Energies 2021 , 14 , 3203
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3.2. Numerical Appraoch for Modelling Crushing The shell stiffnesses ( A k matrix) were computed for five crushing levels of corrugated cardboard considered, i.e., with different amount of crushing: 10%, 20%, 30%, 40% and 50%. The local deterioration factor for region A was assumed 0.5 and for region B was assumed 0.9 (due to severe delamination in these regions as shown by [35,36]). Selected values of A k matrices for different level of crushing were presented in Table 6.
Table6. The stiffnesses of the representative shell element computed for different crushing of corrugated cardboard sample for local deterioration scaling factors, separately for region A and region B.
Stiffness
10% Crushing
20% Crushing
30% Crushing
40% Crushing
50% Crushing
A 11 , ( kPa · m ) A 22 , ( kPa · m ) A 12 , ( kPa · m ) A 33 , ( kPa · m )
2101.4 1591.5
2092.8 1568.1
2088
2085.3 1531.6
2078.7 1477.4
1548.5
371.3 603.0
367.8 586.3
365.5 573.4
363.8 562.3
361.5 543.9
3 3 3 3
5.79
5.19 3.23 0.90 1.34
4.64 2.89 0.81 1.19
4.11 2.57 0.71 1.06
3.61 2.21 0.63 0.92
D 11 , Pa · m D 22 , Pa · m D 12 , Pa · m D 33 , Pa · m
3.6
1.01 1.50
A 44 , ( Pa · m ) A 55 , ( Pa · m )
45.18 82.13
31.12 72.55
22.01 65.58
15.41 60.11
10.14 51.45
Having computed A k stiffnesses which represents the overall material properties of RVE, the values were used to determine the behavior of the corrugated cardboard samples from different tests. Torsion and bending stiffness tests in both directions were considered here, see its analytical formulas derived in [31,32]. The dimension of the torsion sample was 80 × 80 mm, in bending the sample had 100 mm length between the internal supports. The decreases of stiffness of the corrugated sample in the cases of torsion, bending in CD and bending in MD for assumed scaling factors (see non deteriorated values in Table 1) are presented in Table 7. It may be observed that the values are close to the induced crushing values (first column), especially for torsion test (second column), what proves that the method was validated. This method may be used in other applications for modelling crushed corrugated cardboard samples for deriving its material and mechanical properties. Table7. The decreases in stiffness of torsion, bending in CD and bending in MD for scaling factors separately identified for region A and region B – obtained for different input geometry due to induced crushing (see Figure 9).
Induced Crushing (%)
Torsion Stiffness Decrease (%)
CD Bending Stiffness Decrease (%)
MD Bending Stiffness Decrease (%)
10 20 30 40 50
12 21 30 38 46
10 19 28 36 44
12 21 29 37 46
4. Discussion In the first part of our work, we investigated the relationship between the intentional level of flat corrugated cardboard crushing and the drops in the measured parameters of various laboratory tests. The results in Tables 4 and 5 show the coefficients of determination for linear trends (y = x) across all tests. Values closer to 1 indicate the better correlation between the experiment result and the regression lines. It has been observed that the decrease in both bending and torsional stiffness is more or less correlated with the amount of crush.
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