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Stress-strain behavior of paper affected

by the actual contact area

Jian Chen, Edgar Dörsam, Dieter Spiehl, Arash Hakimi Tehrani and Jun Da

Institute of Printing Science and Technology

Technische Universität Darmstadt

Magdalenenstr. 2, 64289, Darmstadt

GERMANY

chen@idd.tu-darmstadt.de

Keywords: actual contact area, actual stress-strain curve, actual modulus

Summary

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.

1. Introduction

The relationship between the stress and strain that a particular material

displays is known as that particular material’s stress–strain curve. Generally,

the stress-strain curve of the material is calculated based on the force-

deformation curve. The force-deformation data obtained from tensile or

compressive tests do not give a direct indication of the material behavior,

because they depend on the specimen geometry.

For paper material, because of the existence of the surface roughness, the

contact areas change continuously during the test. When the surface of the

pressure head is very smooth, the actual contact area A(z) under force is

usually smaller than the nominal contact area A.

The measurement and characterization of the actual contact area are very

important not only for paper materials [1] but also for metal [2, 3] or other

materials [4]. The researches about the actual contact area are very helpful to

further investigate the surface roughness as well as the intrinsic

characteristics of materials.

2. Exerimental setup

The laboratory of the Institute of Printing Science and Technology is

equipped with the universal testing machine Zwick Z050, by which the

deformation performance of specimen can be determined with high accuracy

of the cross head speed (0.0005-2000 mm/min), position repetition accuracy

(± 2 μm), drive system’s travel resolution (27 nm) [5]. The travel sensor

(Heidenhain-Metro MT 2581) is produced by the HEIDENHAIN firm, with the

resolution of 50 nm and the repetition accuracy of 0.2 µm .

3. Materials

The paper selected in this paper for doing the research is the normal copy

paper (copy paper, DIN A4, 210×297 mm, 80 g/m2), produced by the Steinbeis

Paper GmbH. The average thickness is d = 84.7 μm.

For different carbon papers, the force sensitivities are very different. Seven

different types of carbon papers (SH-1, SH-2, SH-3, DL-1, DL-2, DL-3, Geha-

1) were tested. The effects of the ink on the copy paper are shown in Table 1.

Table 1. Sensitivity tests of different carbon papers. SH carbon papers are

produced by Shanghai Huideli Co., Ltd. DL carbon papers are produced by

Deli Group Co., Ltd. Geha carbon papers are produced by Geha Werke

Hannover.

Carbon papers

100 N

20 N

10 N

2 N

SH-1

SH-2

SH-3

DL-1

DL-2

DL-3

Geha-1

It can be seen from Table 1 that the sensitivities of different carbon papers

are quite different, only SH-1, Geha-1 can be used for measuring low

pressure, especially, when the forces are smaller than 20 N. The SH-1 carbon

paper was selected in the following parts to measure the contact areas under

different forces.

4. Method

The method can be summarized as the following three steps [6]:

Carrying out experiments: the forces are changed from 0 N to 100 N, with

the length of the substep 2 N, which means 50 groups of experiments (2 N,

4 N, 6 N, 8 N,…, 96 N, 98 N, 100 N) are carried out. For each group, 20

tests are finished.

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.

Calculating the contact areas: the image processing technique is used to

separate the contact area from the background, then the contact areas can

be calculated.

5. Results

5.1 Actual Contact Area

Figure 1 shows the measured contact areas Amea, the forces are changed

from 0 N to 100 N with the substep of 2 N. The error bars represent the

average (mean) values and the standard deviations of measured contact

areas under different forces. The relationship between the measured contact

area Amea and force F can be drawn by the curve fitting method.

Figure 1. Measured contact areas under different forces. The cubic curve

fitting method (the coefficient of determination: R2 = 0.953) is used in the

above picture, the picture below shows the corresponding residuals.

The cubic curve fitting method was used, the function is provided as follows:

5 3 3 2

3.6 10 5.7 10 0.39 0.24 (1)

mea

A F F F

It is assumed that the measured contact areas Amea between carbon paper

and copy paper are regarded as the actual contact areas A(z) between the

pressure head and copy paper. Based on this assumption, the mechanical

behavior of paper in the out-of-plane direction can be discussed by calculating

the actual stress-strain relation and deducing the actual modulus.

5.2 Actual Stress-strain Curve

It can be seen from Figure 2 that by considering the surface roughness, the

stress-strain curve of paper material is a typical elastic-plastic material, which

is very similar to other engineering materials, such as steel.

Figure 2. Stress-strain curves of paper calculated by using the actual contact

area and the nominal contact area. The actual contact areas under different

forces are calculated by using Equation 1.

Some typical characteristics used for determining the elastic-plastic material,

for example, elastic part, plastic part, the yield stress, ultimate stress, etc., all

of these behaviors can be found in the actual stress-strain curve.

For the nominal stress-strain curve, as we well know, the loading stage

shows a typical J-shaped curve. Based on the results above, we can

reasonably infer that the surface topography has a very big influence on the

compressive behavior of paper materials.

5.3 Actual Modulus

When the changes of the forces are very small, it is reasonable to assume

that the deformation behavior of the material under small forces accord with

the theory of elasticity. Hooke’s law is the law of elasticity under small

deformation, which can be expressed in terms of stress (σ) and strain (ε) [8]:

(2)

E z A z

EA

E F z F z z

dd

Where, A is the nominal contact area, d is the thickness of copy paper, z is

the deformation under the force F (or F(z)). E(z) is the actual modulus, which

is changing with the discrete force F(z). A(z) is the actual contact area.

The actual modulus of paper can be expressed as the product of actual

contact pressure, paper thickness and the inverse of the total deformation.

1 (3)

Fz

E z d

A z z

Then, according to Equation 3, the actual modulus of paper can be

calculated, the result is shown as the pink curve in Figure 3.

Figure 3. Moduli of paper calculated by using different methods. The actual

contact areas are calculated by using Equation 1.

Figure 3 shows the moduli of paper calculated by using different methods.

When the force is changing from 2 N (the pressure is about 0.07 MPa) to 100

N (the pressure is about 3.54 MPa), the actual modulus of paper is decreasing

from 458 MPa to around 40 MPa. The modulus calculated based on the

nominal contact area is changing from 20 MPa to about 25 MPa. With the

contact area approaching to the nominal contact area A, the actual modulus

decreases to a constant value (about 27 MPa), which is close to the result

calculated based on the nominal contact area.

6 Conclusions

Three important concepts of paper materials were proposed in this paper: the

actual contact area, the actual stress-strain curve and the actual modulus.

Firstly, the actual contact areas under different forces were calculated, it can

be seen from the results that the actual contact area of paper is changing with

the change of force, which is not a constant value as the nominal contact area.

Secondly, the concept of actual stress-strain curve was introduced to study

the mechanical behavior of paper materials. The calculation results show the

crucial differences between the actual and nominal stress-stain behaviors.

Thirdly, the concept of actual modulus was presented. The findings indicated

that, with the contact area approaching to the maximum contact area, the

actual modulus of paper is decreasing from about 458 MPa to about 40 MPa.

Literature

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