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Experimental Testing of Elastic Properties of Paper and

WoodEpox® in Honeycomb Panels
Krzysztof Peliński* and Jerzy Smardzewski

The literature lacks comparisons of analytical and numerical calculations

that have been verified experimentally for elastic constants of auxetic cells
in cores manufactured from wood materials. The aim of this study was to
determine the effect of auxetic cell geometry and the type of material used
in their manufacture on elastic properties of the honeycomb panel core.
This paper describes properties of the materials, from which core cells
were modeled and presents mathematical models of cell properties. The
method of numerical optimization of cell shape was specified, and the
numerical calculations concerning modeled cells are given along with the
course and results of experimental tests. Additionally, the results of
analytical, numerical, and experimental tests were compared. Cell
geometry had a considerable effect on elastic properties of honeycomb
panel cores, particularly the angle of the cell wall. Moreover, geometric
imperfections had a significant effect on the results of analytical
calculations. Based on numerical calculations, satisfactory consistency
between these results and experimental tests was obtained.

Keywords: Auxetic; Periodic core structures; WoodEpox®; Experiment; FEM

Contact information: Department of Furniture Design, Faculty of Wood Technology,

Poznan University of Life Sciences, Poznan, Poland; *Corresponding author: pelinski@up.poznan.pl

INTRODUCTION

Composite layered panels have been applied in the automotive, boatbuilding,

military, and aeronautics industries (Gay et al. 2003; Pan et al. 2008). Honeycomb panel
cores are being improved in terms of their geometry and the type of materials used.
Typically they have the matrix molds composed of hexagonal, pyramidal, grid, or egg-box
cells (Wadley et al. 2003). They are commonly manufactured from aluminum, titanium,
stainless steel (Paik et al. 1999; Xue and Hutchinson 2006; Dharmasena et al. 2008;
Wadley et al. 2013), and plastics (Herup and Palazotto 1998; Nilsson and Nilsson 2002).
The cores made from metal alloys can be strengthened by adding glass fibers and carbon
fibers reinforced with plastics (Rejab and Cantwell 2013). Novel solutions are connected
with cores exhibiting auxetic properties and characterized by a negative Poisson's ratio.
They may be applied specifically in light engineering structures (Wang et al. 2018).
Imbalzano et al. (2017, 2018) examined the auxetic character of honeycomb core and found
interesting behavior of auxetics at dynamic loading. The cores effectively adjusted strains
by gradually condensing the material in the loaded zone. In this manner, they increased
impact strength (Wang et al. 2014). In contrast, cores with the honeycomb structure with
hexagonal cells undergo plastic deformations with no local improvement of rigidity. Wu
et al. (2017) indicated trends in the development of layered composites in order to avoid
defects resulting from the non-solid structure of the core. Yang et al. (2015) analytically
modeled auxetic structures; however, they did not obtain any satisfactory correlation

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between theoretical calculations and experimental tests. Significant results of
investigations concerning shape modifications of honeycomb core structures were
recorded by Majewski and Smardzewski (2014, 2015). Those authors focused on the
analytical modeling of delta-shaped auxetic core. At the same time, they indicated the need
to further search for optimal shapes of comparable relative density. There are very few
studies on applications of auxetic structures in wood-based boards. Smardzewski (2013)
designed and described mechanical properties of a three-layer honeycomb panel with the
auxetic core composed of bow-tie shaped cells; the auxetic core enhances strength and
stiffness of furniture panels. Moreover, it facilitates substitution of particleboards and
plywood traditionally used in the furniture industry with thus designed panels. In order to
simplify computations for large-sized structures composed of honeycomb panels, the
current trend is related with their homogenization (Guedes and Kikuchi 1990; Hohe and
Becker 2001; Hartung et al. 2003; Geers et al. 2010; Park et al. 2016; Jeong et al. 2019).
With homogenization of elastic properties in layered boards, the respective models are
simplified, and the time-intensity of computations is reduced. However, the literature lacks
comparisons of analytical and numerical calculations that have been verified
experimentally for elastic constants of auxetic cells in cores manufactured from wood
materials. Previous studies on the subject used only analytical and numerical models with
idealized geometry (Masters and Evans 1996; Pozniak and Wojciechowski 2014; Yang et
al. 2015; Mousanezhad et al. 2016; Wang et al. 2017). Thus, imperfections of cell shape
may affect the quality of computational results. The analyses presented herein verified the
hypothesis that geometric imperfections have a considerable effect on changes in values of
elastic constants calculated analytically or numerically in relation to idealized models.
This study determined the effect of the geometry of auxetic cells and the type of
material used in their manufacture on elastic properties of cores for honeycomb panels.
Moreover, the analytical and numerical calculations for Young's moduli and Poisson's
ratios were compared with results of experimental tests. First, the properties of materials
used to model the core cells were described. The mathematical models describing cell
properties and the numerical optimisation of cell shape were presented. After determining
the numerical computations for the modeled cells, the results of analytical, numerical and
experimental analyses were compared.

EXPERIMENTAL

Materials
Tests were conducted using Testliner 2 paper of 0.15 mm in thickness and
grammage of 123 g/m2 manufactured by HM Technology (Brzozowo, Poland). Moreover,
WoodEpox® Abatron wood-based composite, density: 430 kg/m3 (Abatron, Kenosha,
USA) was also used. According to the manufacturer it is a wood-based material; its
composition is entirely based on natural resins (zero VOC emissions and Greenguard®
certificate). Elastic properties of paper, including Young's moduli MOEx and MOEy,
moduli of rupture MORx (MPa) and MORy (MPa) as well as Poisson's ratios, ϑyx and ϑxy,
for the machine direction (Y) and the cross-machine direction (X) were determined
following the requirements of PN-EN ISO 1924-2 (2010). Tests were conducted on 10
strips of paper of 15 mm in width, from the original length of 180 mm. In order to determine
Young's modulus, MOEy (MPa) and MORy (MPa) in WoodEpox®, tests were conducted
on 10 dumbbell-type samples of 5.0 mm ± 0.1 mm in thickness, gauge length of 24 mm ±

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0.1 mm, and total length of 235 mm. The testing procedures followed the requirements of
PN-EN ISO 527-3 (1998). Samples were subjected to a uni-axial tensile test on a Zwick
1445 testing machine (Zwick GmbH, Ulm, Germany) using a Dantec Dynamics system
optical extensometer (Dantec Dynamics A/S, Skovlunde, Denmark). Table 1 presents the
average properties of tested materials. The recorded properties are consistent with literature
data for papers of comparable grammage (Szewczyk and Łapczyńska 2013). To date, no
alternative results have been published for WoodEpox®. Uniaxial tensile tests made it
possible to record directly the load, elongation, and narrowing of the samples. Based on
this, the stress-strain relation was determined for each material, which was implemented
for numerical calculations. This method was selected to be the most adequate way of
material definitions in the finite element analysis. In further analysis the authors used the
thickness of paper given above as a fixed value, and for the WoodEpox® due to technical
processing limitations, the thickness of cell wall was tA = 1.5 mm.

Table 1. Properties of Materials

MOEY MOEX MORY MORX ϑxy ϑyx
MATERIAL
[MPa] -
Paper 5142 1350 43 12.5 0.55 0.17
WoodEpox® 1029 - 5 - 0.3

Fig. 1. Elementary core cell, where: tA (mm): thickness of cell wall; lA,hA (mm): length of cell wall
side, φA, 𝜀A(˚): interior angles of a cell; Sy, Lx (mm): overall dimensions of cell; l0 (mm): distance
between cell walls

Mathematical Models of Auxetic Cells

Analytical models of elastic constants for auxetic core cells are presented using
geometry as in Fig. 1. Relative density ρA of the analyzed auxetic structure may be
described as the product of core density ρA* and density of the substance forming the
skeleton of the structure ρsA (Eq. 1).
∗
𝜌A
ρA = . (1)
𝜌sA

Following necessary transformations relative density may be written as follows:

𝐹sA
ρA = . (2)
𝐹A∗

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The area of the elementary core section is represented by Eq. 3.
𝐹A∗ = 4(𝑡A + 𝑙A 𝑐𝑜𝑠(𝜑A ))(ℎA − 𝑡A 𝑐𝑡𝑔(𝜀A ) − 𝑙A 𝑠𝑖𝑛(𝜑A )). (3)
The area of the substance forming the skeleton of the structure may be described as follows:
𝐹sA = 4(𝑡A + 𝑙A 𝑐𝑜𝑠(𝜑A ))(ℎA − 𝑡A 𝑐𝑡𝑔(𝜀A ) − 𝑙A 𝑠𝑖𝑛(𝜑A ))
−4𝑙A 𝑐𝑜𝑠(𝜑A )(ℎA − 2 𝑡A 𝑐𝑡𝑔(𝜀A ) − 𝑙A 𝑠𝑖𝑛(𝜑A )), (4)
Thus,
𝑙A 𝑐𝑜𝑠(𝜑A )(ℎA −2 𝑡A 𝑐𝑡𝑔(𝜀A )−𝑙A 𝑠𝑖𝑛(𝜑A )
𝜌A = 1 − (𝑡 , (5)
A +𝑙A 𝑐𝑜𝑠(𝜑A ))(ℎA −𝑡A 𝑐𝑡𝑔(𝜀A )−𝑙A 𝑠𝑖𝑛(𝜑A ))
90°−𝜑A
where 𝜀 A = . (6)
2

The condition critical for the existence of an auxetic cell with the presented geometry is:
ℎ
l0 = 2 ( 2A − 𝑙A 𝑠𝑖𝑛(𝜑A ) − 𝑡A 𝑐𝑡𝑔(𝜀A )) > 0 (7)
and simultaneously:
90° > 𝜑A > 0°. (8)

Elastic constants of cells presented below are given after Masters and Evans (1996). In this
paper, modeled structures and materials were characterized by linear elasticity,
ℎ
𝐾f ( A +𝑠𝑖𝑛(𝜑A ))
𝑙 A
Ey = , (9)
𝐻𝑐𝑜𝑠3 (𝜑A )

𝐾f 𝑐𝑜𝑠(𝜑A )
Ex = ℎ
, (10)
𝑏( A +𝑠𝑖𝑛(𝜑A )) sin2(𝜑A )
𝑙A

ℎ
𝑠𝑖𝑛(𝜑A )( A +𝑠𝑖𝑛(𝜑A ))
𝑙A
υxy = , (11)
cos2 (𝜑A )

cos2 (𝜑A )
υyx = ℎ
, (12)
( A +𝑠𝑖𝑛(𝜑A ))𝑠𝑖𝑛(𝜑A )
𝑙A

ℎ
𝐾f ( A +𝑠𝑖𝑛(𝜑A ))
𝑙A
Gxy = ℎA 2 2ℎ
. (13)
𝐻( ) (1+ A )𝑐𝑜𝑠(𝜑A )
𝑙A 𝑙A

where Ex (MPa) is Young's modulus in the X axis, Ey (MPa) is Young's modulus in the Y
axis, ϑxy is Poisson's ratio in plane XY, ϑyx is Poisson's ratio in plane YX, Gxy (MPa) is the
3
𝐸𝑠 𝐻𝑡A
modulus of elasticity in shear, 𝐾𝑓 = 3 (N/m) is the bending force constant, and Es
𝑙A
(MPa) is Young's modulus of the core substance.

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Monte Carlo Optimization
In engineering computations, statistical methods constituting a component of
numerical methods are used in static optimization (Smardzewski 2015). In this study, the
cell shape was optimized using the Monte Carlo approach, consisting of random searches
for the admissible area and on the basis of results to estimate the optimal value. In this case
the maximization of the moduli of elasticity, Ex (MPa) and Ey (MPa), for the developed
core cells was considered to be a natural optimization criterion. The set of decision
variables comprised outer dimensions of cells, dimensions, and angles of cell walls. The
optimal solution may be reached after a conjunctive satisfaction of numerous limiting
conditions, resulting from cell shape. A significant limiting criterion was connected with
the specific density of a hexagonal cell, most frequently found in honeycomb panels used
in the furniture industry (𝜌A = 0.0249). The mathematical model for the optimization of a
polygonal auxetic cell included the following,
a) decision variables, for which the variables block Kz takes the form in Eq. 14,
𝐾z = {𝑥̅ = (𝑥1 … 𝑥4 ): 𝑥i(min) ≤ 𝑥i ≤ 𝑥i(max) : 𝑖 = 1 … 4} (14)
where i is the number of decision variables for the cell, x1 = tA (mm) cell wall thickness, x2
= hA (mm) length of a common cell wall, x3 = lA (mm) length of a free cell wall, and
x4 = fA (˚) is the angle of a cell wall,
b) parameter Es (MPa) Young's modulus of the material, from which cell walls
were manufactured,
c) admissible set Φ was produced from inequality restrictions Φi(x) > 0, i.e.:
Φ = {𝑥̅ = (𝑥1 … 𝑥4 ): Φi (𝑥̅ ) > 0: 𝑖 = 1 … 4} (15)
The critical condition for the existence of an auxetic cell with the presented geometry was
as follows:
l0 > 0 (mm) (16)
The objective function as maximization of values for Young's moduli of a cell in the
primary orthotropic directions was as follows.
f(opt.) =Ex → 𝑚𝑎𝑥 and Ey → 𝑚𝑎𝑥. (17)
The optimization program was written in the Visual Basic programming language
(Microsoft, Redmond, WA, USA). A total of 30000 samplings were performed, among
which approx. 0.03% were successful. Table 2 presents values of elastic constants
calculated based on formulas 9 through 13 for optimized cells. In turn, Fig. 2 presents
shapes of cells selected through optimization. Cells A through D are characterized by
relative density identical in relation to the reference cell 𝜌A = 0.0249. The shape of cell E
made from WoodEpox® is the result of maintaining technical feasibility of the core at the
maximum values of Young's moduli and minimum relative density 𝜌A = 0.1547. Moreover,
it results from Table 2 that cell D exhibits considerable orthotropy: a high value of Young's
modulus in the direction of the Y axis, as well as a low value of Poisson's ratio ϑyx. Greater
values of elastic properties in comparison to paper cells are recorded for cell E. This is
caused by a significant increase in cell wall thickness at a comparable scale of overall
dimensions of the cell. Cells A through C were at comparable dimensions, Lx and Sy differ
significantly in the cell wall angle. Selected cells were used in further modeling.

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Table 2. Elastic Properties Provided by Mathematical Analysis
Ex Ey ϑxy ϑyx
Cell type
[MPa] -
A 0.033 0.194 -0.42 -2.40
B 0.023 0.458 -0.23 -4.43
C 0.033 0.058 -0.75 -1.32
D 0.0001 1.109 -0.03 -38.0
E 2.049 47.906 -0.21 -4.83

Fig. 2. Elementary core cells: Hexagonal cell, the reference cell with relative density 𝜌A = 0.0249;
A through D: paper cells; E: WoodEpox® cell

As shown, the inclination angle φA differed within range 1 to 25°, cell length lA, hA
respectively within range 5 to 20 mm. The relative density of the paper cells was a fixed
value 𝜌A = 0.0249, whereas relative density of the WoodEpox® 𝜌A = 0.1547.

Numerical Models
Numerical computations were performed using the Finite Element Method (FEM)
applying the Abaqus v6.13 program (Dassault Systemes Simulia Corp., Waltham, MA,
USA) and the resources of the EAGLE computational cluster, the Poznań Supercomputing
and Networking Center, within the framework of the computational grant, “Properties of
furniture panels with the synclastic surface and an auxetic core”. The computational model
is presented in Fig. 3. The HEX (hexahedral elements) grid was applied along with elastic-
plastic strains for selected materials (Table 1). The models were characterized by the mean
number of nodes ranging from 0.6 million to 1.4 million and the mean number of elements
ranging from 0.4 million to 1.2 million. The C3D6 (wedge) and C3D8R (tetrahedral) solid
elements were used.
Calculations were made for loads in the direction of the X axis, and next in the Y
axis for recording respective strains. For the paper models, variable loads were assumed
ranging from 0 N to 0.1 N at an increment of 0.02 N for the direction of the X axis and
from 0 N to 0.3 N at an increment of 0.05 N for the direction of the Y axis. For the

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WoodEpox ® model it was from 0 N to 10 N at an increment of 2 N and from 0 N to 35 N
at an increment of 5 N. Results of calculations are given in Table 3.

Fig. 3. An example model of a core used in numerical computations; Left is a diagram of loads,
and right is a mesh grid for the model

Table 3. Elastic Properties Provided by FEM

Ex Ey ϑxy ϑyx
Cell type
[MPa] -
A 0.0148 0.0224 -0.7056 -0.8645
B 0.0167 0.0393 -0.7777 -1.0227
C 0.0116 0.0198 -0.7760 -0.7393
D 0.0076 0.0875 -0.3465 -2.5368
E 0.1467 4.9693 -0.1774 -5.2667

As shown in Table 3, cores showed negative Poisson's ratios. The highest Young's
moduli, Ey = 4.97 MPa and Ex = 0.15 MPa, were found for the core with cell E. It also
showed the lowest value of Poisson's ratio, ϑyx = -5.23. Among the cores with paper cells,
the highest value of the modulus, Ey = 0.0875 MPa, was found for the core with cell D. At
the same time, it is a model with the greatest orthotropy, since the modulus of elasticity in
the direction was Ex = 0.0076 MPa. The core with cell D also showed a low value of
Poisson's ratio ϑyx = -2.54, while ϑxy = -0.35. Cores with cells A through C are characterized
with evident orthotropy of elastic properties. In the case of C type cells, the values of
Young's moduli and Poisson's ratios were very similar. This results in the core with
mechanical properties close to the isotropic structure. Furthermore, a smaller angle in the
wall led to a greater increase in Young's modulus Ey (MPa), while the difference between
Poisson's ratios ϑxy and ϑyx increased as well.

Experimental Tests
In the next stage of the study, molds facilitating technical manufacture of paper
cores were designed. These molds in the form of 3D solids (Fig. 4.) were designed in the
Autodesk Inventor Professional® environment (Autodesk, San Rafael, CA, USA) and
finished models were exported to the STL format. They were printed on a Stratasys Fortus
400mc small printer (Stratasys, Eden Prairie, MN, USA) using FDM Nylon-12™ filament
(Stratasys, Minnesota, USA).

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Fig. 4. An example mold to manufacture the selected paper core

Sheets of paper were embossed and cut into strips of 10 mm in width using a
Kongsberg-X plotter (Esko, Ghent, Belgium). The next step was to glue single strips into
sets using the PVAC Woodmax FF 12.47 D2 adhesive (Synthos, Oświęcim, Poland). Such
prepared sets were pulled onto the mold. The core shape was fixed using a SLW 15
laboratory dryer (Pol-Eko-Aparatura, Warsaw, Poland) at a temperature of 180 ± 1°C for
300 s. In case of the WoodEpox®, panels of 10 ± 0.2 mm in thickness were manufactured.
Panels were sanded on a Felder FW 1102 perform wide belt sander (Felder Group, Żory,
Poland) and milled on a Kongsberg-X milling cutter using an HM straight shank cutter of
3 mm in diameter (CMT, Poznań, Poland). Five cores each were manufactured to
determine properties in planes XY and YX. Figure 5 presents the manufactured cores.

a) b)
Fig. 5. a) Example paper core type D, b) core E manufactured from WoodEpox®

Cores were subjected to uniaxial compression (Fig. 6). The testing stand was
composed of a support beam, on which the sample and platens were placed. A ZEMIC H3-
C3-25kg-3B sensor (Serial No. P2Z122353) (Zemic, Hanzhong, China) was placed
between the platens, recording the force exerted onto the tested element accurate to 0.001
N. The stand was lighted with two lamps of 630 lumens to ensure proper photographic
recording of the test.

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Fig. 6. The testing stand: 1) pressing screw, 2) sensor platten, 3) ZEMIC H3-C3-25kg-3B sensor,
4) sample platten, 5) tested sample, and 6) support

Samples were tested following the analogical method, which was applied in
numerical calculations. For each applied load a monochromatic image was taken using an
Olympus OM-D camera (Olympus, Tokyo, Japan). Next strains were measured in the core
by analyzing the images using the National Instruments IMAQ Vision Builder 6.1 linear
analysis software (National Instruments, Austin, TX, USA) (Fig. 7).
The edge detection method was applied in digital image analysis, and respective
Poisson's ratios and Young's moduli were calculated from the dependence shown below,
𝑑𝑋∙𝑌
ϑyx = 𝑋∙𝑑𝑌 for loading direction Y (18)
𝑑𝑌∙𝑋
ϑxy = 𝑌∙𝑑𝑋 for loading direction X (19)
where dX (mm) and dY (mm) are displacements in directions X and Y, X (mm) is width,
Y (mm) is length of the core,
𝐹𝑦∙𝑌
Ey = 𝐻∙𝑋∙𝑑𝑌 for loading direction Y (20)
𝐹𝑥∙𝑋
Ex = 𝐻∙𝑌∙𝑑𝑋 for loading direction X (21)
where Fx,y (N) are loads for directions X and Y, H (mm) is height of the core, dX (mm) and
dY (mm) are displacements in directions X and Y, respectively.

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Fig. 7. The method to measure strain: detection of gauge points

Based on the image analysis elastic constants of the tested cores were determined,
as presented in Table 4.

Fig. 8. Images of elementary cells in cores: A through D: paper cores; E: WoodEpox® core

During the paper expanding process, the cells of the core are expanding differently.
The width of the adhesive stream is not repeatable, which has a significant impact on the
length of the free and common cell wall. This affects the inclination angles as shown in
Fig. 9. Measured angles of inclination differ from each other, φA1≠ φA2≠ φA3≠ φA4.

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Fig. 9. Different angles of inclination of tested core D fragment

Scanned images of real cores allowed to remodel cores for finite element method.
With the analytical approach, real cell geometric dimensions were used to recalculate the
elastic properties, with no further corrections to the cell geometry.

Table 4. Elastic Properties of the Core from the Experimental Sample

Ex Ey ϑxy ϑyx
Cell type [MPa] -
A 0.0148 0.0227 -0.7092 -0.8522
B 0.0195 0.0413 -0.9842 -1.0446
C 0.0114 0.0213 -0.7072 -0.7222
D 0.0083 0.0915 -0.3049 -2.7411
E 0.1201 5.4655 -0.1915 -5.9104

Table 4 shows that the greatest values of Young's modulus, Ey = 5.47 MPa
and Ex = 0.12 MPa, were recorded for the core with the E type cells. This core was also
characterized by the lowest Poisson's ratio ϑyx = -5.91, while ϑxy = -0.19. Among paper
cores, the greatest value of the modulus Ey = 0.0915 MPa was recorded for the core with
the D type cells, with Ex = 0.0083 MPa. During tests of this core, it was also most prone to
buckling due to its slenderness. Among cores manufactured from paper, it showed a low
value of Poisson's ratio by ϑyx = -2.74 and ϑxy = -0.30. When analyzing properties of cores
with cell types A through C it needs to be stressed that similar to the previous ones, they
exhibited orthotropic properties. Only the core with type C cells showed comparable values
of ϑxy and ϑyx and slightly varied values of moduli Ex = 0.0213 MPa, Ey = 0.0114 MPa.

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RESULTS AND DISCUSSION

Digital image analysis for the geometry of elementary core cells was used to
determine their actual dimensions with imperfections. Table 5 presents major parameters
of geometry for actual cells and idealized (theoretical) cells. The main imperfections in
geometry are caused by changes in linear dimensions of wall length and angles, while in
the case of paper cores, it was also wall shape. Core type E exhibited the smallest
differences in dimensions, as it has the most accurate manufacturing technology. Changes
in geometry resulted from the uneven structure of the material and technological feasibility
of its manufacture. Dimensions of cells with imperfections were used in the repeated
mathematical modeling of elastic properties of the core using equations 8 through 12.
Figure 10 presents a comparison of values for Young's moduli Ex (MPa), Ey (MPa) of
theoretical cells A, B, C, D, and E, as well as Young's moduli 𝐸xR (MPa), 𝐸yR (MPa) of
analogous actual cells. In turn, Fig. 11 presents a comparison of Poisson's ratios 𝜗xy , 𝜗yx
R R
of theoretical cells A, B, C, D, and E, as well as Poisson's ratios 𝜗xy , 𝜗yx of analogous
actual cells.

Table 5. Dimensions of Actual (R) and Theoretical Cells (T)

A B C D E
Cell Dimension
R T R T R T R T R T
lA [mm] 19.96 17.403 16.23 14.821 19.86 19.746 8.04 7.568 19.15 19.290
hA [mm] 9.64 10.010 9.46 10.460 9.46 10.598 17.97 20.001 19.31 20.685
φA [°] -13.9 -15 -8.9 -10 -22.3 -25 -6.4 -5 -16.4 -16
H [mm] 9.99 10 10.03 10 10.00 10 10.01 10 10.04 10

Fig. 10. Analytical values of Young's moduli for theoretical and actual models

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Fig. 11. Analytical values of Poisson's ratio for theoretical and actual models

As shown in Figs. 10 and 11, cell geometry strongly affects its elastic properties.
A particularly important role is played by the effect of the cell wall angle fA, while as
a result of the manufacturing technology of paper cores it is susceptible to changes. For
paper cores with cells A through C, the obtained interior angles of cells were much more
obtuse than theoretically assumed, while for cell D the angle was less obtuse than it had
been assumed. For cell E, no marked change was observed in Poisson's ratio (6% to 7%).
However, Young's modulus 𝐸xR (MPa) of that cell decreased 400-fold in relation to the
idealized models 𝐸x (MPa) , while in the case of modulus 𝐸yR (MPa) it was over 100-fold.
The greatest changes in values of Poisson's ratio were observed for type C cells. In that
R
case the index 𝜗yx of the physical model increased over 17-fold in relation to the idealized
model 𝜗yx . This was caused by a change in angle fA of the cell wall. For cells A and B, the
change in Poisson's ratio was much smaller. In the case of cell A, an almost 7-fold increase
R
in 𝜗yx compared to 𝜗yx was observed, while in the case of cell B it was almost 3-fold. In
R
turn, for cell D the value of 𝜗yx decreased 9-fold in relation to 𝜗yx . Analogously, for cells
R
A through C values of 𝜗xy decreased in relation to 𝜗xy . Values of Young's moduli 𝐸x (MPa)
for cells A through C turned out to be as much as 11-fold greater in relation to 𝐸xR (MPa).
Cell D exhibited strong orthotropy, and the respective differences exceeded 800%. For
cells A through C, values of 𝐸y (MPa) were 2- to 4-fold greater in comparison to the results
of mathematical analysis based on actual dimensions.
Figure 12 compares Young's moduli Ex (MPa), Ey (MPa) for cells A, B, C, D, and
E that was established based on numerical calculations (FEM) and experimental tests
(EXP), while Fig. 13 gives analogous values of Poisson's ratio 𝜗xy , 𝜗yx .

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Fig. 12. Young's moduli Ex, Ey determined numerically and experimentally

Fig. 13. Poisson's ratios determined numerically and experimentally

When comparing results of FEM analyses and experimental tests of paper cores,
the greatest differences were recorded for modulus Ex (MPa) of type B core. The result of
numerical simulations was by 14% lower than that obtained in experimental tests. In turn,
for calculations of modulus Ey (MPa), a considerable difference was recorded for type C
core. In that case the results were by 7% lower than the value of the laboratory
measurement. For the WoodEpox® core, numerical analysis made it possible to estimate
the modulus Ex (MPa) with a value by 22% lower in comparison to the experimental results.
In turn, the value of modulus Ey (MPa) was by approx. 9% lower. The other results of
numerical calculations for paper cores in comparison to the experimental results differed
by max. 9%. This may be considered a satisfactory agreement. When analyzing the quality
of calculations for Poisson's ratio the greatest difference in relation to laboratory tests was

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PEER-REVIEWED ARTICLE bioresources.com
found for Poisson's ratio 𝜗xy for core B. Obtained values were by approx. 21% lower in
comparison to those recorded in the experiment. For type D core the value of 𝜗yx was by
8% lower in relation to the experimental results, while for core E values from numerical
simulations of 𝜗xy and 𝜗yx were by 8% to11% lower. For the other cell types the differences
in values of Poisson's ratios were calculated numerically in relation to the results recorded
in the experiment and they did not exceed 13%. This may also be considered a satisfactory
consistency with the experimental findings resulting from an accurate representation of
geometry and material properties in numerical models.
In turn, evident and marked differences were found between the results of
experimental tests and analytical calculations. The mathematical simulation shows
significant sensitivity to imperfections of cell geometry. When considering the idealized
mathematical model, free from imperfections, the results of calculations for paper cores
are much lower than the recorded experimental results. Differences exceeded several fold
between the results of the laboratory tests. The greatest difference, over 30-fold, was
recorded for the simulation of Poisson's ratio ϑyx for the core with type C cells. In turn,
smaller, although still considerable differences were recorded for the simulation of elastic
properties of type E core. Although the simulation of values of Young's moduli Ex (MPa)
and Ey (MPa) is far from satisfactory, it needs to be stressed that Poisson's ratio was
determined with a 16% to 25% difference in values. This results from a highly precise
processing of core E and its shape being almost identical to that assumed analytically.
As indicated above, there were significant differences between experimental
research and analytical modeling. Taking into consideration the proximity of the FEM
analysis to the experiment, in most cases not exceeding 9% of the error, it should be noted
that there was no unambiguous tendency for the results to overestimate or underestimate
with use of the FEM analysis. Therefore, it should be pointed out that the results obtained
from the FEM analysis are also significantly different from the analytical approach -
several to several hundred times.

CONCLUSIONS

1. Cell geometry has a considerable effect on elastic properties of cell cores. The change
in the cell wall angle is particularly significant.
2. The WoodEpox® biocomposite is a suitable material for the manufacture of light
layered composites because the imperfections were slight enough.
3. Geometrical imperfections of paper cores have a considerable effect on the results of
analytical calculations. Idealized analytical models indicate very large differences in
comparison to experimental tests or numerical simulations.
4. Implementation of geometrical imperfections in modeling of CAD models used for
FEM method, shows that results differ significantly from analytical approach.
5. Based on numerical calculations a satisfactory consistency of the results was obtained
in relation to the data from experimental tests. This similarity is a result of the
representation of geometrical imperfections in numerical modeling.
6. Investigated cores show orthotropic properties, which justifies their application in
semi-finished elements or elements with a highly precise character of loads.

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ACKNOWLEDGMENTS

Calculations were conducted using the computational resources of the EAGLE

computational cluster, of the Poznań Supercomputing and Networking Center, within the
framework of the computational grant, “Properties of furniture panels with the synclastic
surface and an auxetic core”.
The present work was conducted as part of a research project (No.
2016/21/B/ST8/01016) financed from the funds of the National Science Centre.

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Article submitted: November 22, 2018; Peer review completed: February 3, 2019;
Revised version received: February 14, 2019; Accepted: February 16, 2019; Published:
February 25, 2019.
DOI: 10.15376/biores.14.2.2977-2994

Peliński & Smardzewski (2019). “Honeycomb panels,” BioResources 14(2), 2977-2994. 2994