Materials Sciences and Applicatio ns, 2010, 1, 317-322
doi:10.4236/msa.2010.16046 Published Online December 2010 (
Copyright © 2010 SciRes. MSA
Stress Relaxation of a Paper Sheet under Cyclic
Load: An Experimental and Theoretical Model
Sharon Kao-Walter1, Etienne Mfoumou1,2, Efraim Laksman3
1Dept. of Mech. Eng., School of Eng. Blekinge Institute of Technology, Karlskrona, Sweden; 2Dept. of Aero. and Mech. Eng., Uni-
versity of Arizona, Tucson, USA; 3Dept. of Math. And Sc., School of Eng. Blekinge Institute of Technology, Karlskrona, Sweden.
Received October 21st, 2010; revised November 12th, 2010; accepted November 19th, 2010.
Mechanical experiments have been performed to study the dynamic stress relaxation of a paper sheet material mainly
used in food packaging industry. The material was cyclically tensile-loaded with a strain range between 2.4% and 4%.
The time period for each cycle was 400 seconds. It was found that stress will decrease when the number of cycles in-
creases in the case of upper load and vice versa in the case of lower load. At the same time, the stress to strain curves
followed the same pattern as the one from the previous cycle. The stress relaxation behavior of each cycle has been
analyzed and the dynamic relaxation modulus was derived. An improved model is proposed to describe the dynamic
relaxation behavior of the paper sheet. This model shows a very good fit to the experimental results and trends of pre-
diction are observed. Furthermore, the physical description of this model and the variation by the cycles is discussed.
Keywords: Relaxation, Cyclical Load, Paper Sheet, Improved Relaxation Model
1. Introduction
Paper as a material is subjected to a complicated loading
pattern from its birth in the drying operation to the con-
verting operation and a possible collapse in an end-use
situation. In recent years, several works have been done
to study the mechanical properties of the paper sheet
[1-5]. Corresponding material models have been created
for paper or carton materials in [2-4] and [5]. In liquid
food packaging companies, it is also important to avoid
the swelling of paper-based liquid food packages within
their given storage time. Therefore it is important to
study the relaxation behavior of the paper sheet. Experi-
ments and characterization of the nonlinear viscosity of
some composites and paper fiber composites have been
performed in [6-9] where creep tests and Schapery’s
constitutive law have been used to determine the viscoe-
lastic nonlinearity parameters. Inherently, a more sophis-
ticated testing with regard to the time-dependent behav-
ior of the paper with constant deformation (stress relaxa-
tion) and at cyclic deformation (dynamic behavior),
might give a better understanding of the difference in the
performance of this type of material during a given time
scale. Therefore, our work addresses the stress relaxation
phenomenon under cyclic loading on the paper sheet
used in the food packaging industry.
Thin sheets having no bending stiffness are in general
complex types of material which do not always satisfy
the classical plane stress theory [1-9]. Their unusual
elastic properties behavior is the expression of this com-
plexity. Like in geo-materials [10-12], gels [13], granular
material [11], and even sometimes liquids [13], long-
time recovery relaxation phenomena and anomalous sof-
tening of the resonance frequency with strain are some
examples of the elastic properties of this class of material.
This was demonstrated on thin sheets having extremely
small bending stiffness for the first time by Mfoumou et
al. [14], and is the motivation of a further investigation
of the stress relaxation process on a paper sheet in this
Earlier works have been done to analyze relaxation
and reverse relaxation in synthetic fibers under cyclic
loading [15,16], as well as their potential analytical
models for predi c t i on of rel a x a t i on and reverse relaxati o n
and deformation-recovery process [17]. These works
pointed out that during the dynamic loading of fibers
initially having no bending stiffness there was a viscoe-
lastic effect, which was clear from the increased elonga-
tion following the number of load cycles and from the
hysteresis curve of load versus strain. Vangheluwe [15]
argued that the existence of viscoelastic effects during
Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model
dynamic loading influences the relaxation behavior af-
terwards, and that the existence of reverse relaxation
seems to be due to viscoelastic effects from the previous
dynamic loading of the specimen.
Traditionally, the relaxation behavior of different ma-
terials can be described by several different models as
listed in [18]. The viscoelastic properties of the material
under investigation have recently been illustrated by
Mfoumou [19]. The relaxation curves were fitted to a
series of exponential terms derived from a Maxwellian
model [20,21] which has a similar response to the Prony
series as discussed in [7]:
Yta bt
where Y(t) is the decaying parameter (force, stress or
apparent modulus), t the time and ai and bi constants
characteristic of the material. Based on this model, the
paper sheet investigated clearly exhibits a nonlinear vis-
coelastic behavior since both the number of terms and
the constants depended on the deformation history and
level. The number of terms reported was 2 involving 4
constants; this makes comparison of curves difficult be-
cause all the constants are independent. In this paper, an
improved model will be introduced, analyzed and dis-
cussed. Here, only the relaxation stress at the upper strain
of load cycles is considered in the analysis. We believe
that once the relaxation model and its relation to the cy-
cle are clear, the reversed relaxation can possibly be
modeled in a similar way.
2. Experimental Method and Results
2.1. Sample Geometry and Test Setup
The test specimens are rectangular strips, 250 mm long,
15 mm wide, with a thickness of 100 µm. The samples
were placed in a conditioned environment at 23˚C and an
atmospheric humidity of 40% during at least three days
prio r to the t ests. Th e MTS Tensile Test Machine used is
shown in Figure 1. The load on the sample is recorded
Figure 1. Experime ntal setup: The tensile test machine with
a specimen clamped longitudinally.
by a piezo-electric load cell mounted between the sample
and the crosshead of the machine. This load cell (with
maximum 2.5 kN) is used together with a pair of pneu-
matic clamps. The grip separation is set to the specimen
length. Each specimen is clamped at the upper and lower
ends to create a fixed-fixed boundary condition on a ten-
sile test machine.
2.2. Tensile Test Measurement
A (low-frequency) cyclic loading was applied to the
samples. This conditioning was made of repeated step
perturbations at two different strain levels. Each step
perturbation consisted of first moving the crosshead to 1
mm elongation at a speed of 1 mm/s, holding the cross-
head for 200 seconds, unloading the sample by moving
back the crosshead to 0.6 mm elongation at the same
speed as in the loading process, and again, holding for
200 seconds. The subsequent step perturbations, shown
in Figure 2, were performed with the extension between
0.6 mm and 1 mm. As the loading and unloading proc-
esses last for less than one second, we assume the phe-
nomenon can be considered as a cyclic deformation. The
Figure 2. Cyclic tensile loading strain and measured stress.
Copyright © 2010 SciRes. MSA
Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model 319
relaxation of the sample is therefore monitored and ana-
lyzed at constant strains corresponding to the higher
strain level.
2.3. Stress Relaxation Monitoring
The measurement of stress relaxation is usually made by
placing the specimen in series with a spring of sufficiently
great stiffness so that it undergoes negligible deformation
compared with the specimen. In our experimental setup,
the spring is the element of the load cell transducer, which
enables a direct measurement of stress. Figure 2 shows
the time-dependent variation of the strain and measured
stress for cycles between 2 and 6. Here, the first cycle has
been removed since there is often an intermittent behavior
which is not always repeatable.
3. Analysis and Discussion
3.1. Relaxation Modulus
Figure 3 shows the time-dependent relaxation modulus
of the above mentioned 5 cycles. Relaxations moduli
here are calculated by simply dividing the upper stress
value by the strain. Note that the horizontal axis is a
logarithm of time. Curves are moved so that each upper
strain is supposed to start at time zero. As can been seen,
the relaxation modulus is reduced as the loading time
increases during each load. The same pattern is repeated
for all the following load cycles.
More completely, the experiment time was extended to
one hour in Figure 4 for the purpose of getting a complete
picture of the shape of the relaxation curve for the paper
sheet studied. Irregularities are observed, indicating the
end of the relaxation process. Such fluctuating as shown in
Figure 4 has also been found in other materials and Kol-
seth et al. [22] described it as the presence of stress in
Figure 3. Relaxation modulus of a paper sheet at different
loading histories: Loading period of 400 s.
Figure 4. Relaxation modulus of paper sheet: Loading pe-
riod 1 hour.
homogeneities arising from insufficient annealing.
3.2. Analytical Modeling of the Observed
Figure 5(a) redraws the stress relaxation of 5 cycles
(marked with data) at the constant strain of the upper
level as shown in Figure 2. Again each curve is moved
so that the load cycle starts at time zero. Based on these
results, a mathematical description is discussed below.
An ideal mathematical representation of a physical
phenomenon is based on the following criteria:
The number of constants is kept at a minimum;
The constants and the equation components bear
meaningful physical information;
The equation is sensitive to physical changes in the
system but insensitive to arbitrary parameters;
The mathematical form of the equation is as simple
as possible.
To apply these conditions to our relaxation curves, an
improved model has been suggested as below:
() ()
=+ (2)
The super index i is only used to indicate the cycle.
We have
lim i
. So there is no need to
estimate by statistical methods. a
Thus, the number of parameters that need to be esti-
mated is 12i
, where i is the number of cycles. The
model is thus slightly more complex than the simplified
model, but it will be shown that the fit is sufficiently
better that it is worth the extra complexity. The method
of least squares is used for finding the parameters in Ta-
ble 1 and 0.05d823
As can be seen, d is small, and this means that the
Copyright © 2010 SciRes. MSA
Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model
Figure 5. (a) Stress relaxation curves of paper sheets at
successive loading histories together with the data from the
improved model; (b) The residuals (differences between
measurements and the model) of different cycles. The re-
siduals are very small, and appear rather independent.
model is not very stable for t < d. For example, td in-
creases relatively sharply in the interval 0 to d, as 0d = 0
while . The measurements in each cycle span
over 198.4 seconds, and rather than assuming that the
first measurement is made at t = 0, we have assumed that
it is made at t = 1.6, so that the last measurement is made
at t = 200. If one assumes that our measurements actu-
ally start at time 0, this would result in a large discrep-
ancy between the model and our measurements for small
values of t, and in particular for the first measurement.
Thus one could argue that the improved model for re-
laxation is actually
() ()
Table 1. Coefficients used for the least squares method.
Cycle i = 2 i = 3 i = 4 i = 5 i = 6
b(i) 2.819 2.659 2.612 2.593 2.556
c(i) 0.6741 0.5692 0.5399 0.5324 0.5081
b(i) + c(i)2.145 2.090 2.072 2.061 2.048
where τ is a parameter to be optimized, yielding a total
of 22i
parameters. If one prefers to see the model in
this way, then we simply set 1.6
= rather than opti-
mizing over τ.
Since the model is unstable around , to study the
internal irreversible changes, it m ay be more interesti ng to
use , than to use .
Thus b + c, which steadily decrease, may be an appro-
priate statistic for studying the internal irreversible changes.
() ()
As can be seen in Figure 5(a) where the solid line
shows the values calculated by Equation (3), the im-
proved model follows the measured values satisfactorily.
In Figure 5(b), we have gathered residuals (i. e. the dif-
ferences between values from measurements and the
improved model) from all cycles, and observe that the
residuals are, along the entire timeline, fairly evenly scat-
tered around zero. Cycle 2 clearly has the largest residu-
als, in particular around and . Consider-
ing the data in cycle 2, one can see that the measure-
ments between
and appear to lie on a
different curve than the other data points of cycle 2. Most
likely these measurements are flawed, causing the re-
siduals of cycle 2 to be larger than the residuals of the
other cycles. Also, the removal of some of these
“flawed” data points from cycle 2 would lower and
increase , so that parameter values from cycle 2
would not differ much from the parameter values of the
other cycles.
We also observe that for each cycle, th e residuals soon
after 0t
are among the largest residuals for that cycle,
which is no surprise as measurements soon after 0t
are less accurate than other measured results.
4. Further Discussion of the Improved
Relaxation Model
The empirical formula in Equation (2) can be suggested
to describe an experimentally measured relaxation law.
In order to make it more comparable to the standard
relaxation laws that are known, Equation (2) takes the
=+ or
= (4)
with, ,
and d
It means that the stress
approaches its equilibrium
value a
ac cording to th e tempo ral law that w ill reach
=+ (3)
Copyright © 2010 SciRes. MSA
Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model 321
one at . On the side, at t = 0, t→∞
Notice that this relaxation law in Equation (4) differs
significantly from the standard relaxation law (see
=+ or t
= (5)
which satisfies the ordinary differential equation:
dt T
=− (6)
As it is known,
is the “relaxation time”, a is the
“equilibrium stress” and is the initial stress.
To reconstruct the differential equation of type
Equation (6) by empirical relaxation law, Equation (4)
is differentiated to get:
dt t
Comparing this equation with Equation (6), we can
conclude that formally the governing equation is the
same. But the relaxation time is now time-dependent.
Now define:
Equation (7) can be rewritten by a similar way as Equa-
tion (6).
As discussed before, 1
is the characteristic time
that increases with an increase in “physical time” t. If
formally 1
= in Equation (8), the equation will reduce
to 11.
== st
Consequently, at 1
=, our empirical law transforms
to the usual one as in Equation (5) in which .
TT p
Correspondingly, the Equation (7) transforms to Equa-
tion (6).
We can come to two important conclusions as follows
from this consideration:
11Tt Tp=, because we know 1
. This
means that the relaxation of the paper sheet is much
slower than the usual relaxation process as soon as
Tt increases with time as can be seen by Equa-
tion (8). This means that the more time that has
passed since the beginning of the process, the slower
will the stress/tension of the paper decrease/relax.
The second form of governing equation can now be
derived which does not contain time t in the coefficients.
Rewriting Equation (4) as:
ln lnln
After differentiation on time t, we get:
ln ln
dt a
or lndY Y
= (11)
ln, ln
This logarithmic dependence on time indicates a slow-
ness of the relaxation process.
5. Conclusions
From the experimental tests and theoretical modeling it
can be concluded that:
The paper sheet clearly shows the relaxation be-
havior. This behavior cannot be described by the
well known Maxwell model since its relaxation is
much slower than the usual relaxation process in
materials like metals.
Paper relaxation, moreover, demonstrates the
presence of a wide spectrum of relaxation time.
The relaxation modulus is reduced by the loading
time and loading cycles with the same pattern.
This can be explained by the nonlinearity of the
stress and strain relation of the paper sheet.
An improved relaxation model:
shows very close correspondence with the experi-
mental results and its governing equation is similar
to that of the Maxwell model but with variable re-
laxation time.
This model can also describe the behavior of the
material under a wide range of material condition-
ing (loading levels) without becoming inconsis-
The main characteristic of the improved model is
that the parameters play a major role in the overall
response of the material. One of the important ad-
vantages is the possibility to anticipate the asymp-
Copyright © 2010 SciRes. MSA
Stress Relaxation of a Paper Sheet under Cyclic Load: An Experimental and Theoretical Model
Copyright © 2010 SciRes. MSA
totic limiting value at which the stress will tend at
infinite time.
6. Acknowledgements
The authors would like to thank Professor Oleg Rudenko
and Professor Claes Hedberg for valuable discussions
during the work. The Swedish Foundation for Knowl-
edge and Competence Development and Blekinge Insti-
tute of Technology in Sweden are acknowledged for
their financial support.
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