Energies 2021 , 14 , 3203
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the correct transvers shear stiffnesses of a representative volume element (RVE) as shown by our recent work [25]. Here, the number of segments was doubled due to modelling contact between the top liner vs. fluting and fluting vs. bottom liner. In tangential direction, the frictionless contact was assumed; and in normal direction the Herz type contact was assumed. Boundary conditions allowed to deform the RVE in out of plane direction, blocking from movement the external (side) nodes.
(b)
(c)
(a)
(d) (f) Figure 7. The crushed geometries of the corrugated cardboard obtained from the static finite element analysis: ( a ) 0%, ( b ) 10%, ( c ) 20%, ( d ) 30%, ( e ) 40%and ( f ) 50%. (e)
Based on FEM computations performed to obtain different crushing levels, the alter- native approach may be used to determine the crushing shapes of fluting to reconstruct its crushed geometry (this is valid for different flute amplitudes and periods). The analytical formula is proposed here, which accounts for the vertical coordinates of a half-wave fluting: f ( x )= ts 1 1 + e − 2 wx / L − 1 2 (4) inwhich t is the amplitude of intact fluting, L is the period length of intact fluting, x is the horizontal coordinate, w is the parameter related to inclination and curviness of the fluting vertical wall, while s is the parameter scaling the crushing thickness; w and s shouldbe used to fit the fluting shape to particular level of crushing. The parameters of w and s for cases used in this study are summarized in Table 2. The fluting shapes of half-waves for different crushing levels obtained from the formula proposed are presented in Figure 8a. It should be noted that the fluting length in the analytical approach was preserved by reproducing the geometry from FE analyses. The example of comparison between the fluting shape computed with the FE model and the analytical formula for 40% of crushing is presented in Figure 8b; a perfect agreement can be observed. In the next stage of the study, the output geometries (without any residual stresses) were imported to Abaqus software to build the initial material stiffness matrix of the structure, see Figure 6c–d. Before this, for each case, namely, crushing of 10%, 20%, 30%, 40% and 50%, the geometries were inspected in order to determine which regions of the fluting were actually plasticized. For those finite element models (along CD), all elastic properties (apart Poisson’s ratio) were deteriorated by scaling factor. Two regions of fluting were distinguished for each case; thus, two scaling factors were considered, see Figure 9. The first region is the contacting area of liners and fluting (region A) and the second
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