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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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