Energies 2021 , 14 , 1095
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Table2. Material characteristics for 3E450-1 cardboard quality, which were used in the finite element method (FEM) to determine the critical buckling loads of the panels [44].
Thickness (mm)
D 11 (Nmm)
D 22 (Nmm)
D 33 (Nmm)
A 44 (N/mm)
A 55 (N/mm)
3E450-1
ECT
mean
1.589 1.582 1.596
5.299 5.028 5.574
1261 1206 1318
507.8 469.2 547.3
1036
3.904 3.743 4.077
4.074 3.691 4.498
min max
967
1115
The data used in Table 2 and the selected equations presented earlier may be used to determine the typical set of elastic orthotropic material constants. One of the issue to address is to compute w , which, according to Equation (19), depends on the Poisson’s ratios, v 12 and v 21 . Here, the Baum et al. [45] formula was used to compute v 12 , namely: ν 12 = 0.293 E 1 E 2 , (26) In the above equation, the relation E 1 / E 2 was obtained from Equation (11); thus, we may write the following formula:
D 11 D 22
E 1 E 2
.
(27)
=
On the other hand, the ratio v 21 was computed from the classic formula, namely:
E 2 E 1
,
(28)
ν 21 = ν 12 ×
In the above equation, the following relation, derived again from Equation (11), may beused:
D 22 D 11
E 2 E 1
.
(29)
=
Finally, having w , allows us to compute E 1 and E 2 from the Equation (11). Similarly, G 12 was computed from Equation (13); G 13 and G 23 were obtained from Equation (17) and Equation (18), respectively. The computed material constants are presented in Table 3; mean, minimal, and maximal values are demonstrated.
Table3. Elastic orthotropic parameters for 3E450-1 cardboard quality computed from stiffnesses, see Table 2.
3E450-1
E 1 (MPa)
E 2 (MPa)
v 12 (-) 0.461 0.450 0.470
v 21 (-) 0.184 0.183 0.189
G 12 (MPa) G 13 (MPa) G 23 (MPa)
mean
3451 3340 3560
1390 1301 1478
3099 2931 3291
2.95 2.84 3.07
3.08 2.80 3.38
min max
In Figure 5, the mean data from the physical measurements, and estimated counterpart values were presented by the bar plot. Black bars represent the mean values measured at the laboratory, see the last column in Table 1. The values obtained from the analytical– numerical approach proposed are presented by light blue plot, see Equation (23). In the analytical–numerical approach presented, k and r usedwere k = 0.4 and r = 0.75.
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