ASTM C747-93(2010)e1
(Test Method)Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
SIGNIFICANCE AND USE
This test method is primarily concerned with the room temperature determination of the dynamic moduli of elasticity and rigidity of slender rods or bars composed of homogeneously distributed carbon or graphite particles.
This test method can be adapted for other materials that are elastic in their initial stress-strain behavior, as defined in Test Method E111.
This basic test method can be modified to determine elastic moduli behavior at temperatures from −75°C to +2500°C. Thin graphite rods may be used to project the specimen extremities into ambient temperature conditions to provide resonant frequency detection by the use of transducers as described in 6.1.
SCOPE
1.1 This test method covers the measurement of the fundamental transverse, longitudinal, and torsional frequencies of isotropic and anisotropic carbon and graphite materials. These measured resonant frequencies are used to calculate dynamic elastic moduli for any grain orientations.
1.2 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.3 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety and health practices and determine the applicability of regulatory limitations prior to use.
General Information
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Standards Content (Sample)
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
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Designation: C747 − 93 (Reapproved 2010) An American National Standard
Standard Test Method for
Moduli of Elasticity and Fundamental Frequencies of
Carbon and Graphite Materials by Sonic Resonance
This standard is issued under the fixed designation C747; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
ε NOTE—Updated 9.1 and 9.2 editorially in May 2010.
1. Scope 3.1.2 longitudinal vibrations—when the oscillations in a
slender rod or bar are in a plane parallel to the length
1.1 This test method covers the measurement of the funda-
dimension, the vibrations are said to be in the longitudinal
mental transverse, longitudinal, and torsional frequencies of
mode (Fig. 1(a)).
isotropic and anisotropic carbon and graphite materials. These
measured resonant frequencies are used to calculate dynamic 3.1.3 nodal points—a slender rod or bar in resonance
elastic moduli for any grain orientations. contains one or more points having zero displacement, called
nodal points. In the longitudinal and torsional fundamental
1.2 The values stated in SI units are to be regarded as
resonances of a uniform rod or bar, the mid-length point is the
standard. No other units of measurement are included in this
nodal point (Fig. 1(a) and Fig. 1(d)). For the fundamental
standard.
transverseorflexuralresonance,thenodalpointsarelocatedat
1.3 This standard does not purport to address all of the
0.224 L from each end, where L is the length of the specimen
safety concerns, if any, associated with its use. It is the
(Fig. 1(b) and Fig. 1(c)).
responsibility of the user of this standard to establish appro-
3.1.4 resonance—a slender rod or bar driven into one of the
priate safety and health practices and determine the applica-
above modes of vibration is said to be in resonance when the
bility of regulatory limitations prior to use.
imposed frequency is such that resultant displacements for a
2. Referenced Documents
givenamountofdrivingforce(voltage)areatamaximum.The
resonant frequency is a natural vibration frequency which is
2.1 ASTM Standards:
determined by the elastic moduli, density, and dimensions of
C215 Test Method for Fundamental Transverse,
the test specimen.
Longitudinal, and Torsional Resonant Frequencies of
Concrete Specimens
3.1.5 slender rod or bar—a specimen whose ratio of length
C559Test Method for Bulk Density by Physical Measure-
to minimum cross-sectional dimension is at least 5 but not
ments of Manufactured Carbon and Graphite Articles
more than 20.
C885Test Method for Young’s Modulus of Refractory
3.1.6 transverse vibrations—whentheoscillationsinaslen-
Shapes by Sonic Resonance
der rod or bar are in a horizontal plane normal to the length
E111Test Method for Young’s Modulus, Tangent Modulus,
dimension, the vibrations are said to be in the transverse mode
and Chord Modulus
(Fig. 1(b)). This mode is also commonly referred to as the
flexuralmodewhentheoscillationsareinaverticalplane(Fig.
3. Terminology
1(c)). Either the transverse or flexural mode of specimen
3.1 Definitions of Terms Specific to This Standard:
vibration will yield the correct fundamental frequency, subject
3.1.1 elastic modulus—the initial tangent modulus as de-
to the geometric considerations given in 9.1.
fined in Test Method E111.
3.1.7 torsional vibrations—when the oscillations in each
cross-sectional plane of a slender rod or bar are such that the
This test method is under the jurisdiction of ASTM Committee D02 on
plane twists around the length dimension axis, the vibrations
Petroleum Products, Liquid Fuels, and Lubricants and is the direct responsibility of
Subcommittee D02.F0 on Manufactured Carbon and Graphite Products. are said to be in the torsional mode (Fig. 1(d)).
Current edition approved May 1, 2010. Published May 2010. Originally
approved in 1974. Last previous edition approved in 2005 as C747–93(2005). DOI:
4. Summary of Test Method
10.1520/C0747-93R10E01.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
4.1 The dynamic methods of determining the elastic moduli
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
are based on the measurement of the fundamental resonant
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. frequencies of a slender rod of circular or rectangular cross
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
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C747 − 93 (2010)
4.1.3 Torsional Mode—The equation for the fundamental
resonant frequency of the torsional mode of vibration is as
follows:
2 2
G 5RBf L ρ (3)
where:
G = modulus of rigidity, Pa,
R = ratio of the polar moment of inertia to the shape factor
for torsional rigidity,
B = a constant consistent with the units of G, R, f, L, and ρ,
f = frequency of fundamental torsional mode of vibration,
Hz,
L = length of the specimen, m, and
ρ = density of the specimen as determined by Test Method
C559, kg/m .
5. Significance and Use
5.1 This test method is primarily concerned with the room
temperature determination of the dynamic moduli of elasticity
and rigidity of slender rods or bars composed of homoge-
neously distributed carbon or graphite particles.
5.2 This test method can be adapted for other materials that
are elastic in their initial stress-strain behavior, as defined in
Test Method E111.
5.3 This basic test method can be modified to determine
elastic moduli behavior at temperatures from −75°C to
FIG. 1 Resonance Modes
+2500°C. Thin graphite rods may be used to project the
specimen extremities into ambient temperature conditions to
provide resonant frequency detection by the use of transducers
as described in 6.1.
section. The resonant frequencies are related to the specimen
dimensions and material properties as follows:
6. Apparatus
4.1.1 Transverse or Flexural Mode—The equation for the
6.1 The fundamental resonant frequencies for the different
fundamental resonant frequency of the transverse or flexural
modes of vibration of a test specimen can be determined by
mode of vibration is as follows:
severalestablishedtestingprocedures.Theapparatusdescribed
E 5 CMf (1)
herein uses phonograph record pickup cartridges as a conve-
nient method of generating and detecting these frequencies.A
where:
typical testing apparatus is shown schematically in Fig. 2.
E = elastic modulus, Pa,
6.1.1 Driving Circuit—The driving circuit consists of a
C = a dimensional constant that depends upon the shape
variable-frequency oscillator and a record pickup cartridge
andsizeofthespecimen,andPoisson’sratio.Theunits
assembly. It is recommended that a variable-frequency oscil-
of C are to be consistent with those of E, M, and f,
lator be used in conjunction with a digital-frequency counter.
M = mass of the specimen, kg, and
The oscillator shall have sufficient power output to induce
f = frequency of fundamental transverse or flexural mode
detectable vibrations in the test specimen at frequencies above
of vibration, Hz.
and below the fundamental frequency under consideration.
4.1.2 Longitudinal Mode—The equation for the fundamen-
Means for controlling the output of the oscillator shall be
tal resonant frequency of the longitudinal mode of variation is
provided. The vibrating needle of the driving unit shall be
as follows:
small in mass as compared to the test specimen, and a means
2 2
shallbeprovidedtomaintainaminimalcontactpressureonthe
E 5 Df L ρ (2)
specimen. Either a piezoelectric or magnetic driving unit
where:
meeting these requirements may be used.
E = elastic modulus, Pa,
6.1.2 Pickup Circuit—The pickup circuit consists of a re-
D = a constant consistent with the units of E, f, and L,
cord pickup cartridge, amplifier, optional high-pass filter, and
f = frequency of fundamental longitudinal mode of
an indicating meter or cathode-ray oscilloscope. The pickup
vibration, Hz,
unit shall generate a voltage proportional to the amplitude,
L = length of the specimen, m, and
velocity, or acceleration of the test specimen. Either a piezo-
ρ = density of the specimen as determined by Test Method
electric or magnetic pickup unit meeting these conditions may
C559, kg/m .
be used. The amplifier shall have a controllable output of
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C747 − 93 (2010)
bedifficulttoexciteinthefundamentalmodesofvibration.For
this method, the ratio must be between 5 and 20 (slender rod
limitations).
8. Procedure
8.1 Switch on all electrical equipment and allow to stabilize
in accordance with the manufacturers’ recommendations. (Use
of a metal bar as a calibration standard is recommended to
check equipment response and accuracy. Dimensional mea-
surements and weight shall meet the requirements of 7.2.)
8.2 Transverse Fundamental Resonance Frequency:
8.2.1 Place the specimen on the supports, which are located
at the fundamental transverse nodal points (0.224 L from each
end).Placethedrivingandpickup-unitvibratingneedlesonthe
specimen center line at its extreme opposite ends with a
minimal contact pressure consistent with good response. The
vibrating direction of the driving and pickup needles must be
perpendicular to the length of the specimen (Fig. 1(b)).
8.2.2 Force the test specimen to vibrate at various frequen-
cies and simultaneously observe the amplified output on an
FIG. 2 Schematic Diagram of Typical Dynamic Elastic Modulus
indicating meter or oscilloscope. Record the frequency of
Detection Apparatus
vibration of the specimen that results in a maximum
displacement, having a well-defined peak on the indicator,
where nodal point tracking indicates fundamental transverse
sufficient magnitude to sharply peak out the resonant frequen-
resonance.
cies on the indicating meter or the cathode-ray oscilloscope
8.2.3 A basic understanding of Lissajous patterns as dis-
display tube. It may be necessary to use a high-pass filter in
played on an oscilloscope cathode ray tube (CRT), will aid in
order to reduce room noise and spurious vibrations. The
the proper identification of the modes of vibration and har-
indicating meter may be a voltmeter, microammeter or oscil-
monic frequencies observed. As the oscillator frequency level
loscope. An oscilloscope is recommended because it enables
is increased from a point well below expected resonance, a
the operator to positively identify resonances, including higher
single closed loop Lissajous pattern tilted from the horizontal
order harmonics, by Lissajous figure analysis.
reference plane, will eventually be displayed on the CRT. This
6.1.3 Specimen Supports—The supports shall permit the
pattern denotes a resonance mode. The nodal points dynamic
specimen to oscillate without significant restriction in the
modulus tracking guide template (Fig. 3) may be used to
desired mode. This is accomplished for all modes by support-
identify any resonant mode.
ing the specimen at its transverse fundamental nodal points
8.2.4 Move the pickup cartridge needle slowly toward the
(0.224 L from each end). The supports should have minimal
specimen center and observe the Lissajous pattern loop.
area in contact with the specimen and shall be of cork, rubber,
Fundamental transverse resonance is indicated when the fol-
or similar material. In order to properly identify resonant
lowing conditions prevail:
frequencies, the receiver record pickup cartridge must be
8.2.4.1 Thelooppatternflattenstoahorizontallinewiththe
movable along the total specimen length. Provisions shall be
pickup needle over the specimen support.
made to adjust contact pressures of both record pickup car-
8.2.4.2 The CRT pattern opens up to a full loop in a
tridges in order to accommodate specimen size variations. The
direction normal to its original direction, with the pickup
entire specimen support structure shall be mounted on a
needle over the specimen center.
massive base plate resting on vibration isolators.
8.2.5 Return the pickup needle to its original position at the
specimen end.
7. Test Specimens
8.2.6 Spurious resonating frequency modes may mask or
7.1 Selection and Preparation of Specimens—In the selec-
attenuate the fundamental transverse frequency indication.
tion and preparation of test specimens, take special care to
Investigation of higher order harmonic resonating frequencies
obtain representative specimens that are straight, uniform in
by use of the tracking guide template (Fig. 3) will help to
cross section, and free of extraneous liquids.
identify the correct fundamental frequency mode.Aplot of the
ratio of harmonic to fundamental frequency for transverse
7.2 Measurement of Weight and Dimensions—Determine
mode of vibration (Fig. 4) may then be used to calculate the
the weight and the average length of the specimens within
fundamental transverse resonant frequency mode.
60.5%. Determine average specimen cross-sectional dimen-
sion within 61%.
8.3 Longitudinal Fundamental Resonance Frequency:
7.3 Limitations on Dimensional Ratio— Specimens having 8.3.1 Leave the specimen supported at the fundamental
eitherverysmallorverylargeratiosoflengthtothicknessmay transverse mode nodal points as in 8.2.1. Rotate the driving
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C747 − 93 (2010)
FIG. 3 Nodal Points Dynamic Modulus Tracking Guide Template
The second harmonic longitudinal resonant frequency is twice
the fundamental longitudinal resonant frequency.
8.4 Torsional Fundamental Resonance Frequency:
8.4.1 Leave the specimen supported as in 8.2.1. Rotate the
driving unit and pickup cartridge needles so as to induce
vibrations perpendicular to the length of the sample (Fig. 1
(d)).
8.4.2 Force the specimen to vibrate as in 8.2.2. Record the
frequency of vibration of the test specimen, where nodal point
tracking indicates fundamental torsional resonance. The sec-
ond harmonic torsional resonant frequency is twice the funda-
mental torsional resonant frequency.
9. Calculation
9.1 Calculate the dynamic modulus of elasticity for the
transverse or flexural mode of vibration from the fundamental
transverse frequency, weight, and dimensions of the test
specimen as follows:
Dynamic E 5 CMf (4)
where units are as defined in 4.1.1. The evaluation of the
constant C, because of the complexity of its determination, is
in tabular form. Eq 4 may be rewritten in the forms:
Dynamic E pascals 5 A Mf /d forrodswith (5)
~ !
c
NOTE1—TakenfromPickett,Gerald,“EquationsforComputingElastic
circularcrosssections
Constants from Flexural andTorsional Resonant Frequencies ofVibration
of Prisms and Cylinders,” Proceedings, AS
...
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