CORRELATIONS OF INDENTATION HARDNESS AND YIELD STRESS IN
POLYMERS
T. Koch and S. Seidler
Institute of Materials Science and Technology
Vienna University of Technology
Vienna, A-1040 Austria
ABSTRACT
In mechanical testing often empirical correlations are used for estimation of yield stress values from hardness values. Such
empirical correlations are only valid within particular classes of materials and in the case of polymers they might be dependent
on deformation rate, temperature, and kind of deformation (tensile or compression). For a better understanding of these
complex correlations, instrumented indentation tests, tensile tests and compression tests were performed on different
semicrystalline and amorphous thermoplastic polymers. Indentation hardness and Martens hardness values were compared
to tensile and compression yield stresses. The ratio of hardness values to tensile yield stress depends very strongly from the
basic deformation mechanism (crazing, voiding, or shear yielding) of the polymers. Therefore, under tensile load a general
linear relation between hardness and yield stress cannot be stated. In contrast to this, in the case of compression loading
hardness and yield stress are much closer to a linear dependence from each other.
Introduction
Microhardness testing methods have been long established in polymer testing and polymer characterisation to determine
mechanical properties of small volumina of the materials. After the research on methodical aspects [1, 2] the method was used
for the detection of the influence of morphology parameters, i.e. lamellar thickness, orientation, texture, or blend composition
[3]. More information were obtained by instrumented indentation testing [2], which could be applied also for the polymeric
materials if the specific mechanical behaviour of this materials group is considered. Hardness values, indentation modulus,
strain hardening exponents and viscoelastic properties can be measured with the instrumented indentation test. Also
measurable are the fracture toughness of brittle materials as well as the influence of residual stresses in solid materials or thin
layers, or the elastic behaviour (spring constant) of miniaturized components. The presence of orientations can also be
detected [3].
From the viewpoint of application the easy correlation of microhardness to macroscopic mechanical parameters is of growing
interest in the polymer field. The best correlation should exist between microhardness and yield stress as it was shown by
Tabor [4] for nearly full plastically deformable metals. In the literature there exist few works in which hardness values vs. yield
stresses of polymers is plotted [2, 5, 6]. The relationships are linear but the slopes are different (see Fig. 1). When considering
such relationships, fundamental methodological and material-related aspects have to be kept in mind, e.g the reason for the
different relations given in the literature are different testing conditions and methods of evaluation. In [6] it is reported for
polyethylene materials that there is a ratio of hardness to yield stress of about 3 for tensile and of about 2 for compression
loading (Fig. 1b). This difference can be ascribed to a hydrostatic component during compression.
In the presented work tensile and compression yield stresses of different semicrystalline (PE-HD, PE-LD, PA12, PVDF, PP,
PP/EPR blends, PET, PEEK, and POM) and amorphous (PC, PS) thermoplastic polymers, which cover a wide range of
stiffness and strength, are compared with their indentation hardness and Martens hardness.
a)
b)
120
100
MHV (MPa)
80
MHV = 3σy
60
MHV = 2σcy
40
20
0
0
10
20
30
40
50
60
σy , σcy (MPa)
Figure 1. Vickers hardness vs. tensile yield stress (taken from [5]) (a) and
Microhardness vs. tensile and compression yield stress for different PE materials (taken from [6]) (b)
Experimental
Semicrystalline (PE-HD, PE-LD, PA12, PVDF, PP, PP/EPR blends, PET, PEEK, and POM) and amorphous (PC, PS)
thermoplastic polymers were investigated.
For the tensile tests specimens of type 5A and 5B according to EN ISO 527-2 were prepared. Deviant from this standard a
reduced thickness of 1 mm was used. Compression testing was carried out on cylindrical specimens having a diameter of
5 mm and a height of 6 mm. According to [7] between the specimen and the compression plates of the testing machine a
0.1 mm thick PTFE tape was positioned on the top and bottom faces and between the tape and the plates a soap solution was
given to reduce friction. A continuous observation of the specimen did not show bulging until the compression yield point was
reached. A small bulging first occurred at very high deformations.
Instrumented indentation testing was done with a Nanoindenter XP, MTS Systems. A maximum indentation depth of 2 µm and
an indentation velocity of 100 nm/s was chosen. At maximum load a holding time of 30 s was applied and after this the
specimens were unloaded. Martens hardness HM and Indentation hardness HIT were determined. HM is determined from the
values given by the force–indentation depth curve during the increasing of the test force and is defined as the applied test
force F divided by the surface area As (h) of the indenter penetrating beyond the zero-point of the contact [8]. It includes the
plastic, elastic and the time dependent deformation. For a Berkovich-Indenter it is:
HM =
F
As (h )
=
F
(1)
26.43 h2
The indentation hardness HIT is a measure of the resistance to permanent deformation or damage. It is defined as the
maximum applied force Fmax divided by the projected contact area of the indenter with the specimen. For a Berkovich-Indenter
follows:
H IT =
Fmax
(2)
24.5 hc
The contact depth hc can be calculated from the unloading curve using the tangent depth and the maximum displacement with
a correction for elastic displacement [8, 9].
Results
Figure 2 shows the stress–strain curves, determined in the tensile tests. As expected, there are large differences between the
investigated materials. The deformation behaviour ranges from ductile to brittle, the yield stresses are different pronounced.
PE-HD
100
PE-LD
PA 12
σ (MPa)
80
PVDF
PP
60
PP/EPR 80/20
PP/EPR 55/45
40
PET
PEEK
20
POM
PC
0
0
10
20
30
PS
40
ε (%)
-1
Figure 2. Stress–strain behaviour under tensile loading for different thermoplastics; strain rate 0.001 s
200
200
150
150
σc, true (MPa)
σc (MPa)
The stress–strain behaviour of the polymers under compression loading is shown in Fig. 3. Most of the materials do not show
a compression yield stress as defined in EN ISO 604. In such cases it is established to line up a tangent against the kink
region of the curves. This procedure is much easier and more reproducible if the true compression stress–strain curves are
plotted (Fig. 3).
The curves of the PE materials have two distinct kinks, a so-called double yield point (Fig. 4). This effect can be described with
two different deformation processes. At the point σcy1 a slight slip of polymer chains ("fine slip") occurs whereas at the point
σcy2 lamellas were destroyed ("coarse slip") [10]. Here the question has to be answered which yield point has to be chosen for
the relation to the hardness. For solving the problem, the explanation in [11] is used. In [11] it is shown that the mechanisms of
-3
deformation in PE during a microhardness test depend on the density of the material. At densities smaller than 0.92 gcm the
-3
compression of amorphous regions dominates, whereas at densities higher than 0.92 gcm the deformation is dominated by
-3
the destruction of crystalline regions. The materials investigated in this work have densities of 0.95 gcm (PE-HD) and
-3
0.935 gcm (PE-LLD). So, they belong to the last criteria and the yield point σcy2 was chosen.
100
50
0
0.0
100
50
-0.2
-0.4
εc
0
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
εc, true
-1
Figure 3. Stress–strain (left) and true stress–true strain curves (right) under compression loading; compression rate 0.001 s ;
symbols as in Figure 2
40
σcy2
σc, true (MPa)
30
PE-HD
σcy1
PE-LLD
σcy2
20
10
σcy1
0
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
εc, true
Figure 4. Occurrence of so-called double yielding in PE
In Figs. 5 and 6 hardness values are plotted against the appropriate yield stresses. With the help of the plotted lines, which
represent different ratios of hardness and yield stress, it can be seen that the ratios of hardness and yield stress vary strongly.
For polymers showing shear dominated deformation behaviour under tensile loading the ratio HIT /σy is about 3 and the ratio
HM/σy is about 2.5. These ratios are much higher for polymers showing crazing or strong voiding (Fig. 5), the correlation
becomes non-linear. Because of that it is not possible to give a general linear correlation between the hardness and tensile
yield stress.
In Fig. 6 the correlations between compression yield stress and the both microhardness parameters, indentation hardness and
Martens hardness, are shown. The dependencies nearly follow a linear function. In the case of HIT against σcy the slope is 2.5
and it is 2 for HM against σcy. Under compression loading the deformation is shear dominated, no crazing or voiding is visible.
That is the reason for a better linearity compared to the plots of hardness and tensile yield stress. Nevertheless there is a nonnegligible scatter to the fitted linear function. This can be reduced to a non satisfactorily determination of compression yield
stress if it is not well developed and perhaps a different loading rate dependency of the indentation and the compression tests.
400
5
4.5
4
3.5
400
3
4.5
4
2.5
3
2.5
300
300
2
2
HM (MPa)
HIT (MPa)
3.5
200
200
100
100
0
0
0
50
100
σy (MPa)
150
0
50
100
σy (MPa)
Figure 5. Indentation Hardness HIT and Martens hardness HM vs. tensile yield stress;
full symbols: no crazing or stress whitening visible
open symbols: crazing or stress whitening visible
150
400
400
3
3
2.5
2.5
300
300
2
200
1.5
200
HM (MPa)
HIT (MPa)
2
1.5
100
100
0
0
50
100
0
150
0
σcy (MPa)
50
100
150
σcy (MPa)
Figure 6. Indentation hardness HIT and Martens hardness HM vs. compression yield stress
Therefore, in future extensive tests at different testing velocities have to be realized. Simultaneously, a calculation of the real
strain rate in the region beneath the indenter has to be done. This is a requirement for the correlation of tests with comparable
strain rates, that should reduce the scatter especially in polymer materials with a strong rate dependency in mechanical
properties.
Summary
In the presented work, tensile and compression yield stresses are compared with indentation hardness and Martens hardness
values, which were determined by instrumented microindentation testing.
In the case of polymers showing a shear dominated deformation behaviour under tensile loading the correlation between
hardness values and yield stress is nearly linear. For polymers showing clear crazing or voiding there is no linear correlation.
If they were compressed, shear deformation processes dominate in all investigated polymers. Therefore the scatter of the
ratios of hardness to compression yield stress is smaller than the scatter of the ratios of hardness to tensile yield stress.
Nevertheless the variations are relatively high. One experimental problem, which should influence this, is the determination of
the compression yield stress of materials not showing a distinct yield point. Additionally, the influence of the loading rate hat to
be considered.
In polymers, the validity of a general correlation between hardness and yield stress is strongly connected with the deformation
process. In the case of shear deformation under tensile and compression loading, such correlations might be possible,
comparable with the Tabor’s relation that is often used for metals. In polymers with different deformation behaviour under
tensile and compression loading such correlations are not useful due to the possible under- and overestimation of the resulting
yield stress.
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