Fibre cross section modelling

The cross-section of the earlywood fibres is assumed to be collapsed during the pulping and paper manufacturing processes. Different collapse modes can occur.

In this study, the shape of the cross section of fibres was determined by means of simulations which imitated the loads applied to fibres in the refining and paper-making processes. The purpose is to determine the final shape of the cross-section of the earlywood fibres found in a paper sheet. In a refiner, wood chips are fed between two discs rotating in opposite directions. The cross section of the fibres is pressed between the two refiner discs and is assumed to be loaded in a state of combined compression and shear. In the paper-making process the cross-sectional deformations that develop in the refining process are further increased when the paper sheet is pressed between the various rollers in the paper machine.

The cross-sectional shape of the thick-walled transition- and latewood fibres is assumed here to not be altered during mechanical and chemical processes that oc-cur. On the other hand, the cross-sectional shape of the earlywood fibres is assumed to be collapsed, which was determined by means of the finite element method for two-dimensional undeformed models of the fibre cross-section. A simplified, unde-formed initial cross-sectional shape of the studied fibre was assumed, as shown in Figure 8.1. In that figure, a dashed reference line is drawn in the mid-plane of the thickness direction in the fibre wall. The thickness of the cell wall is different for the earlywood than the latewood. The simplified shape was determined by studying such micrographs as those shown in Chapter 3.

m 28

µm R=5

µ

µm R=65

µm 34

Figure 8.1: Assumed initial geometry of the cross section of an individual fibre model, the mid-plane of the fibre being the reference of measure.

Loading direction α Rigid surface

Loading direction α

Moving rigid surface Moving rigid surface

Fixed surfaces

Figure 8.2: Finite element models of the fibre cross-section with rigid surfaces and with the loading direction defined.

To determine the shape in collapsed form, the undeformed fibre model was pressed between two rigid surfaces. Four collapsed cross-sectional shapes were deter-mined by applying different loading conditions. The loading cases used for analysing collapse of the fibre are shown in Figure 8.2, in which two major loading cases are shown, one case being modelled with the loading boundary planes oriented horizon-tally and the other case with the loading planes oriented at a 45^◦ angle in order to simulate the fibre being rotated during collapse. The ratio of the compression to the shear load applied is determined by the angle α, see Figure 8.2. The upper rigid surface in Figure 8.2 was moved along a straight line in the direction determined by the loading angle α, whereas the lower rigid surface was fixed. For each of the two major loading situations, two load cases were constructed where, α = 0^◦ resulting in pure compression, and α = 45^◦ resulting in combined compression and shear. To determine the residual deformations of the fibres after loading, an unloading phase was simulated by moving the upper rigid surface towards its original position.

To simulate the collapse of the fibre cross section, the finite element method was employed, the problem being analysed as a two-dimensional static problem under plane strain conditions. First-order bilinear 4-noded quadrilateral solid elements were employed in the fibre walls. The rigid surfaces were modelled using interface elements on the outward boundary of the fibre to describe the contact forces devel-oped during loading. To describe the contact forces that develop when the fibre is totally collapsed and opposing fibre walls meet, interface elements were also used on the inside boundary, i.e. the boundary facing the lumen. A Coulumb friction model was employed, the coefficient of friction at the contact surfaces being set to

Table 8.1: Microfibril angles and thicknesses of the earlywood cell wall layers.

Layer Microfibril angle, ϕ^◦ Thickness, µm

Middle layer +45 0.45

S2 -10 0.78

S3 +75 0.05

0.25. Three cell wall layers were assumed to be present in the fibre wall, their elas-tic properties being selected on the basis of the modelling results shown in Tables 5.3 and 5.4, where the medium set of the material parameters was chosen. The assumed thickness and orientation of the material in these layers are given in Table 8.1. Since large strains of the fibre walls develop during loading, nonlinear material behaviour of the layers in the fibre wall is required. Moreover, after the fibres in the refining process are unloaded, permanent deformations remain. The nonlinearities and permanent deformations that develop in the material are due to plastic and vis-coelastic deformations, as well as to various types of material degradation such as microcracking and damage. In the simulations, this was accounted for by adopting the elastic-plastic material model proposed by Hill, Eq.(4.53) for the various layers in the fibre wall. The six parameters r_i,j and the reference stress σ_ref in Eq.(4.53), which defines the yield surface, were chosen as

r11 = 0.1, r22 = 0.1, r33= 1.0, r12= 0.15, r13= 0.25, r23= 0.25

and for the middle lamella σref=400 MPa and for the S2- and S3-layers σref=800 MPa. The values chosen for σref were low as compared with the values employed in the simulations presented in Chapter 7. They were selected so as to suppress the springback effect after the unloading of the fibre. In the material definition, the 3-axis here is in the longitudinal microfibril direction, the 2-axis in the thickness direction of the fibre and the 1-axis is perpendicular to the other two.

For the model with the horizontally oriented loading planes, successively de-formed shapes obtained from the two combinations of compressive loading and shear loading are shown in Figure 8.3. In Figure 8.4, the successively deformed shapes obtained are shown for the model for which the loading planes are oriented at a 45^◦ angle. In each figure, the last subfigure shows the residual deformations after unloading. Since plastic deformations occurred, some stresses remains in the cell walls after unloading. The final deformed shapes after unloading are similar for all the four loading cases. However, the forces required to deform the fibres are differ-ent. Figure 8.5 shows the force-displacement curves for the two pure compressional loading cases, i.e. the two cases for which α = 0^◦. The force required to deform the fibre for which the loading planes are oriented at a 45^◦ angle are smaller than for the fibre for which the loading planes are oriented horizontally.

A A A

A

A

A A

A

Figure 8.3: Deformation of the fibre cross-section for loading directions of α=0^◦, top figure and α=45^◦, bottom figure. The last subfigure in each figure shows the residual deformations after unloading of the fibre.

A

A A

A

A A

A A

Figure 8.4: Deformation of the fibre cross-section for loading directions of α=0^◦, top figure and α=45^◦, bottom figure. The last subfigure in each figure shows the residual deformations after unloading of the fibre.

0 5 10 15 20 25 30 35 40 45 0

0.2 0.4 0.6 0.8 1 1.2x 10⁻⁴

Deformation, µm

Force, N

Orientation 0^° Orientation 45 ^°

Figure 8.5: Load-displacement curves obtained from simulations of two loading situations of fibre collapse.