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M. Nygårds
Fig. 5 ZD strength of whole paperboards plotted versus the density of middle plies
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Density [kg/m 3 ]
assume that the following average fiber lengths apply to the different plies:
that the ratio tensile stiffness/fiber length also has almost a linear correlation with density for the free-lied plies. There is a deviation for the top ply at high densities. This should be due to the long fiber length that cause greater entanglement between the fibers, and lead to fiber failure rather than bond failure where the fibers are torn out of the fiber network. To make a simple model of stiffness in relation to strength it can be assumed that
• Bottom ply L fiber = 2.0 mm, • Middle ply L fiber = 1.3 mm, • Top ply L fiber = 1.5 mm.
This is essentially a curve-fitting of the data, but it has a physical relevance. The bottom plies were made of softwood fibers, the top plies had different mixtures of hardwood and softwood fibers, while the middle plies had mixtures of soft- wood, hardwood, broke, and CTMP. All these measurements were also in agreement with measurement of average fiber length for headbox pulp samples. However, due to the large number of paperboards tested, there are samples the deviate from these fiber lengths. Then one gets the plot in Fig. 3, where the ration σ 0 / L fiber has been plotted for three plies. In Fig. 3 it was obvious that the density and σ 0 / L fiber were dif- ferent for the different plies. By doing a linear regression it became evident that the failure stress, 0 , of the ply data in Fig. 3 can be simplified into a simple equation
(24)
E = 3.0 L fiber A .
Typically, ZD tensile tests are performed on the whole paperboard. This is since testing on free-laid middle plies can show smaller ZD strength values, because damage can be initiated during the splitting operation. Therefore, we will present ZD strength tested on whole paperboard, but it will be presented against the middle ply density, since this is most physically relevant since the ZD tensile fail- ures occurs in the middle ply. In Fig. 5, the ZD strength density can be seen, and a linear trend can be observed, yet there is variation. A linear trend would be expected since ZD tensile strength correlates with the number of bonds in the fiber network. Formations effects that give local variation of density, as well of ply strength in rela- tion to interface strength will, however, contribute to the observed variation. For the modeling purpose it will how- ever be relevant to assume that denser middle plies give higher ZD strength, that based on the data in Fig. 5 can be assumed to follow
(22)
0 = 0.03 L fiber .
The aim of plotting the tensile properties of the free-laid plies was to find relations to different parameters that easily can be controlled during machine trials. This was identi- fied to be density, fiber length, and fiber orientation (here expressed as a function of the MD/CD strength ratio). Based on this, simple expression was found that can be imple- mented in the derived stress state during folding. Here we found that the tensile strength can be expressed as:
(25)
0.7 ∗ 10 − 3 MPa.
f
ZD =
Based on properties of the free-laid sheets it has been possible to develop models that express the failure stresses as function of the papermaking parameters: density, fiber orientation, and fiber length, which are parameters that can be altered during the papermaking process. The derived
(23)
= 0.03 L fiber A .
Normally the tensile strength and tensile stiffness is correlated in paper materials. In Fig. 4, it should be noted
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