Conference PaperPDF Available

Investigation on deformation behavior of paper in Z-direction

Investigation on deformation
behavior of paper in Z-
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
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 [1].
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 [1].
After that Schaffrath built the models for the
surface and internal structures. According to
Hooke’s Law
EF x
 
Figure 4. Paper structure and model under
compression [1]
The force-deformation relationship of surface and
internal structures could be easily obtained.
1101 303
() ()
4( ) 4( )
()2 ( ) ( )
EAx Ex A x Ex A x
Ex R R L m
EAx n
Fx x
  
 
  
 
Rr Radius of curvature at the point (line)
where the fibers contact each other [1]
L Contact length of the fibers
RA Average value of surface roughness plus
standard deviation of caliper [1]
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
into account
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 [1].
  
01222 2
2 2
(1) ()
xFx x
Ad R
xi xi Fi
 
 
 
22 22
mRL mL
B and B
Then using a curve fitting method, the relational
expression between x1 and x can be obtained.
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
1 (7)pxpxx
 
   
01222 2
2 2
fx p px
Ad R
 
    
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
Deformation (mm)
Pressure (N/mm
2 sheets 4 sheets
1 sheet
(b) Calculation result for big deformation.
Figure 8. Calculation results for multiple sheets.
3.3 Relationship between Contact Area
and Force
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
areas [5].
According to Schaffrath’s model [1], the surface
structure is characterized by pyramid units (Fig. 3).
The following relation express can be obtained
11 1
() (9)
2 A
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.
4.1 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
Deformation (mm)
Pressure (N/mm
1 sheet 2 sh eets 4 sheets
00.5 11.5 22.5 33.5 4
Pressure (N/mm
Contact area / Nominal area (%)
00.002 0.004 0.006 0.008 0.01 0.012
Deformation (mm)
Pressure (N/mm
average value
00.02 0.04 0.06 0.08 0.1 0.12
Deformation (mm)
Pressure (N/mm
2 sheets
1 sheet
4 sheets
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
contact areas
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
contact area.
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.
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
was presented.
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.
[1] 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
[2] 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).
Journal Reference
[3] N. Stenberg. A model for the through-
thickness elastic-plastic behavior of paper.
International Journal of Solid and
Structures. 40, 7483-7498 ( 2003).
Journal Reference
[4] N. Stenberg, C. Fellers and S. Östlund.
Measuring the stressstrain properties of
paperboard in the thickness direction.
Journal of Pulp and Paper Science. 27 (6),
213221 (2001).
Journal Reference
[5] Introduction to Engineering and Statics.
Internet Reference
[6] T. Kaulitz. Bilden von Schneidlagen unter
Ausnutzung des Nipinduzierten Effekts für
die Druckweiterverarbeitung (in German).
PhD thesis, TU Darmstadt (2009).
Thesis Reference
0 1 2 3 4
Pressure (N/mm
Contact area / Nominal area (%)
Experimental result
Calculation result (Fig.10)
... The influence of surface roughness was also discussed in some papers, for example, the paper surface topography under compression was studied by Teleman, et al. (2004). According to the surface topography, the paper body was considered as being composed of two rough surfaces and an internal structure (Schaffrath and Göttsching, 1991;Chen, Neumann and Dörsam, 2014), the force-deformation relationship of paper was derived by using the Newton formula. ...
Full-text available
The mechanical behavior of paper materials under compression in the out-of-plane direction is highly nonlinear. If the influence of the surface topography is not taken into account, the stress-strain curve of paper materials in the loading process is a typical example of materials with J-shaped compressive curves. When compression is released, the stress-strain curve in the unloading process is also nonlinear. The main purpose of this paper is to establish a suitable mathematical model and actualize the description of the compression curve for paper and paper stacks. The loading and unloading nonlinearities of paper stress-strain relations can be approximated by using different equations. In this paper, the loading curve of paper is calculated by using the sextic polynomial equation and the unloading curve is described by using the modified exponential function. All the used coefficients for determining the functions are expressed as the functions of the stress at the start point of unloading. The compressive behavior of paper under some given forces are also calculated by using the identified equation and verified by means of the experimental data. For multiple sheets, it is assumed that when the force is the same, the deformation of the paper stack is directly proportional to the number of sheets. Based on this assumption, the force-deformation relation of the paper stack is derived. The comparative analysis of the experimental results demonstrates the effectiveness of the description model.
... The method can be summarized as the following three steps [6]:  Enlarging and transferring the pictures: the surface of the specimen is magnified 25 times under a binocular microscope and captured by a camera with pixels of 1200×1600. With the aid of MATLAB 8.1 [7], all pictures can be transferred into binary images. ...
Conference Paper
Full-text available
The surface topography of paper can range from very rough to extremely smooth, which has significant influences on mechanical properties of paper materials, especially the compressive behavior of paper in the out-of-plane direction. Normally, the stress-strain relations of most of the materials are calculated by using the nominal contact area, which is the whole area of the pressure head. The difference between actual and nominal contact area is ignored, but they are very different, and cannot be neglected in all situations. In this paper, a new experimental method for evaluating the relationship between the actual contact area and the normal load is proposed. A carbon paper is introduced in this method, and it is assumed that the measured contact areas between carbon paper and the actually tested paper are the same as the actual contact areas between the pressure head and the tested paper. Based on this assumption, the mechanical behavior of paper in the out-of-plane direction could be discussed by calculating the actual stress-strain relation and deducing the actual modulus. In addition, the force sensitivities of different carbon papers used for showing the actual contact areas were also compared. The calculation results show the crucial differences between the actual and nominal stress-stain behaviors.
... In order to achieve high-yield and high-performing systems on paper, various efforts on characterizing paper at macro-, micro-and nanoscales abide ( [7], [8], [9]). Besides, new paper types are being fabricated to conform to specialized engineering applications. ...
Full-text available
Driven by low-cost, resource abundance and distinct material properties, the use of paper in electronics, optics and fluidics is under investigation. Considering sensor systems based on magneto-resistance principles (anisotropic, giant, tunnel) that are conventionally manufactured onto inorganic semiconductor materials, we propose the use of paper substrates for cost reduction purposes primarily. In particular, we studied the magneto-resistance sensitivity of permalloy (Py:Ni81Fe19) onto paper substrates. In this work, we report on our findings with clean room paper (80 g/m2, Rrms = 2.877 {\mu}m, 23% surface porosity, latex impregnation, no embossed macro-structure). Here, the Py:Ni81Fe19 coating was manufactured by means of a dry process, sputter deposition, and spans an area of 10x10 mm2 and a thickness of 70 nm. Employing a four-point-probe DC resistivity measurement setup, we investigated the change of electrical resistance of Py:Ni81Fe19 under the presence of an oriented external magnetic field. In particular, we investigate the magneto-resistive change at two configurations: (1) the direction of the magnetic field is parallel to the nominal induced electric current and (2) the direction of the magnetic field is perpendicular to the electric current. Due to the stochastic orientation of the fibers interplaying with the Py:Ni81Fe19 coating, the change in magneto-resistance of the overall system at both measurement configurations closely corresponds to the classical response of Py:Ni81Fe19 at a +/-45{\deg} angle between the direction of electrical current and magnetic field. Using the magneto-optic kerr effect, we observed the formation of domain walls at the fiber bending locations. Future work will focus on the impact of layer thickness, fiber dimensions and structure of magnetic coating on the performance of the paper-based Py:Ni81Fe19 magneto-resistors.
Full-text available
An elastic–plastic material model for the out-of-plane mechanical behaviour of paper is presented. This model enables simulation the elastic–plastic behaviour under high compressive loads in the through-thickness direction (ZD). Paper does not exhibit a sharp transition from elastic to elastic–plastic behaviour. This makes it advantageous to define critical stress states based on failure stresses rather than yield stresses. Moreover, the failure stress in out-of-plane shear is strongly affected by previous plastic through-thickness compression. To cover these two features, a model based on the idea of a bounding surface that grows in size with plastic compression is proposed. Here, both the bounding and the yield surfaces are suggested as parabolas in stress space. While the bounding surface is open for compressive loads, the yield surface is bordered by the maximum applied through-thickness compression.
In this paper, a device for measuring stress-strain properties in the thickness direction of paperboard is presented. The device is used for out-of-plane tensile, compression and shear loading. In order to measure stress and strain accurately, the deformation of the test piece is restricted to two directions in the desired plane of deformation by means of an attached fixture. The papaperboard is first glued to metal blocks with a high-viscosity adhesive, and these blocks are then attached to the device using a fast-curing epoxy adhesive. To find the true strain in the material, knowledge of the penetration of the adhesive into the surface of the paper structure is important. A method for determining the penetration of the adhesive, based on a comparison stress-deformation curves for glued and non-glued test pieces, is presented. Finally, true stress-strain curves in tension, compression, and shear presented together with an analysis of the accuracy of the method.
Modellierung der Kompression von Papier in z-Richtung bei niedriger Flächenpressung
  • H.-J Schaffrath
  • L Göttsching
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). Journal Reference
Bilden von Schneidlagen unter Ausnutzung des Nipinduzierten Effekts für die Druckweiterverarbeitung
  • T Kaulitz
T. Kaulitz. Bilden von Schneidlagen unter Ausnutzung des Nipinduzierten Effekts für die Druckweiterverarbeitung (in German). PhD thesis, TU Darmstadt (2009). Thesis Reference
  • Flächenpressung
Flächenpressung (in German). Das Papier, Heft 7, S. 350-355 (1992). Journal Reference