Stress-strain curve of paper revisited - Innventia.com

PAPER PHYSICS 

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 60 s 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. 

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

PAPER PHYSICS 

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 mm 2 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 F bs is the 

bond strength expressed through force, K c is the contact 

stiffness which describes the compliance of the bonding 

regions, d f is the fracture distance, and d s 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, 

E 

E 

n 

cn 

Et 

+ = 1 

(1) 

E 

ct 

where E n and E t 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 E cn and E ct . When a bond fails, the 

contact between corresponding fibers is described with a 

frictional contact. This is probably the first threedimensional 

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/m 2 , 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. 

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

PAPER PHYSICS 

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. 

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

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] 

3.20±1.38 32.45±2.45 20 10 100 

Yield 

stress 

[MPa] 

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] 

25.5 4.2 0.73 0.60 40 8 

Tang 

CS 

[GPa] 

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 

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 mm 2 

samples with a basis weight up to 30 g/m 2 . 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 mm 2 networks have somewhat lower elastic 

modulus than 10x4 mm 2 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 mm 2 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 mm 2 , 

27 g/m 2 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/m 2 ). 

Fig 7. Experimental results from the tensile test. Effect of increase 

in sample width for laboratory handsheets (27 g/m 2 ). 

Fig 8. Experimental results from the tensile test. Effect of increased 

grammage for 10x4 mm 2 laboratory handsheets (27 g/m 2 ). 


PAPER PHYSICS 

a) 

a) 

b) 

Fig 9. Tensile test on 10 x 4 mm 2 samples, cut from laboratory 

handsheets (27 g/m 2 ). (a) Initial state prior to loading and loading 

direction. (b) Post-failure localization. 

b) 

c) 

d) 

Fig 10. Stress-strain curve from the tensile test on 10 x 4 mm 2 , 27 

g/m 2 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. 


e) 

Fig 11. Digital speckle photography results. Development of 

fracture during tensile test on 10 x 4 mm 2 , 27 g/m 2 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

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Fig 12. Comparison of the experimental and simulated results from 

the tensile test on 10 x 4 mm 2 , 27 g/m 2 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 Tangent 

[mN] 

[mN] 

Strong 42.5 7 

Control 25.5 4.2 

Weak 8.5 1.4 

BS 

Fig 13. Stress-strain curves for different bond strengths from the 

simulated tensile test on 10 x 4 mm 2 , 27 g/m 2 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 

damage were plotted. 

towards the end. It demonstrates again that the hardening 

a) 

b) 

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 


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

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energy cannot dissipate in highly strained areas by other 

contribute to the response of a dry network. 

a) 

Fig 17. Stress-strain curves for different fiber material models 

utilized on fiber level in 10 x 4 mm 2 , 27 g/m 2 network: (a) bilinear 

isotropic hardening plasticity; (b) linear elastic fiber material model. 

b) 

Fig 19.Normalized strain in the network with: (a) linear elastic; (b) 

bilinear isotropic hardening plasticity fiber material model at 

network fracture. 

a) 

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 

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. 


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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 mm 2 , 27 g/m 2 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 mm 2 , 27 g/m 2 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 Normal CS Tangent CS 

(CS) 

[GPa] 

[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 Tangent SD 

[μm] 

[μm] 

a (reference) 0.73 0.60 

b 0.67 0.55 

c 0.80 0.66 


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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 


PAPER PHYSICS 

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. 

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