Calculation of Honeycomb Paperboard Resistance to Edge Crush Test

The article presents the method of calculating the edge crush test (ECT) of honeycomb paperboard. Calculations were made on the basis of mechanical properties of paper raw materials used for the production of cellular paperboard and geometrical parameters describing cellular paperboard. The presented method allows ECT calculation of honeycomb paperboard in the main directions in the paperboard plane; i.e., machine direction (MD) and cross direction (CD). The proposed method was verified by comparing the results of calculations with the results of ECT measurements of paperboard with different geometrical parameters made of different fibrous materials.


Introduction
Honeycomb board is manufactured on the basis of fibrous materials, most often from coniferous trees. Due to the spatial structure, it has low specific weight and good strength properties [1,2]. Its advantages also include recyclability; excellent energy absorption properties; and insulation, thermal, and acoustic properties [3]. Honeycomb paper cores are used in many multilayer and packaging products and they successively replace this type of plastic or aluminium products due to the lower manufacturing cost and lower specific weight. This organic and biodegradable raw material is gaining great popularity in various industries. It is used as a fillings for doors, countertops, furniture boards, partition walls in construction, and sandwiched multilayer structures in the aviation and automotive industries [4][5][6][7]. The production and use of honeycomb paper panels in the furniture industry are developing rapidly in Europe [8]. The demand for lighter furniture elements is increasing, which is contributing to lower transport costs and easier assembly, alongside the reduction of formaldehyde emissions, which is also an important issue in the modern world [9].
In the packaging industry, cellular paperboard is used to produce large-size boxes and their fillings. In most applications, in which cellular paperboard is treated as a construction material, it carries compressive loads caused by forces acting in its plane. To assess strength at this type of load, its edge crush test, ECT, can be used. In a certain direction along the paperboard plane, the ECT value is defined as the maximum compressive force transferred by the crushed paperboard until it is destroyed, related to the length of the side of the sample perpendicular to the direction of the force.
This indicator is particularly useful in cases wherein the strengths of panels with large values of panel thickness in the direction of load application are considered. This indicator is of decisive importance in, e.g., determining the resistance of boxes to the static pressure of boxes.
The study of cellular paperboard was carried out by Wen [10], who compared the results of measuring 5 mm thick cellular paperboard with the results of corrugated board. The comparison Materials 2020, 13, 1706 2 of 14 concerned both ECT edge crushing resistance in both directions in the paperboard plane, and the FCT flat crushing test.
Chen et al. [11] presented studies on lightweight multilayer panels with different honeycomb core structures made of paper, and wood cladding composite. By using experimental tests and finite element modelling methods, the authors presented the impacts of the construction parameters of honeycomb and the properties of the core material and cladding on the mechanical properties of light laminated panels.
Borsellino and Di Bella [12] conducted tests of laminates with different core structures at different load methods, including resistance to edge crushing of the paper honeycomb. The purpose of the work was based on experimental research to assess the relationship between stresses and deformations at uniform compressive static load.
Smardzewski et al. [13] conducted a study to determine the effect of a rectangular cellular paper core on the mechanical properties of three-layer furniture panels.
Smardzewski and Prekrat [14] presented modelling of mechanical properties of cellular wooden panels with a paper honeycomb core. The subjects of the study were the thin panels of a paper honeycomb with hexagonal cells. The research was carried out using numerical models; the results of numerical calculations were compared with the results of experimental measurements. As a result of the research, it was confirmed that the cores of cellular wood panels show strong orthotropic properties.
The authors of the work decided to develop a simple method for determining the ECT of honeycomb boards and to verify them in experimental studies. Another verification method was to develop complex numerical models in the finite element method (FEM), which should still be compared with experimental research.
Despite the fact that the literature provides information on ECT of honeycomb boards, none of the presented studies presents a simple, analytical method for determining the value of the edge crush resistance of cellular cardboard.
The purpose of the work is to present mathematical relationships that allow calculating the resistance of honeycomb paperboard to edge crushing in the machine and cross directions, based on its geometric parameters and mechanical properties of the materials from which it was made.
Cellular paperboard consists of two outer layers and a honeycomb core (see Figure 1). concerned both ECT edge crushing resistance in both directions in the paperboard plane, and the FCT flat crushing test. Chen et al. [11] presented studies on lightweight multilayer panels with different honeycomb core structures made of paper, and wood cladding composite. By using experimental tests and finite element modelling methods, the authors presented the impacts of the construction parameters of honeycomb and the properties of the core material and cladding on the mechanical properties of light laminated panels.
Borsellino and Di Bella [12] conducted tests of laminates with different core structures at different load methods, including resistance to edge crushing of the paper honeycomb. The purpose of the work was based on experimental research to assess the relationship between stresses and deformations at uniform compressive static load.
Smardzewski et al. [13] conducted a study to determine the effect of a rectangular cellular paper core on the mechanical properties of three-layer furniture panels.
Smardzewski and Prekrat [14] presented modelling of mechanical properties of cellular wooden panels with a paper honeycomb core. The subjects of the study were the thin panels of a paper honeycomb with hexagonal cells. The research was carried out using numerical models; the results of numerical calculations were compared with the results of experimental measurements. As a result of the research, it was confirmed that the cores of cellular wood panels show strong orthotropic properties.
The authors of the work decided to develop a simple method for determining the ECT of honeycomb boards and to verify them in experimental studies. Another verification method was to develop complex numerical models in the finite element method (FEM), which should still be compared with experimental research.
Despite the fact that the literature provides information on ECT of honeycomb boards, none of the presented studies presents a simple, analytical method for determining the value of the edge crush resistance of cellular cardboard.
The purpose of the work is to present mathematical relationships that allow calculating the resistance of honeycomb paperboard to edge crushing in the machine and cross directions, based on its geometric parameters and mechanical properties of the materials from which it was made.
Cellular paperboard consists of two outer layers and a honeycomb core (see Figure 1). The length of the regular hexagon's side can be determined from Equation (1) using parameter D given by the paperboard manufacturer (see Figure 2): Materials 2020, 13, x FOR PEER REVIEW 3 of 13

= /√3
(1) Cellular cardboard has the characteristics of orthotropic bodies. This is due to the core structure and the distribution of mechanical properties of the flat layers, which is characteristic of orthotropic bodies. In the plane of cellular paperboard, two main directions of orthotropy can be distinguished. The first one coincides with the direction of manufacture and it is called machine direction, MD. The second main direction, perpendicular to the machine direction, is referred to as cross direction, CD. The main directions of the CD and MD of the paperboard coincide with the main directions of the paper used for the flat layers of the CDO and MDO paperboard (see Figure 1). In the case of a paperboard core, the machine direction of the paper used for the MDR core is parallel to the height of the core and the cross direction CDR is perpendicular to the height of the core.

Materials and Methods
Sixteen honeycomb paperboards with different geometrical parameters and made of different fibrous materials were tested, and 4 types of paper used for their production. The following papers are used in the remainder of the article: • T135-testliner, 135 g/m 2 basis weight; • T160-testliner, 160 g/m 2 basis weight; • T200-testliner, 200 g/m 2 basis weight; • F140-fluting, 140 g/m 2 basis weight. Table 1 presented material constants of the papers. The honeycomb paperboard markings contain information about their material composition according to the following record (material of the first cover layer/core material/material of the Cellular cardboard has the characteristics of orthotropic bodies. This is due to the core structure and the distribution of mechanical properties of the flat layers, which is characteristic of orthotropic bodies. In the plane of cellular paperboard, two main directions of orthotropy can be distinguished. The first one coincides with the direction of manufacture and it is called machine direction, MD. The second main direction, perpendicular to the machine direction, is referred to as cross direction, CD. The main directions of the CD and MD of the paperboard coincide with the main directions of the paper used for the flat layers of the CD O and MD O paperboard (see Figure 1). In the case of a paperboard core, the machine direction of the paper used for the MD R core is parallel to the height of the core and the cross direction CD R is perpendicular to the height of the core.

Materials and Methods
Sixteen honeycomb paperboards with different geometrical parameters and made of different fibrous materials were tested, and 4 types of paper used for their production.
The following papers are used in the remainder of the article: • T135-testliner, 135 g/m 2 basis weight; • T160-testliner, 160 g/m 2 basis weight; • T200-testliner, 200 g/m 2 basis weight; • F140-fluting, 140 g/m 2 basis weight. Table 1 presented material constants of the papers. The honeycomb paperboard markings contain information about their material composition according to the following record (material of the first cover layer/core material/material of the second cover layer). For example: honeycomb paperboard with the core made of T160 paper and two cover layers of T135 paper has the mark T135/T160/T135.
To identify each paperboard, thickness H, the diameter of the circle inscribed in the regular hexagon of the core D cells, and the markings of the paper from which it was made, are provided.
To avoid the impact of climatic conditions on the results of strength tests of papers and boards, the testing pieces were conditioned before the test in accordance with PN-EN 20187: 2000 [27], and the tests were carried out in an air-conditioned room with the same climatic conditions as during the conditioning of the testing pieces; i.e., temperature 23 ± 1 • C and relative air humidity 50% ± 2%.
In the case of paper from which paperboard was made, the basis weight was measured in accordance with PN-EN ISO 536: 2012 [28], thickness in accordance with PN-EN ISO 534: 2012 [29], and the Young's modulus in machine and cross directions was determined based on a tensile test at constant speed stretching performed in accordance with PN-EN ISO 1924-2: 2010 [30].
ECT measurements were made using a universal Zwick testing machine (Ulm, Germany) with a load range up to 20 kN using the tooling shown in Figure 3. The tooling consists of two square plates of 144 cm 2 each. Both plates are rigidly attached to the lower frame of the machine, and the upper to the movable traverse (see Figure 3a). Supporting blocks (see Figure 3b) were used to test cardboard with small thickness, maintaining the tested piece in a vertical position until reaching the initial force.
Materials 2020, 13, x FOR PEER REVIEW 4 of 13 second cover layer). For example: honeycomb paperboard with the core made of T160 paper and two cover layers of T135 paper has the mark T135/T160/T135. To identify each paperboard, thickness H, the diameter of the circle inscribed in the regular hexagon of the core D cells, and the markings of the paper from which it was made, are provided.
To avoid the impact of climatic conditions on the results of strength tests of papers and boards, the testing pieces were conditioned before the test in accordance with PN-EN 20187: 2000 [27], and the tests were carried out in an air-conditioned room with the same climatic conditions as during the conditioning of the testing pieces; i.e., temperature 23 ± 1 °C and relative air humidity 50% ± 2%.
In the case of paper from which paperboard was made, the basis weight was measured in accordance with PN-EN ISO 536: 2012 [28], thickness in accordance with PN-EN ISO 534: 2012 [29], and the Young's modulus in machine and cross directions was determined based on a tensile test at constant speed stretching performed in accordance with PN-EN ISO 1924-2: 2010 [30].
ECT measurements were made using a universal Zwick testing machine (Ulm, Germany) with a load range up to 20 kN using the tooling shown in Figure 3. The tooling consists of two square plates of 144 cm 2 each. Both plates are rigidly attached to the lower frame of the machine, and the upper to the movable traverse (see Figure 3a). Supporting blocks (see Figure 3b) were used to test cardboard with small thickness, maintaining the tested piece in a vertical position until reaching the initial force. One-hundred-millimetre squared tested pieces were crushed; only for paperboard less than 10 mm thick were the tested pieces reduced to 50 mm to protect against global buckling.
Before starting the measurement, the tested piece was subjected to a 10 N initial force. During the measurements, the plates approached each other at a speed of 12.5 mm/min.
The measurements were carried out in two main directions in the plane of the paperboard; and on their basis its resistance to edge crushing in the machine direction ECTMD and cross direction ECTCD was determined respectively. The result of the determination in each direction is given as the average value obtained after testing ten pieces.
The ECT value was calculated from the equation: where: F-value of destructive force, kN; l-length of the loaded edge of the tested piece, m. One-hundred-millimetre squared tested pieces were crushed; only for paperboard less than 10 mm thick were the tested pieces reduced to 50 mm to protect against global buckling.

Calculation Methodology
Before starting the measurement, the tested piece was subjected to a 10 N initial force. During the measurements, the plates approached each other at a speed of 12.5 mm/min.
The measurements were carried out in two main directions in the plane of the paperboard; and on their basis its resistance to edge crushing in the machine direction ECT MD and cross direction ECT CD was determined respectively. The result of the determination in each direction is given as the average value obtained after testing ten pieces.
The ECT value was calculated from the equation: where: F-value of destructive force, kN; l-length of the loaded edge of the tested piece, m.

Calculation Methodology
The resistance of cellular board, both in the machine direction and in the cross direction, was calculated as the sum of loads carried by the core and both cover layers: where: ECT RCD and ECT OCD -edge crushing resistance towards CD, core and both cover layers, respectively, ECT RMD and ECT OMD -resistance to edge crushing in the MD direction of core and both cover layers, respectively.
It was assumed that the calculation model of the resistance to edge crushing of the core will describe the destruction of a repeating element of the paperboard core structure. The ABCE periodic cell was separated from the honeycomb core structure (see Figure 4). The resistance of cellular board, both in the machine direction and in the cross direction, was calculated as the sum of loads carried by the core and both cover layers: where: ECTRCD and ECTOCD-edge crushing resistance towards CD, core and both cover layers, respectively, ECTRMD and ECTOMD-resistance to edge crushing in the MD direction of core and both cover layers, respectively.
It was assumed that the calculation model of the resistance to edge crushing of the core will describe the destruction of a repeating element of the paperboard core structure. The ABCE periodic cell was separated from the honeycomb core structure (see Figure 4). It has been assumed that the cross-section of the core cell with a plane parallel to the cover layer has the shape of a regular hexagon with side a.   It has been assumed that the cross-section of the core cell with a plane parallel to the cover layer has the shape of a regular hexagon with side a. Figure 5 shows the dimensions of the periodic cell.
Materials 2020, 13, x FOR PEER REVIEW 5 of 13 The resistance of cellular board, both in the machine direction and in the cross direction, was calculated as the sum of loads carried by the core and both cover layers: where: ECTRCD and ECTOCD-edge crushing resistance towards CD, core and both cover layers, respectively, ECTRMD and ECTOMD-resistance to edge crushing in the MD direction of core and both cover layers, respectively.
It was assumed that the calculation model of the resistance to edge crushing of the core will describe the destruction of a repeating element of the paperboard core structure. The ABCE periodic cell was separated from the honeycomb core structure (see Figure 4). It has been assumed that the cross-section of the core cell with a plane parallel to the cover layer has the shape of a regular hexagon with side a.   The dimensions of the periodic cell can be determined from the equation: In the case of a regular hexagon cell that has been taken into account in the calculations, the angle γ is 60 • . It was assumed that only single-thickness walls are responsible for cell destruction in the ECT test, t R marked in black in Figure 4. The double walls t R , which are formed by gluing two layers of core material marked in red in Figure 4, are not damaged, as observed in preliminary ECT tests of honeycomb panels.
The load schemes used to calculate the Q forces transmitted through the periodic core cell in the MD and CD directions are illustrated in Figure 6. The maximum QMD and QCD forces transferred by the periodic core cell in the MD and CD direction were calculated from: where: S-maximum force transmitted during compression in the cross direction of the core material by a wall of a single thickness.
The force S can be determined from the following relationship: where: tR-thickness of paper used for core production; h-core thickness of cellular paperboard; Ϭcr-critical stress; α-coefficient of elastic restoration of a single cell wall.
The α factor is respectively: Critical stress Ϭcr can be determined from the equation: The maximum Q MD and Q CD forces transferred by the periodic core cell in the MD and CD direction were calculated from: Q MD = 2 ·S · sin γ (7) where: S-maximum force transmitted during compression in the cross direction of the core material by a wall of a single thickness.
The force S can be determined from the following relationship: where: t R -thickness of paper used for core production; h-core thickness of cellular paperboard; σ cr -critical stress; α-coefficient of elastic restoration of a single cell wall.
The α factor is respectively: Critical stress σ cr can be determined from the equation: where: Substituting Equation (14) into Equation (13) gets: where: E RCD , E RMD -Young's paper moduli used to produce the cellular paperboard core in cross and machine directions, respectively.
The ECT RMD and ECT RCD core crush test was calculated from the relationship: The resistance of two honeycomb board cover layers to ECT OMD and ECT OCD edge crushing was determined from the following equation: where: E OCD , E OMD -Young's moduli of paper used to produce cover layers of cellular paperboard in cross and machine directions, respectively; t o -the thickness of the paper used for flat layers of cellular paperboard; β OMD , β OCD -elastic support coefficients, determined experimentally.
It should be taken into account that E OMD E OCD , and this implies that β OMD β OCD . The β OMD factor is assumed as a function of the a/h ratio: Materials 2020, 13, 1706 8 of 14 The coefficient β OCD is constant, and it is: The cellular board's resistance to edge crushing in the machine direction ECT MD and cross direction ECT CD can be determined from the Equations (23) and (24), obtained after substituting Equations (16) and (18) into Equations (3) and (17), and Equation (19) to Equation (4):

Symbol
The largest discrepancy between the measured and calculated values was in the MD direction: 20% of the actual ECTMD value and in the CD direction 24%. The largest ECTMD discrepancy was found in the case of cardboard with mesh size D = 15 mm, thickness H = 60 mm, made of T200 and F140 papers. The largest discrepancy between the calculated and measured ECTCD values was in the case of paperboard with mesh size D = 15 mm, thickness H equal to 30 mm, made of T200 and F140 papers. The mean value of the discrepancy between the measured and calculated ECT values in all the cases examined was 11% of the actual value in both MD and CD. Figure 9 summarizes the ECT measurements of paperboard with the same mesh size made of the same materials. In almost all cases examined, the differences between the measured and calculated ECT values are within the variability of the results of measurements and calculations. For two honeycomb paperboards, the minimum calculation value is greater than the maximum value obtained from measurements by less than 1.2%.
The largest discrepancy between the measured and calculated values was in the MD direction: 20% of the actual ECT MD value and in the CD direction 24%. The largest ECT MD discrepancy was found in the case of cardboard with mesh size D = 15 mm, thickness H = 60 mm, made of T200 and F140 papers. The largest discrepancy between the calculated and measured ECT CD values was in the case of paperboard with mesh size D = 15 mm, thickness H equal to 30 mm, made of T200 and F140 papers. The mean value of the discrepancy between the measured and calculated ECT values in all the cases examined was 11% of the actual value in both MD and CD. Figure 9 summarizes the ECT measurements of paperboard with the same mesh size made of the same materials. In both cases, at low core heights, up to 20 mm, there is an increasing tendency for greater edge crush resistance as the cardboard thickness increases. Then, the ECT value remains at the same level. This is due to the fact that as the value of the a/h decreases, the influence of the core on the ECT of the paperboard decreases, striving for a constant value. Figure 10 shows ECT values of cardboard with the same geometrical parameters made of various raw fibres. In both cases, at low core heights, up to 20 mm, there is an increasing tendency for greater edge crush resistance as the cardboard thickness increases. Then, the ECT value remains at the same level. This is due to the fact that as the value of the a/h decreases, the influence of the core on the ECT of the paperboard decreases, striving for a constant value. Figure 10 shows ECT values of cardboard with the same geometrical parameters made of various raw fibres. Figure 9. Results of ECTMD and ECTCD measurements of paperboard with 15 mm mesh diameter, made of T135 paper.
In both cases, at low core heights, up to 20 mm, there is an increasing tendency for greater edge crush resistance as the cardboard thickness increases. Then, the ECT value remains at the same level. This is due to the fact that as the value of the a/h decreases, the influence of the core on the ECT of the paperboard decreases, striving for a constant value. Figure 10 shows ECT values of cardboard with the same geometrical parameters made of various raw fibres. The measurement results illustrated in Figure 10 show the effect of physical properties of the papers on the ECT of cellular board. The use of a paper with a higher basis weight, and thus higher thickness and better mechanical properties, causes a significant increase in the ECT value in both machine and cross directions. Figures 11 and 12 show ECT values of paperboard made from the same raw materials of the same thickness, differing in mesh size. The measurement results illustrated in Figure 10 show the effect of physical properties of the papers on the ECT of cellular board. The use of a paper with a higher basis weight, and thus higher thickness and better mechanical properties, causes a significant increase in the ECT value in both machine and cross directions. Figures 11 and 12 show ECT values of paperboard made from the same raw materials of the same thickness, differing in mesh size.    The very large decrease in ECT of cellular board visible in Figures 11 and 12 is associated with a decrease in the force transmitted by the cover layers, which buckle more easily with an increase in mesh size.
The relationships between the individual parameters used to calculate ECT and its value show a consistent nature with the mathematical relationships presented in Equations (23) and (24); e.g., an increase in cardboard thickness causes an increase in ECT value, while an increase in mesh size causes a decrease in ECT value.
The big impact on the differences between the real values of edge crush resistance and the values calculated in theoretical way result from the fact that the paperboard was produced in different periods of time, and during the production of the core, the papers are unwound simultaneously from several turns, and thus for their production, materials from various supplies are used, the mechanical properties of which may differ significantly. It happens that the actual values of mechanical properties differ by up to 20% from the nominal values given in the specification. In addition, switching machines for the production of paperboard of a different thickness can cause a different degree of stretching of the core or a different arrangement of the cell wall gluing lines, and thus the deviation of the dimensions and shape of the cell from the cells shaped like a hexagon, which is the shape adopted in the mathematical description. Very often during production, the core is slightly crushed, which also has a significant impact on the value of edge crush resistance [31].
During the tests, no global buckling of paperboard samples was found. Shibao Wen [10], who tested much thinner (and thus more vulnerable to global buckling) cellular cardboard with a thickness of about 5 mm did not find this phenomenon. The very large decrease in ECT of cellular board visible in Figures 11 and 12 is associated with a decrease in the force transmitted by the cover layers, which buckle more easily with an increase in mesh size.

Conclusions
The relationships between the individual parameters used to calculate ECT and its value show a consistent nature with the mathematical relationships presented in Equations (23) and (24); e.g., an increase in cardboard thickness causes an increase in ECT value, while an increase in mesh size causes a decrease in ECT value.
The big impact on the differences between the real values of edge crush resistance and the values calculated in theoretical way result from the fact that the paperboard was produced in different periods of time, and during the production of the core, the papers are unwound simultaneously from several turns, and thus for their production, materials from various supplies are used, the mechanical properties of which may differ significantly. It happens that the actual values of mechanical properties differ by up to 20% from the nominal values given in the specification. In addition, switching machines for the production of paperboard of a different thickness can cause a different degree of stretching of the core or a different arrangement of the cell wall gluing lines, and thus the deviation of the dimensions and shape of the cell from the cells shaped like a hexagon, which is the shape adopted in the mathematical description. Very often during production, the core is slightly crushed, which also has a significant impact on the value of edge crush resistance [31].
During the tests, no global buckling of paperboard samples was found. Shibao Wen [10], who tested much thinner (and thus more vulnerable to global buckling) cellular cardboard with a thickness of about 5 mm did not find this phenomenon.

Conclusions
The proposed method allows one to calculate the ECT of cellular paperboard both in machine and cross directions on the basis of the paperboard's geometric parameters and the mechanical properties of materials used for its production. It is much easier and much faster in practical application than numerical methods such as the finite element method or the finite difference method. In the examined range of a/h values in each of the main directions in the board plane, the theoretically calculated values differed from real values by an average of 11% of the actual value, and the maximum difference that occurred in the cross direction was 24%. Considering that the mechanical properties of raw materials and the geometrical parameters of paperboard can differ significantly from the nominal values assumed during the calculations, the obtained calculation accuracy can be considered satisfactory. In the future, the authors plan to compare the results obtained with the proposed calculation method and the results obtained by numerical calculations. However, they do not expect significant differences in the accuracy of the calculations due to the fact that the calculation errors result from the variability of strength properties of materials and heterogeneity of the geometry of the core due to changes in the parameters of the production process.