4
LINDBERG AND KULACHENKO
2.3
Elastic material properties
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and shapes correctly. The grammage is 330 g/m 2 and the thickness of the paperboards, including the thin (30 μ m) and compliant PET coating, is 0.5 mm. Figure 3 shows the studied tray after a successful forming operation, that is, without detectable failure. The linear dimensions of the formed tray are 185 125 25mm.
In the following theory, the principal material directions MD, CD and ZD are described with indices 1, 2 and 3, respectively. The elastic part of the paperboard is described using Hooke's law ε ¼ C σ : For an orthotropic material the full expression reads
1 E 1
ν 21 E 2
ν 31 E 3 ν 32 E 3
2 666 666 666 666 666 666 664
3 777 777 777 777 777 777 775
0 0 0
2.2
Material data
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1 E 2
ν 12 E 1 ν 13 E 1
0 0 0
ε 11 ε 22 ε 33 ε 23 ε 31 ε 12 2 666 666 664
σ 11 σ 22 σ 33 σ 23 σ 31 σ 12
2 666 666 664
3 777 777 775 ¼
3 777 777 775 :
Paperboard is an anisotropic material, which may be approximated as an orthotropic material. The material shows different responses in tension and compression which may not always be captured by the materials models available in the standard libraries in the commercial finite element tools. The source for the input data was the physical tensile tests of the two paperboard types considered in this study. The tensile tests are performed under ISO standard conditions in three in-plane directions, MD, CD and 45 , and are shown in Figure 4. As observed, the biggest difference between the boards is the tensile properties in the 45 0 -direction. The tensile strain in the MD is about 1.9% for Board A, and 2.2% for Board B. The tensile stress in the MD is about 60 MPa for Board A and 70 MPa for Board B. For the CD, Board A has a tensile strain of about 5.0% at 30 MPa, while for Board B the tensile strain is 6.4% at 32 MPa. The distribution of the tensile test results is discussed later in the paper. Paperboard exhibit a reduction in yield limit and strength in com- pression compared to the corresponding values in tension. 17,18 This is taken into account by assuming the yield stress in compression being 70% of that in tension. For the failure evaluation, the compressive strength is reduced by 50% from the tensile value. The chosen values for yield and failure stress levels in compression are based on the reported values in Xia et al. 18
1 E 3
ν 23 E 2
0 0 0
ð 1 Þ
1 2 G 23
0 0
0 0 0
1 2 G 31
0
0 0 0 0
1 2 G 12
0 0 0 0 0
The paperboards are modelled with 3D shells with plane stress assumption. In the case of plane stress, the expression in Equation 1 reduces to
1 E x
ν 21 E y
2 666 666 64
3 777 777 75 2 64
0
ε 11 ε 22 ε 12
σ 11 σ 22 σ 12
2 64
3 75 ¼
3 75 :
1 E y
ν 12 E x
ð 2 Þ
0
1 2 G 12
0 0
The two in-plane values for the Young's modulus E i in Equation 2 are determined by the fitting procedure, and the in-plane shear modulus G 12 and Poisson's ratio ν 12 are then calculated using the two separate relations for commercially produced papers 19
FIGURE 4
Uni-axial tensile tests in the three directions for the two paperboards as received for this study: (A) Board A; (B) Board B
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