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M. Nygårds
constructed a paperboard, where data was taken from Fig. 3. This is a asymmetric paperboard with different densities 600, 500, and 750 kg/m 3 for the bottom, middle, and top plies, respectively, and fiber lengths 2, 1.3, and 1.5 mm in the plies. The grammages (83, 250, and 50 g/m 2 ) were cho- sen such that the paperboard thickness became h =400 μm and based on the properties of the plies the failure stress during bending was estimated and put into the stress state Eqs. (14–15). With the chosen material parameters, the nor- mal stress is still dominating, but it should be seen that the largest normalized stress can be found in the middle ply, close to the interfaces. This indicates that tensile failure can occur in the middle ply before it can be seen in the outer plies. The normalized shear stress was smaller than the normal stress, but it was not neglectable. This can be seen when the proposed failure criterion was plotted (Fig. 7) it was noted that the f max is located along the bottom interface, where we hence expect failure to occur when the maximum bending moment has been reached. If a higher bending moment is desired for the paperboard, then the highest pri- oritization should be to increase the strength of the bottom interface, thereafter the top interface should be strengthened. The fact that the failure criterion f is largest in the mid- dle ply suggest that the failure will occur there during bending, and that modifications of the middle ply will alter the value and position of the failure criterion. In paper- board making, it is common to work with the middle ply to alter the properties of the paperboard. If the density of
the middle ply is changed the in-plane and ZD strength of the middle ply will change, which will alter the failure criterion. In Fig. 8, the middle ply density has been varied from 250 to 750 kg/m 3 . It should be noted when the den- sity is around 500 kg/m 3 the failure criterion is roughly the same in the middle ply and the outer plies with this param- eter set. This must be optimal; the middle ply has then been made bulkier without compromising the maximum bending moment. As the density is decreased the risk of failure in the middle ply increases, which will also lower the maximum bending moment. And when the middle ply is denser the largest risk of failure is within the outer plies and dominated by the normal component. In the previous plots (Figs. 7, 8) the failure criterion has been shown for A = 1, i.e., when the paperboard is in-plane isotropic. At a paperboard machine more fibers are often oriented in MD. This will alter the shape of the failure criterion. In fact, the orientation of different plies can be altered to minimize the risk of failure, and hence increase the maximum bending moment. If the orientation of the outer plies varies from A = 1 to A = 2 one can see that it is the CD direction that will have the largest risk of failure, see Fig. 9. This is since the failure stress in MD will increase more when the fibers are oriented in MD. In Fig. 9, it can also be seen that a similar effect occurs when the orientation of the middle ply is changed, the risk of failure during folding increase with increased MD orientation.
Fig. 7 Failure criterion based on the unsymmetrical paperboard that was constructed
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