Stress-strain curve of paper revisited Svetlana Borodulina, Artem Kulachenko, Sylvain Galland, and Mikael Nygårds

KEYWORDS: Network simulation, Mechanical properties,

Fibers, Bonds, Paper properties, Damage

SUMMARY: We have investigated a relation between

micromechanical processes and the stress-strain curve of a

dry fiber network during tensile loading. By using a detailed

particle-level simulation tool we investigate, among other

things, the impact of “non-traditional” bonding parameters,

such as compliance of bonding regions, work of separation

and the actual number of effective bonds. This is probably

the first three-dimensional model which is capable of

simulating the fracture process of paper accounting for

nonlinearities at the fiber level and bond failures.

The failure behavior of the network considered in the study

could be changed significantly by relatively small changes in

bond strength, as compared to the scatter in bonding data

found in the literature. We have identified that compliance of

the bonding regions has a significant impact on network

strength. By comparing networks with weak and strong

bonds, we concluded that large local strains are the

precursors of bond failures and not the other way around.

ADDRESS OF THE AUTHORS: Svetlana Borodulina (svebor@kth.se), Artem Kulachenko (artem@kth.se),

Mikael Nygårds (nygards@kth.se): KTH, BiMaC

Innovation, Department of Solid Mechanics, SE-100 44,

Stockholm, Sweden. Sylvain Galland (galland@kth.se):

KTH, Wallenberg Wood Science Center, SE-100 44

Stockholm, Sweden.

Corresponding author: Svetlana Borodulina

The mechanism behind the hardening behavior in the

in-plane stress-strain curve of paper has been a matter of

discussion during the 60s and 70

s last century. The theme for

discussion was “fibers versus bonds”. As never-dried fibers

with a low micro-fibril angle show very little or no

hardening behavior until failure during tensile testing, it was

speculated that debonding and consequent fiber-fiber

friction is controlling the behavior of paper in the plastic

regime.

Seth and Page 1983 presented the experimental study

showing that bonding parameters such as measured relative

bonded area and bonding strength affect the paper strength

but do not influence the appearance of the stress-strain curve

at a given density. At the same time, the non-linearity of the

curves was tracked to the non-linear response of the fibers,

which is amplified during papermaking by drying and

mechanical treatments. This means that the measured

response of the fibers to tensile loading prior to papermaking

differs from that exhibited once the fibers are in the paper.

Other experimental results support the argument of

insignificant influence of bond failures on the shape of the

stress-strain curve. For example, using acoustic emission,

Gradin et al. 2008 recorded rather limited number of

acoustic events during the hardening path of the stress-strain

curve, which suggests that only few bonds fail completely

before rupture.

Although the presented evidence clearly establishes that

the shape of the stress-strain curve and bond failure are

unrelated, the limited impact of bonds is counterintuitive.

Reducing the number of bonds obviously increases stress

variation inside the network. This means that fibers reach

yield stress at lower network stress level. At the same time,

the non-linear behavior of the fiber is explained by the

deformation of its non-crystalline parts. Yet, the paper made

of nanofibrilated cellulose with crystallinity up to 70-80%

show similar hardening behavior, which in that case was

attributed to frictional sliding of fibers (Henriksson et al.

2008). Since the effect of the bonds is invisible in the

stress-strain curve, it is still unknown what the processes are,

which precede the final rupture and control the strength. As

so much effort is being invested in improving the bond

strength to increase the strength of paper, it is vital to

understand how the bonds contribute to it.

Several attempts have been made to reproduce paper

fracture numerically. Heyden (2000) modeled the tensile

behavior of paper with elastic fibers. The fracture was

captured by a series of linear computations in which the

force was incrementally increased and the failed bonds were

eliminated at each subsequent step. This modeling technique

could not account for inelastic deformation of fibers since

the deformation history was not accounted for.

Räisänen et al. 1996 showed with two-dimensional fiber

network simulations without bond failures that the

non-linear features of the stress-strain curve of individual

fibers were reflected in the response of the network.

Niskanen 1999 modeled two-dimensional sparse networks

with elastic fibers introducing fiber and bond failures.

Through comparison between energies released through

fiber and bond breakage, they showed a specific relation

between the number of failed bonds and failed fibers.

Moreover, Hägglund and Isaksson 2008 used a

two-dimensional isotropic fiber network to model the

fracture process in paper by accounting solely for

fiber-to-fiber bond failure. They observed that the network

stiffness reduction was proportional to the number of

collapsed bonds prior to peak load. They were able to

capture the softening behavior through the bond failure

mechanism.

The objective of this study was to describe a stress-strain

curve of the network of fibers and parameters that dominate

its shape as well as development of failure. By using detailed

particle level simulation tools we investigated, among other

things, the impact of “non-traditional” bonding parameters,

such as compliance of bonding regions, work of separation

and the actual number of effective bonds.

PAPER PHYSICS

318 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

Material and Methods Network modeling

In our model, we considered a three-dimensional network of

fibers. The network was created with the help of a deposition

technique described elsewhere (Kulachenko and Uesaka

2010). Each fiber is represented as a series of Timoshenko

quadratic beam elements with a tubular cross-section. The

beam element has three translational and three rotational

degrees of freedom at each node.

The following parameters can be varied during

construction of the network: fiber length, width, wall

thickness, curl, basic weight and other material properties of

fibers. All the listed parameters can vary according to a

specified distribution law.

We accounted for large deflections, rotations and strains.

A 4 x 4 mm2 snippet of a typical fiber network is shown in

Fig 1.

Fiber-to-fiber bonds were modeled by a point-wise contact

with friction (Zavarise and Wriggers 2000).

Fiber bonds can break and separate during loading. This

requires a specific description. A relation between force and

displacement of an individual bond is schematically

demonstrated in Fig 2.

There are three principal characteristics of the curve above:

stiffness, strength and work of separation (dark area below

the descending part of the curve). We captured this behavior

with a bilinear cohesive zone model, Fig 3, where Fbs is the

bond strength expressed through force, Kc is the contact

stiffness which describes the compliance of the bonding

regions, df is the fracture distance, and ds is the separation

distance.

A mixed I + II debonding mode was assumed, in which the

bond separation depends on both normal and tangential

contact forces. A power law energy criterion was used to

define the completion of debonding,

n t

cn ct

1E E

E E+ =

(1)

where En and Et are the computed normal and tangent

fracture energies respectively. The inputs for the model are

the maximum traction in both directions, and the critical

energy release rates Ecn and Ect. When a bond fails, the

contact between corresponding fibers is described with a

frictional contact. This is probably the first three-

dimensional model which is capable of simulating the

fracture process of paper accounting for nonlinearities at the

fiber level and bond failures.

Material characterization

Nonlinear stress-strain behavior of a single fiber was

described by bilinear isotropic hardening plasticity, Fig 4.

Selecting reasonable values for the fiber material properties

was a difficult task. Most of the experiments reported in the

literature were performed on the untreated fibers with

relatively low micro-fibril angles. These fibers behave

almost elastically before failure. It was shown that fibers

dried under compressive strains or having a high fibril

Fig 1. A typical simulated fiber network, 4 mm x 4 mm with 27 g/m2, where MD is machine direction, CD – cross-machine and ZD – thickness directions (ZD coming out of the network plane).

Fig 2. Schematic diagram of a force-displacement curve measured during fiber joint testing in either normal or tangent direction. Joint strength is the maximum point of the curve, joint stiffness is the tangent of the linear region of the curve and work of separation is the area below the curve from the point of maximum load to the separation distance.

Fig 3. Bilinear cohesive zone model used for modeling of contact debonding.

PAPER PHYSICS

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 319

Fig 4. Bilinear isotropic hardening plasticity fiber material model, where σy is yield stress, E is Young’s modulus and Etan is tangent modulus. For numerical values see Table 1.

angles exhibit stronger nonlinear behavior with a distinct

hardening region (Page and El-Hosseiny 1983; Groom et al.

2002). In this respect, drying inside the sheet affects the

fibril angles, reduces the elastic modulus of the fiber and

promotes distinct non-linear hardening response through

introducing compressive strains. Fig 5 shows the material

data found in the literature on a fiber dried under

compressive restraint (Seth and Page 1983) and an untreated

fiber (Groom et al. 2002).

These two measurements exemplify probably the upper

and lower bounds for the fiber behavior inside the network.

For our numerical experiments, we assigned the following

average fiber properties, Table 1, which were measured for

the pulp used in the experimental sheets preparation;

described in the next section. The fiber width and length

were varied according to a Gaussian distribution with cut-off

values of 6 μm in fiber width and 100 μm in fiber length.

Fiber bonding properties are widely available in the

literature (Lindström 2005). However, it is often left

unspecified whether normal or shear strength is measured

since a clear distinction is difficult to make in the

experimental analysis. There are at least eight experimental

methods including in-plane, shear, peel or z-directional

loading modes for determination of fiber-fiber bond

strengths, which are often reported in stress units

(Retulainen 1993). There has been a debate about the

relative bonded area, as it is involved in calculations. Based

on the previously published results of measured bond

strengths, Joshi 2007 and Batchelor 2010 stated that most of

the reported shear bond strengths are far too low compared

to reality. However, since we specify the bonding strength in

force units, we stay unaffected by the uncertainty related to

the relative bonded area.

For the reference case, we selected the data for

summerwood fibers (Aulin et al. 2010; Niskanen 1998)

summarized in Table 2.

The friction coefficient which acted between the fibers

after debonding was assumed to be 0.5.

Fig 5. Stress-strain curve for a latewood fiber, adopted from Groom et al. 2002 and for holocellulose summerwood fiber, adopted from Seth and Page 1983.

Table 1: Fibers properties used during network construction.

Fiber length mean±std [mm]

Fiber width mean±std [μm]

Young’s modulus [GPa]

Tangent modulus [GPa]

Yield stress [MPa]

3.20±1.38 32.45±2.45 20 10 100

Table 2 Parameters used for the control case in simulation: separation distance (SD), bond strength (BS), contact stiffness (CS).

Norm BS [mN]

Tang BS [mN]

Norm SD [μm]

Tang SD [μm]

Norm CS [GPa]

Tang CS [GPa]

25.5 4.2 0.73 0.60 40 8

Experimental setup

The purpose of the experimental setup was to get a

qualitative comparison between the numerical network

model and the physical fiber network. With the help of a

sharp die, we extracted and tested small rectangular pieces

from isotropic handsheets made of unbleached softwood

sulfate pulp with removed fines. Fines were removed by a

screening technique. Fiber length analysis was performed

prior to fractionation. The measured fiber length distribution

followed a bimodal Gaussian mixture distribution.

Analyzing the mode which corresponds to longer fiber

fraction retained after screening gave the mean fiber length

of 3.2 mm with standard deviation of 1.38 mm. This fiber

length data was used during numerical network generation.

Laboratory handsheets of the specified pulp were prepared

according to ISO 5269/2 using Rapid Köthen sheet former

equipment in a climate-controlled room. Sheets were dried

under restraints at 93°C in the dryer for 15 minutes. After

preparation, the handsheets were conditioned at 50% RH

and 23 °C until testing.

Tensile tests were performed on a INSTRON® 5944 system

PAPER PHYSICS

320 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

equipped with a 50 N load cell at 1 mm/min loading rate and

at a standard climate 50% RH and 23 °C. We checked for the

presence of jaw slip by looking for the traces of zigzagging

behavior in the force measurements and concluded that the

jaw slip did not take place during testing. Digital speckle photography was performed with the

deformation measurement system Vic 2D (LIMESS®).

Results and Discussion Size effects in the physical fiber networks

The size and the basis weight of the simulated specimen are

critical for collecting meaningful results and being able to

compute the network on the available computing resources.

With the fiber properties described earlier, we are currently

able to compute the strength of dry three-dimensional

networks with a number of fibers equivalent to 10x4 mm2

samples with a basis weight up to 30 g/m2. The main

limitation arises from the necessity to have a fine mesh along

the fibers to achieve convergence with respect to mesh

density.

In order to assess the implication of computational

limitations on the planned analyses, we investigated how the

size of the specimen influences the average values of

network stiffness and strength on a limited set of networks.

We do not draw any conclusions from the experimental data

alone.

Fig 6-Fig 8 show the stress-strain curves recorded in

tensile tests experimentally. Tensile force was applied in the

length (longest) direction of paper sample in Fig 9. Fig 6

shows that 4x4 mm2 networks have somewhat lower elastic

modulus than 10x4 mm2 networks, presumably because the

sample cutting procedure reducing the average length of the

fibers and the effect of boundary conditions. Shorter network

had a long softening region which can be explained by the

fact that few long fibers can extend from one constrained end

of the network to another and being stretched even after

losing all the contacts.

At the same time, increasing the width from the chosen

reference level of 10x4 mm2 to 10x10 mm

2 did not affect the

stiffness but increased the strength and strain to failure, Fig

7.

Finally, increasing the basis weight did not change the

tensile stiffness index and tensile index significantly, but

increased the strain to failure, Fig 8.

From these experiments, we can that the size of 10x4 mm2,

27 g/m2 is sufficiently large and thick in the view of

representing elastic properties and the overall shape of the

stress-strain curve.

Evolution of damage. Experiments.

A stress-strain curve provide limited information even for

alitative comparison with simulation. In order to extract

more data, we observed the evolution of the strain field in the

physical specimen with the help of the digital speckle

photography. There was no need to apply a speckle pattern

onto the network as the fibers themselves created a sufficient

Fig 6. Experimental results from the tensile test. Effect of increased sample length for laboratory handsheets (27 g/m2).

Fig 7. Experimental results from the tensile test. Effect of increase in sample width for laboratory handsheets (27 g/m2).

Fig 8. Experimental results from the tensile test. Effect of increased grammage for 10x4 mm2 laboratory handsheets (27 g/m2).

PAPER PHYSICS

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 321

a)

b) Fig 9. Tensile test on 10 x 4 mm2 samples, cut from laboratory handsheets (27 g/m2). (a) Initial state prior to loading and loading direction. (b) Post-failure localization.

Fig 10. Stress-strain curve from the tensile test on 10 x 4 mm2, 27 g/m2 sample. The points a to e correspond to instances of strain field measurement (Fig 11a to Fig 11e).

contrast pattern for analyses. Fig 9 shows the initial state of

the physical network and the post-failure damage

localization, which was on a millimeter length-scale. In all

the observed samples, damage localization occurred at an

angle to the direction of loading typical for semi-ductile

behavior with shear-band formation. Since dry paper usually

fails more abruptly, it also suggests that the handsheets did

not have a sufficient level of interfiber bonding due to

relatively low pressure applied during sheet making.

Fig 9b shows the localization path at the network

separation point. The localization path is not clearly visible

before that moment. At the separation point, the measured

force is already zero and there are some considerable

out-of-plane deformations.

By analyzing Fig 10 and Fig 11, we can conclude that

nucleation of damages, manifested through increased local

strain on the millimeter scale, which develops already during

the hardening region. In this particular sample, there are two

“competing” localization regions of which only one will be

preferred for the final rupture already along the softening

part of the curve.

a)

b)

c)

d)

e) Fig 11. Digital speckle photography results. Development of fracture during tensile test on 10 x 4 mm2, 27 g/m2 sample. The color bar indicates local in-plane strains in tension direction with range from 0 to 2%.

Evolution of damage. Simulations.

We used the acquired fiber length distribution data to create

an isotropic fiber network numerically. Table 1 and Table 2

summarize the fiber and bonding data used in the reference

(control) numerical experiment. During fiber deposition, we

assumed the maximum level of conformability, which gave

us an average number of contacts per fiber of 156. This

number corresponds to a distance between the bonds of

21 µm, which is comparable to the average diameter of

PAPER PHYSICS

322 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

Fig 12. Comparison of the experimental and simulated results from the tensile test on 10 x 4 mm2, 27 g/m2 sample.

fibers and agrees with the prediction by Alava and Niskanen

2006. Fibers not connected to the network were removed

during computations.

We performed numerical simulations until the stress-strain

curves showed the persistent signs of softening. Capturing

the entire softening part of the curve would require a fine

time-step and consume a significant amount of computation

time without providing much additional information.

Fig 12 shows the comparison of the computed stress-strain

curve with the experimental ones. Although the initial part of

the curve was captured, the strength and the hardening

regions were not. The fact that the strength was considerably

overestimated can be attributed to an overestimated number

of contacts or other contact parameters. We have

investigated these factors on the very same network and

mesh in order to exclude the influence of the variations due

to disordered network structure.

Effect of the bond strength

We considered four cases: (a) bond failure with the reference

bond strength “Control”; (b) with the reference bond

strength divided by 3 “Weak”; (c) with the reference bond

strength multiplied by a factor of 5/3 “Strong”, as defined in

Table 3; (d) no bond failure “No debonding”. The

separations distance was 15% longer than the fracture

distance in all the cases.

Fig 13 shows the stress-strain curves for different bond

strengths. The appearance of the curves is similar to the

results of Seth and Page 1983, who varied the bonding

strength by addition of bonding agents – the networks show

identical response until the point close to failure. The higher

the bond strength is, the longer the curve follows the case

Table 3 Different bond strengths used in parametric study.

Bond strength (BS) Normal BS [mN]

Tangent BS [mN]

Strong 42.5 7 Control 25.5 4.2

Weak 8.5 1.4

Fig 13. Stress-strain curves for different bond strengths from the simulated tensile test on 10 x 4 mm2, 27 g/m2 network.

with no bond failure. Remarkably, a factor of 3 in bond

strength, which is lower than the variation found in the

literature (partially with data reported in stress units),

changed the strength of the network dramatically with a

given bond compliance and relation between fracture and

separation distances.

The intensity of damage can be related to the number of

broken bonds. Using acoustic emission, Gradin et al. 2008

showed that during the elastic deformation of paper only a

very limited number of acoustic events was registered. It

remained unclear, however, whether all the fiber bond

breakage could be detected with the utilized equipment,

since some of the broken bonds could not emit sufficient

amount of energy at failure to be registered. For example,

similar acoustic emission testing applied to cellulose

nanopaper by Henriksson et al. 2008 showed only very few

events registered until the rupture.

By observing the number of bonds failed in the network,

Fig 14, we can conclude that the growth rate in the number

of contacts follows an exponential law. At the same time,

relatively few bonds, namely, 3% of the total number, failed

completely in the network with strong bonds before the

catastrophic failure. Although more contacts failed in the

network with weaker bonds, the total number of them stayed

below 5%. By observing the difference between the number

of fractured and fully separated bonds at a given substep, Fig

14, we can conclude that there is no observable delay

between the damage and complete separation of the bonds

(dashed and solid lines almost coincide).

Let us now examine the processes on the network level

which preceded the failure and compare to what was

measured in the experiments. We looked first how the failed

bonds were distributed in the network. Fig 15a shows initial

contact positions in the network while Fig 15b and Fig 15c

show the fractured bonds at the time of onset of softening.

The color scale indicates the state of damage where 0

corresponds to undamaged state and 1 to separated state

according to Fig 3. Only the bonds having some degree of

PAPER PHYSICS

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 323

damage were plotted.

Fig 14. Total amount of fractured (solid line) and separated (dashed line) bonds for networks with different bond strengths: (a) Control, (b) Weak and (c) Strong bonds.

Although the stronger network has fewer failed bonds, the

failure pattern remains similar. There is a clear localization

zone which indicates the failure path. The fibers which were

pulled out were clearly indicated by the continual chain of

separated contacts.

Fig 16 shows the strain fields in the direction of loading. It

was calculated as a continuous field after mapping the

displacement field onto a 2D mesh. The calculated strains

fields agree well with the measured strain fields, Fig 11, in

terms of size and magnitude of variations.

The strain field in the Fig 16a was output when the network

with weak bonds reached the maximum stress, that is,

around 0.7% strain. Fig 16b shows the strain field in the

network with no debonding also at 0.7% strain. Apart from

greater strain variations in the network with weaker bonds,

both strain fields have similar features. The locations of

maximum strains were the same for both cases. At the same

time, the locations of the separated bonds, Fig 15b,

correlated with the areas where the local strains were larger,

Fig 16a. This means that the bond failures are largely

affected by the local strains and not the other way around. In

other words, large local strains are the precursors of bond

failures. The local strain variations depend, in their turn, on

the initial details of the network structure, such as local fiber

orientations, number of bonds and density. This is different

from wet networks, where strain variations are largely

affected by the stick-slip behavior of the bonds.

Effect of plasticity

We investigated how plasticity on the fiber level affects the

stress strain curve and micromechanical processes during

straining. We compared the simulation with and without

invoking bilinear plasticity for the fibers.

Fig 17 shows that the network with elastic fibers follows a

straight line up until the failure showing some deviation

towards the end. It demonstrates again that the hardening

a)

b)

c) Fig 15. (a) The initial contact positions prior to loading. Total number of contacts at different levels of fracture, represented by the contour legend, where 0 and 1 corresponds to reaching fracture and separation distances respectively: (b) weak bonds; (c) strong bonds.

a)

b) Fig 16. Strain field in the network with (a) weak bonds prior to failure at a global strain of 0.7%, (b) no debonding at a global strain of 0.7%.

behavior of the network is mainly controlled by the fibers

and not by the bonds. Interestingly, the strength of these two

networks was almost identical. Assigning a linear elastic material model for fibers resulted

in almost 20% fewer fiber-fiber bonds completely separated

prior to network rupture compared to the case when a plastic

behavior of fibers was assumed, Fig 18.

This can be explained by the fact that a network consisting

of elastic fibers sustains sharper strain variations as the

PAPER PHYSICS

324 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

energy cannot dissipate in highly strained areas by other

Fig 17. Stress-strain curves for different fiber material models utilized on fiber level in 10 x 4 mm2, 27 g/m2 network: (a) bilinear isotropic hardening plasticity; (b) linear elastic fiber material model.

Fig 18. Total amount of fractured (solid line) and separated (dashed line) bonds for networks with different fiber material models: (a) bilinear isotropic hardening plasticity; (b) linear elastic fiber material model.

means than bond failure. The strain variations can be

observed in Fig 19a, in which we plot the strain fields at the

moment of failure normalized with the global strain to

failure recorded at this point.

It is also reflected in the map of the failed bonds, Fig 20,

which shows more clustered concentrations of bond failures

in the elastic fiber networks.

Note on the effect of friction

Numerical experiments showed that friction coefficient

varied in a reasonable range had virtually no effect on either

stress-strain curve or strength. The observed frictional forces

were two orders of magnitude lower than the forces

developed in the bonds and thus could not significantly

contribute to the response of a dry network.

a)

b) Fig 19.Normalized strain in the network with: (a) linear elastic; (b) bilinear isotropic hardening plasticity fiber material model at network fracture.

a)

b) Fig 20. Total number of contacts at different levels of fracture, represented by the contour legend, where 0 and 1 corresponds to reaching fracture and separation distances respectively: (a) linear elastic; (b) bilinear isotropic hardening plasticity fiber material model.

The role of unconventional contact parameters

As mentioned earlier, the bond strength is one of three

parameters characterizing a bond failure under the adopted

bilinear cohesive zone model. Measuring the two remaining

properties, namely, the compliance of bond regions and the

work of separation have not been possible with reported

techniques. During experiments, the fiber-fiber bonds were

not isolated, and the measured force encompassed the

response from both the bonds and from global deformation

of the fibers.

We look at the effect of the bond compliance and the work

of separation on the stress-strain curve with the help of

modeling on a single realization of the network.

PAPER PHYSICS

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 325

Effect of compliance of contact regions

Beam elements that we use to represent the fibers have no

through-thickness normal strains, which makes their

cross-section rigid against the load in the normal direction.

In reality, the fibers deform locally due to reciprocal forces

in the bond regions. We express the compliance of the bond

regions through the contact stiffness, which is usually a

numerical parameter in a penalty-based contact algorithm. In

our case, when the cross-sections of the fibers are rigid

against point-wise loads, the contact stiffness alone is

appropriate enough to represent the compliance of the

bonding regions. Considering its influence in the frames of

the cohesive zone model, it is nothing but the slope of the

elastic region in Fig 3. Consequently, with a given bond

strength, the bond with a lower stiffness can accumulate

greater critical fracture energy (the area under the curve up

to the fracture distance).

We varied the contact stiffness in the normal and tangent

directions according to Table 4.

Stress-strain curves plotted for these cases (Fig 21) show

that the bond compliance has a very limited effect on the

elastic part of the curve. Decreasing the bond stiffness in

both directions by a factor of two did not change the initial

slope of the curve simply because the amount of elastic

energy stored in the bonds is relatively small. Having softer

bond regions delays, however, the point of failure. On the

contrary, the stiffer the bond regions the lower the strength,

which together with the results on the impact of bond

strength means that the critical fraction energy accounting

for both the strength and compliance is a better measure for

relating the bonds and network strength properties.

Effect of work of separation

The work of separation is the amount of energy needed to

separate completely the bond from the point of fracture. In

practice this means creating a delay between failure and full

separation in a displacement-controlled test of individual

bonds. We varied the separation distance, Fig 22, according

to Table 5 leaving bond strength and contact stiffness intact.

The work of separation was set to be 15 percent of the

fracture energy in the reference and was changed to 5 and 25

percents in two selected cases.

Fig 22 shows that the work of separation varied in a

reasonable range has an expected but rather limited effect on

the strength of the fiber network. Increasing the work of

separation from 5 to 25 percents of the critical fracture

energy increased the strength by about 12 percent.

Effect of the number of bonds

The number of bonds in the network can be related to a

commonly used term “the level of interfiber bonding”. The

latter is usually changed by wet pressing or beating. In

numerical simulations, we reduced the number of bonds by

randomly removing contacts all over the network.

Along with the stress-strain curves, we computed the

efficiency factors introduced by Seth and Page 1983. The

efficiency factor Ф is the ratio of the initial Young’s

modulus (for all bonds) to the current Young’s modulus for

Fig 21. Stress-strain curves for different bonding compliance in 10 x 4 mm2, 27 g/m2 network: (a) control compliance; (b) softer bond regions; (c) stiffer bond regions.

Fig 22. Stress-strain curves for different separation distance the simulated tensile test on 10 x 4 mm2, 27 g/m2 network at normal 25.5 mN and tangent 4.2 mN bond strength kept constant: (a) reference case; (b) minimized; (c) maximized separation distance (see Table 5).

Table 4. Varying contact stiffness (CS) in the numerical tests.

Contact stiffness (CS)

Normal CS [GPa]

Tangent CS [GPa]

Soft bonds 20 4

Control 40 8

Stiff bonds 80 16

Table 5 Effect of varying separation distances (SD) keeping bond strength constant (25.5 mN in the normal and 4.2 mN in the tangent directions).

Separation distance (SD) Normal SD

[μm] Tangent SD

[μm]

a (reference) 0.73 0.60

b 0.67 0.55

c 0.80 0.66

PAPER PHYSICS

326 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012

a)

b) Fig 23. (a) Stress-strain curves for different number of fiber-fiber bonds in the simulated tensile test on 10 x 4 mm2, 27 g/m2 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

PAPER PHYSICS

Nordic Pulp and Paper Research Journal Vol 27 no.2/2012 327

fibers. Plasticity on the fiber level reduced the strain

variations since the energy dissipated through plastic

deformations in the areas of high strain.

The influence of the bond strength was significant. A factor

of 2-3 in bond strength, which is relatively low in the view of

the scatter encountered throughout the literature, changed

the strength of the network dramatically. At the same time, a

"non-traditional" bonding parameter, namely, the

compliance of the bond regions showed a comparably strong

effect on the strength as the bond strength. More compliant

bond regions increase the strength by accumulating more

energy prior to failure. This suggests that the critical fracture

energy, which account for both the strength and compliance

of the bonds, is a better measure for relating bond and

network strength. The work of separation (another

unconventional bonding property) showed a relatively small

impact on the stress-strain curve and can be discarded in

practical applications.

Decreasing the number of bonds in the network by 50%,

did not change the elastic stiffness significantly but

decreased the strength. It also increased stress variation

inside the network as well as the mean axial stress in the

load-bearing fiber segments.

Acknowledgement

The authors appreciate WoodWisdom-NET and BiMaC Innovation together with their industrial partners for the financial support.

Literature

Alava, M. and Niskanen, K. (2006): The physics of paper, Reports on Progress in Physics 69(3), 669-723.

Aulin, C., Gällstedt, M. and Lindström, T. (2010): Oxygen and oil barrier properties of microfibrillated cellulose films and coatings, Cellulose 17(3), 559-574.

Batchelor, W. (2010): Application of an analitical method to calculate the load distribution along a fibre in a loaded fibre network, Appita Journal 63(4), 287-293.

Gradin, P., Graham, D., Nygård, P. and Vallen, H. (2008): The Use of Acoustic Emission Monitoring to Rank Paper Materials with Respect to Their Fracture Toughness, Experimental Mechanics 48(1), 133-137.

Groom, L., Mott, L. and Shaler, S. (2002): Mechanical properties of individual southern pine fibres. Part I. Determination and variability of stress-strain curves with respect to tree height and juvenility, Wood and Fibre Science 34(1), 14-27.

Henriksson, M., Berglund, L. A., Isaksson, P., Lindström, T. and Nishino, T. (2008): Cellulose Nanopaper Structures of High Toughness, Biomacromolecules 9(6), 1579-1585.

Heyden, S. (2000): Network Modelling for the Evaluation of Mechanical Properties of Cellulose Fluff, Lund University, Lund.

Hägglund, R. and Isaksson, P. (2008): On the coupling between macroscopic material degradation and interfiber bond fracture in an idealized fiber network, International journal of solids and structures 45, 868-878.

Joshi, K., Batchelor, W., Parker, I., Nguyen, L. (2007): A new method for shear bond strength measurement, International Paper Physics conference, Queensland, Australia, pp. 7-13.

Kulachenko, A. and Uesaka, T. (2010): Simulation of Wet Fiber Network Deformation, Progress in Paper Physics, Montreal, Canada.

Lindström, T., Wågberg, L., Larsson, T. (2005): On the nature Advances in Paper Science and Technology, Cambridge, The Pulp and Paper Fundamental Research Society, pp. 457-562.

Niskanen, K. (1998): Paper Physics, Fapet Oy.

Niskanen, K. J., Alava, M.J., Seppäla, E.T., Åström, J. (1999): Fracture Energy in Fibre and Bond Failure, Journal of Pulp and Paper Scince 25(5), 167-169.

Page, D. H. and El-Hosseiny, F. (1983): The mechanical properties of single wood pulp fibers. Part IV. Fibril angle and the shape of the stress-strain curve, Journal of Pulp and Paper 9(4), 99-100.

Retulainen, E., Ebeling, K. (1993): Fibre-fibre bonding and ways of characterizing bond strength, Appita Journal 46(4), 282-288.

Räisänen, V. I., Alava, M. J., Nieminen, R. M. and Niskanen, K. J. (1996): Elastic-plastic behaviour in fibre networks, Nordic Pulp and Paper Research Journal 11(4), 243-248.

Seth, R. S. and Page, D. H. (1983): The Stress Strain Curve of Paper, The Role of Fundamental Research in Paper Making, Mechanical Engineering Publication, London, pp. 421-452.

Zavarise, G. and Wriggers, P. (2000): Contact with friction between beams in 3-D space, International Journal for Numerical Methods in Engineering 49, 977-1006.

PAPER PHYSICS

328 Nordic Pulp and Paper Research Journal Vol 27 no.2/2012