Stress-strain curve of paper revisited - Innventia.com

PAPER PHYSICS
a)
b)
Fig 23. (a) Stress-strain curves for different number of fiber-fiber
bonds in the simulated tensile test on 10 x 4 mm 2 , 27 g/m 2 network:
(A) control network, 100% bonds; (B) 25% of removed bonds; (C)
50% of removed bonds. (b) Transposed curves.
a respective curve. Stress-strain curves are transposed by
diving them by the efficiency factor Ф, which has a
maximum value of 1.0 for a network with all bonds. The
results of reducing the number of bonds by 25 and 50 percent
are presented in Fig 23a and the transposed curves in Fig
23b.
The curves coincide in the linear region and up to 1.4%
strain. Seth and Page 1983 reported corresponding
experimental findings by modifying the number of bonds by
wet-pressing and beating. Since the wet-pressing affects the
thickness and we removed the bonds at given thickness, we
referred to the experiment with beating for comparison. The
results agree rather well for the range of efficiency factors
considered in the study. It should be noted, however, that
even with 50% removed bonds we could not reach as low
values of the efficiency factor as reported by Seth
(Ф=0.686). Even with 50% bonds removed, the efficiency
factor dropped to 0.9 only. This can indicate that fiber
properties were modified by beating in the physical
experiments but remained unchanged in our numerical
simulations.
Decreasing the number of bonds by 50% resulted in nearly
Fig 24. Fiber axial stress distribution for different number of
contacts at specific stress level of 13 kN·m/kg: (a) control network,
100% bonds; (b) 75% of bonds; (c) 50% of bonds.
30% lower strength. The stiffness of the network was not
significantly affected.
The reduction in strength can be explained by a simple fact
that a lower number of contacts would have a higher average
stress at a given global strain level, since the elastic stiffness
of the network was not affected, and the global stress at a
given strain level is the same. Lower number of contacts also
imposes a greater stress variation as well as higher mean
stress along the fibers. Fig 24 shows that distribution of axial
stress in the beam element expressed through probability
density function. This data was collected along the linear
slope of the curve at a specific stress of 13 kN·m/kg. It
shows that the distribution is bimodal. The left mode
represents unstressed fiber segments and the right mode
corresponds to the fiber segments bearing the load. With
50% removed bonds, the average stress in the right-side
mode is greater. It explains the fact the network with fewer
bonds deviates from the straight line sooner.
Conclusions
We used a three-dimensional fiber network model that
encompasses fiber nonlinearities and bond failures to
examine the phenomena, which take place in a network of
fibers along the stress-strain curve. The main outcome is that
the original strain inhomogeneities due to the structure are
transferred to the local bond failure dynamics.
The results show that failed bonds are located in the places
with high local strain. By comparing networks with weak
and unbreakable bonds, we concluded that strain
concentrations are the precursors of bond failures and not the
other way around. The width of strain concentrations regions
have a size on a millimeter scale and obviously depend on
the initial details of the network structure, such as local fiber
orientations and bond density.
The network with elastic fibers showed no sign of
softening up to the point close to failure. It confirmed again
that non-linear response of the network has its origin in the
Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 327