Energies 2021 , 14 , 3203
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the section moment of inertia) was expected. Less load capacity reduction was expected in the edge crush test. However, due to the high resilience of the corrugated board, in some cases, especially for a small amount of crushing, the sample recovers its original thickness (because of the relaxation), which allow the observation of interesting findings. All issues will be discussed in the Results section. In the second step, a numerical model based on the homogenization that can easily be used to predict all the stiffness parameters of the corrugated board with different degrees of crushing was proposed. The numerical model, based on the finite element method (FEM), was used to build a global stiffness matrix of the full detailed 3D finite element (FE) model of the corrugated board. The model (based on the numerical homogenization procedure presented in [25]) takes into account the crushing of the corrugated board. This extension of the homogenization procedure allows to determine the degraded corrugated board stiffness matrix, which ultimately enables a robust simulation of the real laboratory tests using the simplified analytical formulas. The results obtained from the simulation of four-point bending and torsion (twist) tests [31,32] are in good agreement with the results obtained from the laboratory tests.
2. Materials and Methods 2.1. Mechanical Tests of Corrugated Cardboard
The strength and stiffness of a corrugated board can be determined by performing various types of tests. The most commonly used include: (a) BNT—stiffness in the four- point bending test; (b) ECT—edge crush test, (c) SST—shear stiffness testing and (d) TST— torsional stiffness test. The measurement of the bending stiffness is a laboratory test which is based on the four-point bending method (BNT), see Figure 1. This test is usually carried out on a sample with a dimension of 50 × 250 mm. It is important that in this test there is a constant moment and zero shear force in the sample between the internal supports. However, there is still a shear force between the outer and inner supports—this allows to take into consideration the shear stiffness aspect. It is worth noting that a bending stiffness measurement is very sensitive to sample damage (crushing or creasing), so the results for samples crushed by more than 50% are usually unreliable.
( a ) ( c ) Figure 1. The four-point bending test: ( a ) cardboard testing device; ( b ) loaded sample; ( c ) de- formed sample. ( b ) The sample compressive strength in the Edge Crush Test (ECT) is obtained for rela- tively stocky samples (conventionally thicker than 1 mm) with dimensions of 25 × 100mm, see Figure 2. In the case of slender specimens, the main failure mechanism is the loss of stability, and not the crushing of a sample. The ECT of the corrugated cardboard is one of the most known and important (from the practical point of view) parameter, often used in an analytical [1,10] or an analytical-numerical [27–29] determination of the load capacity of a corrugated board packaging. The shear stiffness (SST) of a corrugated board is measured on a sample 80 × 80mm loaded with a pair of forces at opposite corners, see Figure 3. Measuring the displace- ments and reaction forces at the other two corners allows to determine the cardboard shear stiffness. Only the linear part of the load-displacement curve is used in an iden- tification of the sample shear stiffness. The SST parameter is very sensitive to a sample
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