ASTM E1875-20a

Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Sonic Resonance

SIGNIFICANCE AND USE
5.1 This test method has advantages in certain respects over the use of static loading systems for measuring moduli.  
5.1.1 This test method is nondestructive in nature. Only minute stresses are applied to the specimen, thus minimizing the possibility of fracture.  
5.1.2 The period of time during which measurement stress is applied and removed is of the order of hundreds of microseconds. With this test method it is feasible to perform measurements at elevated temperatures, where delayed elastic and creep effects would invalidate modulus of elasticity measurements calculated from static loading.  
5.2 This test method is suitable for detecting whether a material meets the specifications, if cognizance is given to one important fact in materials are often sensitive to thermal history. Therefore, the thermal history of a test specimen must be considered in comparing experimental values of moduli to reference or standard values. Specimen descriptions should include any specific thermal treatments that the specimens have received.
SCOPE
1.1 This test method covers the determination of the dynamic elastic properties of elastic materials. Specimens of these materials possess specific mechanical resonant frequencies that are determined by the modulus of elasticity, mass, and geometry of the test specimen. Therefore, the dynamic elastic properties of a material can be computed if the geometry, mass, and mechanical resonant frequencies of a suitable test specimen of that material can be measured. The dynamic Young's modulus is determined using the fundamental flexural resonant frequency. The dynamic shear modulus, or modulus of rigidity, is found using the fundamental torsional resonant frequency. Dynamic Young's modulus and dynamic shear modulus are used to compute Poisson's ratio.  
1.2 This test method is specifically appropriate for materials that are elastic, homogeneous, and isotropic (1).2  
1.3 Materials of a composite character (particulate, whisker, or fiber reinforced) may be tested by this test method with the understanding that the character (volume fraction, size, morphology, distribution, orientation, elastic properties, and interfacial bonding) of the reinforcement in the test specimen will have a direct effect on the elastic properties. These reinforcement effects shall be considered in interpreting the test results for composites.  
1.4 This test method shall not be used for determination of Poisson’s ratio of anisotropic materials.
Note 1: For anisotropic materials, Poisson’s ratio can have different values in different directions. Due to the lack of symmetry in anisotropic materials, the elasticity tensor cannot be reduced to only two independent numbers, and the simplified relation between E, G, and µ is not valid.  
1.5 This test method should not be used for specimens that have cracks or voids that are major discontinuities in the specimen.  
1.6 The test method should not be used when materials cannot be fabricated in a uniform rectangular or circular cross section.  
1.7 An elevated-temperature furnace and cryogenic chamber are described for measuring the dynamic elastic moduli as a function of temperature from –195 °C to 1200 °C.  
1.8 This test method may be modified for use in quality control. A range of acceptable resonant frequencies is determined for a specimen with a particular geometry and mass. Any specimen with a frequency response falling outside this frequency range is rejected. The actual modulus of each specimen need not be determined as long as the limits of the selected frequency range are known to include the resonant frequency that the specimen must possess if its geometry and mass are within specified tolerances.  
1.9 There are material-specific ASTM standards that cover the determination of resonant frequencies and elastic properties of specific materials by sonic resonance or by impulse excitation of vibration. Test Methods C215, C623, C74...

General Information

• Status: Published
• Publication Date: 30-Nov-2020
• ICS: 81.060.20 - Ceramic products
• Technical Committee: E28 - Mechanical Testing
• Drafting Committee: E28.04 - Uniaxial Testing

Relations

• Refers: ASTM C747-23 - Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• Effective Date: 01-Oct-2023
• Refers: ASTM C1198-20 - Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Sonic Resonance
• Effective Date: 01-Jan-2020
• Refers: ASTM C747-16 - Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• Effective Date: 01-Oct-2016
• Refers: ASTM C215-14 - Standard Test Method for Fundamental Transverse, Longitudinal, and<brk/> Torsional Resonant Frequencies of Concrete Specimens
• Effective Date: 15-Dec-2014
• Refers: ASTM E177-14 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2014
• Refers: ASTM C1259-14 - Standard Test Method for Dynamic Young&rsquo;s Modulus, Shear Modulus, and Poisson&rsquo;s Ratio for Advanced Ceramics by Impulse Excitation of Vibration
• Effective Date: 01-Jan-2014
• Refers: ASTM C1198-09(2013) - Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio for Advanced Ceramics by Sonic Resonance
• Effective Date: 01-Aug-2013
• Refers: ASTM E177-13 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2013
• Refers: ASTM E691-13 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-May-2013
• Refers: ASTM E691-11 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-Nov-2011
• Refers: ASTM C848-88(2011) - Standard Test Method for Young's Modulus, Shear Modulus, and Poisson's Ratio For Ceramic Whitewares by Resonance
• Effective Date: 01-Mar-2011
• Refers: ASTM E177-10 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-Oct-2010
• Refers: ASTM C747-93(2010)e1 - Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• Effective Date: 01-May-2010
• Refers: ASTM C623-92(2010) - Standard Test Method for Young's Modulus, Shear Modulus, and Poisson's Ratio for Glass and Glass-Ceramics by Resonance
• Effective Date: 01-Apr-2010
• Refers: ASTM C1198-09 - Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio for Advanced Ceramics by Sonic Resonance
• Effective Date: 04-Nov-2009

Overview

ASTM E1875-20a is the standard test method developed by ASTM International for determining the dynamic Young's modulus, shear modulus, and Poisson’s ratio of elastic materials using sonic resonance techniques. This nondestructive test method is essential in evaluating the dynamic elastic properties of materials by measuring their mechanical resonant frequencies, mass, and geometry. The method is highly valued in industries where accurate, fast, and noninvasive evaluation of material stiffness and structural integrity is required, especially for quality control, materials research, and manufacturing.

Key Topics

• Nondestructive Measurement: The sonic resonance method applies only minute stresses during testing, reducing the risk of specimen damage or fracture, and allows repeated evaluations on the same specimen.
• Dynamic Moduli Evaluation: The standard outlines the determination of dynamic Young’s modulus from the fundamental flexural resonant frequency and the dynamic shear modulus from the fundamental torsional resonant frequency. These are then used to calculate Poisson’s ratio for isotropic materials.
• Material Suitability: Best suited for materials that are elastic, homogeneous, and isotropic. The standard also applies to some composite materials-such as particulate, whisker, or fiber-reinforced specimens-although the interpretation of results must account for the specifics of the reinforcement.
• Temperature Variability: The method is applicable for a wide range of tests, including those at elevated and cryogenic temperatures (−195 °C to 1200 °C), enabling the analysis of dynamic elastic properties as materials undergo thermal cycling.
• Quality Control Integration: The flexibility of this ASTM method allows adaptation for routine quality control. Specimens are quickly screened based on whether their resonant frequencies fall within acceptable predefined ranges, with or without complete modulus calculations.

Applications

• Material Specification Compliance: Industries can use this test method to check if materials comply with mechanical property requirements, vital for safety-critical components.
• Research & Development: Provides researchers and engineers with reliable measurements of dynamic elastic properties for new material development or comparative studies.
• Quality Control in Manufacturing: Enables rapid inspection and sorting of manufactured items, identifying defective batches or materials with improper elastic moduli before they proceed further in the production chain.
• High-Temperature and Cryogenic Applications: Useful for characterizing materials required to maintain mechanical performance under extreme thermal conditions in aerospace, automotive, and energy sectors.
• Composite and Advanced Materials Testing: Though mainly designed for isotropic materials, ASTM E1875-20a is also referenced in interpreting results for composites, provided reinforcement characteristics are accounted for.

Related Standards

ASTM E1875-20a is closely aligned with other ASTM standards covering resonance or impulse excitation-based measurement of dynamic elastic properties for specific material classes. These include:

• ASTM C215: Resonant Frequencies of Concrete
• ASTM C623: Young’s Modulus and Shear Modulus for Glass and Glass-Ceramic Materials
• ASTM C747: Moduli of Elasticity for Carbon and Graphite by Sonic Resonance
• ASTM C848: Young’s Modulus, Shear Modulus, and Poisson’s Ratio of Ceramic Whitewares by Resonance
• ASTM C1198/C1259/C1548: Dynamic Modulus Evaluation for Advanced Ceramics and Refractories
• ASTM E1876: Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation

For comprehensive materials testing programs, practitioners frequently use ASTM E1875 in conjunction with these material-specific standards to ensure adherence to industry and application requirements.

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Keywords: ASTM E1875-20a, dynamic Young’s modulus, shear modulus, Poisson’s ratio, sonic resonance, elastic modulus testing, nondestructive testing, high-temperature modulus testing, composite materials testing, quality control, mechanical resonance frequencies, ASTM standards, materials characterization.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E1875 − 20a
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio by Sonic Resonance
This standard is issued under the fixed designation E1875; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope* 1.6 The test method should not be used when materials
cannot be fabricated in a uniform rectangular or circular cross
1.1 This test method covers the determination of the dy-
section.
namic elastic properties of elastic materials. Specimens of
these materials possess specific mechanical resonant frequen-
1.7 An elevated-temperature furnace and cryogenic cham-
cies that are determined by the modulus of elasticity, mass, and
ber are described for measuring the dynamic elastic moduli as
geometry of the test specimen. Therefore, the dynamic elastic
a function of temperature from –195 °C to 1200 °C.
propertiesofamaterialcanbecomputedifthegeometry,mass,
1.8 This test method may be modified for use in quality
and mechanical resonant frequencies of a suitable test speci-
control. A range of acceptable resonant frequencies is deter-
men of that material can be measured. The dynamic Young’s
mined for a specimen with a particular geometry and mass.
modulus is determined using the fundamental flexural resonant
Any specimen with a frequency response falling outside this
frequency.The dynamic shear modulus, or modulus of rigidity,
frequency range is rejected. The actual modulus of each
is found using the fundamental torsional resonant frequency.
specimen need not be determined as long as the limits of the
Dynamic Young’s modulus and dynamic shear modulus are
selected frequency range are known to include the resonant
used to compute Poisson’s ratio.
frequency that the specimen must possess if its geometry and
1.2 This test method is specifically appropriate for materials
mass are within specified tolerances.
that are elastic, homogeneous, and isotropic (1).
1.9 There are material-specific ASTM standards that cover
1.3 Materials of a composite character (particulate, whisker,
thedeterminationofresonantfrequenciesandelasticproperties
or fiber reinforced) may be tested by this test method with the
of specific materials by sonic resonance or by impulse excita-
understanding that the character (volume fraction, size,
tion of vibration. Test Methods C215, C623, C747, C848,
morphology, distribution, orientation, elastic properties, and
C1198, C1259, and C1548 differ from this test method in
interfacial bonding) of the reinforcement in the test specimen
several areas (for example; specimen size, dimensional
will have a direct effect on the elastic properties. These
tolerances, specimen preparation). The testing of these mate-
reinforcementeffectsshallbeconsideredininterpretingthetest
rials shall be done in compliance with these material specific
results for composites.
standards. Where possible, the procedures, specimen
1.4 This test method shall not be used for determination of
specifications, and calculations are consistent with these test
Poisson’s ratio of anisotropic materials. methods.
NOTE 1—For anisotropic materials, Poisson’s ratio can have different
1.10 A separate standard, Test Method E1876, governs
values in different directions. Due to the lack of symmetry in anisotropic
determination of dynamic elastic moduli by impulse excitation
materials, the elasticity tensor cannot be reduced to only two independent
instead of sonic resonance.
numbers, and the simplified relation between E, G, and µ is not valid.
1.11 The values stated in SI units are to be regarded as
1.5 This test method should not be used for specimens that
standard. No other units of measurement are included in this
have cracks or voids that are major discontinuities in the
standard.
specimen.
1.12 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
This test method is under the jurisdiction of ASTM Committee E28 on
responsibility of the user of this standard to establish appro-
Mechanical Testing and is the direct responsibility of Subcommittee E28.04 on
priate safety, health, and environmental practices and deter-
Uniaxial Testing.
Current edition approved Dec. 1, 2020. Published March 2021. Originally mine the applicability of regulatory limitations prior to use.
approved in 1997. Last previous edition approved in 2020 as E1875-20. DOI:
1.13 This international standard was developed in accor-
10.1520/E1875-20A.
dance with internationally recognized principles on standard-
The boldface numbers in parentheses refer to a list of references at the end of
this standard. ization established in the Decision on Principles for the
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1875 − 20a
–2
Development of International Standards, Guides and Recom- 3.1.3 dynamic shear modulus, G [FL ],n—the value of
d
mendations issued by the World Trade Organization Technical the shear modulus determined using an oscillatory applied
force or displacement and in conformance with this test
Barriers to Trade (TBT) Committee.
method.
–2
2. Referenced Documents
3.1.4 elastic limit [FL ],n—the greatest stress that a
material is capable of sustaining without permanent strain
2.1 ASTM Standards:
remaining upon complete release of the stress.
C215 Test Method for Fundamental Transverse,
3.1.4.1 Discussion—Due to practical considerations in de-
Longitudinal, and Torsional Resonant Frequencies of
termining the elastic limit, measurements of strain using a
Concrete Specimens
small force, rather than zero force, are usually taken as the
C623 Test Method for Young’s Modulus, Shear Modulus,
initial and final reference.
and Poisson’s Ratio for Glass and Glass-Ceramics by
–2
Resonance 3.1.5 modulus of elasticity [FL ],n—the ratio of stress to
C747 Test Method for Moduli of Elasticity and Fundamental corresponding strain below the proportional limit.
Frequencies of Carbon and Graphite Materials by Sonic
3.1.5.1 Discussion—The stress-strain relationships of many
Resonance
materialsdonotconformtoHooke’slawthroughouttheelastic
C848 Test Method for Young’s Modulus, Shear Modulus, range, but deviate therefrom even at stresses well below the
and Poisson’s Ratio For Ceramic Whitewares by Reso- elastic limit. For such materials, the slope of either the tangent
nance to the stress-strain curve at the origin or at a low stress, the
secant drawn from the origin to any specified point on the
C1198 Test Method for Dynamic Young’s Modulus, Shear
Modulus, and Poisson’s Ratio for Advanced Ceramics by stress-strain curve, or the chord connecting any two specified
points on the stress-strain curve is usually taken to be the
Sonic Resonance
C1259 Test Method for Dynamic Young’s Modulus, Shear “modulus of elasticity.” In these cases, the modulus should be
designated as the “tangent modulus,” the “secant modulus,” or
Modulus, and Poisson’s Ratio for Advanced Ceramics by
Impulse Excitation of Vibration the “chord modulus,” and the point or points on the stress-
strain curve described. Thus, for materials where the stress-
C1548 Test Method for Dynamic Young’s Modulus, Shear
strain relationship is curvilinear rather than linear, one of the
Modulus, and Poisson’s Ratio of Refractory Materials by
four following terms may be used:
Impulse Excitation of Vibration
–2
(a) initial tangent modulus [FL ], n—the slope of the
E6 Terminology Relating to Methods of Mechanical Testing
stress-strain curve at the origin.
E177 Practice for Use of the Terms Precision and Bias in
–2
(b) tangent modulus [FL ], n—the slope of the stress-
ASTM Test Methods
strain curve at any specified stress or strain.
E691 Practice for Conducting an Interlaboratory Study to
–2
(c) secant modulus [FL ], n—the slope of the secant
Determine the Precision of a Test Method
drawnfromtheorigintoanyspecifiedpointonthestress-strain
E1876 Test Method for Dynamic Young’s Modulus, Shear
curve.
Modulus, and Poisson’s Ratio by Impulse Excitation of
–2
(d) chord modulus[FL ], n—theslopeofthechorddrawn
Vibration
between any two specified points on the stress-strain curve
below the elastic limit of the material.
3. Terminology
3.1.5.2 Discussion—Modulus of elasticity, like stress, is
3.1 Definitions:Terms common to mechanical testing that
expressed in force per unit of area (pounds per square inch,
appear inTerminology E6 and are listed in this section apply to
etc.).
this test method. In addition, the terms indicated temperature
3.1.6 Poisson’s ratio, µ,n—the negative of the ratio of
and specified temperature are used as defined in Terminology
transverse strain to the corresponding axial strain resulting
E6.
from an axial stress below the proportional limit of the
3.1.1 dynamic mechanical measurement, n—a technique in
material.
which either the modulus or damping, or both, of a substance
3.1.6.1 Discussion—Poisson’s ratio can be negative for
under oscillatory applied force or displacement is measured as
some materials, for example, a tensile transverse strain will
a function of temperature, frequency, or time, or a combination
result from a tensile axial strain.
thereof.
3.1.6.2 Discussion—Poisson’sratiowillhavemorethanone
–2
3.1.2 dynamic Young’s modulus, E [FL ],n—the value of
d
value if the material is not isotropic.
the Young’s modulus determined using an oscillatory applied
–2
3.1.7 proportional limit [FL ],n—the greatest stress that a
force or displacement and in conformance with this test
material is capable of sustaining without deviation from
method.
proportionality of stress to strain (Hooke’s law).
3.1.7.1 Discussion—Many experiments have shown that
values observed for the proportional limit vary greatly with the
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
sensitivity and accuracy of the testing equipment, eccentricity
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
of loading, the scale to which the stress-strain diagram is
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. plotted, and other factors. When determination of proportional
E1875 − 20a
limit is required, the procedure and the sensitivity of the test 3.2.6.1 Discussion—For the fundamental flexural
equipment should be specified. resonance, the nodes are located at 0.224 L from each end,
–2
where L is the length of the specimen.
3.1.8 shear modulus, G [FL ],n—the ratio of shear stress
to corresponding shear strain below the proportional limit, also
3.2.7 resonance, n—state of slender rectangular or cylindri-
called torsional modulus and modulus of rigidity.
cal specimen driven into one of the modes of vibration
3.1.8.1 Discussion—The value of the shear modulus can
described in 3.2.3 or 3.2.9 when the imposed frequency is such
depend on the direction in which it is measured if the material
that the resultant displacements for a given amount of driving
is not isotropic. Wood, many plastics and certain metals are
force are at a maximum.
markedly anisotropic. Deviations from isotropy should be
3.2.7.1 Discussion—The resonant frequencies are natural
suspected if the shear modulus differs from that determined by
vibration frequencies that are determined by the modulus of
substituting independently measured values of Young’s
elasticity, mass, and dimensions of the test specimen.
modulus, E, and Poisson’s ratio, µ, in the relation:
3.2.8 slender rectangular or cylindrical specimen, n—in
E
G 5
dynamic mechanical measurement, a specimen whose ratio of
2~11µ!
length to minimum cross-sectional dimension is at least five
3.1.8.2 Discussion—When reporting values of shear
and should be in the range from 20 to 25.
modulus, the range of stress over which it is measured should
be stated.
3.2.9 torsional vibrations, n—oscillations that occur in each
–2
3.1.9 Young’s modulus, E [FL ],n—the ratio of tensile or cross-sectional plane of a slender rectangular or cylindrical
compressive stress to corresponding strain below the propor-
specimen, such that the plane twists around the length dimen-
tional limit of the material.
sion axis.
3.2 Definitions of Terms Specific to This Standard:
4. Summary of Test Method
3.2.1 anti-nodes, n—two or more locations in an uncon-
strained slender rectangular or cylindrical specimen in reso-
4.1 This test method measures the resonant frequencies of
nance that have local maximum displacements.
test specimens of suitable geometry by exciting them at
3.2.1.1 Discussion—For the fundamental flexural
continuously variable frequencies. Mechanical excitation of
resonance, the anti-nodes are located at the two ends and the
the specimen is provided through the use of a driving trans-
center of the specimen.
ducer that transforms a cyclic electrical signal into a cyclic
3.2.2 elastic, adj—the property of a material such that an
mechanical force on the specimen. A detecting transducer
application of stress within the elastic limit of that material
senses the resulting mechanical vibrations of the specimen and
making up the body being stressed will cause an instantaneous
transforms them into an electrical signal. The amplitude and
and uniform deformation that will be eliminated upon removal
frequency of the signal are measured by an oscilloscope or
of the stress, with the body returning instantly to its original
other means to detect resonance. The resonant frequencies,
size and shape without energy loss.
dimensions, and mass of the specimen are used to calculate
3.2.2.1 Discussion—Most elastic materials conform to this
dynamic Young’s modulus and dynamic shear modulus.
definition well enough to make this resonance test valid.
3.2.3 flexural vibrations, n—oscillations that occur in a 5. Significance and Use
slender rectangular or cylindrical specimen in a vertical plane
5.1 This test method has advantages in certain respects over
normal to the length dimension.
the use of static loading systems for measuring moduli.
3.2.4 homogeneous, adj—the condition of a specimen such
5.1.1 This test method is nondestructive in nature. Only
that the composition and density are uniform, such that any
minute stresses are applied to the specimen, thus minimizing
smaller specimen taken from the original is representative of
the possibility of fracture.
the whole.
5.1.2 The period of time during which measurement stress
3.2.4.1 Discussion—Practically, as long as the geometrical
is applied and removed is of the order of hundreds of
dimensions of the test specimen are large with respect to the
microseconds. With this test method it is feasible to perform
size of individual grains, crystals, or components, the body can
measurements at elevated temperatures, where delayed elastic
be considered homogeneous.
and creep effects would invalidate modulus of elasticity
3.2.5 isotropic, adj—the condition of a specimen such that
measurements calculated from static loading.
the values of the elastic properties are the same in all directions
in the material.
5.2 This test method is suitable for detecting whether a
3.2.5.1 Discussion—Materials are considered isotropic on a
material meets the specifications, if cognizance is given to one
macroscopic scale, if they are homogeneous and there is a
important fact in materials are often sensitive to thermal
random distribution and orientation of phases, crystallites, and
history. Therefore, the thermal history of a test specimen must
components.
be considered in comparing experimental values of moduli to
reference or standard values. Specimen descriptions should
3.2.6 nodes, n—one or more locations of a slender rectan-
includeanyspecificthermaltreatmentsthatthespecimenshave
gular or cylindrical specimen in resonance that have a constant
zero displacement. received.
E1875 − 20a
depending on the type of coupling chosen for use between the
driving transducer and the specimen.
6.4.2 The detecting transducer may be a crystal or magnetic
reluctance type of phonograph cartridge or a capacitive pickup.
An electromagnetic coupling system with an attached metal
foil may also be used, with due consideration for effects of the
foilonthenaturalvibrationofthetestspecimen.Thefrequency
responseofthedetectingtransduceracrossthefrequencyrange
ofinterestshallhaveatleasta6.5kHzbandwidthbefore–3dB
power loss occurs.
6.5 Detecting-transducer Amplifier, in the detector circuit
shall be impedance matched with the type of detecting trans-
ducer selected and shall serve as a preamplifier for the
ocsilloscope.
6.6 Oscilloscope, any model suitable for general laboratory
work.
6.7 Frequency Meter, should be digital, and shall be able to
measure frequencies to within 61 Hz.
6.8 Furnace—For data at elevated temperature, a furnace
shall be used that is capable of controlled heating and cooling.
It shall have a specimen zone large enough for the specimen to
be uniform in temperature within 65 °C along its length
through the range of specified temperatures.
6.8.1 An independent thermocouple should be placed in
FIG. 1 Block Diagram of a Typical Test Apparatus
close proximity to (within 5 mm), but not touching, the center
of the specimen to accurately measure temperature during
heating and cooling. Ensure that the indicated temperature
6. Apparatus
from the remote thermocouple and the specimen temperature
6.1 The test apparatus is shown in Fig. 1. It consists of a
do not differ.
variable-frequency audio oscillator, used to generate a sinusoi-
6.9 Cryogenic Chamber—For data at cryogenic
dal voltage, and a power amplifier and suitable driving trans-
temperatures, the cryogenic chamber shall be capable of
ducer to convert the electrical signal to a mechanical driving
controlled heating and cooling, frost-free, and uniform in
vibration.Afrequencymeter,whichshouldbedigital,monitors
temperature within 65 °C over the length of the specimen at
the audio oscillator output to provide an accurate frequency
any specified temperature. A suitable cryogenic chamber is
determination.Asuitable suspension-coupling system supports
shown in Fig. 2 (2).
the test specimen.Adectecting transducer senses the mechani-
6.9.1 An independent thermocouple should be placed in
cal vibration in the specimen and to convert it into an electrical
close proximity to (within 5 mm), but not touching, the center
signal that is passed through an amplifier and displayed on an
of the specimen to accurately measure temperature during
indicating meter. The meter may be a voltmeter,
heating and cooling. Ensure that the indicated temperature
microammeter, or oscilloscope. An oscilloscope should be
from the remote thermocouple and the specimen temperature
used, because it enables the operator to positively identify
do not differ.
resonances, including higher order harmonics, by Lissajous
figureanalysis.IfaLissajousfigureisdesired,theoutputofthe 6.10 Specimen Suspension—The method of specimen sus-
audio oscillator should be displayed on the horizontal axis of
pension shall be adequate for the specified temperatures and
the oscilloscope. If temperature-dependent data are desired, a allow the specimen to vibrate without significant restriction.
suitable furnace or cryogenic chamber shall be used. Details of
Thread suspension should be used for cryogenic and elevated-
the equipment are as follows: temperature testing. (See Fig. 1 and Fig. 3.) Common cotton
thread, silica-glass fiber thread, oxidation-resistant nickel (or
6.2 Audio Oscillator, having a continuously variable fre-
platinum) alloy wire, or platinum wire may be used. The
quencyoutputfromabout100Hztoatleast30kHz.Frequency
specimen should be initially suspended at distances of approxi-
drift shall not exceed 1 Hz/min for any given setting.
mately 0.1 L from each end. The specimen should not be
6.3 Driving-transducer Amplifier, having a power output
suspended at its fundamental flexural node locations (0.224 L
sufficient to ensure that the driving transducer can excite any
from each end). The suspension point distances may be
specimen the mass of which falls within a specified range.
adjusted experimentally to maximize the vibrational deflection
6.4 Transducers—Two transducers shall be used. and resulting signal. For torsional vibration, the axes of
6.4.1 Thedrivingtransducermaybeaspeakerofthetweeter suspension shall be off-center from the longitudinal axis of the
type, or a magnetic cutting head, or other similar device specimen (shown in Fig. 3).
E1875 − 20a
FIG. 4 Specimen Positioned for Measurement of Flexural and Tor-
sional Resonant Frequencies Using Direct Support and Direct
Contact Transducers
NOTE 1—Legend:
1 = Cylindrical glass jar
to oscillate without significant restriction in the desired mode.
2 = Glass wool
In flexural modes, the specimen should be supported at its
3 = Plastic foam
transverse fundamental node locations (0.224 L from each
4 = Vacuum jar
5 = Heater disk
end). In torsional modes the specimen should be supported at
6 = Copper plate
its center point. The supports should have minimal area in
7 = Thermocouple
contact with the specimen and shall be cork, rubber, or similar
8 = Specimen
material. In order to properly identify resonant frequencies, the
9 = Suspension wires
10 = Fill port for liquid transducers should be movable along the total specimen length
FIG. 2 Detail Drawing of a Typical Cryogenic Chamber
and width. See Fig. 4. The transducer contact pressure should
be consistent with good response and minimal interference
with the free vibration of the specimen.
7. Test Specimen
7.1 Prepare the specimens so that they are either rectangular
or circular in cross section. Either geometry may be used to
measure both dynamic Young’s modulus and dynamic shear
modulus.
NOTE 3—Experimental difficulties in obtaining torsional resonant
frequencies for a cylindrical specimen usually preclude its use in
determining dynamic shear modulus, although the equations for comput-
ing dynamic shear modulus with a cylindrical specimen are both simpler
and more accurate than those used with a rectangular specimen.
7.2 Select the size so that, for an estimated dynamic
modulus of elasticity, the resonant frequencies measured will
fall within the range of frequency response of the transducers
used. A slender specimen with a ratio of length to minimum
cross-sectional dimension greater than 5 and approximately 25
should be used for ease in calculation. For dynamic shear
modulus measurements of rectangular specimens, a ratio of
width to thickness of five sho
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E1875 − 20 E1875 − 20a
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio by Sonic Resonance
This standard is issued under the fixed designation E1875; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope*
1.1 This test method covers the determination of the dynamic elastic properties of elastic materials. Specimens of these materials
possess specific mechanical resonant frequencies that are determined by the modulus of elasticity, mass, and geometry of the test
specimen. Therefore, the dynamic elastic properties of a material can be computed if the geometry, mass, and mechanical resonant
frequencies of a suitable test specimen of that material can be measured. Dynamic The dynamic Young’s modulus is determined
using the resonant frequency in the flexural mode of vibration. fundamental flexural resonant frequency. The dynamic shear
modulus, or modulus of rigidity, is found using the fundamental torsional resonant vibrations.frequency. Dynamic Young’s
modulus and dynamic shear modulus are used to compute Poisson’s ratio.
1.2 This test method is specifically appropriate for materials that are elastic, homogeneous, and isotropic (1).
1.3 This test method is specifically appropriate for materials that are elastic, homogeneous, and isotropic (1).Materials of a
composite character (particulate, whisker, or fiber reinforced) may be tested by this test method with the understanding that the
character (volume fraction, size, morphology, distribution, orientation, elastic properties, and interfacial bonding) of the
reinforcement in the test specimen will have a direct effect on the elastic properties. These reinforcement effects mustshall be
considered in interpreting the test results for composites. This test method is not satisfactory for specimens that have cracks or
voids that are major discontinuities in the specimen. Neither is the test method satisfactory when these materials cannot be
fabricated in a uniform rectangular or circular cross section.
1.4 This test method shall not be used for determination of Poisson’s ratio of anisotropic materials.
NOTE 1—For anisotropic materials, Poisson’s ratio can have different values in different directions. Due to the lack of symmetry in anisotropic materials,
the elasticity tensor cannot be reduced to only two independent numbers, and the simplified relation between E,G, and μ is not valid.
1.5 This test method should not be used for specimens that have cracks or voids that are major discontinuities in the specimen.
1.6 The test method should not be used when materials cannot be fabricated in a uniform rectangular or circular cross section.
1.7 A high-temperatureAn elevated-temperature furnace and cryogenic cabinetchamber are described for measuring the dynamic
elastic moduli as a function of temperature from –195–195 °C to 1200 °C.
This test method is under the jurisdiction of ASTM Committee E28 on Mechanical Testing and is the direct responsibility of Subcommittee E28.04 on Uniaxial Testing.
Current edition approved April 15, 2020Dec. 1, 2020. Published May 2020March 2021. Originally approved in 1997. Last previous edition approved in 20132020 as
E1875-13.-20. DOI: 10.1520/E1875-20.10.1520/E1875-20A.
The boldface numbers in parentheses refer to a list of references at the end of this standard.
*A Summary of Changes section appears at the end of this standard
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E1875 − 20a
1.8 Modification of this This test method may be modified for use in quality control is possible. control. A range of acceptable
resonant frequencies is determined for a specimen with a particular geometry and mass. Any specimen with a frequency response
falling outside this frequency range is rejected. The actual modulus of each specimen need not be determined as long as the limits
of the selected frequency range are known to include the resonant frequency that the specimen must possess if its geometry and
mass are within specified tolerances.
1.9 There are material-specific ASTM standards that cover the determination of resonanceresonant frequencies and elastic
properties of specific materials by sonic resonance or by impulse excitation of vibration. Test Methods C215, C623, C747, C848,
C1198, C1259and, C1259 mayand C1548 differ from this test method in several areas (for example; samplespecimen size,
dimensional tolerances, samplespecimen preparation). The testing of these materials shall be done in compliance with these
material specific standards. Where possible, the procedures, samplespecimen specifications, and calculations are consistent with
these test methods.
1.10 A separate standard, Test Method E1876, governs determination of dynamic elastic moduli by impulse excitation instead of
sonic resonance.
1.11 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.12 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, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.13 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
C215 Test Method for Fundamental Transverse, Longitudinal, and Torsional Resonant Frequencies of Concrete Specimens
C623 Test Method for Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Glass and Glass-Ceramics by Resonance
C747 Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
C848 Test Method for Young’s Modulus, Shear Modulus, and Poisson’s Ratio For Ceramic Whitewares by Resonance
C1198 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Sonic
Resonance
C1259 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Impulse
Excitation of Vibration
C1548 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio of Refractory Materials by Impulse
Excitation of Vibration
E6 Terminology Relating to Methods of Mechanical Testing
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
E691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
E1876 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation of Vibration
3. Terminology
3.1 Definitions:Terms common to mechanical testing.testing that appear in Terminology E6 and are listed in this section apply
to this test method. In addition, the terms indicated temperature and specified temperature are used as defined in Terminology E6.
3.1.1 dynamic mechanical measurement, n—a technique in which either the modulus or damping, or both, of a substance under
oscillatory applied force or displacement is measured as a function of temperature, frequency, or time, or a combination thereof.
–2
3.1.2 dynamic Young’s modulus, E [FL ],n—the value of the Young’s modulus determined using an oscillatory applied force or
d
displacement and in conformance with this test method.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
E1875 − 20a
–2
3.1.3 dynamic shear modulus, G [FL ],n—the value of the shear modulus determined using an oscillatory applied force or
d
displacement and in conformance with this test method.
–2
3.1.4 elastic limit [FL ],n—the greatest stress that a material is capable of sustaining without permanent strain remaining upon
complete release of the stress.
3.1.4.1 Discussion—
Due to practical considerations in determining the elastic limit, measurements of strain using a small force, rather than zero force,
are usually taken as the initial and final reference. (E6)
–2
3.1.5 modulus of elasticity [FL ],n—the ratio of stress to corresponding strain below the proportional limit.
3.1.5.1 Discussion—
The stress-strain relationships of many materials do not conform to Hooke’s law throughout the elastic range, but deviate therefrom
even at stresses well below the elastic limit. For such materials, the slope of either the tangent to the stress-strain curve at the origin
or at a low stress, the secant drawn from the origin to any specified point on the stress-strain curve, or the chord connecting any
two specified points on the stress-strain curve is usually taken to be the “modulus of elasticity.” In these cases, the modulus should
be designated as the “tangent modulus,” the “secant modulus,” or the “chord modulus,” and the point or points on the stress-strain
curve described. Thus, for materials where the stress-strain relationship is curvilinear rather than linear, one of the four following
terms may be used:
–2
(a) initial tangent modulus [FL ], n—the slope of the stress-strain curve at the origin.
–2
(b) tangent modulus [FL ], n—the slope of the stress-strain curve at any specified stress or strain.
–2
(c) secant modulus [FL ], n—the slope of the secant drawn from the origin to any specified point on the stress-strain curve.
–2
(d) chord modulus [FL ], n—the slope of the chord drawn between any two specified points on the stress-strain curve below
the elastic limit of the material.
3.1.5.2 Discussion—
Modulus of elasticity, like stress, is expressed in force per unit of area (pounds per square inch, etc.).
3.1.6 Poisson’s ratio, μ,n—the negative of the ratio of transverse strain to the corresponding axial strain resulting from an axial
stress below the proportional limit of the material.
3.1.6.1 Discussion—
Poisson’s ratio maycan be negative for some materials, for example, a tensile transverse strain will result from a tensile axial strain.
3.1.6.2 Discussion—
Poisson’s ratio will have more than one value if the material is not isotropic. (E6)
–2
3.1.7 proportional limit [FL ]], , n—the greatest stress that a material is capable of sustaining without deviation from
proportionality of stress to strain (Hooke’s law).
3.1.7.1 Discussion—
Many experiments have shown that values observed for the proportional limit vary greatly with the sensitivity and accuracy of the
testing equipment, eccentricity of loading, the scale to which the stress-strain diagram is plotted, and other factors. When
determination of proportional limit is required, the procedure and the sensitivity of the test equipment should be specified. (E6)
–2
3.1.8 shear modulus (G)modulus, G [FL ],n—the ratio of shear stress to corresponding shear strain below the proportional limit,
also called torsional modulus and modulus of rigidity.
3.1.8.1 Discussion—
The value of the shear modulus maycan depend on the direction in which it is measured if the material is not isotropic. Wood, many
plastics and certain metals are markedly anisotropic. Deviations from isotropy should be suspected if the shear modulus differs
from that determined by substituting independently measured values of Young’s modulus, E, and Poisson’s ratio, μ, in the
relation:relation:
G 5 E/ 2 11μ
@ ~ !#
E
G 5
2~11μ!
3.1.8.2 Discussion—
In general, it is advisable in When reporting values of shear modulus to state modulus, the range of stress over which it is
measured.measured should be stated.
E1875 − 20a
–2
3.1.9 Young’s modulus (E)modulus, E [FL ]], , n—the ratio of tensile or compressive stress to corresponding strain below the
proportional limit of the material. (E6)
3.2 Definitions of Terms Specific to This Standard:
3.2.1 anti-nodes, n—two or more locations in an unconstrained slender rodrectangular or bar cylindrical specimen in resonance
that have local maximum displacements.
3.2.1.1 Discussion—
For the fundamental flexureflexural resonance, the anti-nodes are located at the two ends and the center of the specimen.
3.2.2 elastic, adj—the property of a material such that an application of stress within the elastic limit of that material making up
the body being stressed will cause an instantaneous and uniform deformation that will be eliminated upon removal of the stress,
with the body returning instantly to its original size and shape without energy loss.
3.2.2.1 Discussion—
Most elastic materials conform to this definition well enough to make this resonance test valid.
3.2.3 flexural vibrations, n—oscillations that occur in a slender rodrectangular or bar cylindrical specimen in a vertical plane
normal to the length dimension.
3.2.4 homogeneous, adj—the condition of a specimen such that the composition and density are uniform, such that any smaller
specimen taken from the original is representative of the whole.
3.2.4.1 Discussion—
Practically, as long as the geometrical dimensions of the test specimen are large with respect to the size of individual grains,
crystals, or components, the body can be considered homogeneous.
3.2.5 isotropic, adj—the condition of a specimen such that the values of the elastic properties are the same in all directions in the
material.
3.2.5.1 Discussion—
Materials are considered isotropic on a macroscopic scale, if they are homogeneous and there is a random distribution and
orientation of phases, crystallites, and components.
3.2.6 nodes, n—one or more locations of a slender rodrectangular or bar cylindrical specimen in resonance that have a constant
zero displacement.
3.2.6.1 Discussion—
For the fundamental flexural resonance, the nodes are located at 0.224 L from each end, where L is the length of the specimen.
3.2.7 resonance, n—state of slender rodrectangular or bar cylindrical specimen driven into one of the modes of vibration described
in 3.2.3 or 3.2.9 when the imposed frequency is such that the resultant displacements for a given amount of driving force are at
a maximum.
3.2.7.1 Discussion—
The resonant frequencies are natural vibration frequencies that are determined by the modulus of elasticity, mass, and dimensions
of the test specimen.
3.2.8 slender rodrectangular or bar, cylindrical specimen, n—in dynamic elastic property testing, mechanical measurement, a
specimen whose ratio of length to minimum cross-sectional dimension is at least five and preferably should be in the range from
20 to 25.
3.2.9 torsional vibrations, n—oscillations that occur in each cross-sectional plane of a slender rodrectangular or bar, cylindrical
specimen, such that the plane twists around the length dimension axis.
4. Summary of Test Method
4.1 This test method measures the resonant frequencies of test specimens of suitable geometry by exciting them at continuously
variable frequencies. Mechanical excitation of the barsspecimen is provided through the use of a driving transducer that transforms
a cyclic electrical signal into a cyclic mechanical force on the specimen. A seconddetecting transducer senses the resulting
mechanical vibrations of the specimen and transforms them into an electrical signal. The amplitude and frequency of the signal
E1875 − 20a
FIG. 1 Block Diagram of a Typical Test Apparatus
are measured by an oscilloscope or other means to detect resonance. The resonant frequencies, dimensions, and mass of the
specimen are used to calculate dynamic Young’s modulus and dynamic shear modulus.
5. Significance and Use
5.1 This test method has advantages in certain respects over the use of static loading systems for measuring moduli.
5.1.1 This test method is nondestructive in nature. Only minute stresses are applied to the specimen, thus minimizing the
possibility of fracture.
5.1.2 The period of time during which measurement stress is applied and removed is of the order of hundreds of microseconds.
With this test method it is feasible to perform measurements at highelevated temperatures, where delayed elastic and creep effects
would invalidate modulus of elasticity measurements calculated from static loading.
5.2 This test method is suitable for detecting whether a material meets the specifications, if cognizance is given to one important
fact in materials are often sensitive to thermal history. Therefore, the thermal history of a test specimen must be considered in
comparing experimental values of moduli to reference or standard values. Specimen descriptions should include any specific
thermal treatments that the specimens have received.
6. Apparatus
6.1 The test apparatus is shown in Fig. 1. It consists of a variable-frequency audio oscillator, used to generate a sinusoidal voltage,
and a power amplifier and suitable driving transducer to convert the electrical signal to a mechanical driving vibration. A frequency
meter (preferably digital) meter, which should be digital, monitors the audio oscillator output to provide an accurate frequency
determination. A suitable suspension-coupling system supports the test specimen. Another transducer acts to detect A dectecting
transducer senses the mechanical vibration in the specimen and to convert it into an electrical signal that is passed through an
amplifier and displayed on an indicating meter. The meter may be a voltmeter, microammeter, or oscilloscope. An oscilloscope is
recommended should be used, because it enables the operator to positively identify resonances, including higher order harmonics,
by Lissajous figure analysis. If a Lissajous figure is desired, the output of the oscillator is also coupled toaudio oscillator should
E1875 − 20a
be displayed on the horizontal platesaxis of the oscilloscope. If temperature-dependent data are desired, a suitable furnace or
cryogenic chamber is shall be used. Details of the equipment are as follows:
6.2 Audio Oscillator, having a continuously variable frequency output from about 100 Hz to at least 30 kHz. Frequency drift shall
not exceed 1 Hz/min for any given setting.
6.3 AudioDriving-transducer Amplifier, having a power output sufficient to ensure that the type of driving transducer used can
excite any specimen the mass of which falls within a specified range.
6.4 Transducers—Two are required; one used as a driver may be a speaker of the tweeter type or a magnetic cutting head or other
similar device depending on the type of coupling chosen for use between the transducer and the specimen. The other transducer,
used as a detector, may be a crystal or magnetic reluctance type of phonograph cartridge. A capacitive pickup may be used if
desired. An electromagnetic coupling system with an attached metal foil may also be used, with due consideration for effects of
the foil on the natural vibration of the test bar. The frequency response of the transducer across the frequency range of interest shall
have at least a 6.5 kHz bandwidth before –3 dB power loss occurs.transducers shall be used.
6.4.1 The driving transducer may be a speaker of the tweeter type, or a magnetic cutting head, or other similar device depending
on the type of coupling chosen for use between the driving transducer and the specimen.
6.4.2 The detecting transducer may be a crystal or magnetic reluctance type of phonograph cartridge or a capacitive pickup. An
electromagnetic coupling system with an attached metal foil may also be used, with due consideration for effects of the foil on the
natural vibration of the test specimen. The frequency response of the detecting transducer across the frequency range of interest
shall have at least a 6.5 kHz bandwidth before –3 dB power loss occurs.
6.5 PowerDetecting-transducer Amplifier, in the detector circuit shall be impedance matched with the type of detectordetecting
transducer selected and shall serve as a prescope amplifier.preamplifier for the ocsilloscope.
6.6 Oscilloscope, any model suitable for general laboratory work.
6.7 Frequency Counter,Meter, preferably should be digital, and shall be able to measure frequencies to within 61 Hz.
6.8 Furnace—IfFor data at an elevated temperature are desired, elevated temperature, a furnace shall be used that is capable of
controlled heating and cooling. It shall have a specimen zone large enough for the specimen to be uniform in temperature within
65 °C along its length through the range of temperatures encountered in testing. It is recommended that an independent
thermocouple be placed in close proximity to (within 5 mm), but not touching, the center of the specimen to accurately measure
temperature during heating and cooling.specified temperatures.
6.8.1 An independent thermocouple should be placed in close proximity to (within 5 mm), but not touching, the center of the
specimen to accurately measure temperature during heating and cooling. Ensure that the indicated temperature from the remote
thermocouple and the specimen temperature do not differ.
6.9 Cryogenic Chamber—For data at cryogenic temperatures, any the cryogenic chamber shall suffice that shall be capable of
controlled heating/cooling, frost-free and heating and cooling, frost-free, and uniform in temperature within 65 °C over the length
of the specimen at any selectedspecified temperature. A suitable cryogenic chamber is shown in Fig. 2 (2). It is recommended that
an independent thermocouple be placed in close proximity to (within 5 mm), but not touching, the center of the specimen to
accurately measure temperature during heating and cooling.
6.9.1 An independent thermocouple should be placed in close proximity to (within 5 mm), but not touching, the center of the
specimen to accurately measure temperature during heating and cooling. Ensure that the indicated temperature from the remote
thermocouple and the specimen temperature do not differ.
6.10 Specimen Suspension—AnyThe method of specimen suspension shall be used that is adequate for the temperatures
encountered in testing and that allows specified temperatures and allow the specimen to vibrate without significant restriction.
Thread suspension is the system of choice should be used for cryogenic and high-temperatureelevated-temperature testing. (See
Fig. 1 and Fig. 3.) Common cotton thread, silica-glass fiber thread, oxidation-resistant nickel (or platinum) alloy wire, or platinum
E1875 − 20a
NOTE 1—Legend:
1 = Cylindrical glass jar
2 = Glass wool
3 = Plastic foam
4 = Vacuum jar
5 = Heater disk
6 = Copper plate
7 = Thermocouple
8 = SampleSpecimen
9 = Suspension wires
10 = Fill port for liquid
FIG. 2 Detail Drawing of a Typical Cryogenic Chamber
FIG. 3 Specimen Positioned for Measurement of Flexural and Torsional Resonant Frequencies Using Thread or Wire Suspension
wire may be used. If metal wire suspension is used in the furnace, coupling characteristics will be improved if, outside the
temperature zone, the wire is coupled to cotton thread, and the thread is coupled to the transducer. The specimen should be initially
suspended at distances of approximately 0.1 L from each end. The specimen should not be suspended at its fundamental flexural
node locations (0.224 L from each end). The suspension point distances canmay be adjusted experimentally to maximize the
vibrational deflection and resulting signal. For torsional vibration, the axes of suspension shall be off-center from the longitudinal
axis of the specimen (shown in Fig. 3).
E1875 − 20a
FIG. 4 Specimen Positioned for Measurement of Flexural and Torsional Resonant Frequencies Using Direct Support and Direct Contact
Transducers
NOTE 2—If metal wire suspension is used in the furnace, coupling characteristics will be improved if, outside the temperature zone, the wire is coupled
to cotton thread, and the thread is coupled to the transducer.
6.11 Specimen Supports—If the specimen is supported on direct contact supports, the supports shall permit the specimen to
oscillate without significant restriction in the desired mode. This is accomplished for flexural modes by supporting the specimen
In flexural modes, the specimen should be supported at its transverse fundamental node locations (0.224 L from each end). In
torsional modes the specimen should be supported at its center point. The supports should have minimal area in contact with the
specimen and shall be cork, rubber, or similar material. In order to properly identify resonant frequencies, the transducers should
be movable along the total specimen length and width. See Fig. 4. The transducer contact pressure should be consistent with good
response and minimal interference with the free vibration of the specimen.
7. Test Specimen
7.1 Prepare the specimens so that they are either rectangular or circular in cross section. Either geometry may be used to measure
both dynamic Young’s modulus and dynamic shear modulus. However, experimental difficulties in obtaining torsional resonant
frequencies for a cylindrical specimen usually preclude its use in determining shear modulus, although the equations for computing
shear modulus with a cylindrical specimen are both simpler and more accurate than those used with a rectangular bar.
NOTE 3—Experimental difficulties in obtaining torsional resonant frequencies for a cylindrical specimen usually preclude its use in determining dynamic
shear modulus, although the equations for computing dynamic shear modulus with a cylindrical specimen are both simpler and more accurate than those
used with a rectangular specimen.
7.2 Resonant frequencies for a given specimen are functions of the bar dimensions as well as its mass and moduli; therefore,
dimensions should be selected with this relationship in mind. Make selection of Select the size so that, for an estimated dynamic
modulus of elasticity, the resonant frequencies measured will fall within the range of frequency response of the transducers used.
A slender rodspecimen with a ratio of length to minimum cross-sectional dimension greater than 5 and approximately 25 is
preferred should be used for ease in calculation. For dynamic shear modulus measurements of rectangular bars,specimens, a ratio
of width to thickness of five is recommended for minimizing should be used to minimize experimental difficulties.
NOTE 4—Resonant frequencies for a given specimen are functions of the specimen dimensions as well as its mass and moduli.
7.2.1 These specimen sizes should produce a fundamental flexural resonant frequency in the range from 10001000 Hz to
10 000 Hz and a fundamental torsional resonant frequency in the range from 10 00010 000 Hz to 30 000 Hz. Specimens shall have
a minimum mass of 5 g to avoid coupling effects; any size of specimen that has a suitable length-to-cross section ratio in terms
of frequency response and meets the mass minimum mass may be used. Maximum specimen size and mass are determined
primarily by the power of the test system and physical space capabilities.
NOTE 5—Maximum specimen size and mass are determined primarily by the power of the test system and physical space capabilities.
7.3 All surfaces on the rectangular specimen shall be flat. Opposite surfaces across the length, thickness, and width shall be
parallel to within 0.1 %. The cylindrical specimen shall be round and constant in diameter to within 0.1 %.
E1875 − 20a
TABLE 1 Effects of Variable Error on Dynamic Modulus of
Elasticity Calculation
Variable Exponent
Measurement in Calculation
Variable
ErrorUncertaintly Dynamic ErrorUncertainty
Modulus Equation
Frequency (f) 0.1 % f 0.2 %
Frequency (f) 0.1 % f 0.2 %
Length (L) 0.1 % L 0.3 %
Mass (m) 0.1 % m 0.1 %
-1
Width (b) 0.1 % b 0.1 %
−1
Width (b) 0.1 % b 0.1 %
-3
Thickness (t) 0.1 % t 0.3 %
−3
Thickness (t) 0.1 % t 0.3 %
-4
Diameter (D) 0.1 % D 0.4 %
−4
Diameter (D) 0.1 % D 0.4 %
7.4 Specimen mass shall be determined Measure specimen mass to within 0.1 % 0.1 %.
7.5 Specimen length shall be measured Measure specimen length to within 0.1 %. The Measure the thickness and width of the
rectangular specimen shall be measured to within 0.1 % at three locations and an average determined. Thedetermine an average.
Measure the diameter of the cylindrical specimen shall be measured to with
...