2.1. Sample Preparation
Corrugated paperboard is an orthotropic sandwich structure consisting of liners and flute that provide bending stiffness and shear stiffness, respectively (
Figure 1). Depending on the number of flutes, corrugated paperboard is classified into single-wall (SW), a double-wall (DW), and a triple-wall (TW) corrugated paperboard [
18]. Flute is also categorized into A/F, B/F, C/F depending on flute height and take-up factor [
19].
Corrugated paperboard is an orthotropic material that has three symmetry planes for the elastic properties. It has different mechanical properties in each direction (machine direction (MD, x), the cross direction (CD, y), the thickness direction (TD, z) due to the shape of the flute. Alternatively, corrugated paperboard components such as linerboard and corrugated medium exhibit orthotropy due to the fiber orientation. The paper fibers are oriented in MD while forming the paper sheets of machine-made paper board.
The types of corrugated paperboard used in this study were DW-AB/F, DW-BB/F and SW-A/F. The board combinations and main specifications for the corrugated paperboard are summarized in
Table 1.
2.3. Theoretical Consideration for Modeling
The constitute equation of the stress-strain relationship, which represents the mechanical properties of orthotropic material, is given by:
where
is the tensile strain in the
x direction.
and
are the Poisson shrinkage strain in the
y and
z directions, respectively.
is elastic modulus (Pa),
are shear modulus (Pa),
are shear strain at a different direction (unit),
is Poisson’s ratio at
ith number.
Since and Therefore, the number of engineering constants of constitute equation for describing the behavior of orthotropic material is 9. For the evaluation of the equivalent mechanical properties, 9 constants should be calculated from Equation (1).
To evaluate the equivalent mechanical properties of a structure, it is important to determine the unit cell representing the average response of the structure. The unit cell be selected as a repeating minimum basic unit in terms of the geometry and boundary conditions. Equivalent mechanical properties are evaluated through numerical experiments on the selected unit cell. In other words, the i-directional equivalent elastic modulus Ei and the equivalent shear modulus Gij can be obtained by simulating the uniaxial tensile and in-plane shear load states that resemble the actual test, respectively.
To obtain the equivalent elastic modulus
E1, it is assumed that a uniaxial tensile load is applied as shown in
Figure 3. The uniaxial tensile load state is simulated by dividing into three sub-problems and then superimposing the results linearly. The solid line and dotted lines indicate the shape before and after deformation, respectively. Here, the tensile strain
in the
x-direction and Poisson shrinkage strain
and
in the
y- and
z- direction be obtained using the superposition method.
In Sub-problem (1), the strain on the other directions is constrained to zero except for the tensile strain in the
x-direction. For the Sub-problems (2) and (3), displacement boundary conditions are applied so that the deformation of the other side was 0, except for the tensile deformation in the
y- and
z-directions. The reaction force at the interface can be obtained when FEA is performed on the sub-problems. To simulate the uniaxial tensile state, the obtained reaction forces are multiplied by weight coefficient α for Sub-problems (1), (2), and (3), respectively, and are combined as the following equation:
where
F is reaction force (N), α is weight (kg), P is applied load (kg)
i indicates
x,
y, and
z direction,
j indicates sub-problems numbers.
By using the determinant, the coefficient matrix [
α]
ji can be obtained, and the strain under an applied load [
F]
ij can be obtained using the matrix, as indicated in Equation (5).
The work
Wx by the tensile load
Px, and the internal energy
Ux stored in the unit structure are estimated from the following equations:
Since the work
Wx by the tensile load
Px, is equal to the internal energy
Ux stored in the unit structure, the equivalent elastic modulus
E1 for Sub-problem (1) can be calculated from Equation (8).
E2 and
E3 for Sub-problem (2) and (3) are also obtained from Equations (9) and (10).
where Δ
x, Δ
y, and Δ
z represent the dimensions of the unit cell.
Equivalent Poisson’s ratio is given by:
The calculation of the equivalent shear coefficients for orthotropic materials should be conducted separately because the shear stress only produces shear deformation of the component. After performing FEA under the boundary condition in which the unit cell is in the state of pure shear deformation, the equivalent shear modulus can be calculated from Equation (12) derived from the relation between the work
W by the shear load and the energy
U stored in the unit cell (
Figure 4).
With the same calculation method for
G12, the equivalent shear modulus
G23 and
G31 can be derived from Equations (15) and (16), respectively.
2.4. Model Development
Through the preliminary analysis of various repetition models, modeling for the unit cell in the MD of the corrugated paperboards used in this study was three times the minimum repetition level based on the integer multiples of the A/F and B/F wavelength. This accounted for the dimension that can overcome the excessive reaction force at the constraint points of both ends in the x-direction. The dimensions were 18 mm for AB/F-DW and BB/F-DW corrugated paperboard, and 27 mm for A/F-SW corrugated paperboard. However, the dimension of the unit cells in the CD was taken as unit lengths on all corrugated paperboard samples and modeled as solid state considering the thickness direction.
MiDAS NFX (2018R2, MiDAS IT) software was used for FE modeling [
26]. The interface between the flute and liner of the corrugated paperboard was joined to simplify modeling. The geometrical shape of the flute was modeled as a sine function based on the data shown in
Table 1. The FE modeling for the unit cells of the corrugated paperboard samples is indicated in
Figure 5. The hexahedral mesh was used due to the shape of corrugated papers. The computational domains for AB/F-DW, BB/F-DW, and A/F-SW were discretized into 156,825 nodes and 102,400 mesh elements, 148,767 nodes and 97,200 mesh elements, and 144,228 nodes and 94,200 mesh elements, respectively.
The boundary conditions applied when analyzing the three vertical displacements and shear displacements were applied according to the methods shown in
Figure 3 and
Figure 4. As a representative example, the AB/F-DW case is shown in
Figure 6 and
Figure 7.