ASTM C747-93(1998)
(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
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 This standard does not purport to address all of the safety problems, 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
Relations
Standards Content (Sample)
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
An American National Standard
Designation: C 747 – 93 (Reapproved 1998)
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 C 747; 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 (e) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This test method covers the measurement of the funda-
mental 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 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 appro-
priate safety and health practices and determine the applica-
bility of regulatory limitations prior to use.
2. Referenced Documents
2.1 ASTM Standards:
C215 Test Method for Fundamental Transverse, Longitu-
dinal, and Torsional Frequencies of Concrete Specimens
C559 Test Method for Bulk Density by Physical Measure-
ment of Manufactured Carbon and Graphite Articles
C885 Test Method for Young’s Modulus of Refractory
Shapes by Sonic Resonance
E111 TestMethodforYoung’sModulus,TangentModulus,
and Chord Modulus
3. Terminology
3.1 Definitions of Terms Specific to This Standard:
FIG. 1 Resonance Modes
3.1.1 elastic modulus—the initial tangent modulus as de-
fined in Test Method E111.
3.1.2 slender rod or bar—a specimen whose ratio of length
3.1.4 transverse vibrations—whentheoscillationsinaslen-
to minimum cross-sectional dimension is at least 5 but not
der rod or bar are in a horizontal plane normal to the length
more than 20.
dimension, the vibrations are said to be in the transverse mode
3.1.3 longitudinal vibrations—when the oscillations in a
(Fig. 1(b)). This mode is also commonly referred to as the
slender rod or bar are in a plane parallel to the length
flexuralmodewhentheoscillationsareinaverticalplane(Fig.
dimension, the vibrations are said to be in the longitudinal
1(c)). Either the transverse or flexural mode of specimen
mode (Fig. 1(a)).
vibration will yield the correct fundamental frequency, subject
to the geometric considerations given in 9.1.
3.1.5 torsional vibrations—when the oscillations in each
This test method is under the jurisdiction of ASTM Committee D02 on
cross-sectional plane of a slender rod or bar are such that the
Petroleum Products and Lubricantsand is the direct responsibility of Subcommittee
plane twists around the length dimension axis, the vibrations
D02.Fon Manufactured Carbon and Graphite Products.
Current edition approved Jan. 15, 1993. Published March 1993. Originally
are said to be in the torsional mode (Fig. 1(d)).
e1
published as C747–74. Last previous revision C747–74 (1988) .
3.1.6 resonance—a slender rod or bar driven into one of the
Annual Book of ASTM Standards, Vol 04.02.
above modes of vibration is said to be in resonance when the
Annual Book of ASTM Standards, Vol 05.05.
Annual Book of ASTM Standards, Vol 03.01. imposed frequency is such that resultant displacements for a
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
C 747
givenamountofdrivingforce(voltage)areatamaximum.The
L = length of the specimen, m, and
resonant frequency is a natural vibration frequency which is
r = density of the specimen as determined by Test Method
determined by the elastic moduli, density, and dimensions of
C559, kg/m .
the test specimen.
3.1.7 nodal points—a slender rod or bar in resonance 5. Significance and Use
contains one or more points having zero displacement, called
5.1 This test method is primarily concerned with the room
nodal points. In the longitudinal and torsional fundamental
temperature determination of the dynamic moduli of elasticity
resonances of a uniform rod or bar, the mid-length point is the
and rigidity of slender rods or bars composed of homoge-
nodal point (Fig. 1(a) and Fig. 1(d)). For the fundamental
neously distributed carbon or graphite particles.
transverse or flexural resonance, the nodal points are located at
5.2 This test method can be adapted for other materials that
0.224 L from each end, where L is the length of the specimen
are elastic in their initial stress-strain behavior, as defined in
(Fig. 1(b) and Fig. 1(c)).
Test Method E111.
5.3 This basic test method can be modified to determine
4. Summary of Test Method
elastic moduli behavior at temperatures from −75°C to
4.1 The dynamic methods of determining the elastic moduli
+2500°C. Thin graphite rods may be used to project the
are based on the measurement of the fundamental resonant
specimen extremities into ambient temperature conditions to
frequencies of a slender rod of circular or rectangular cross
provide resonant frequency detection by the use of transducers
section. The resonant frequencies are related to the specimen
as described in 6.1.
dimensions and material properties as follows:
4.1.1 Transverse or Flexural Mode—The equation for the
6. Apparatus
fundamental resonant frequency of the transverse or flexural
6.1 The fundamental resonant frequencies for the different
mode of vibration is as follows:
modes of vibration of a test specimen can be determined by
E 5 CMf (1) severalestablishedtestingprocedures.Theapparatusdescribed
herein uses phonograph record pickup cartridges as a conve-
where:
nient method of generating and detecting these frequencies.A
E = elastic modulus, Pa,
typical testing apparatus is shown schematically in Fig. 2.
C = a dimensional constant that depends upon the shape
6.1.1 Driving Circuit—The driving circuit consists of a
and size of the specimen, and Poisson’s ratio. The
variable-frequency oscillator and a record pickup cartridge
units of C are to be consistent with those of E, M, and
assembly. It is recommended that a variable-frequency oscil-
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.
Means for controlling the output of the oscillator shall be
4.1.2 Longitudinal Mode—The equation for the fundamen-
tal resonant frequency of the longitudinal mode of variation is
as follows:
2 2
E 5 Df L r (2)
where:
E = elastic modulus, Pa,
D = a constant consistent with the units of E, f, and L,
f = frequency of fundamental longitudinal mode of vibra-
tion, Hz,
L = length of the specimen, m, and
r = density of the specimen as determined by Test Method
C559, kg/m .
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 r (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 r,
FIG. 2 Schematic Diagram of Typical Dynamic Elastic Modulus
f = frequency of fundamental torsional mode of vibration,
Detection Apparatus
Hz,
C 747
provided. The vibrating needle of the driving unit shall be 8.2.1 Place the specimen on the supports, which are located
small in mass as compared to the test specimen, and a means at the fundamental transverse nodal points (0.224 L from each
shallbeprovidedtomaintainaminimalcontactpressureonthe
end).Placethedrivingandpickup-unitvibratingneedlesonthe
specimen. Either a piezoelectric or magnetic driving unit
specimen center line at its extreme opposite ends with a
meeting these requirements may be used.
minimal contact pressure consistent with good response. The
6.1.2 Pickup Circuit—The pickup circuit consists of a
vibrating direction of the driving and pickup needles must be
recordpickupcartridge,amplifier,optionalhigh-passfilter,and
perpendicular to the length of the specimen (Fig. 1(b)).
an indicating meter or cathode-ray oscilloscope. The pickup
8.2.2 Force the test specimen to vibrate at various frequen-
unit shall generate a voltage proportional to the amplitude,
cies and simultaneously observe the amplified output on an
velocity, or acceleration of the test specimen. Either a piezo-
indicating meter or oscilloscope. Record the frequency of
electric or magnetic pickup unit meeting these conditions may
vibration of the specimen that results in a maximum displace-
be used. The amplifier shall have a controllable output of
ment,havingawell-definedpeakontheindicator,wherenodal
sufficient magnitude to sharply peak out the resonant frequen-
point tracking indicates fundamental transverse 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 postively 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
(0.224 L from each end). The supports should have minimal 8.2.4 Move the pickup cartridge needle slowly toward the
area in contact with the specimen and shall be of cork, rubber,
specimen center and observe the Lissajous pattern loop.
or similar material. In order to properly identify resonant Fundamental transverse resonance is indicated when the fol-
frequencies, the receiver record pickup cartridge must be
lowing conditions prevail:
movable along the total specimen length. Provisions shall be
8.2.4.1 Thelooppatternflattenstoahorizontallinewiththe
made to adjust contact pressures of both record pickup car-
pickup needle over the specimen support.
tridges in order to accommodate specimen size variations. The
8.2.4.2 The CRT pattern opens up to a full loop in a
entire specimen support structure shall be mounted on a
direction normal to its original direction, with the pickup
massive base plate resting on vibration isolators.
needle over the specimen center.
7. Test Specimens 8.2.5 Return the pickup needle to its original position at the
specimen end.
7.1 Selection and Preparation of Specimens—In the selec-
8.2.6 Spurious resonating frequency modes may mask or
tion and preparation of test specimens, take special care to
obtain representative specimens that are straight, uniform in attenuate the fundamental transverse frequency indication.
cross section, and free of extraneous liquids. Investigation of higher order harmonic resonating frequencies
7.2 Measurement of Weight and Dimensions—Determine by use of the tracking guide template (Fig. 3) will help to
the weight and the average length of the specimens within identify the correct fundamental frequency mode.Aplot of the
60.5%. Determine average specimen cross-sectional dimen-
ratio of harmonic to fundamental frequency for transverse
sion within 61%. mode of vibration (Fig. 4) may then be used to calculate the
7.3 Limitations on Dimensional Ratio— Specimens having
fundamental transverse resonant frequency mode.
eitherverysmallorverylargeratiosoflengthtothicknessmay
8.3 Longitudinal Fundamental Resonance Frequency:
bedifficulttoexciteinthefundamentalmodesofvibration.For
8.3.1 Leave the specimen supported at the fundamental
this method, the ratio must be between 5 and 20 (slender rod
transverse mode nodal points as in 8.2.1. Rotate the driving
limitations).
unit and pickup cartridge needles so as to induce vibrations
parallel to the specimen length (Fig. 1(a)).
8. Procedure
8.3.2 Force the test specimen to vibrate as in 8.2.2. Record
8.1 Switch on all electrical equipment and allow to stabilize
the frequency of vibration of the test specimen, where nodal
in accordance with the manufacturers’ recommendations. (Use
point tracking indicates fundamental longitudinal resonance.
of a metal bar as a calibration standard is recommended to
The second harmonic longitudinal resonant frequency is twice
check equipment response and accuracy. Dimensional mea-
the fundamental longitudinal resonant frequency.
surements and weight shall meet the requirements of 7.2.)
8.2 Transverse Fundamental Resonance Frequency: 8.4 Torsional Fundamental Resonance Frequency:
C 747
FIG. 3 Nodal Points Dynamic Modulus Tracking Guide Template
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 3.1.1. 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 /dforrodswith
c
circularcrosssections (5)
where d is the diameter of the rod in metres, and
Dynamic E ~pascals!5 A Mf /wforbarswith
R
squareorrectangularcrosssections (6)
where w is the width dimension of the bar in metres.
9.1.1 Values of A and A are shown in Annex A1 under
c R
Table A1.1 and Table A1.2. The value of A is given as a
c
function of the diameter-to-length ratio of the sample. The
NOTE—Taken from Pickett, Gerald, “Equations for Computing Elastic
Constants from Flexural andTorsional R
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.