Energies 2021 , 14 , 3203
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Figure 6. The scheme representing the steps of numerical study conducted in this study: ( a ) unde- formed RVE, ( b ) loaded and deformed RVE, ( c ) crushed geometry extracted, ( d ) material stiffness matrix ofRVE, ( e ) the representative shell stiffness matrix and ( f ) tests outcomes from analytical estimation. Initial geometry of the corrugated cardboard used in the numerical study represents an intact (i.e., unconverted or uncrushed) geometry of the cardboard and it was assumed from the literature [19,25]. The fraction of a single wall corrugated cardboard was simulated, namely, the in-plane section of 8 × 8 mm. The fluting period was also 8 mm; the fluting wave “starts” from the liner. The thickness of the liners and fluting are 0.29 mm and 0.30 mm, respectively. The axial spacing between the liners is 3.51 mm. The material parameters of intact corrugated cardboard were also taken from the literature [19,25]. The classical orthotropic constitutive law was assumed for each layer with a perfect plasticity (no hardening). The orthotropic material data ( E 1 , E 2 , v 12 , G 12 , G 13 and G 23 , i.e., Young moduli in both directions, Poisson’s ratio and 3 shear moduli, respectively) and yield stress, σ 0 , for liners and fluting are presented in Table 1.
Table1. Material properties of liners and fluting of intact corrugated cardboard.
E 1
E 2
G 12
G 13
G 23
ν 12
σ 0
Layers
(MPa)
(MPa)
(-)
(MPa)
(MPa)
(MPa)
(MPa)
liners fluting
3326 2614
1694 1532
0.34 0.32
859 724
429.5
429.5
2.5 2.5
362
362
To acquire the crushed geometry of the corrugated cardboard the static FE analysis was performed. In the numerical study, five cases were considered, see Figure 7, in which the induced crushing of the cardboard were 10%, 20%, 30%, 40% and 50%. For instance, 10% of crushing means here that the corrugated cardboard was enforced by kinematic constraint to decrease its thickness to be 90% of the intact geometry (see Figure 6a,b). In the numerical model (to the upper and lower liner surfaces) the kinematic constraints were applied assuming that 50% of the crushed deformation is elastic and the other 50% comes from the plastic and/or damage deformations. Therefore, in the numerical analysis, to obtain the geometry from plastic deformation only (i.e., after unloading), the actual kinematic constraints were 5%, 10%, 15%, 20% and 25%. The output geometries of the FE analysis using those constraints were later considered to be the ones coming from the crushing of 10%, 20%, 30%, 40% and 50%. For the FE analysis the Abaqus Unified FEA from Dassault Systems was used, in which 4-node general-purpose shell finite elements were utilized (S4 according to [34]). Single model had about 3280 shell elements with linear shape functions and about 3160 nodes. The fluting was represented by 64 segments, since this value is important to retrieve
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