Materials 2022 , 15 , 663
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openings, ventilation holes and perforations or indentations [9–14], shifted creases on the flaps [15] or imprinting on packaging cardboard [16], e.g., product or seller logos. Much of the research presented in the literature is devoted to the assessment of the load-bearing capacity of the cardboard. Analytical methods were described 70 years ago [17], where simple and fast solutions for the assessment of the strength of simple stan- dard boxes was presented. The proposed formulae have evolved over the decades and have been enriched, expanded and improved, i.e., by introducing the Poisson’s ratio, dimensions of the box, the buckling influence or modification of constants and exponents [18–23]. A conventional numerical approach engaged for the assessment of load-bearing capacity of a cardboard is the finite element method (FEM). The numerical strength estimation of the paperboard tubes was discussed in [24] while consideration on the corrugated board pack- ages load-bearing capacity was presented in [25–28] and bending stiffness (BS) estimation in [29,30]. Buckling and post-buckling phenomena while applying FEM have been taken into account in [31], and torsional and transversal stiffness of orthotropic paper materials in- fluence on the strength of cardboard in [32–36]. The acquisition of mechanical properties of the paperboard during the simulation of its creasing involving FEM is discussed in [37–42]. FEM can also be utilized to perform a numerical homogenization [43]. Homogenization is a method that enables to simplify a multi-layer model to a single-layered one and ascertain the equivalent stiffnesses and effective thicknesses of the model. This procedure requires the determination of material parameters of individual cardboard layers; however, it al- lows for a significant saving of computation time while maintaining accurate results. This approach is being intensively developed [44–52], as are analytical [53], asymptotic [54] and multiple scales homogenization methods [55]. Experimental methods are very common and frequently used to assess the load capacity of corrugated boards. The box compression test (BCT) and the edge crush test (ECT) are the most prevalent. The bending test (BNT), which allows to define the bending stiffness, the shear stiffness test (SST), the torsional stiffness test (TST) and humidity testing are also pertinent to the assessment of the mechanical properties of the cardboard box. Non-contact measurement methods are increasingly used to measure displacements or strains, even in routine laboratory tests. A technique that allows to gather the data from the outer surface of the specimen, in accordance with the measurement of the relative distances between pairs of points tracked across images acquired at various load values, is a video extensometry [56,57] which is similar to the digital image correlation (DIC) that is a full-field non-contact optical measurement routine [36,58–64]. The two most significant mechanical parameters of corrugated board are the bending stiffness (BS) and the edge crush resistance (ECT). They are exploited in analytical formulae to estimate the load-bearing capacity of corrugated cardboard boxes. The paper presents the analytical determination of BS of five-layer corrugated cardboard in four-point bend- ing test basing on the known parameters of the constituent papers and the geometry of the corrugated layers. It was assumed that only flat layers, without the participation of corrugated layers are taken into account in the calculations. In the analytical model the presence of initial imperfections in compressed segments of the corrugated board was assumed. In addition, FEM numerical models have been built to validate the aforemen- tioned assumptions. Two cases have been discussed—in the first one, both liners and fluting were taken into account to determine BS and in the second one, the stiffness of the corrugated layers was reduced to imitate a situation in which they are excluded from the computation. The method has also been validated by means of experimental data taken from the literature [29]. The obtained compliance of the computational model with the experimental model was very satisfactory. The optimal selection of the arrangement of corrugated cardboard layers is funda- mental for the load-bearing capacity of packages. For that reason, sensitivity analysis with respect to mechanical properties of liners and the flute geometric parameters was conducted to answer the question of which of the parameters have the greatest impact on BS. The main contribution of this study was the derivation of analytical relationships that
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