795
Relating papermaking process parameters to properties of paperboard with special attention…
models are now inserted in the failure envelope (Eq. 18), hence
of anisotropy, A = 1 in all plies. In the simulations the effect of different grammage in the plies was investigated, and the impact of density of the middle ply as well as fiber length in the middle ply. When the density of the middle ply is low the paperboard thickness is high (Fig. 6). When L fiber = 1.0 mm was used, a rather weak middle ply was simulated; a local maximum was found, where the bend- ing stiffness can be optimized with respect to grammage in the outer plies. The simulations showed that the maximum bending stiffness have about 25–30% of the grammage in the outer plies. However, simulations with L fiber = 2.0 mm, which represent a middle ply with high elastic modulus. For this case, it was observed that the middle ply contributes considerably to the bending stiffness of the paperboard, and therefore the bending stiffness increase with increas- ing grammage in the middle plies. Hence, the effect of the increased thickness due to more grammage in the middle ply was larger than the effect of separating the denser outer plies. This was an interesting observation since many paperboard products are within the property range used. It is hence not necessarily true that maximum bending stiffness is achieved with 30% outer plies. The bending stiffness analysis showed that the bend- ing stiffness can be optimized by altering the grammages of outer plies, and the maximum ratio between bending stiffness and grammage of the paperboard can be found. However, there are also other parameters that depend on the grammage of the outer plies, such as printing and surface properties. Therefore, other parameters than bending stiff- ness also contribute to optimal paperboard performance. For the upcoming simulation of paperboard folding, we have
L
z 2 t 2
PL wt 3
833 t 2
400 L fiber A
= 1.
(26)
1 − 4
z −
f =
Or expressed in MD and CD
z
PL wt 3 PL wt 3
z 2 t 2
L
b CD
833 t 2
400 L fiber
(27)
= 1,
−
1 − 4
f MD =
b
MD
z
L
z 2 t 2
b MD
833 t 2
400 L fiber
(28)
1 − 4
−
= 1.
f CD =
b
CD
Modeling of bending of paperboard structures
Paperboard structures do not have uniform through-thick- ness (ZD) profiles, instead the idea is to have a ZD- profile to optimize performance. In papermaking one can easily change the density, fiber length, and orientation of each ply, which was also observed in Figs. 2, 3, 4. Based on this information some artificial paperboard structures will be constructed to visualize potential problems that can arise during bending. For this purpose, the laminate theory and bending analysis equations were implemented in an Excel sheet. To show how the concept works, the laminate theory was first used to optimize the bending stiffness of an artifi- cial symmetric paperboard with grammage 250 g/m 2 , degree
Fig. 6 Optimization of bending stiffness simulating paperboards with different densities in the middle ply, as well as different fiber lengths L fiber = 1 mm (solid lines) corresponding to E MP = 900 MPa and L fiber = 2 mm (dotted lines) corresponding to E MP = 1800 MPa
1 3
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