Energies 2021 , 14 , 1095
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These stiffnesses are used in the constitutive equations that describe shell FE [21]:
0 0
A 11 A 12 A 12 A 22
A = ⎡ ⎣ D = ⎡ ⎣
⎤ ⎦ ⎤ ⎦
,
(8)
0 0
A 33
0 0
D 11 D 12 D 12 D 22
,
(9)
0 0
D 33
0
A 44
Γ =
,
(10)
0
A 55
and are represented by the following formulas:
t 3 12 w
t 3 12 w
,
,
(11)
D 22 = E 2 ×
D 11 = E 1 ×
D 12 = D 21 = D 22 ν 12 = D 11 v 21 ,
(12)
t 3 12
,
(13)
D 33 = G 12 ×
A 11 = E 1 t ,
A 22 = E 2 t ,
(14)
A 12 = A 21 = A 11 ν 21 = A 22 ν 12 ,
(15)
(16)
A 33 = G 12 t
5 6 5 6
G 13 t ,
(17)
A 44 =
G 23 t ,
(18)
A 55 =
w = 1 − ν 12 ν 21
(19)
where: •
t —effective cardboard thickness;
E 1 —effective stiffness modulus in the MD; E 2 —effective stiffness modulus in the CD;
• •
• ν 12 = 0.293 ( E 1 / E 2 ) —effective Poisson’s ratio in the 1–2 (x–y) plane; • v 21 = v 12 E 2 / E 1 —effective Poisson’s ratio in the 1–2 (x–y) plane, • G 12 —effective shear modulus in 1–2 (x–y) plane, • G 13 —effective transverse shear modulus in the 1–3 (x–z) plane, • G 23 —effective transverse shear modulus in the 2–3 (y–z) plane. Having the geometry of all panels, the adopted boundary conditions and the corru- gated board constitutive model defined above, all that remains is to define the behavior of perforation lines. In order to include the mechanical properties of perforations in the numerical model, two techniques are used here, namely: (a) spring connection between the parts separated by perforation, see Figure 2b, and (b) thickness reduction of the ele- ments adjacent to the perfection, see Figure 2c. The first technique is to divide the panel with the perforation line, duplicate the mesh nodes, and add spring joints at each of the 6 degrees of freedom at each duplicated node. The second technique is to search for all elements adjacent to the perforation line, sort them by the number of nodes lying on the perforation line, and reduce the thickness of these elements (due to creasing techniques used to produce perforations). Figure 2c shows the elements, in which the thickness was reduced. Element color indicates the amount of reduced thickness; the darker the color, the greater the reduction. Both spring stiffness and thickness reduction depends on the definition of perforation, its geometrical pattern, and shape. In each case, the influence
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