corrugated board sample placed with the E wave upwards or vice versa—B wave. 3. Results In the first step, the theoretical assumption, in which only liners affect the stiffness of the entire section, was validated. For this purpose, two simple numerical models of a five-layer corrugated cardboard in a plane state (i.e., a beam model) were built (see Figure 9). Both models consist of classic Bernoulli 2-node beam elements and were implemented in Matlab software (Mathworks Inc., Natick, MA, USA) [65]. Small rotation φ was applied in both ends and the corresponding reaction moments M were determined in order to calculate BS from Equations (13) and (16). In all cases, displacements resulting from φ rotation wrt neutral axis were applied on both ends of the model (in external nodes on the left and right sides of the model).
( a )
( b )
Figure9. Numerical model of corrugated board: ( a ) 2-period FE model ( b ) 4-period FE model.
In the first model all layers were modeled according to their geometry and mechanical parameters, while in the second model, the stiffness of the corrugated layers was signifi- cantly reduced (by 100 times) to mimic a situation where only liners are active. The results are shown in Figure 10. Naturally, this assumption is not valid if one would like to derive the BS in CD, where all liners as well as both corrugated layers are equally important. In order to eliminate a possible error related to the discretization of numerical models, the influence of the number of finite elements and the number of waves in the model was also checked. The results are summarized in Table 4. All FE models consist of 2-node linear beam elements with a seed equal to 0.1 mm, which generated the following number of nodes and elements in four models: 1. FEM-1 (1-wave), number of nodes: 375, number of elements: 377; 2. FEM-2 (2-waves), number of nodes: 746, number of elements: 754; 3. FEM-3 (3-waves); number of nodes: 1118, number of elements: 1131; 4. FEM-4 (4-waves); number of nodes: 1489, number of elements: 1508.
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