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Relating papermaking process parameters to properties of paperboard with special attention…
stress state. Here the difference is however smaller than a factor 2 in density, hence the effect should be neglecta- ble compared to other assumptions. It also has a large dependency of thickness. In thick paperboard, and paper- boards with low τ f or high σ f the shear stress component will become larger than the normal stress component. This means that it will contribute to failure, basically the paperboard will shear and cause failure. L = 10 m is an idealized lab setting, bending of paperboards can occur, with shorter bending arm, L , the shear component will be even more important. The conclusion from this is that the out-of-plane shear properties affect the folding behavior for paperboard grades, yet it can be disregarded for non- paper materials. Because of the biaxial stress state, a failure criterion needs to be formulated. The idea here is to keep it simple to enable predictions of failure. It should also cover the simple mechanisms that have been observed, which are: • Normalized stress components should be used. • If the normal stress is positive, shearing will be more difficult since the fibers are stretched. • If the normal stress is negative, shearing will be easier since the paperboard ply can buckle. • For simplicity no mixed terms are accounted for. Using the assumptions above it was assumed that
Maximum bending moment during folding
Two-point bending, or a cantilever configuration is often used to measure the bending stiffness of paper and paper- board. Folding of a paperboard will give an initial linear response associated with the bending stiffness. Thereafter, a peak bending moment will be reached, which is followed by a non-linear response as the paperboard is folded 90°. To evaluate bending stiffness the standard span is L = 50 mm. This ensures that the shear component during folding is small. On the other hand, a span of only L = 10 mm is used when folding both uncreased and creased paper- boards. To analyze the stress state in the paperboard during folding a Timoshenko beam analysis [14] can be applied to uncreased paperboard samples, see further Nygårds [16]. The stress state then consists of two components, an in-plane tensile/compressive stress, σ x , and an out-of-plane shear stress, τ xz , which is expressed as
M b w
z t 3
PL wt 3
PL I
(12)
z = 12
x =
z = 12
,
and
P 2 I
− z 2 =
P w
z 2 t 2
d 2 4
3 2 t
(13)
1 − 4
xz =
,
where t is the sample thickness, w the width, and z the dis- tance from the neutral line of the specimen. For most materi- als, the shear component will be small, and can be neglected. However, due to the orthotropy in paper materials, where the in-plane stiffness and strength are much greater than the out-of-plane stiffness and strength, the shear component will impact the behavior. For this purpose, it will be of interest to evaluate the stress components in relation to its measured failure stresses, σ f and τ f , accordingly
f =
f ( z )
(16)
= 1,
−
f ( z )
where σ f ( z ) and τ f ( z ) are the failure stresses that are different in the different plies. Out-of-plane shear strength is needed to evaluate the folding behavior. Normally out-of-plane shear strength can be associated with breaking of bonds within the sheets. In the literature out-of-plane shear prop- erties are often lacking. Instead, ZD strength is most often measured strength value. A fair approximation base on the work by Stenberg et al. [17] is
x f
PL w
z f t
(14)
= 12
,
3
and
(17)
P w
z 2 t 2
f ( z ) = 3 ∗
ZD ( z ) .
xz f
3 2 t
(15)
=
1 − 4
.
Hence, the failure criterion can be express as
f
z 2 t 2
( z ) tL
w
In the analysis, it was assumed that the failure stress is homogenous in the thickness direction, hence the paper- board does not have a property gradient. This is an obvious simplification in the case of multiply paperboard, which will not have a homogeneous stress distribution in the thickness direction. It shows that within the normal limit of failure properties there is shift of dominating stress component. However, it should also be emphasized that if there is large difference in properties of the outer plies and the middle ply, this will also affect the parabolic shear
12 f ( z ) t 3
3
PL
= 1.
(18)
z −
1 − 4
f ( z ) =
6 f
ZD
The failure criterion has been plotted in Fig. 1 for some different combinations of uniform failure stresses σ f and f ZD . This show that as f ZD decreases the failure mecha- nism is changing from in-plane tension/compression, i.e., failure close to the outer surfaces, that is dominated by the network structure to out-of-plane shear, i.e., failure within the paperboard, that is dominated by bond strength.
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