Use of the efficiency factor to account for previous straining on the tensile behavior of paperDouglas W Coffin

KEYWORDS: Tensile Properties, Efficiency Factor,

Viscoelasticity, Plasticity, Strain Hardening

SUMMARY: The presented results suggest that the

concept of efficiency factor previously used to

demonstrate that changes in inter-fiber bonding in paper

do not change the shape of the stress-strain curve can be

extended to describe the changes that are observed in the

tensile response of paper subjected to previous straining.

It is found that the pre-yielding response for samples that

have fully recovered from previous straining scales with

changes in maximum tangent modulus. This deformation

is mainly recoverable. When the scaling holds, one can

extract a reasonable approximation of the initial

recoverable deformation, which is separate from the

plastic deformation. In essence, the efficiency factor acts

as a stress magnification factor that easily can be

incorporated into a constitutive equation. Tracking the

change in efficiency factor with straining allows one to

account for the loss of observed compliance for the entire

range of recoverable deformation.

ADDRESS OF THE AUTHOR: Douglas W. Coffin

(coffindw@muohio.edu), Department of Chemical and

Paper Engineering, Miami University, Oxford, OH

45056, USA

Corresponding author: Douglas W. Coffin

The tensile response of paper manifests from a complex

interaction of materials and structure on size-scales

ranging from molecular, cell wall, fiber, and network.

Still, the response is repeatable and systematic enough

that continuum-based constitutive modeling can be

successful at modeling the response. Many different

approaches to modeling the constitutive behavior of paper

have been applied including for example: hyperelasticity

(Johnson 1986), damage accumulation, (Isaksson et al.

2006), elastic-plastic behavior (Mäkelä and S. Östlund

2003), viscoelasticity (Lif, Fellers 1999), and nonlinear

viscoelastic-plastic behavior (Coffin, 2009). Although

these models can be fit to data and can be useful for

modeling specific situations, they cannot be used for

general loading cases. In addition, a model that only

considers limited contributing factors, will over-attribute

observed behavior to the accounted for feature. For

example, if one considers only damage it is attractive to

account for all decreases in tangent modulus with damage

accumulated in the sheet. On the other hand, a

viscoplastic model will attribute the drop in tangent

modulus entirely to relaxation and yielding, while

ignoring any contribution due to damage. Only after

accounting for all the main contributing factors, will a

model be adequate. The aim of the present work is to

determine how to account for both loss of properties

attributed to damage and strain hardening due to

structural alignment or activation while maintaining the

essential time-dependent and yielding aspects that are

inherent in the cellulosic materials from which paper is

made.

Craven (1962) studied work hardening in paper, which

he defined as an increase in the measured slope of the

initial de-straining part of a stress-strain cycle with

increased peak strain. In general, Craven (1962) found a

significant increase in this slope with increased

magnitude of strain. Skowronski and Robertson (1985)

also observed the increase in de-loading modulus with

strain level. In addition, they noted a decrease in initial

modulus for samples loaded to a previous strain and then

given time to recover under no load. Skowronski and

Robertson (1985) suggest that there are two competing

effects, (1) structural or material failure causing the

decrease in modulus and (2) activation of new material

with strain level. Skowronski and Robertson (1986)

further studied the recovery process for samples

previously loaded and allowed to recover under no

applied load. They again noted the decrease in initial

modulus with previous straining and the increase in local

modulus with level of straining. Whereas in the previous

paper they reported that little additional plastic strain

developed with stress-relaxation, in the latter paper they

reported that stress-relaxation did cause significant

increase in plastic strain when one allowed enough time

for the relaxation.

Isaksson et al. (2005) observed changes in modulus with

straining and correlated it with acoustic emissions from

the sheet. The acoustic emissions suggest damage is

occurring within the material and it was correlated with

damage observed with microscopy. Much of the damage

occurred as the sample approached the maximum load

and was especially increased post peak-load straining.

Tryding (1996) found that for straining beyond the post-

peak stress, very large decreases in modulus occurred.

The significant damage occurring near and after peak

stress is most likely dominated by the localized

progression to failure at some weak point in the system.

The damage that occurs prior to this is more likely

distributed through out the entire structure and an

inherent feature that changes the observed tensile

response.

Associating these observations of damage with inter-

and intra- fiber mechanisms has been a long-standing

debate. Ebeling (1973) summarized the early work and

added new information based on his measurements of

heat flow into and out of the sample during tensile load

cycles. He concludes that while inter-fiber bond breaking

occurs during straining, it is not required to explain the

plastic flow. Seth and Page (1981) showed that

differences in levels of bonding did not change the shape

of the stress-strain curve. They introduced the concept of

an efficiency factor, = Ei/E0, where Ei is the elastic

modulus of a sheet with low bonding and E0 is the elastic

modulus of a well-bonded sheet. Decreased bonding may

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cause the efficiency factor to decrease from unity and will

lower strength and stretch by moving the failure point

along the stress-strain curve. For sheets with insufficient

bonding for full activation of the network, the efficiency

factor is less than one and the observed stress-strain curve

is only a scaled version of the well-bonded sheet except

that the failure point is also moved to lower strain. Coffin

(2005) and DeMaio and Patterson (2005) showed that

this same efficiency factor could be used to explain the

changes in the measured creep response due to changes in

bonding created during papermaking. The implication of

this observation is that the factor 1/ acts like a stress

magnification factor to explain the role of bonding on the

development of the stress-strain curve.

It is well established that there are changes in structure

that occur during tensile straining. New surfaces are

created: material can re-align. Changes in the structure

can result in decreased initial elastic modulus, while at

the same time increasing the yield stress. Considering

that the degree of bonding in papermaking is associated

with modulus and strength development, the increase in

surface area and loss of modulus during straining should

be considered as inter-fiber interactions. The increased

yield and plastic deformation are likely to be more

associated with intra-fiber actions.

The influence of papermaking process changes and

previous straining on the development and subsequent

changes in material properties should be consistent. An

adequate model will have to account for multiple features

to give a more representative prediction. The following

results and discussion try to provide some insights on

how this reconciliation can be achieved by extending the

concept that the efficiency factor can decrease with

straining.

Materials and Methods Testing was conducted with the papers specified in Table

1. The tensile results in Table 1 were conducted using an

Instron 3344 in an environment held constant at 50% RH

and 22°C. Specimens had a gage length of 178 mm and a

width of 25.4 mm. The samples were clamped with line

clamps, and the rate of loading was 25.4 mm/min. The

standard deviation of the measurements is given in

parenthesis. The seventh column gives the ratio of the

tensile strength index to the tensile stiffness index. The

two-ply paperboard is a commercially laminated product

made from two-side coated bleached paperboard. The

hardwood market pulp is in the form of a dried

fiberboard.

All specialized tensile tests sections were conducted on

the Instron 3344 universal testing machine in a lab

conditioned to 50% RH and 22 °C. These specimens were

clamped using flat clamps with serrated edges. Masking

tape was used as tabs to reinforce the paper under the

clamps. The rate of displacement was 10 mm/min.

Dimensions of the tensile specimens varied with widths

from 15.0 mm to 76.2 mm and gage lengths from 100

mm to 535 mm. All the results presented here are for

cross direction, CD, testing. Preliminary results showed

that similar behavior as reported here for CD is observed

in MD, but to a much lesser extent.

Specimens were cut using a manual guillotine cutter.

Preliminary tests were conducted to determine the

difference between using the guillotine cutter and a

manual rotary trimmer and no significant variations were

found.

Several cases of tensile testing were completed as listed

below:

1) Loading to failure at a constant rate of elongation

2) Loading to a specified strain followed by cyclic

unloading-loading between specified limits.

3) Loading to a specified strain followed by rapid

unloading.

4) Loading to a specified elongation followed by a

period of stress relaxation.

Loading and unloading was conducted at constant rates

of elongation. Some samples were subjected to multiple

loading and unloading events. After any loading event,

the sample was allowed to recover under no load for

extended periods (up to 7 days). Examples of stress-

strain curves illustrating the four loading cases are

provided in Fig 1.

For the stress-strain curves given in the results section,

the stress was normalized to the tensile stiffness ratios

given in Table 1 and expressed as a percentage. The

strain is also shown as a percentage. Thus, the maximum

slope of the initial loading curve is equal one. The

deviation from linear-elastic behavior is observed in

deviation of the curve from the one-to one proportional

line.” All the samples tested have significant deviation

from a linear-elastic response

Results Fig 1 illustrates several important aspects of how the

response of paper changes with previous loadings. First,

note that that although there is not a significant change in

stretch with various loading cycles, samples that undergo

many cycles of loading tend to have lower stretch (Loss

Table 1. Properties of tested papers (standard deviation in parentheses).

Paper Grammage g/m2

Density kg/m3

MD/CD ratio of Tensile Strength Index

CD Tensile Stiffness Index MNm/kg

CD Tensile Strength Index kNm/kg

Scaled Tensile Strength, %

CD Stretch

2-Ply paperboard 300 900 2.4 3.80 (.1) 31.1 (0.6) 0.82 6.2 (0.4) linerboard 168 616 1.7 3.74 (.4) 43 (3.5) 1.15 4.3 (0.6) Copy paper 78 746 2.2 2.95 (0.1) 25.5 (0.5) 0.86 3.9 (0.3) Bond paper 75 750 2.1 3.40 (0.1) 32.9 (0.7) 0.97 6.1 (0.4) Newsprint 46 625 3.8 1.74 (0.2) 14.4 (0.4) 0.83 2.3 (0.2) Hardwood market pulp

990 990 1.2

3.60 (0.1) 24.4 (0.3) 0.68 3.8 (0.2)

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

Fig 1. Example of four types of tensile testing completed in study for 2-ply paperboard.

of stretch observed in Fig 3). Second, the slope of the

recovered stress-strain curve (case 3) is lower than the

original curve for a large portion of the pre-yielding

curve. Third, the relaxation (case 4) causes additional

plastic deformation, implying time dependence on plastic

deformation. Fourth, the slope of the unloading and re-

loading (case 2) tend to be higher than the initial loading

from a recovered state. Fig 2 displays the slope or

tangent modulus for the stress-strain curves provided in

Fig 1. Note, that the maximum initial slope decreases

when the sample is reloaded after recovery. The peak

slope for immediate re-load cycle reaches higher limits,

and the initial unloading slope is higher than the re-

loading slope.

Four regimes of deformation appear after study of the

case 4 line (blue dash-dot line) in Figs 1 and 2.

(I) Initial activation of the network where the slope

increases rapidly to a peak,

(II) Recoverable stress-relaxation occurs where the

slope decays towards a middle plateau,

(III) Transition to additional plastic yielding where the

slope begins to drop rapidly

(IV) Plastic yielding dominates where the slope tends

toward a lower plateau.

This four–regime behavior was observed for many

different papers and loading schemes. For the results of

Fig 2, it appears that for the recoverable relaxation the

slope relaxes to about a third of its peak value. The

transition from elastic dominated behavior to plastic

dominated behavior gets shorter as the yield point

increases.

Fig 3 provides the stress-strain curves for a specimen

subjected to multiple events of case 2 loadings. After

loading to a specified strain, the sample was cycled

between the maximum strain and a lower load limit of

1 N for three cycles. Then recovery at zero load was

allowed before loading to the next event of strain. Nine

loading and unloading events were conducted before the

sample failed. Fig 3 also provides a loading to failure

Fig 2. Specific tangent modulus for stress-strain curves shown in Fig 1.

curve, case 1, for a similar sample. Note in loading event

6, a larger strain increment was taken. The strain at

failure for this sample is much lower than that of the

straight loading to failure. Presumably, this is due to

fatigue from the multiple loadings to which it was

subjected.

The maximum tangent modulus continues to decrease

for each subsequent reloading after recovery. Thus, an

efficiency factor, , can be defined for each re-loading

curve as the ratio of maximum tangent modulus to initial

maximum tangent modulus. Fig 4 shows the shifted and

scaled re-loading curves corresponding to Fig 3. Curve 1

is the same in Figs. 3 and 4. The other curves are shifted

by the plastic strain, taken as the amount of strain

required to shift the curves to curve one, and then scaled

by the inverse of the efficiency factor. In other words,

curves 2 to 9 in Fig 3 are divided by the corresponding

efficiency factor, The efficiency and the plastic strain

are given in the table inserted in Fig 4. The convergence

of the curves is good. The last loading event nine appears

to be stiffer than seven, and eight, but when one considers

the fact that the modulus has dropped to 68% of its

original value, the agreement is quite impressive. In fact,

when one looks at the scaling using the third re-loading

curves previously shown in Fig 3, the superposition is

quite good. The superimposed third reloading is given in

Fig 5, where the same scaling factors from Fig 4 are used

but shifted up one curve because the additional plastic

deformation and loss of efficiency occurred with the first

loading. For these third reloading curves, the strain is

fully recoverable.

The results in Fig 4 show that the scaling factor

remained one for strains up to 0.7% and the subsequent

scaling allowed the superposition of the curves to

increase to about 1.5% strain. This new level of

superposition would work for stress levels up to the

tensile strength, which has a stress to modulus ratio of

0.82.

The shifting by the plastic strain and the scaling by 1/,

appears to hold for all the papers tested. Fig 6 provides

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Fig 3. Stress-strain curves for a 2-ply paperboard subjected to nine load cycles.

Fig 4. Superposition of reloading from a recovered state of subsequent stress-strain curves using the efficiency factor.

shifted and scaled curves for tensile tests using the other

papers listed in Table 1. For the bond paper, Fig 6a, nine

events of loading case 3, rapid unloading, were

conducted. These curves appear to scale well. Figs 6b-e

show examples of multiple loading events of loading case

2, where only one cycle of loading and unloading was

completed. Each subsequent loading event was to a

higher strain except curve 3 in Fig 6c, where the loading

was to the same maximum strain of curve 2. It appears

that scaling the reloading curves with the factor 1/

provides a method to account for the loss of modulus as

the sample isstrained. Obviously, the initial linear pro-

portional part of the stress-strain curve must superimpose,

Fig 5. Third reloading stress-strain curves from Fig 3 shifted and superimposed with the efficiency factor (Note x-axis has range of 1.6% strain compared to 3% strain for Fig 4).

but Figs 3-6 show that the superposition can work for the

entire re-loading curves. A comparison of Figs 4 and 5

reveal that the relaxed loading curve is much more

compliant than the immediate re-loading curve. The

difference is due to relaxation and emphasizes that time-

dependence cannot be ignored.

Fig 7 provides a graph of efficiency factor versus plastic

strain for the papers shown in Figs. 4 and 6. Although,

there is not a universal trend, for each paper there appears

to be a systematic drop in efficiency with plastic

straining. The newsprint is the most sensitive and the

copy and bond paper are the least sensitive. The lines

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

(a)

(b) (c)

(d) (e)

Fig 6. Superimposed re-loading curves for various papers.

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Fig 7. Efficiency factor versus plastic strain for samples shown in Figs 4 and 6.

Fig 8. Third unloading curves shifted along strain axis and scaled by efficiency factor (third loading curve for event 8 provided for reference.)

Fig 9. Increase in modulus with applied strain.

shown in Fig 7 are included only to suggest that a trend

exists.

After investigating the unloading curves, it was found

that they do not superimpose with the efficiency factors

given in Figs. 4 and 5. This is because the paper has a

stiffer response at higher levels of straining. When the

sample is unloaded from a higher strain the initial slope is

higher. As the load approaches zero load level

components of the sheet are put into compression. Fig 8

shows that the unloading curves do form an envelope that

is comparable to the re-loading curves. The unloading

curves were inverted by taking the peak stress as the

origin. The curves were scaled by the same efficiency

factors given in Fig 5. For the larger peak strains (curves

7 and 8), the initial unloading has a significantly stiffer

response.

Fig 9 illustrates the increase in stiffness that occurs with

loading. A sample of linerboard that was previously

strained to 1.5% was strained to 3.5% followed

immediately by one minute of relaxation and three and

one half cycles of saw tooth strain with a magnitude of

0.19% strain (curve 1 in Fig 9). Then the sample was

allowed time for recovery. Eight more events of loading

to a specified strain followed by 1 minute of relaxation,

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

and the 0.19% amplitude strain cycles were completed.

After the cycling and unloading, a recovery period was

given (curves 2, 3, and 5-8 in Fig 9). Fig 9 shows the

modulus normalized to the initial modulus for these

events. At the end of event 4, the sample was not

unloaded, and one observes that the initial slope of curve

5 matches that of curve 4. Event 10 also occurred after

holding the sample at the end of event 9. One sees that

the slope rapidly approaches that of the slope in the

plastic flow regime. Fig 9 shows that for the 2% strain

reloading range, the stiffness increases about 20%. Event

2 in Fig 9 shows that at the peak slope, the network is just

being fully activated and the loading portion of the cycle

does not experience elevated stiffness, but the unloading

portion does.

The curves provided in Fig 9, demonstrates how the

slope changes during the small-scale loading and this is

typical of a viscoelastic response. Curve 1 in Fig 9 also

shows the four regimes of deformation previously defined

above.

Discussion When previously strained samples are re-loaded, the yield

point tends to be much better defined and the deformation

below yield is mostly recoverable. When plastic yielding

occurs, one assumes that any increase in load is

accompanied by an increase in recoverable deformation.

Separating the recoverable deformation from the plastic

deformation can be achieved only if one knows the shape

of the recoverable curve. The use of the Seth and Page

(1981) efficiency factor, , allows one to account for the

loss of load carrying ability due to previous strain

loading. It also allows for an estimate of the original

recoverable deformation from the subsequent re-loadings

of recovered samples. The scaling of 1/ magnifies the

stress and allows for a systematic extrapolation to higher

stresses. One only needs to know the shape up to the

tensile strength of the sample. For example, one could

use the loading curves labeled 7 or 8 in Fig 4 as the shape

of the recoverable curve, which is needed up to a scaled

value of /E=0.82, which is the scaled failure stress. In a

model, one would take this as the shape of the

recoverable deformation curve and then modify it by

multiplying it by the efficiency factor as straining

progresses and efficiency is lost.

Given that significant relaxation influences the

recoverable response, the stress-strain curves could be

predicted from a viscoelastic model. Just scaling the

initial modulus by the efficiency factor would not suffice.

Coffin (2005, 2009) showed that if 1/ is considered a

load magnification factor the effect of initial bonding

level could be incorporated into a nonlinear viscoelastic

model for paper. If the constitutive equation for the paper

is written as a function of stress, strain, and their

derivatives,

F(, d/dt, d2/dt

2 …, , d/dt, d

2/dt2 )=0. [1] (1)

The scaling results presented here suggest that one

would simply modify it by dividing all stress terms by the

factor .

F(, d( )/dt, d2( )/dt

2 …, , d/dt, d

2/dt2 )=0 [2]

The efficiency factor would be a specified function of

the plastic strain. This approach provides a solid

connection to the previous observations of the effect of

initial bonding level on the stress strain curve given by

Seth and Page (1981) and the loss of bonding observed

during plastic strain accumulation, (Ebeling 1973) and

(Isaksson et al 2006). This new result would suggest that

lack or loss of bonding influences the tensile behavior in

the same manner regardless of whether it was initially

low bonding or a later loss of bonding. For strained

sheets, this loss of bonding can be considered as damage

to the sheet. This damage is not localized in one region

but dispersed though the entire sample. Linerboard

samples (583 mm x 75 mm) that were strained to failure

in tension were cut and tensile and compressive tests

were conducted in regions away from the failure. In

tension, the samples exhibited a yield stress at the level of

the previous failure stress and displayed more yielding

before failure. The compressive strength dropped 8%.

Clearly, damage was induced in the sheet. It also

suggests that while yielding and plastic behaviors are

more likely attributed to intra-fiber behavior, there is a

significant loss of inter-fiber interactions resulting in a

reduction of both elastic modulus and compressive

strength.

The data suggests that the efficiency factor changes

mainly when plastic deformation occurs. Thus, a

functional dependence of efficiency factor versus plastic

deformation can be established. The plot of efficiency

factor versus plastic strain for the papers tested (Fig 7),

gave multiple curves. Some papers appeared to follow

the same trend, but newsprint for example is much more

sensitive than the other papers. Presumably, this is

because the newsprint is a low-density sheet and made

from a mechanical pulp so it would be expected to be

more sensitive to bonding changes. The shape of the

curves does not appear to be linear, but rather after the

efficiency factor begins to drop the slope decreases as

plastic strain accumulates.

Even though tensile strains induce damage in the sheet,

strain hardening still occurs. This strain hardening can be

significant and must be accounted for in a model. It

would seem likely that this hardening is due to re-

alignment of stiff components with the direction of the

strain. When loads are removed, the material recovers

and the modulus drops. Since the third reloading curves

can be scaled, the effect of damage appears to influence

the degree of strain hardening uniformly. For unloading,

the strain hardening at high strains prevents the same

scaling as the loading curve. As the sample approaches

peak load and failure, there is additional observed

hardening, curve 9 in Fig 4, which prevents the scaling.

The strain hardening could be accounted for in a model

by prescribing the stiffness associated with the

recoverable deformation to be functions of recoverable

train level. It appears that a linear function would suffice.

Damage and strain hardening can be reconciled with

each other. One effect of straining on subsequent tensile

loading is that it induces damage and opens the structure

up. In the recovered state, the paper is then more

compliant requiring additional deformation for full

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activation of the network. At first, the loads are not

sufficient for plastic deformation of load-carrying

components. Once the load carrying components are

activated, some structural alignment will occur. Once

load levels are sufficiently high, yielding occurs and

material self-stiffening will occur. Along with this plastic

yielding, more damage accumulates, most likely in

regions where load is transferred between stiff elements.

Upon subsequent recovery and reloading, the load

transfer is less efficient and treated as a magnification of

actual stress.

Conclusions The effect of pre-straining on the tensile response of

paper can be accounted for by scaling the initial response

by an efficiency factor and allowing the network

efficiency to decrease with the accumulation of plastic

strain in the sheet. Subsequent loading curves from a

recovered state can be obtained by multiplying the initial

response by the efficiency factor. For recoverable

deformation, the scaling appears to be adequate for load

levels up to the tensile strength of the material. The

decrease in modulus with subsequent loading is not

negligible and a previously strained paper can possess

significantly more compliance in the recoverable

deformation zone. If the damage is not accounted for, too

much compliance may be associated with relaxation or

plastic flow.

Paper also exhibits strain hardening when under tension.

The hardening is inherently active during constant rate of

loading but is masked by relaxations. If hardening is not

accounted for, the elastic deformation or relaxation at

higher loads may be under predicted.

A significant portion of the load can be relaxed from the

sample, especially if enough time is provided for plastic

deformation, which is time dependent. Much of the

curvature of the recoverable deformation curve also

appears to be associated with relaxations. If the time

dependence was ignored too much emphasis would be

placed on either plastic flow or damage.

Thus, to have a more representative model for the

tensile response of paper, time dependence, plastic

yielding, damage evolution, and strain hardening should

all be accounted for. Time-dependence and yielding can

be adequately handled with viscoelastic-plastic models,

damage can be accounted for by tracking the efficiency

factor, and strain hardening will require tracking stiffness

as a function of some loading parameter.

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