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Investigation on deformation

behavior of paper in Z-

direction

JIAN CHEN, JANN NEUMANN,

EDGAR DÖRSAM

Institute of Printing Science and Technology

Darmstadt University of Technology

Magdalenenstr. 2, 64289, Darmstadt

chen@idd.tu-darmstadt.de

1 INTRODUCTION

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

materials.

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 [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

0

0

(1)

EA

EF x

l

Figure 4. Paper structure and model under

compression [1]

The force-deformation relationship of surface and

internal structures could be easily obtained.

33

1101 303

122

1

22

2

2

2

22

2

() ()

4( ) 4( )

(2)

()2 ( ) ( )

2

4

AA

rr

A

EAx Ex A x Ex A x

Fx

hRR

x

Ex R R L m

EAx n

Fx x

hdR

Where

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

amoun

t

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

[1]

In addition, the following equation has to be taken

into account

123 (3)xx x x

Also

123

13

() ( ) ()

(4)

Ex Ex Ex

xx

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].

2

2

32

01222 2

3

2 2

0

22

16

2

4(5)

(1) ()

'

A

A

x

x

RBBxBxx

xFx x

Ad R

Fi

xi xi Fi

Where

22 22

12

22

(6)

4

r

mRL mL

B and B

nn

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

12

1 (7)pxpxx

And

32

01222 2

3

2 2

0

16

1(8)

4

A

A

RBBxBxx

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

0

0.5

1

1.5

2

2.5

3

3.5

Deformation (mm)

Pressure (N/mm

2

)

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

22

11 1

01

() (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 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)

[6].

(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

0

5

10

15

20

25

30

Deformation (mm)

Pressure (N/mm

2

)

1 sheet 2 sh eets 4 sheets

00.5 11.5 22.5 33.5 4

0

20

40

60

80

Pressure (N/mm

2

)

Contact area / Nominal area (%)

00.002 0.004 0.006 0.008 0.01 0.012

0

0.5

1

1.5

2

2.5

3

3.5

Deformation (mm)

Pressure (N/mm

2

)

average value

00.02 0.04 0.06 0.08 0.1 0.12

0

5

10

15

20

25

30

Deformation (mm)

Pressure (N/mm

2

)

2 sheets

1 sheet

4 sheets

paper could be transferred on the contact area (Fig.

13(b)).

(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

image.

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.

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

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

discussed.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the financial

support from China Scholarship Council.

REFERENCES

[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 stress–strain properties of

paperboard in the thickness direction.

Journal of Pulp and Paper Science. 27 (6),

213–221 (2001).

Journal Reference

[5] Introduction to Engineering and Statics.

http://www.brown.edu/Departments/Engin

eering/Courses/En3/Notes/Statics/friction/f

riction.htm

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

0

20

40

60

80

Pressure (N/mm

2

)

Contact area / Nominal area (%)

Experimental result

Calculation result (Fig.10)