ram
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with
strain gages, taking transverse sensitivity into account
Katrin Baumann,
TU Darmstadt;
Introduction
This undesirable property is characterized by
Jochen Naumann,
Carbon fiber reinforced plastics (CRP) are
so-called transverse sensitivity and gener-
TU Chemnitz,
increasingly being used for lightweight con-
ally invalidates the measurement results. So
structions, as their structure ensures good
this article is not just about identifying the
strength at low specific weight, reduced wall
material parameters of characteristic CRP
thickness and low temperature dilatation. It
experimentally, but also about examining
is essential to know the material parameters
the effect of the transverse sensitivity of the
when designing CRP components. Methods
strain gage on the errors that occur when
of parameter identification are described
identifying the material constants.
Germany
many times in the references. Scharr [1], for
determining elastic constants and Dudescu
Theoretical basis
[2] for determining thermal expansion coef-
Hooke’s Law generalized for orthotropy
ficients, are just two examples.
With carbon fiber reinforced plastics, the
spatial arrangement of the fibers embedded
Currently, the modulus of elasticity and the
in the plastic matrix causes a pronounced
shear modulus, as well as the Poisson’s ratios,
anisotropy of the material behavior. With
are defined in experimental investigations on
unidirectional and bidirectional reinforce-
simple stress states by measuring the active
ment (see Fig. 1), the material properties are
loading and the resultant strains and associ-
orthotropic, that is, they are symmetrical
ating them via Hooke’s Law.
in the material-specific 1-2-3 principal axis
system with regard to planes 1-2 , 1-3 and
Strain gages (SGs) are frequently used in
2-3.
tests to identify material characteristics
2
and also to identify stresses experimentally.
The correlation between the stresses and
But the strain gages are affected not only
the distortions in the material is described
by the strain to be measured in the longitu-
by Altenbach [3] in the elastic deformation
dinal direction, but also by the strains that
range by Hooke’s Law for orthotropic mate-
occur across the direction of measurement.
rial behavior as
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
2
Fibers
3
1
unidirectional
bidirectional
Fig. 1: The arrangement of the fibers for unidirectionally and bidirectional
reinforced CRPs
(1)
and in the shortened matrix notation by
(2)
The compliance matrix S includes twelve material characteristics for this general situation.
Because of the requisite symmetry of S, [3], there are three additional conditions, which
link the moduli of elasticity with the Poisson’s ratios. Consequently only nine of the twelve
parameters are independent.
ram no. 2/2006
3
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Because of their good strength properties, CRPs are often used as thin-walled components,
in which there is virtually an even stress state (ESS) with
(3)
From this, Hooke’s Law for the ESS for orthotropic material behavior in a reduced form is
(4)
By inverting (4), the representation
(5)
of the stresses is obtained subject to the distortions.
The restriction to an even stress state means that the material law, equation (4) and (5), only
includes five material parameters, that are interlinked by the symmetry condition
(6)
Consequently, only four of the five material characteristics are independent.
Should the material behavior not be formulated in the direction of the 1-2-3 principal axis
system, but with regard to a 1-2-3 system rotated as required, the material law must be transformed accordingly. According to [3], this produces for equation (2)
(7)
4
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
In this, the transformation matrix T « causes the transformation of compliance matrix S. The
following applies for equation (4) of the even stress state and a rotation of w = 45° around
axis 3
(8)
Determining material parameters in a 4-point bending test
To determine material parameters in a 4-point bending test, the sample is loaded symmetrically between two supports as shown in Figure 2, in each case with F/2. The strain gages
for measuring the longitudinal and transverse strain on the tension and pressure side of the
sample are prepared in the pure bending range.
F
2
y
Strain gages
(tension or pressure)
Fig. 2:
4-point bending test,
progression of bending
moment M
F
2
x
d
b
z
l
l
M
The x-y-z coordinate system in Figure 2 always relates to the geometry of the bending sample
and its loading and does not depend on the material-specific 1-2-3 system.
The geometrically defined deformation state of the sample causes a uniaxial stress state in
the longitudinal direction of sample x with edge stresses,
(9)
ram no. 2/2006
5
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Hooke’s Law, equation (4), then produces the equations for defining the unknown material
parameters Ex and nxy, from the measured quantities of force F and the longitudinal and
transverse strains «x and « y on the surface of the sample,
(10)
As orthotropic material has different moduli of elasticity Ei and Poisson ratios nij in the two
principal directions 1 and 2, two samples with longitudinal axes coinciding with each of the
two principal axes have to be examined to identify the two moduli of elasticity and the two
Poisson‘s ratios of a material.
A third bending test is suited to determine the shear modulus G12, where the longitudinal axis
x of the sample lies below the angle of w=45° to directions 1 and 2. For this sample, equations
(4) of Hooke’s Law are transformed by means of the transformation matrix T « in accordance
with equation (8) to
(11)
By subtracting the first two equations from (11),
(12)
follows as the conditional equation for the shear modulus Gxy . As well as the longitudinal
and transverse stresses «x and «y during the bending test with a 45° sample, bending stress sx
simultaneously causes a slip gxy, see equation (11).
6
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
The effect of transverse sensitivity
Electrical strain gages often also experience a change in resistance if a strain is acting exclusively across their direction of measurement. This property is designated transverse sensitivity q and is described many times in the references, for example, by Keil [4]. In [5], Hoffmann
states the basic ways to affect transverse sensitivity by the layout of the end loops. Stockmann undertakes extensive experimental and theoretical analysis in [6], which continues with
Naumann in [7], to present a method to identify transverse sensitivity and a method for
designing strain gages that are not susceptible to transverse sensitivity.
Transverse sensitivity q is defined as the ratio of the strain sensitivities of the strain gage in
the longitudinal and transverse directions k1 and kq,
(13)
This can be either positive or negative, as determined by the type of strain gage construction
and depending on the resistance material. Values in the range -1 to +3 % are typical.
The determination of the gage factor of a strain gage is standardized in [8] and is performed
for a uniaxial stress state with a material Poisson’s ratio of n0= 0.285. With this, a strain gage
with existing transverse sensitivity measures without error for any load, if the strain ratio
exactly matches that of the calibration.
The error that occurs when identifying the material constants is examined below, when during the analysis of the relevant tests, the transverse sensitivity becomes negligible or can only
be inaccurately considered.
The absolute error of a quantity is described by the difference between the incorrect and the
correct value. If a material characteristic is determined directly from the display values
and
of the strain gage, without taking the transverse sensitivity of the strain gage into
account, the error that then arises comes from the difference between the value defined by
the display and the actual value of the particular characteristic quantity,
(14)
ram no. 2/2006
7
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
The absolute errors are produced with relations (10) and (12) for the dependency of the material characteristics on the measured quantities of stress sx and the longitudinal and transverse strain «x and « y
(15)
(16)
(17)
The display values for the longitudinal and transverse stress
and
contained here
can be eliminated with the relations for the usual transverse sensitivity correction
(18)
Using relations (18) in equations (15), (16) and (17) gives the absolute errors of the particular
material parameter. For better reproducibility, the relative error of each characteristic value
is also defined.
The absolute and relative errors of the modulus of elasticity Ex are
(19)
The absolute and relative errors of Poisson‘s ratio nxy are
(20)
The errors of the shear modulus Gxy are calculated to be
(21)
8
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Experimental determination of material parameters
Materials and sampling
Investigations were carried out on a total of four different CRP materials internally designated HT090, HT0, HM090 and HM0. The CRP composites were made at the Institute for
Design and Composite Structures of the University of Technology TU Chemnitz from two
different types of fiber (marked “HT” and “HM”) in both the unidirectional and bidirectional
fiber arrangement (marked “0” and “090”), see Table 1. For the sheet material, 10 prepregs
were stacked one on top of the other, so that the bidirectional fiber arrangement of the material can also be regarded as a good approximation of homogeneous from the continuum
mechanics viewpoint.
Material
Fiber arrangement,
composite
HT090
bidirectional
HT0
Fiber type
Fiber property
SIGRAFIL prepreg
CE 1201
high tensile strength
unidirectional
HM090
bidirectional
HAUFLER prepreg
high modulus of elasticity
HM0
unidirectional
HM DU 450 FT109 38 %
(640 GPa)
Tab. 1:
The properties
of the sample
materials
The unidirectional reinforcement of a plastic by means of a parallel fiber arrangement causes
pronounced anisotropy in the material behavior. In contrast to this, the bidirectional CRPs
behave quasi-isotropically, because the fibers are aligned perpendicularly to one another.
In Figure 3, the blanks of the three differently aligned samples are shown for material HM090
as an example. The samples are a = 145 mm long, b = (12 60.1) mm wide and d = 2 mm thick.
y
x
y
x
y
2
x
1
Fig. 3: Blank and designation of the samples relative to the direction of the fibers
for HM090
ram no. 2/2006
9
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Experimental techniques and test
implementation
The test is implemented on a computer-driven
test machine made by Zwick/Roell in a specially
made bending test jig, see Figure 4. All the bearings, which mechanically are roller bearings, are
executed as loose bearings.
The stress loading and relief of the sample were
displacement-controlled at a traverse rate of
1mm / min, with a short hold time at maximum
load.
Fig. 4: View of the test stand, section of a sample prepared
with the 6/120XY11 strain gage (HBM)
T-rosettes of type 6/120XY11, made by Hottinger
Baldwin Messtechnik GmbH (HBM), with a
gage factor of 2.05 and a transverse sensitivity
Bearing
of 0.6 % were used to measure the strains.
Sample
According to the equations (11), with unidirectional samples at 45°, there is a certain twisting
of the sample during the test. So as not to prevent this and to ensure that it is only the bending stress sx that develops, three half-pins are
Fig. 5: Schematic sketch of the point bearings
affixed to each of the samples, which interact
with the rest of the test equipment as shown
in Figure 5 to produce three point bearings.
The fourth bearing remains a linear bearing,
to exclude the possibility of rigid-body sample
motion (Fig. 6).
Fig. 6: Sample with point bearing, separately and in the installed state
10
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Measurement results
Figures 7 and 8 show two sets of measured values as graphs, as an example. These clearly
show the strain load and relief arms, which are separated by a hold time at the turning point.
With the 45° samples – in contrast to the samples in and across the direction of the fibers
– the longitudinal and transverse strain are of the same order of magnitude. Also, with the
45° samples, a clear residual stress remains, see Figure 7.
Transverse strain
Longitudinal strain
Force
6000
75
5000
4000
Strain [μm/m]
2000
25
1000
0
0
Force [N]
50
3000
-1000
-25
-2000
-3000
-50
-4000
-5000
0
100
200
300
400
500
-75
700
600
Time [s]
Fig. 7: Measured value progressions, HT090, sample 30 at 45°, tension side
Transverse strain
Longitudinal strain
Force
1000
6000
5000
750
4000
Strain [μm/m]
2000
250
1000
Force [N]
500
3000
0
0
-1000
-250
-2000
-3000
-500
0
100
200
300
400
500
600
700
Time [s]
Fig. 8: Measured value progressions, material HT0, sample 50 in the direction
of the fibers, tension side
ram no. 2/2006
11
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Determining material parameters
There were two stages to determining the material characteristics of the sample materials. First the
measured strain values were treated with the usual correction (18) for the transverse sensitivity q
of the strain gage and the progressions |sx ( «x )|, |«y ( «x )| and |sx ( 2(«x – «y ))| were visualized,
see Figures 9 and 10. In the charts, the material parameters are shown in accordance with the
conditional equations (10) and (12) as the rising straight lines.
The plots shown have an approximately linear progression. The existing deviations are largely
random measurement inaccuracies and deviations of the actual material behavior from the
linear-elastic material model. In virtually all the tests, the strain load and relief arms do not
coincide exactly, but with the same rise, lie parallel to one another.
E1 – Pressure
E1 – Tension
E2 – Pressure
E2 – Tension
300
250
Edge stress [N/mm²]
200
150
100
50
0
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Amount of the q -corrected longitudinal strain
Fig. 9: Progressions |sx ( «x )| to identify the moduli of elasticity, HT090
QDZ 12 – Pressure
QDZ 12 – Tension
QDZ 21 – Pressure
QDZ 21 – Tension
Amount of the q -corrected transverse strain
0.00025
0.0002
0.00015
0.0001
0.00005
0
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Amount of the q -corrected longitudinal strain
Fig. 10: Progressions |« y ( «x )| to identify the Poisson’s ratios, HT090
12
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz
The identification of the material parameters in the form of the rising arms of the plots in
Figures 9 to 11 is done by regression analysis. The calculated standard deviation testifies to
the accuracy of the characteristic values. These results are entered in the first row of cells of
Table 2.
The moduli of elasticity and the Poisson’s ratios were harmonized by means of a special
algorithm, in accordance with the symmetry condition, in equation (6). These final material
parameters, that are suitable for FEM calculations, are entered in bold in the cells of Table 2.
G12 – Pressure
G12 –Tension
80
70
Edge stress [N/mm²]
60
50
40
30
20
10
0
0
0.005
0.01
0.015
0.02
0.025
Amount of 2*(longitudinal strain – transverse strain, q-corrected)
Fig. 11: Progressions |sx ( 2(«x – «y ))| to identify the shear modulus, HT090
Material
HT090
HT0
HM090
HM0
54300 ± 22.7
8300 ± 5.5
144000 ± 331.4
5500 ± 45.7
53900
8300
144300
4100
38500 ± 10.7
98500 ± 33.0
101900 ± 113.0
190100 ± 1360.7
38600
98500
101800
213700
0.040 ± 5.2·10 -5
0.030 ± 2.4·10 -4
0.041 ± 9.1·10 -4
0.006 ± 2.5·10 -5
0.043
0.036
0.010
0.006
0.032 ± 3.1·10 -5
0.430 ± 2.6·10 -4
0.006 ± 8.0·10 -5
0.400 ± 3.0·10 -3
0.031
0.043
0.007
0.322
3900 ± 10.0
3700 ± 2.5
3600 ± 9.1
3900 ± 6.1
Characteristic
Tab. 2: Material parameters and their standard deviation
ram no. 2/2006
13
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
Discussion and conclusions
In summary, it is possible to establish that
The determined material parameters
the experimental-theoretical methodology
• As can be seen from the material charac-
presented here that is based on the data
teristics in Table 2, it is to be expected that
from bending tests instrumented with strain
with the bidirectional, that is the quasi-
gages, is eminently suitable for identifying
isotropic CRPs HT090 and HM090, both the
the material parameters of carbon fiber rein-
moduli of elasticity and both the Poisson’s
forced plastics. It provides a complete set of
ratios are of the same order of magnitude.
reliable material parameters that can be used
Compared to those of the structurally
immediately for the subsequent FEM calcula-
comparable material HT090, the Poisson’s
tions of component design.
ratios of material HM090 are somewhat
further apart.
• The unidirectional, that is highly aniso-
Effect of transverse sensitivity
on the determination of material
parameters
tropic materials HT0 and HM0 on the
• According to equation (19), the relative
other hand, show moduli of elasticity and
error of the modulus of elasticity only
Poisson’s ratios of clearly different orders
depends on the transverse sensitivity q of
of magnitude.
the strain gage and the Poisson’s ratios n0
and nxy . For nxy= n0= 0.285 , the relative
• With the HM materials, the moduli of elas-
error is zero, as strain gage calibration takes
ticity in the direction of the fibers are 2 to
place at a Poisson’s ratio of n0= 0.285 and
2.6 times greater than for the HT materials.
thus a transverse strain of this magnitude
The reason for this is the far higher modulus
is already taken into consideration in the
of elasticity of the HM fibers.
gage factor. For an assumed, relatively
large transverse sensitivity of q=0.03 and
• It is possible, using the magnitude of the
nxy= 0 to 0.5,
DEx
= -0.0086
Ex
to 0.0065. So
Poisson’s ratios, to establish that, for the
the effect of the transverse sensitivity on
bidirectional material, the transverse strain
the identification of the modulus of elas-
is impeded in both directions by verti-
ticity remains minimal.
cal arrangement of the fibers. In contrast
the corresponding unidirectional materials show an extremely high transverse
strain when loaded in the direction of the
fibers and a relatively low one when loaded
across the fibers.
14
ram no. 2/2006
reports in applied measurement
Identifying the material parameters of carbon fiber reinforced plastics with strain gages,
taking transverse sensitivity into account
Katrin Baumann, TU Darmstadt; Jochen Naumann, TU Chemnitz, Germany
• As the Poisson’s ratio can assume low values, it is useful to specify its absolute error,
which according to equation (20) only depends on transverse sensitivity q and the
Poisson’s ratio nxy itself. For q=0.03 and
nxy=0 to 0.5, Dnxy= − 0.003 to − 0.022.
Without taking transverse sensitivity into
account, the Poisson’s ratios that are determined are generally lower than the true
values. Errors must be rated as critical.
• According to equation (21), the relative
error of the shear modulus is solely defined by the transverse sensitivity q of the
strain gage and n0 = 0.285. For q=0.03,
DGxy
Gxy
= 0.0221 is obtained, which admit-
tedly is not negligible, but is still small.
Finally, it must be established that for reasons
of accuracy, it is extremely useful to take the
effect of the transverse sensitivity of the
References
[1] Scharr, G: Experimentelle Prüfverfahren zur
Bestimmung des kompletten Stoffgesetzes von
anisotropen faserverstärkten Kunststoffen,
Messtechnische Briefe, 21 (1985), H1, S.7-11.
[2] Dudescu, C.; Naumann, J.; Stockmann, M.; Nebel, S.: Characterisation of Thermal Expansion
Coefficient of Anisotropic Materials by ESPI,
Strain, 42(2006) S. 197-205
[3] Altenbach, H.; Altenbach, J.; Rikards, R.:
Einführung in die Mechanik der Laminat- und
Sandwichtragwerke – Modellierung und Berechnung von Balken und Platten aus Ver-bundwerkstoffen. Stuttgart: Deutscher Verlag für
Grundstoffi ndustrie, 1996.
[4] Keil, S.: Beanspruchungsanalyse mit Dehnungsmessstreifen, Cuneus-Verlag, 1995.
[5] Hofmann, K.: Zur Herstellung moderner FolienDehnungsmessstreifen und den dabei gegebenen Korrekturmöglichkeiten für Kriechen
und Querempfi ndlichkeit. Messtechnische
Briefe 22(1986)2, S. 41-46.
[6] Stockmann, M.: Mikromechanische Analyse der Wirkungsmechanismen elektrischer
Dehnungsmessstreifen. Habilitationsschrift
Technische Universität Chemnitz, Institut für
Mechanik, Bericht 3/2000 und http://archiv.
tu-chemnitz.de/pub/2000/0049.
strain gage into account and to correct the
displayed strains by means of equations (18).
[7] Stockmann, M.; Naumann, J.: The transverse
sensitivity of strain gages – determination and
compensation. Transactions of Famena XXVII
(2004), Heft 1, S. 43-50, ISSN1333-1124.
[8] VDI/VDE/GESA-Richtlinie 2635, Blatt1:
Dehnungsmessstreifen mit metallischem Messgitter – Kenngrößen und Prüf bedingungen.
Düsseldorf: VDI-Verlag, 2006.
ram no. 2/2006
15