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Investigation on deformation
behavior of paper in Z-
JIAN CHEN, JANN NEUMANN,
Institute of Printing Science and Technology
Darmstadt University of Technology
Magdalenenstr. 2, 64289, Darmstadt
Paper, is a composite made up of fibers, moisture,
voids and chemical additives that are in the form of
discrete fibers cross-linked in a complex network.
The mechanical behavior of paper has a very close
relationship with many operations in the printing
production, such as printing, paper counting, folding,
creasing, calendaring, cutting, etc.
A number of studies have been made with the
objective of predicting the stress-strain behavior of
paper materials under the forces in MD and CD-
directions, but the research in ZD-direction is
limited. The purpose of this paper is to improve the
mathematical model proposed by Schaffrath and
extend the model to multiple sheets. Additionally,
based on image processing technique, a new
approach for measuring the actual contact area and
calculating the relationship between force and actual
contact area is presented.
In the model proposed by Schaffrath [1,2], who
divided paper body into three parts, developed the
models respectively and derived the force-
deformation relationship of paper materials by using
the Newton formula.
Also, Stenberg [3,4] published some articles
between 2001 and 2003, in which he summarized the
literature, developed a new device to measure the
stress-strain properties of paperboard in ZD-
direction, and built an elastic-plastic model for paper
2 THE MATHEMATICAL MODEL
PROPOSED BY SCHAFFRATH
The topography of paper surface could be
measured today very precisely with different
measurement methods, e. g., a NanoFocus μSCAN
system. For further processing, these physical data
have to be transferred in an analysis model.
Figure 1. Abstraction of the paper surface.
As mentioned above, Stenberg and Schaffrath
proposed mathematical models of paper materials in
ZD-direction. Compared with the model proposed
by Stenberg, the parameters needed in Schaffrath’s
model are much easier to obtain.
Figure 2. Schematic diagram of the model .
In this model, the paper body was divided into three
parts: two surface structures and one internal
structure, which can be described by using the
following units (Fig. 3).
Figure 3. Elementary units of paper structure .
After that Schaffrath built the models for the
surface and internal structures. According to
Figure 4. Paper structure and model under
The force-deformation relationship of surface and
internal structures could be easily obtained.
4( ) 4( )
()2 ( ) ( )
EAx Ex A x Ex A x
Ex R R L m
Rr Radius of curvature at the point (line)
where the fibers contact each other 
L Contact length of the fibers
RA Average value of surface roughness plus
standard deviation of caliper 
m, n Parameters used to determine the
of internal units
E(x1),E(x2) Elastic modulus of different structure
A0 Nominal contact area
x1, x2 Deformation of each structure
B0 Initial internal area to withstand load
In addition, the following equation has to be taken
123 (3)xx x x
() ( ) ()
Ex Ex Ex
There are two equations (Equ. 2 and Equ. 3) but
three unknown variables (x1, x2, x). The relationship
between x1(x2) and x can be calculated by using
Newton-Raphson method .
xi xi Fi
B and B
Then using a curve fitting method, the relational
expression between x1 and x can be obtained.
3 THEORETICAL CALCULATION
3.1 Further Discussion about One Sheet
The specimen used here is the normal copy paper
(A4, 80 g/m2, average thickness d=84.7 μm). The
curve fitting method used by Schaffrath is linear. A
more complicate curve can be obtained by using
more compression data and which can also be
described as quadratic or cubic equation (Fig. 5).
Figure 5. Relation expresses between x and x1.
In Equ. 2, x1 can be replaced with the relation
expresses showed in Fig. 5, which leads to the
function of the relationship between press force F
and total deformation x.
Figure 6. Pressure-deformation relationship based
on different curve fitting methods.
Fig. 6 shows the force-deformation relationship of
different curve fitting methods. The result of
quadratic method cannot be used for big deformation
and the force-deformation relationship between
linear and cubic methods is a little different.
3.2 Discussion about Multiple Sheets
Figure 7. The model of multiple sheets.
Supposing that the number of paper is p, the
relationship as follows
fx p px
Combining with Equ. 2, the force-deformation
relationship of multiple sheets can be calculated.
(a) Calculation result for small deformation.
00.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
2 sheets 4 sheets
(b) Calculation result for big deformation.
Figure 8. Calculation results for multiple sheets.
3.3 Relationship between Contact Area
Generally, the actual contact area of the surface is
much smaller than the nominal contact area because
of the roughness of the contact surfaces.
Figure 9. Description of nominal and actual contact
According to Schaffrath’s model , the surface
structure is characterized by pyramid units (Fig. 3).
The following relation express can be obtained
Ax x xActual contact area
Nominal contact area A h R
Figure 10. Pressure- contact area relationship.
Combined with the force-deformation relation
express, the relationship between pressure and
contact area is showed in Fig. 10.
4 EXPERIMENTAL VERIFICATION
4.1 One Sheet
The equipment used here is ZWICK Z050, which
can be utilized for strain, shear and bending tests
with different substrates and machine components
with high accuracy of the cross head speed (0.0005-
2000 mm/min), position repetition accuracy (± 2μm)
(a) Experimental setup.
(b) Experimental results.
Figure 11. Experimental setup and results.
As showed in Fig. 11(a), for one sheet, the force-
deformation results are quite different in different
positions because of the different densities, so the
average value (Fig. 11(b)) is used as the actual value.
The result of cubic curve fitting is better for this
sample (compared with Fig. 6).
4.2 Multiple Sheets
Figure 12. Experimental results of multiple sheets
With regard to multiple sheets, for small
deformation, the calculation results are acceptable,
but when the force is very big, some further
discussions are still needed (compared with Fig. 8).
4.3 Actual Contact Area
4.3.1 Experimental setup
A carbon paper was put above the test paper, and
then the load was imposed on the carbon paper.
When the force was removed, the ink of the carbon
00.05 0.1 0.15 0.2
1 sheet 2 sh eets 4 sheets
00.5 11.5 22.5 33.5 4
Contact area / Nominal area (%)
00.002 0.004 0.006 0.008 0.01 0.012
00.02 0.04 0.06 0.08 0.1 0.12
paper could be transferred on the contact area (Fig.
(a)The setup used to show the contact areas.
(b) The test results under five different forces.
Figure 13. The setup and test results of showing
4.3.2 Pictures enlarging and transferring
The image processing technique was used to
separate the contact area from the background. The
surface of the specimen was magnified 25 diameters
under binocular microscope. Then all pictures were
transferred into binary images.
Figure 14. Transfer the original picture into binary
A binary image has two possible values for each
pixel. Numerically, the two values are often 0 for
black, and 255 for white. The Otsu method is used
here to perform image thresholding.
Figure 15. A diagram of the Otsu method.
4.3.3 Calculating the contact area
In the examples of Fig. 16, the contact area was
calculated by using different threshold values. The
threshold values provided here are 0.5, 0.25 and
calculated by the Otsu method.
Figure 16. Examples of calculating contact area.
For different pictures, the Otsu method will
produce different threshold values. The average
threshold value was calculated and used to obtain the
whole black area.
Figure 17. Example of calculating contact area
(Average threshold value=0.4514, force=100N).
4.3.4 Results and analysis
Figure 18. Relationship between pressure and
The application shows that this experimental
method is very practical, it can be well used to
calculate the relationship between force and contact
area. When the pressure is 4 N/mm2, the ratio of
contact area and nominal area is around 45%, it is
much less than the value obtained in Fig. 10.
5 CONCLUSION AND DISCUSSION
Different curve fitting methods were discussed in
this paper, the quadratic method cannot be used to
calculate the force-deformation relationship and the
result of cubic method is better for this sample. For
small deformation, this model can be used for
multiple sheets, but for big deformation, some
further discussions are still needed. Additionally, a
new approach for measuring the actual contact area
Further studies are still needed, for example: (1) E-
modulus of paper material is a very special
parameter, the depth study in this aspect is still
necessary, (2) How to extend the model to multiple
sheets under big deformation can also be further
The authors gratefully acknowledge the financial
support from China Scholarship Council.
 H.-J. Schaffrath and L. Göttsching. The
behaviour of paper under compression in z-
direction. International paper physics
conference, Hawaii,Tappi (1991).
Conference Proceedings Reference
 H.-J. Schaffrath and L. Göttsching.
Modellierung der Kompression von Papier
in z-Richtung bei niedriger
Flächenpressung (in German). Das Papier,
Heft 7, S. 350-355 (1992).
 N. Stenberg. A model for the through-
thickness elastic-plastic behavior of paper.
International Journal of Solid and
Structures. 40, 7483-7498 ( 2003).
 N. Stenberg, C. Fellers and S. Östlund.
Measuring the stress–strain properties of
paperboard in the thickness direction.
Journal of Pulp and Paper Science. 27 (6),
 Introduction to Engineering and Statics.
 T. Kaulitz. Bilden von Schneidlagen unter
Ausnutzung des Nipinduzierten Effekts für
die Druckweiterverarbeitung (in German).
PhD thesis, TU Darmstadt (2009).
0 1 2 3 4
Contact area / Nominal area (%)
Calculation result (Fig.10)