Abstract
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.
Download PDF
Full Article
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 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 ± 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
Fig. 1. Elementary core cell, where: tA (mm): thickness of cell wall; lA,hA (mm): length of cell wall side, φA, 
: 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).
(1)
Following necessary transformations relative density may be written as follows:
(2)
The area of the elementary core section is represented by Eq. 3.
(3)
The area of the substance forming the skeleton of the structure may be described as follows:
(4)
Thus,
(5)
where 
(6)
The condition critical for the existence of an auxetic cell with the presented geometry is:
(7)
and simultaneously:
(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,
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 modulus of elasticity in shear 
(N/m) is the bending force constant, and Es (MPa) is Young’s modulus of the core substance.
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 (= 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,
(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 was produced from inequality restrictions i(x) > 0, i.e.:
(15)
The critical condition for the existence of an auxetic cell with the presented geometry was as follows:
(16)
The objective function as maximization of values for Young’s moduli of a cell in the primary orthotropic directions was as follows.
(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 
. 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 
. 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.
Table 2. Elastic Properties Provided by Mathematical Analysis
Fig. 2. Elementary core cells: Hexagonal cell, the reference cell with relative density = 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
, whereas relative density of the WoodEpox® 
.
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 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
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).
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.
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.
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,
where dX (mm) and dY (mm) are displacements in directions X and Y, X (mm) is width, Y (mm) is length of the core,
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.