3D network simulations of paper structure - Innventia.com

PAPER PHYSICS
Table 1. Properties of fibers and simulated 60 gsm sheets, 1x2 mm sample (20F represents fines, fiber objects consisting of single
elements).
Parameter
Softwood pulp Mix 1
Mix 2
Mix 3 Hardwood pulp
0HW+100SW 50HW+50SW 64HW+16SW+20F 80HW+20SW 100HW+0SW
Fiber Length (Avg.), mm 3.0 2.0 1.2 1.4 1.0
Fiber Width (Avg.), μm 24 24 24 24 24
Elastic Modulus Fiber, GPa 10 10 10 10 10
Caliper, μm 100 100 100 100 100
Number of fibers 612 783 1162 952 1111
Elastic Modulus of sheet, GPa 6.433 5.920 4.309 4.278 2.959
Stiffness (mN·m), simulated 0.536 0.493 0.359 0.356 0.246
Stiffness (mN·m), experiment
(Lavrykov et al., 2010)
0.763 0.703 0.693
Wet Pressing Pressure, MPa
60,0
50,0
40,0
30,0
20,0
10,0
Softwood
Hardwood
0,0
0,4 0,5 0,6 0,7 0,8
Density, g/cm 3
Fig 17. Pressure as a function of mat density
Table 1 below shows the results of simulations of 60
gsm sheets composed of a mix of hardwood and
softwood pulps in different ratios along with a portion of
the fines. Note that the mixes are shown in number
fractions (not mass) as is usual for experimental data. The
simulation results show that the sheet modulus is a strong
function of the softwood content, increasing in magnitude
as this increases. An important parameter, the sheet
stiffness was also evaluated and shown to be a strong
function of the softwood content in the pulp. The final
rows in this table present stiffness estimates using the
simulations as compared to data obtained from
experiments. It appears that when the elastic modulus of
the fibers is assumed to be the same, the longer fibers
yield a higher stiffness value, a trend that corresponded to
the experimental results. The magnitudes of the
stiffnesses observed experimentally are much higher than
the simulation predictions though. The modulus of the
fibers and other parameters were obtained from published
values in the literature rather than measurements or
estimates on the furnish itself. This would account for the
difference in magnitude of the estimates.
Simulation of Wet Pressing of Sheets
The most significant use of simulation is to predict the
structure of the paper sheets with particular reference to
their z-dimensions i.e. the caliper or analogously, their
density. This is not possible with 2D simulations and also
simplistic constructions allowing the fibers to bend
according to external rules. The best method of
simulation is to determine the fiber reaction force during
compression, the so-called compressive stress and also
simultaneously track the drag force exerted by the
moving fluid. This is possible with a particle level
simulation such as the present one. The fiber
displacements were simulated using LS-DYNA. At each
step of pressing, the fiber reaction force was summed and
used as the total compressive stress borne by the fibrous
structure. The combination of the drag, translated into
permeability and the compressive stress was applied to a
homogenized wet pressing model (Lavrykov et al., 2009)
to determine the sheet caliper at different values of
applied pressure. The hydraulic stress was estimated
using an effective medium type approximation with
periodic cells, adjusted for local change in porosity.
These were combined to generate the structure in a timeexplicit
method. Fillers (or equivalent fines) were
considered as single discrete elements of approximately 2
to 4µm in size. This allowed the calculation of the sheet
caliper and density at given levels of applied pressure and
the as a result, the sheet structure.
Discussion
As was mentioned before, the main objective of this work
was to provide a new method for the generation of fiber
networks which can be applicable to numerical solution
of mechanical problems. This means fibers in an artificial
mat should consist of set of finite elements connected in
nodes. During mat formation fibers should not be bent
only but compressed too taking into account possible
fiber collapsing. In this situation all previously reported
methods (Nilsen et al., 1998; Provatas et al., 2000;
Heyden 2000; Heyden, Gustafsson 2002; Vincent et al.,
2009; Vincent et al., 2010) were found not correspondent
to our goals.
To provide a simulation of fibers settling and
compression the explicit analysis software was used.
There are several important reasons for using explicit
analysis instead of the implicit one for solution of this
type of problems. First, the explicit methods perform this
analysis much faster. For example, for the sample in Fig
7, which contains about 2000 fibers with 120,000
elements and 500,000 nodes, the solution of the static
elastic problem using the implicit method on a PC
computer with two Xeon Quad Core 3 GHz processors
and 24 GB RAM under 64-bit Windows 7 operational
system requires 4 Hrs. During solution of nonlinear
elastic-plastic contact problem with large strains and
Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 261