ASTM E1876-22

Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Impulse Excitation of Vibration

SIGNIFICANCE AND USE
5.1 This test method may be used for material development, characterization, design data generation, and quality control purposes.  
5.2 This test method is specifically appropriate for determining the dynamic elastic modulus of materials that are elastic, homogeneous, and isotropic (3).  
5.3 This test method addresses the room temperature determination of dynamic elastic moduli of elasticity of slender bars (rectangular cross section) rods (cylindrical), and flat disks. Flat plates may also be measured similarly, but the required equations for determining the moduli are not presented.  
5.4 This dynamic test method has several advantages and differences from static loading techniques and from resonant techniques requiring continuous excitation.  
5.4.1 The test method is nondestructive in nature and can be used for specimens prepared for other tests. The specimens are subjected to minute strains; hence, the moduli are measured at or near the origin of the stress-strain curve, with the minimum possibility of fracture.  
5.4.2 The impulse excitation test uses an impact tool and simple supports for the test specimen. There is no requirement for complex support systems that require elaborate setup or alignment.  
5.5 This technique can be used to measure resonant frequencies alone for the purposes of quality control and acceptance of test specimens of both regular and complex shapes. A range of acceptable resonant frequencies is determined for a specimen with a particular geometry and mass. The technique is particularly suitable for testing specimens with complex geometries (other than parallelepipeds, cylinders/rods, or disks) that would not be suitable for testing by other procedures. Any specimen with a frequency response falling outside the prescribed frequency range is rejected. The actual dynamic elastic 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...
SCOPE
1.1 This test method covers determination of the dynamic elastic properties of elastic materials at ambient temperatures. Specimens of these materials possess specific mechanical resonant frequencies that are determined by the elastic modulus, mass, and geometry of the test specimen. The dynamic elastic properties of a material can therefore be computed if the geometry, mass, and mechanical resonant frequencies of a suitable (rectangular or cylindrical geometry) test specimen of that material can be measured. Dynamic Young's modulus is determined using the resonant frequency in either the flexural or longitudinal mode of vibration. The dynamic shear modulus, or modulus of rigidity, is found using torsional resonant vibrations. Dynamic Young's modulus and dynamic shear modulus are used to compute Poisson's ratio.  
1.2 Calculations are valid for materials that are elastic, homogeneous, and isotropic. Anisotropy can add additional calculation errors. See Appendix X1 for details.  
1.3 The use of mixed numerical-experimental techniques (MNET) is outside the scope of this standard.  
1.4 This test method may be used for determining dynamic Young’s modulus for materials of a composite character (particulate, whisker or fiber reinforced) or other anisotropic materials only after the effect of the reinforcement in the test specimen has been considered. Examples of the characteristics of the reinforcement that can affect the measured dynamic Young’s modulus are volume fraction, size, morphology, distribution, orientation, elastic properties, and interfacial bonding.  
1.4.1 The effect of the character of the reinforcement shall be considered in interpreting the test results for these types of materials.
Note 1: The properties of the reinforcement will directly affect measured elastic properties. Data shown in (1)2 indicates the possibility of underestimating the dynamic Young’s modulus by as much as 20 % due to anisotropy...

General Information

• Status: Published
• Publication Date: 31-Mar-2022
• ICS: 81.060.20 - Ceramic products
• Technical Committee: E28 - Mechanical Testing
• Drafting Committee: E28.04 - Uniaxial Testing

Relations

• 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 E177-14 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-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 C1161-13 - Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
• 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 C372-94(2012) - Standard Test Method for Linear Thermal Expansion of Porcelain Enamel and Glaze Frits and Fired Ceramic Whiteware Products by the Dilatometer Method
• Effective Date: 01-Apr-2012
• 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 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
• Refers: ASTM E6-09b - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 15-May-2009
• Refers: ASTM E6-09be1 - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 15-May-2009
• Refers: ASTM E6-09a - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 01-Apr-2009
• Refers: ASTM E6-09 - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 01-Jan-2009
• Refers: ASTM E6-08a - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 01-Oct-2008
• Refers: ASTM E177-08 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-Oct-2008
• Refers: ASTM E6-08 - Standard Terminology Relating to Methods of Mechanical Testing
• Effective Date: 01-Feb-2008

Overview

ASTM E1876-22, "Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Impulse Excitation of Vibration," is an internationally recognized standard developed by ASTM International. This test method provides procedures for non-destructively determining the dynamic elastic properties of materials, specifically Young's modulus, shear modulus, and Poisson's ratio, through impulse excitation of vibration at ambient temperatures.

The method is applicable to elastic, homogeneous, and isotropic materials, using precise measurements of specimen geometry, mass, and mechanical resonant frequencies. These dynamic properties are crucial for material development, characterization, design data generation, and quality control.

Key Topics

• Dynamic Elastic Property Measurement: The standard describes how to determine mechanical properties such as Young’s modulus (elastic modulus), shear modulus, and Poisson’s ratio by analyzing the resonant vibration frequencies of a material specimen.
• Non-destructive Testing: ASTM E1876-22 emphasizes non-destructive evaluation, allowing for the assessment of mechanical properties without damaging the material.
• Specimen Requirements: The method applies to slender bars (rectangular or cylindrical), rods, and flat disks. Uniform geometry and precise measurement of dimensions and mass are essential for valid results.
• Impulse Excitation: The technique uses an impact tool and simple supports to excite the specimen, with responses measured using either contact or non-contact transducers. This allows for quick setup and repeatability.
• Material Suitability: The method is most accurate for elastic, homogeneous, isotropic materials but may be applied to composites and anisotropic materials with caution and proper interpretation.
• Resonant Frequency Analysis: Fundamental and higher-order resonant frequencies are measured and used in standard equations to calculate dynamic moduli.

Applications

• Material Development and Characterization: Provides fast, non-destructive evaluation of mechanical properties, supporting the development and optimization of metals, ceramics, polymers, and composites.
• Quality Control: Enables manufacturers to verify material integrity and uniformity by quickly measuring the dynamic modulus and identifying specimens that fall outside preset frequency ranges.
• Design Data Generation: Supplies reliable elastic property data for engineers in structural design, ensuring accurate modeling and prediction of material behavior under dynamic loads.
• Detecting Material Defects: Serves as a diagnostic tool to identify cracks, voids, or structural discontinuities by comparing measured frequencies against defect-free specimens.
• Testing Complex Geometries: Suitable for components with regular or irregular shapes that cannot be evaluated easily by other resonance or static test methods.

Related Standards

ASTM E1876-22 references several other ASTM standards relevant to dynamic elastic property measurement:

• ASTM C215: Test Method for Fundamental Transverse, Longitudinal, and Torsional Resonant Frequencies of Concrete Specimens
• ASTM C623: Test Method for Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Glass and Glass-Ceramics by Resonance
• ASTM C747: Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• ASTM C848: Test Method for Young’s Modulus, Shear Modulus, and Poisson’s Ratio For Ceramic Whitewares by Resonance
• ASTM C1198 & C1259: Related to dynamic elastic moduli determination in advanced ceramics
• ASTM E6 & E177: Terminology and practice standards for mechanical testing

ASTM E1876-22 plays a vital role in modern materials engineering, offering a reliable, non-destructive approach to determining dynamic elastic properties, supporting robust material evaluation for a wide range of industries.

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: E1876 − 22
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio by Impulse Excitation of Vibration
This standard is issued under the fixed designation E1876; 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.
measured elastic properties. Data shown in (1) indicates the possibility of
1. Scope*
underestimating the dynamicYoung’s modulus by as much as 20 % due to
1.1 This test method covers determination of the dynamic
anisotropy
elastic properties of elastic materials at ambient temperatures.
1.5 This test method should not be used for establishing
Specimens of these materials possess specific mechanical
accurate dynamic Young’s modulus, dynamic shear modulus,
resonant frequencies that are determined by the elastic
or Poisson’s ratio for specimens that have cracks, voids, or
modulus, mass, and geometry of the test specimen. The
other major structural discontinuities.
dynamic elastic properties of a material can therefore be
1.6 This test method may be used for determining whether
computed if the geometry, mass, and mechanical resonant
structural discontinuities exist in a specimen by comparing
frequencies of a suitable (rectangular or cylindrical geometry)
results with a specimen that is defect free.
test specimen of that material can be measured. Dynamic
Young’s modulus is determined using the resonant frequency
1.7 This test method shall not be used for establishing
in either the flexural or longitudinal mode of vibration. The
accuratedynamicYoung’smodulus,dynamicshearmodulusor
dynamic shear modulus, or modulus of rigidity, is found using
Poisson’s ratio for materials that cannot be fabricated in
torsional resonant vibrations. Dynamic Young’s modulus and
uniform rectangular or cylindrical cross section.
dynamic shear modulus are used to compute Poisson’s ratio.
1.8 The values stated in SI units are to be regarded as
1.2 Calculations are valid for materials that are elastic,
standard. No other units of measurement are included in this
homogeneous, and isotropic. Anisotropy can add additional
standard.
calculation errors. See Appendix X1 for details.
1.9 This standard does not purport to address all of the
1.3 The use of mixed numerical-experimental techniques
safety concerns, if any, associated with its use. It is the
(MNET) is outside the scope of this standard.
responsibility of the user of this standard to establish appro-
priate safety, health, and environmental practices and deter-
1.4 This test method may be used for determining dynamic
mine the applicability of regulatory limitations prior to use.
Young’s modulus for materials of a composite character
1.10 This international standard was developed in accor-
(particulate, whisker or fiber reinforced) or other anisotropic
dance with internationally recognized principles on standard-
materials only after the effect of the reinforcement in the test
ization established in the Decision on Principles for the
specimen has been considered. Examples of the characteristics
Development of International Standards, Guides and Recom-
of the reinforcement that can affect the measured dynamic
mendations issued by the World Trade Organization Technical
Young’s modulus are volume fraction, size, morphology,
Barriers to Trade (TBT) Committee.
distribution, orientation, elastic properties, and interfacial
bonding.
2. Referenced Documents
1.4.1 The effect of the character of the reinforcement shall
2.1 ASTM Standards:
be considered in interpreting the test results for these types of
C372 Test Method for Linear Thermal Expansion of Porce-
materials.
NOTE 1—The properties of the reinforcement will directly affect lainEnamelandGlazeFritsandFiredCeramicWhiteware
Products by Dilatometer Method
C1161 Test Method for Flexural Strength of Advanced
1 2
This test method is under the jurisdiction of ASTM Committee E28 on The boldface numbers in parentheses refer to a list of references at the end of
Mechanical Testing and is the direct responsibility of Subcommittee E28.04 on this standard.
Uniaxial Testing. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved April 1, 2022. Published July 2022. Originally contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
approved in 1997. Last previous edition approved in 2021 as E1876 – 21. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/E1876-22. the ASTM website.
*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
E1876 − 22
Ceramics at Ambient Temperature 3.1.6.1 Discussion—Poisson’s ratio may be negative for
C1198 Test Method for Dynamic Young’s Modulus, Shear some materials, for example, a tensile transverse strain will
Modulus, and Poisson’s Ratio for Advanced Ceramics by result from a tensile axial strain.
Sonic Resonance 3.1.6.2 Discussion—Poisson’sratiowillhavemorethanone
E6 Terminology Relating to Methods of Mechanical Testing
value if the material is not isotropic. E6
–2
E177 Practice for Use of the Terms Precision and Bias in
3.1.7 proportional limit [FL ],n—the greatest stress that a
ASTM Test Methods
material is capable of sustaining without deviation from
proportionality of stress to strain (Hooke’s law). E6
3. Terminology
3.1.7.1 Discussion—Many experiments have shown that
3.1 Definitions:
values observed for the proportional limit vary greatly with the
3.1.1 The definitions of terms relating to mechanical testing
sensitivity and accuracy of the testing equipment, eccentricity
appearing in Terminology E6 and C1198 should be considered
of loading, the scale to which the stress-strain diagram is
as applying to the terms used in this test method.
plotted, and other factors. When determination of proportional
3.1.2 dynamic elastic modulus, n—the elastic modulus,
limit is required, the procedure and the sensitivity of the test
either Young’s modulus or shear modulus, that is measured in
equipment should be specified.
a dynamic mechanical measurement. –2
3.1.8 shear modulus, G [FL ],n—the ratio of shear stress
3.1.3 dynamic mechanical measurement, n—a technique in
to corresponding shear strain below the proportional limit, also
which either the modulus or damping, or both, of a substance
called torsional modulus and modulus of rigidity.
under oscillatory applied force or displacement is measured as
3.1.8.1 Discussion—The value of the shear modulus may
a function of temperature, frequency, or time, or combination
depend on the direction in which it is measured if the material
thereof.
is not isotropic. Wood, many plastics and certain metals are
–2
markedly anisotropic. Deviations from isotropy should be
3.1.4 elastic limit [FL ],n—the greatest stress that a
suspected if the shear modulus differs from that determined by
material is capable of sustaining without permanent strain
substituting independently measured values of Young’s
remaining upon complete release of the stress. E6
modulus, E, and Poisson’s ratio, µ, in the relation:
–2
3.1.5 modulus of elasticity [FL ],n—the ratio of stress to
corresponding strain below the proportional limit.
E
3.1.5.1 Discussion—The stress-strain relationships of many
G 5
2 11µ
~ !
materialsdonotconformtoHooke’slawthroughouttheelastic
3.1.8.2 Discussion—In general, it is advisable in reporting
range, but deviate therefrom even at stresses well below the
values of shear modulus to state the range of stress over which
elastic limit. For such materials, the slope of either the tangent
it is measured. E6
to the stress-strain curve at the origin or at a low stress, the
–2
secant drawn from the origin to any specified point on the
3.1.9 Young’s modulus, E [FL ],n—the ratio of tensile or
stress-strain curve, or the chord connecting any two specified compressive stress to corresponding strain below the propor-
points on the stress-strain curve is usually taken to be the
tional limit of the material. E6
“modulus of elasticity.” In these cases, the modulus should be
3.2 Definitions of Terms Specific to This Standard:
designated as the “tangent modulus,” the “secant modulus,” or
3.2.1 anti-nodes, n—two or more locations in an uncon-
the “chord modulus,” and the point or points on the stress-
strained slender rod or bar in resonance that have local
strain curve described. Thus, for materials where the stress-
maximum displacements.
strain relationship is curvilinear rather than linear, one of the
3.2.1.1 Discussion—For the fundamental flexure resonance,
four following terms may be used:
the anti-nodes are located at the two ends and the center of the
–2
(a) initial tangent modulus [FL ], n—the slope of the
specimen.
stress-strain curve at the origin.
–2
3.2.2 elastic, adj—the property of a material such that an
(b) tangent modulus [FL ], n—the slope of the stress-
application of stress within the elastic limit of that material
strain curve at any specified stress or strain.
–2
making up the body being stressed will cause an instantaneous
(c) secant modulus [FL ], n—the slope of the secant
and uniform deformation, which will be eliminated upon
drawnfromtheorigintoanyspecifiedpointonthestress-strain
removal of the stress, with the body returning instantly to its
curve.
–2
original size and shape without energy loss. Most elastic
(d) chord modulus [FL ], n—the slope of the chord drawn
materials conform to this definition well enough to make this
between any two specified points on the stress-strain curve
resonance test valid.
below the elastic limit of the material.
3.1.5.2 Discussion—Modulus of elasticity, like stress, is 3.2.3 flexural vibrations, n—the vibrations that occur when
expressed in force per unit of area (pounds per square inch, the oscillations in a slender rod or bar are in a plane normal to
etc.). the length dimension.
3.1.6 Poisson’s ratio, µ,n—the negative of the ratio of 3.2.4 homogeneous, adj—the condition of a specimen such
transverse strain to the corresponding axial strain resulting that the values of the elastic properties are uniform throughout,
from an axial stress below the proportional limit of the so that any smaller specimen taken from the original is
material. representative of the whole.
E1876 − 22
3.2.4.1 Discussion—Practically, as long as the geometrical
E = dynamicYoung’s modulus; defined in Eq 1 and Eq 4,
dimensions of the test specimen are large with respect to the
and Eq A1.4
size of individual micro-constituents, the body can be consid-
E = first natural calculation of the dynamic Young’s
ered homogeneous. Interferences and guidelines pertaining to
modulus, used in Eq A1.2
micro-constituent sizing are discussed in 6.1.5.
E = second natural calculation of the dynamic Young’s
modulus. used in Eq A1.3
3.2.5 in-plane flexure, n—for rectangular parallelepiped
G = dynamic shear modulus, defined in Eq 12, Eq 14, and
geometries, a flexure mode in which the direction of displace-
Eq A1.5
ment is in the major plane of the test specimen.
K = correction factor for the fundamental longitudinal
3.2.6 isotropic, adj—the condition of a specimen such that
mode to account for the finite diameter-to-length ratio
the values of the elastic properties are the same in all directions
and Poisson’s Ratio, defined in Eq 8
in the material.
K = geometric factor for the resonant frequency of order i,
i
3.2.6.1 Discussion—Materials are considered isotropic on a
see Table A1.2 and Table A1.3
macroscopic scale, if they are homogeneous and there is a
L = specimen length
random distribution and orientation of phases, crystallites, M = dynamic elastic modulus at temperature T (either the
T
components, pores, or microcracks. dynamic Young’s modulus E, or the dynamic shear
modulus G)
3.2.7 longitudinal vibrations, n—the vibrations that occur
M = dynamic elastic modulus at room temperature (either
O
when the oscillations in a slender rod or bar are parallel to the
the dynamicYoung’s modulus E or the dynamic shear
length of the rod or bar.
modulus G)
3.2.8 micro-constituents, n—Grains, crystals, components,
R = correction factor the geometry of the bar, defined in
pores, microcracks, or aggregates.
Eq 13
T = correction factor for fundamental flexural mode to
3.2.9 nodes,n—oneormorelocationsofaslenderrodorbar
account for finite thickness of bar and Poisson’s ratio;
in resonance that have a constant zero displacement.
defined in Eq 2.
3.2.9.1 Discussion—For the fundamental flexural
'
T = correction factor for fundamental flexural mode to
resonance, the nodes are located at 0.224 L from each end,
account for finite diameter of rod, Poisson’s ratio;
where L is the length of the specimen.
defined in Eq 5.
3.2.10 out-of-plane flexure, n—for rectangular parallelepi-
b = specimen width
ped geometries, a flexure mode in which the direction of
f = frequency
displacement is perpendicular to the major plane of the test
f = resonant frequency at room temperature in furnace or
specimen.
cryogenic chamber
f = first natural resonant frequency; used in Eq A1.2
3.2.11 resonant frequency, n—naturally occurring frequen-
f = second natural frequency; used in Eq A1.3
cies of a body driven into flexural, torsional, or longitudinal
f = fundamental resonant frequency of bar in flexure;
f
vibration that are determined by the elastic modulus, mass, and
used in Eq 1
dimensions of the body.
f = fundamental longitudinal resonant frequency of a
l
3.2.11.1 Discussion—The lowest resonant frequency in a
slender bar; used in Eq 7 and Eq 9
given vibrational mode is the fundamental resonant frequency
f = resonant frequency measured in the furnace or cryo-
T
of that mode.
genic chamber at temperature T, used in Eq 16
3.2.12 slender rod or bar, n—in dynamic elastic property
f = fundamental resonant frequency of bar in torsion;
t
testing, a specimen whose ratio of length to minimum cross-
used in Eq 12 and Eq 14
sectional dimension is at least five and preferably in the range
m = specimen mass
n = the order of the resonance (n=1,2,3,.)
from 20 to 25.
r = radius of the disk, used in Eq A1.1
3.2.13 torsional vibrations, n—the vibrations that occur
t = specimen, disk or bar, thickness
when the oscillations in each cross-sectional plane of a slender
∆T = temperature difference between the test temperature T
rod or bar are such that the plane twists around the length
and room temperature Eq 16
dimension axis.
α = average linear thermal expansion coefficient
3.2.14 RVE, n—The representative volume element for (mm/mm/°C) from room temperature to test tempera-
effective physical properties characterization.
ture; used in Eq 16
µ = Poisson’s ratio
3.2.14.1 Discussion—The RVE must be large enough to
ρ = density of the disk; used in Eq A1.1
contain a sufficient number of micro-constituents in order to be
macroscopically representative of the material (2).
4. Summary of Test Method
3.3 Symbols:
4.1 This test method measures the fundamental resonant
A = plate constant; used in Eq A1.1
frequency of test specimens of suitable geometry by exciting
D = diameter of rod or diameter of disk
them mechanically by a singular elastic strike with an impulse
D = effective diameter of the bar; defined in Eq 10 and
e
tool. A transducer (for example, contact accelerometer or
Eq 11
non-contacting microphone) senses the resulting mechanical
E1876 − 22
vibrations of the specimen and transforms them into electric 6. Interferences
signals. Specimen supports, impulse locations, and signal
6.1 The relationships between resonant frequency and dy-
pick-up points are selected to induce and measure specific
namic elastic modulus presented herein are specifically appli-
modesofthetransientvibrations.Thesignalsareanalyzed,and
cable to homogeneous, elastic, isotropic materials.
the fundamental resonant frequency is isolated and measured
by the signal analyzer, which provides a numerical reading that NOTE 2—Appendix X1 provides discussion on anisotropic materials
and the reasons why this test method will not provide correct results.
is (or is proportional to) either the frequency or the period of
the specimen vibration. The appropriate fundamental resonant
6.1.1 This method of determining the moduli is applicable
frequencies, dimensions, and mass of the specimen are used to
to composite and inhomogeneous materials only with careful
calculate dynamic Young’s modulus, dynamic shear modulus,
consideration of the effect of inhomogeneities and anisotropy.
and Poisson’s ratio.
The character (volume fraction, size, morphology, distribution,
orientation, elastic properties, and interfacial bonding) of the
5. Significance and Use
reinforcement and inhomogeneities in the specimens will have
5.1 This test method may be used for material development,
a direct effect on the elastic properties of the specimen as a
characterization, design data generation, and quality control
whole.These effects must be considered in interpreting the test
purposes.
results for composites and inhomogeneous materials.
6.1.2 The procedure involves measuring transient elastic
5.2 This test method is specifically appropriate for deter-
vibrations. Materials with very high damping capacity may be
mining the dynamic elastic modulus of materials that are
difficult to measure with this technique if the vibration damps
elastic, homogeneous, and isotropic (3).
out before the frequency counter can measure the signal
5.3 This test method addresses the room temperature deter-
(commonly within three to five cycles).
mination of dynamic elastic moduli of elasticity of slender bars
6.1.3 If specific surface treatments (coatings, machining,
(rectangular cross section) rods (cylindrical), and flat disks.
grinding, etching, and so forth) change the elastic properties of
Flat plates may also be measured similarly, but the required
the near-surface material, there will be accentuated effects on
equations for determining the moduli are not presented.
the properties measured by this flexural method, as compared
5.4 This dynamic test method has several advantages and
to static/bulk measurements by tensile or compression testing.
differences from static loading techniques and from resonant
6.1.4 This test method is not satisfactory for specimens that
techniques requiring continuous excitation.
have major discontinuities, such as large cracks (internal or
5.4.1 The test method is nondestructive in nature and can be
surface) or voids.
used for specimens prepared for other tests. The specimens are
6.1.5 Although inhomogeneity is also dependent on mate-
subjected to minute strains; hence, the moduli are measured at
rial selection, a minimum of 10 microconstituents per smallest
or near the origin of the stress-strain curve, with the minimum
edge, or the maximum micro-constituents’ size less than or
possibility of fracture.
equal to one tenth of the smallest edge, is generally recom-
5.4.2 The impulse excitation test uses an impact tool and
mended to consider a material homogenous. Otherwise, the
simple supports for the test specimen. There is no requirement
elastic properties are best represented by an appropriate statis-
for complex support systems that require elaborate setup or
tical distribution where the number of samples to be tested
alignment.
must be increased according to the user’s acceptable error.
5.5 Thistechniquecanbeusedtomeasureresonantfrequen-
NOTE 3—Reference (2) results show that 1000 grains for representative
cies alone for the purposes of quality control and acceptance of
volume element (RVE) of isotropic copper polycrystals give a maximum
test specimens of both regular and complex shapes.Arange of
relative error of 1.5%. The RVE for an Impulse Excitation Technique
acceptable resonant frequencies is determined for a specimen
specimen may be considered as a cube with edges equal to the specimen
smallest edge. For example, a cubic RVE with 1000 micro-constituents
with a particular geometry and mass. The technique is particu-
has 10 micro-constituents per edge.
larly suitable for testing specimens with complex geometries
(other than parallelepipeds, cylinders/rods, or disks) that would
6.2 This test method for determining moduli is limited to
not be suitable for testing by other procedures. Any specimen
specimens with regular geometries (rectangular parallelepiped,
with a frequency response falling outside the prescribed
cylinders, and disks) for which analytical equations are avail-
frequency range is rejected. The actual dynamic elastic modu-
able to relate geometry, mass, and modulus to the resonant
lus of each specimen need not be determined as long as the
vibration frequencies. This test method is not appropriate for
limits of the selected frequency range are known to include the
determining the elastic properties of materials that cannot be
resonant frequency that the specimen must possess if its
fabricated into such geometries.
geometry and mass are within specified tolerances.
6.2.1 The analytical equations assume parallel and concen-
tric dimensions for the regular geometries of the specimen.
5.6 If a thermal treatment or an environmental exposure
Deviations from the specified tolerances for the dimensions of
affects the elastic response of the test specimen, this test
the specimens will change the resonant frequencies and intro-
methodmaybesuitableforthedeterminationofspecificeffects
duce error into the calculations.
of thermal history, environment exposure, and so forth. Speci-
men descriptions should include any specific thermal treat- 6.2.2 Edge treatments such as chamfers or radii are not
ments or environmental exposures that the specimens have considered in the analytical equations. Edge chamfers change
received. the resonant frequency of the test bars and introduce error into
E1876 − 22
the calculations of the dynamic elastic modulus. It is recom- ment should have most of its mass concentrated at the point of
mendedthatspecimensforthistestmethodnothavechamfered impact and have mass sufficient to induce a measurable
or rounded edges. mechanical vibration, but not so large as to displace or damage
6.2.3 For specimens with as-fabricated and rough or uneven the specimen physically. In practice, the size and geometry of
surfaces, variations in dimension can have a significant effect the impulser depends on the size and weight of the specimen
in the calculations. For example, in the calculation of dynamic and the force needed to produce vibration. For commonly
elastic modulus, the modulus value is inversely proportional to tested geometries (small bars, rods, and disks) an example of
the cube of the thickness. Uniform specimen dimensions and such an impulser is a steel sphere 0.5 cm in diameter glued to
precise measurements are essential for accurate results. the end of a flexible 10-cm long polymer rod. (See Fig. 2.)An
alternate impulser is a solid metal, ceramic, or polymer sphere
6.3 This test method assumes that the specimen is vibrating
(0.1 to 1.0 cm in diameter) dropped on the specimen through a
freely, with no significant restraint or impediment. Specimen
guide tube to ensure proper impulse position.
supports should be designed and located properly in accor-
7.2.1 An automatic electromagnetic impulser may be used
dance with the instructions so the specimen can vibrate freely
instead of the implements described listed in 7.2, where a solid
in the desired mode. In using direct-contact transducers, the
metal, ceramic, or polymer projectile (0.1 cm to 1.0 cm in
transducershouldbepositionedawayfromanti-nodesandwith
diameter and 1 cm to 10 cm in length) is shot against the test
minimal force to avoid interference with free vibration.
specimen. The impulser shall not displace or damage the
6.4 Proper location to the impulse point and transducer is
specimen and shall be at free flight at moment of impact.
important in introducing and measuring the desired vibration
7.3 Signal Detection—Signal detection may be by means of
mode.Thelocationsoftheimpulsepointandtransducershould
direct-contact detecting transducers or by non-contact detect-
not be changed in multiple readings; changes in position may
ing transducers. Direct contact detecting transducers are com-
develop and detect alternate vibration modes. In the same
monly accelerometers using piezoelectric or strain gage meth-
manner, the force used in impacting should be consistent in
ods to measure the vibration. Non-contact detecting
multiple readings.
transducers are commonly acoustic microphones, but they may
6.5 If the frequency readings are not repeatable for a
alsouselaser,magnetic,orcapacitancemethodstomeasurethe
specific set of impulse and transducer locations on a specimen,
vibration.Thefrequencyrangeofthedetectingtransducershall
it may be because several different modes of vibration are
be sufficient to measure the expected frequencies of the
being developed and detected in the test. The geometry of the
specimens of interest. A suitable range would be from 100 Hz
test bar and desired vibration mode should be evaluated and
to 50 kHz for most advanced ceramic test specimens. (Smaller
used to identify the nodes and anti-nodes of the desired
and stiffer specimens vibrate at higher frequencies.) The
vibrations. More consistent measurements may be obtained if
frequency response of the detecting transducer across the
the impulse point and transducer locations are shifted to induce
frequency range of interest shall have a bandwidth of at least
and measure the single desired mode of vibration.
10 % of the maximum measured frequency before –3 dB
power loss occurs.
7. Apparatus
7.4 Electronic System—The electronic system consists of a
7.1 Apparatus suitable for accurately detecting, analyzing,
signal conditioner/amplifier, signal analyzer, and a frequency
andmeasuringthefundamentalresonantfrequencyorperiodof
meter.
a vibrating free-free beam is used. The test apparatus is shown
7.4.1 Acomputer with dedicated software and signal acqui-
in Fig. 1. It consists of an impulser, a suitable detecting
sition board may be used instead of the discrete devices listed
transducer to convert the mechanical vibration into an electri-
in 7.4.
cal signal, an electronic system (consisting of a signal
conditioner/amplifier, a signal analyzer, and a frequency read-
outdevice),andasupportsystem.Commercialinstrumentation
is available that measures the frequency or period of the
vibrating specimen.
7.2 Impulser—The exciting impulse is imparted by lightly
striking the specimen with a suitable implement. This imple-
FIG. 1 Block Diagram of Typical Test Apparatus FIG. 2 Diagram of Typical Impulser for Small Specimens
E1876 − 22
7.4.2 The signal conditioner/amplifier, signal analyzer, and be parallel to within 0.1 %. The cylindrical specimen shall be
frequency meter may be combined into a single device that round and constant in diameter to within 0.1 %.
displays the fundamental frequency.
8.4 Measure the specimen mass to within 0.1 %.
7.4.3 The accuracy and precision of the electronic system or
8.5 Measure the specimen length to within 0.1 %. Measure
computerwithdedicatedsoftwareshallbesufficienttomeasure
the thickness and width of the rectangular specimen to within
the frequencies of interest to an accuracy of 0.1 %.
0.1 % at three locations and determine an average. Measure the
7.4.4 The signal conditioner/amplifier or acquisition board
diameter of the cylindrical specimen to within 0.1 % at three
shall be suitable to power the detecting transducer and shall
locations and determine an average. Take special care when
provide an appropriate amplified signal to the signal analyzer.
measuring specimen dimensions that are less than 3 mm.
7.4.5 The signal analyzer may be a frequency meter with
storage capability, a digital storage oscilloscope with a fre- NOTE 6— The accuracy requirements of 8.4 and 8.5 set practical
requirements for the minimum specimen dimensions. For example, the
quency meter, or a computer with dedicated software.
accuracy of a typical laboratory micrometer is a = 0.003 mm; meeting
m
7.4.5.1 With the digital storage oscilloscope or with dedi-
the requirement of 8.5 requires that the minimum width, thickness, or
catedsoftware,aFastFourierTransformsignalanalysissystem
diameterofthetestspecimenbe3mm.Theaccuracyofatypicalprecision
or other dedicated frequency analyzing algorithm may be used laboratory balance is 0.001 g; meeting the requirement of 8.4 requires that
the minimum mass of the test specimen be 1 g. Given these minimum
for analyzing more complex waveforms and identifying the
dimensions and mass, meeting the requirements of 8.4 and 8.5 requires
fundamental resonant frequency.
that the minimum length of a metal alloy test specimen with density of
0.008g/mm be13.9mm.Atestspecimenofatypicalceramic,withlower
7.5 Support System—Thesupportshallisolatethespecimen
density, will be correspondingly longer
from extraneous vibration without restricting the desired mode
NOTE 7—Table 1 illustrates how uncertainties in the measured param-
of specimen vibration. Appropriate materials should be stable
eters influence the calculated dynamic elastic modulus. It shows that
atthetesttemperatures.Supportmaterialsmaybeeithersoftor
calculations are most sensitive to uncertainty in the measurement of the
rigid for ambient conditions. An example of a soft material is thickness, diameter, and length.
a compliant elastomeric material, such as a polyurethane foam
9. Procedure
strip. Such foam strips should have simple flat surfaces for the
specimen to rest on. Rigid materials, such as metal or ceramic,
9.1 Activate all electrical equipment, and allow it to stabi-
should have sharp knife edges or cylindrical surfaces on which
lize according to the manufacturer’s recommendations.
the specimen should rest. The rigid supports should rest on
9.2 Use a test specimen established as a verification/
isolation pads to prevent ambient vibrations from being picked
calibration standard to verify the equipment response and
up by the detecting transducer. Wire suspension may also be
accuracy.
used. Specimens shall be supported along node lines appropri-
9.3 Fundamental Flexural Resonant Frequency (Out-of-
ate for the desired vibration in the locations described in
Section 8. Plane Flexure):
9.3.1 Place the specimen on the supports located at the
fundamental nodes (0.224 L from each end; see Fig. 3).
8. Test Specimen
9.3.2 Determine the direction of maximum sensitivity for
8.1 Prepare the specimens so that they are either rectangular
the detecting transducer. Orient the detecting transducer so that
or circular in cross section. Either geometry may be used to
it will detect the desired vibration.
measure both dynamic Young’s modulus and dynamic shear
9.3.2.1 Direct-Contact Transducers—Place the detecting
modulus.
transducer in contact with the test specimen at point P1 in
NOTE 4—Although the equations for computing shear modulus with a Fig. 3 to pick up the desired vibration. If the transducer is
cylindrical specimen are both simpler and more accurate than those used
placed at an anti-node (location of maximum displacement), it
with a rectangular bar, experimental difficulties in obtaining torsional
may mass load the specimen and modify the natural vibration.
resonantfrequenciesforacylindricalspecimenusuallyprecludeitsusefor
The transducer should be placed only as far from the node as
determining dynamic shear modulus.
necessary to obtain a reading (see Fig. 3). This location will
8.2 Select the size so that, for an estimated dynamic elastic
minimize the damping effect from the direct-contact detecting
modulus,theresonantfrequenciesmeasuredwillfallwithinthe
transducer. The transducer contact force should be consistent,
range of frequency response of the transducers and electronics
used. For a slender test specimen, the ratio of length to
minimum cross-sectional dimension shall have a value of at
TABLE 1 Effects of Variable Uncertainty on Dynamic Elastic
least five (5). However, a ratio of approximately 20 to 25
Modulus Uncertainty
should be used for ease in calculation. For dynamic shear
Variable Exponent in
modulusmeasurementsofrectangulartestspecimens,aratioof Measurement Calculation
Variable Dynamic Elastic
Uncertainty Uncertainty
width to thickness should be five (5) or greater to minimize
Modulus Equation
experimental difficulties.
Frequency (f) 0.1 % f 0.2 %
Length (L) 0.1 % L 0.3 %
NOTE 5—Resonant frequencies for a given specimen are functions of
Mass (m) 0.1 % m 0.1 %
–1
the specimen dimensions as well as its mass and moduli.
Width (b) 0.1 % b 0.1 %
–3
Thickness (t) 0.1 % t 0.3 %
8.3 All surfaces on the rectangular specimen shall be flat. –4
Diameter (D) 0.1 % D 0.4 %
Opposite surfaces across the length, thickness, and width shall
E1876 − 22
FIG. 3 Rectangular Specimens Tested for In-Plane and Out-of-Plane Flexure
with good response and minimal interference with the free to introduce and detect vibrations in the major plane (see Fig.
vibration of the specimen. 3). In the alternate method, rotate the test specimen 90° around
9.3.2.2 Non-Contact Transducers—Place the non-contact its long axis and reposition it on the specimen supports.
detecting transducer over an anti-node at point M1.1 or M1.2 Transpose the width and thickness dimensions in the calcula-
in Fig. 3 and close enough to the test specimen to pick up the tions. For homogeneous, isotropic materials, the calculated
desired vibration, but not so close as to interfere with the free moduli should be the same as the moduli calculated from the
vibration (see Fig. 3). out-of-plane frequency. The comparison of in-plane and out-
9.3.3 Strikethespecimenlightlyandelastically,eitheratthe of-plane frequency measurements can thus be used as a cross
center of the specimen, point X1 in Fig. 3, or at the opposite check of experimental methods and calculations.
end of the specimen from the detecting transducer (see Fig. 3).
9.5 Fundamental Torsional Resonant Frequency:
9.3.4 Record the resultant reading, and repeat the test until
9.5.1 Support the specimen at the midpoint of its length and
five consecutive readings are obtained that lie within 1 % of
width (the torsional nodes) (see Fig. 4).
each other. Use the average of these five readings to determine
9.5.2 Locate the detecting transducer at one quadrant of the
the fundamental resonant frequency in flexure.
specimen, preferably at approximately 0.224 L from one end
9.4 Fundamental Flexural Resonant Frequency (In-Plane and toward the edge at point P3 (direct-contact detecting
Flexure): transducer) or M3 (non-contact detecting transducer) in Fig. 4.
9.4.1 This procedure is the same as 9.3, except that the This location is a node point of flexural vibration and will
direction of vibration is in the major plane of the specimen. minimize the possibility of detecting a spurious flexural
This measurement may be performed in two ways. In one case, vibration mode (see Fig. 4).
move the detecting transducer and impulser 90° around the 9.5.3 Strike the specimen on the quadrant diagonally oppo-
long axis of the test specimen to points X2 and M2.1 and M2.2 site the detecting transducer, again at 0.224 L from the end and
FIG. 4 Rectangular Specimen Tested for Torsional Vibration
E1876 − 22
2 4
near the edge at point X3 in Fig. 4. Striking at a flexural t t
T 5 116.585 ~110.0752 µ10.8109 µ ! 2 0.868
S D S D
vibration node will minimize the possibility of exciting a L L
flexural vibration mode (see Fig. 4).
(2)
9.5.4 Record the resultant reading, and repeat the test until
t
five consecutive readings are obtained that lie within 1 % of
8.340 110.2023 µ12.173 µ
~ ! S D
L
each other. Use the average of these five readings to determine
t
the fundamental resonant frequency in torsion.
3 4
1.00016.338 110.1408 µ11.536 µ
~ ! S D
L
9.6 Fundamental Longitudinal Resonant Frequency:
9.6.1 Support the specimen at the midpoint of its length and where:
width(thesameasfortorsion),orbracethespecimenatitsmid
µ = Poisson’s ratio.
length, the fundamental longitudinal node.
NOTE8—Inthedynamicelasticmodulusequations,themassandlength
9.6.2 Locate the detecting transducer at the center of one of
terms are given in units of grams and millimetres. However, the defined
equations can also be used with mass and length terms in units of
the end faces of the specimen at point P4 or M4 in Fig. 5.
kilograms and metres with no changes in terms or exponents.
9.6.3 Strike the end face of the specimen at point X4 in
Fig. 5, opposite to the face where the detecting transducer is
10.1.1.1 If L/t ≥ 20, T can be simplified to the following:
located.
t
9.6.4 Record the resultant reading, and repeat the test, until
T 5 1.00016.585 (3)
F S D G
L
five consecutive readings are obtained that lie within 1 % of
and E can be calculated directly.
each other. Use the average of these five readings to determine
the fundamental longitudinal resonant frequency.
10.1.1.2 If L/t < 20 and Poisson’s ratio is known, then T
can be calculated directly from Eq 2 and then used to calculate
10. Calculation
E.
10.1 Dynamic Young’s Modulus (3,4): 10.1.1.3 If L/t < 20 and Poisson’s ratio is not known,
10.1.1 For the fundamental flexure resonant frequency of a
assume an initial Poisson’s ratio to begin the computations.
rectangular bar (4), UseaniterativeprocesstodetermineavalueofPoisson’sratio,
2 3 basedonexperimentaldynamicYoung’smodulusanddynamic
mf L
f
E 5 0.9465 T (1)
S DS 3D shear modulus. The iterative process is flowcharted in Fig. 6
b t
and described in (1) through (5),
where:
(1) Determine the fundamental flexural and torsional reso-
E = Dynamic Young’s modulus, Pa, nantfrequencyoftherectangulartestspecimen,asdescribedin
m = mass of the bar, g (see Note 8),
Section 9. Using Eq 12, calculate the dynamic shear modulus
b = width of the bar, mm (see Note 8),
of the test specimen for the fundamental torsional resonant
L = length of the bar, mm (see Note 8),
frequency.
t = thickness of the bar, mm (see Note 8),
(2) Using Eq 1 and Eq 2, calculate the dynamic Young’s
f = fundamental resonant frequency of bar in flexure, Hz,
f
modulusoftherectangulartestspecimenfromthefundamental
and
flexural resonant frequency, dimensions and mass of the
T = correction factor for fundamental flexural mode to
specimen, and initial/iterative Poisson’s ratio. Exercise care in
account for finite thickness of bar, Poisson’s ratio, and
using consistent units for all of the parameters throughout the
so forth.
computations.
FIG. 5 Rectangular Specimen Tested for Longitudinal Vibration
E1876 − 22
FIG. 6 Process Flow Chart for Iterative Determination of Poisson’s Ratio for Isotropic Materials
'
(3) Substitute the dynamic shear modulus and Young’s 10.1.2.2 If L/D < 20 and Poisson’s ratio is known, then T
modulus values calculated in steps (1) and (2) into Eq 15 for
can be calculated directly from Eq 5 and then used to calculate
Poisson’s ratio satisfying isotropic conditions. Calculate a new
E.
value for Poisson’s ratio for another iteration beginning at Step
10.1.2.3 If L/D < 20 and Poisson’s ratio is not known,
(2).
assume an initial Poisson’s ratio to start the computations.
(4) Repeat Steps (2) and (3) until no significant difference
Determine final values for Poisson’s ratio, dynamic Young’s
(2 % or less) is observed between the last iterative value and
modulus, and dynamic shear modulus using the same method
the final computed value of the Poisson’s ratio.
showninFig.6anddescribedin(1)through(5)in10.1.1.3,but
(5) Self-consistent values for the moduli are thus obtained.
using the dynamic modulus equations for circular bars (Eq 4,
10.1.2 For the fundamental flexural resonant frequency of a
and Eq 14).
rod of circular cross section (4) :
3 10.1.3 For the fundamental longitudinal resonant frequency
L
2 '
E 5 1.6067 mf T (4)
S D ~ !
4 f 1
of a slender bar with circular cross-section:
D
L
where:
E 5 16mf (7)
F 2 G
l
π D K
D = diameter of rod, mm (see Note 8), and
'
T = correction factor for fundamental flexural mode to
where:
account for finite diameter of rod, Poisson’s ratio, and
f = fundamental longitudinal resonant frequency of bar, Hz
l
so forth.
D = the diameter of the bar, mm
'
T =
2 K = correction factor for the fundamental longitudinal mode
D
114.939 110.0752 µ10.8109 µ
~ ! S D
to account for the finite diameter-to-length ratio and
L
Poisson’s Ratio:
D
2 0.4883
S D
2 2 2
L
π µ D
K 5 1 2 (8)
F 2 G
8 L
D
4.691 110.2023 µ12.173 µ
~ ! S D
L
where:
2 (5)
D
3 4
1.00014.754 110.1408 µ11.536 µ µ = Poisson’s ratio
~ ! S D
L
'
10.1.4 For the fundamental longitudinal resonant frequency
10.1.2.1 If L/D ≥ 20, then T can be simplified to the
of a slender bar with square or rectangular cross-section:
following:
L
D 2
' E 5 4mf (9)
F G
l
T 5 1.00014.939 (6)
F S D G
btK
L
E1876 − 22
where: where:
f = Fundamental longitudinal frequency of bar, Hz µ = Poisson’s ratio,
l
b = the width of the square cross section, mm E = Dynamic Young’s modulus, and
t = the thickness of the cross-section, mm G = Dynamic shear modulus.
K = correction factor for the fundamental longitudinal mode
If Poisson’s ratio is not known or assumed, use the iterative
to account for the finite diameter-to-length ratio and
process described in 10.1.1.3 to determine an experimental
Poisson’s Ratio:
Poisson’s ratio, using the appropriate equations for dynamic
2 2 2
π µ D Young’s modulus and dynamic shear modulus and the experi-
e
K 5 1 2 (10)
F G
8L mental geometry (round, square, or rectangular cross section)
(Fig. 7).
where:
10.4 If measurements are made at elevated or cryogenic
µ = Poisson’s ratio
temperatures, correct the calculated moduli for thermal expan-
D = the effective diameter of the bar:
e
sion effects using Eq 16.
2 2
b 1t
2 2
D 5 2 (11) f 1
e T
M 5 M (16)
F G F G
T O
f ~1 1α∆T!
O
10.2 Dynamic Shear Modulus (5):
where:
10.2.1 For the fundamental torsional resonant frequency of
M = Dynamic elastic modulus at temperature T (either
a rectangular bar (3):
T
dynamic Young’s modulus E or dynamic shear modu-
4 Lmf
t
G 5 R (12) lus G),
bt
M = Dynamic elastic modulus at room temperature (either
O
where: dynamic Young’s modulus E or dynamic shear modu-
lus G),
G = dynamic shear modulus, Pa,
f = resonantfrequencyinfurnaceorcryogenicchamberat
f = fundamental torsional resonant frequency of bar Hz.
T
t
temperature T,
b
f = resonant frequency at room temperature in furnace or
S D O
2 2
t 0.00851n b
cryogenic chamber,
R 5 11
F G
t 1.991 L
α = average linear thermal expansion (mm/mm·°C) from
3 4 2 2.521 1 2 4
b
S D
b π
e t 11
room temperature to test temperature (Test Method
C372 is recommended), and
nb 2 b
2 0.060 2 1 (13) ∆T = temperature differential in °C between test tempera-
S D S D
L t
ture T and room temperature.
n= the order of the resonance (n=1,2,3,.). For the funda-
mental resonant frequency, n=1 Eq 13 should be accurate to
11. Report
within ~0.2 % for b/L ≤0.3 and b/t ≤10 in the fundamental
mode of vibration, otherwise the errors are estimated to be
11.1 Report the following information:
≤1%.
11.1.1 Identification of specific tests performed, a detailed
10.2.2 For the fundamental torsion resonant frequency of a description of apparatus used (impulser, transducer, electrical
system, and support system), and an explanation of any
cylindrical rod (3):
deviations from the described test method.
L
G 5 16mf (14)
11.1.2 Complete description of material(s) tested stating
S 2D
t
πD
composition, number of specimens, specimen geometry and
10.3 Poisson’s Ratio for isotropic materials:
mass, specimen history, and any treatments to which the
E specimens have been subjected. Include comments on dimen-
µ 5 21 (15)
S D
2G sional variability, surface finish, edge conditions, observed
FIG. 7 Rectangular Specimen Tested for Longitudinal Vibration
E1876 − 22
changes after cryogenic or high-temperature testing, and so mean measured fundamental flexural resonant frequency fo
...

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: E1876 − 21 E1876 − 22
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio by Impulse Excitation of Vibration
This standard is issued under the fixed designation E1876; 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 determination of the dynamic elastic properties of elastic materials at ambient temperatures.
Specimens of these materials possess specific mechanical resonant frequencies that are determined by the elastic modulus, mass,
and geometry of the test specimen. The dynamic elastic properties of a material can therefore be computed if the geometry, mass,
and mechanical resonant frequencies of a suitable (rectangular or cylindrical geometry) test specimen of that material can be
measured. Dynamic Young’s modulus is determined using the resonant frequency in either the flexural or longitudinal mode of
vibration. The dynamic shear modulus, or modulus of rigidity, is found using torsional resonant vibrations. Dynamic Young’s
modulus and dynamic shear modulus are used to compute Poisson’s ratio.
1.2 Calculations are valid for materials that are elastic, homogeneous, and isotropic. Anisotropy can add additional calculation
errors. See Appendix X1 for details.
1.3 The use of mixed numerical-experimental techniques (MNET) is outside the scope of this standard.
1.4 Although not specifically described herein, this test method can also be performed at cryogenic and high temperatures with
suitable equipment modifications and appropriate modifications to the calculations to compensate for thermal expansion. This test
method may be used for determining dynamic Young’s modulus for materials of a composite character (particulate, whisker or fiber
reinforced) or other anisotropic materials only after the effect of the reinforcement in the test specimen has been considered.
Examples of the characteristics of the reinforcement that can affect the measured dynamic Young’s modulus are volume fraction,
size, morphology, distribution, orientation, elastic properties, and interfacial bonding.
1.4.1 The effect of the character of the reinforcement shall be considered in interpreting the test results for these types of materials.
NOTE 1—The properties of the reinforcement will directly affect measured elastic properties. Data shown in (1) indicates the possibility of
underestimating the dynamic Young’s modulus by as much as 20 % due to anisotropy
1.5 This test method should not be used for establishing accurate dynamic Young’s modulus, dynamic shear modulus, or Poisson’s
ratio for specimens that have cracks, voids, or other major structural discontinuities.
1.6 This test method may be used for determining whether structural discontinuities exist in a specimen by comparing results with
a specimen that is defect free.
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 Nov. 1, 2021April 1, 2022. Published February 2022July 2022. Originally approved in 1997. Last previous edition approved in 20152021 as
E1876 – 15.E1876 – 21. DOI: 10.1520/E1876-21.10.1520/E1876-22.
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
E1876 − 22
1.7 There are material specific ASTM standards that cover the determination of resonance frequencies and elastic properties of
specific materials by sonic resonance or by impulse excitation of vibration. Test Methods This C215, C623, C747, C848, C1198,
and C1259 may differ from this test method in several areas (for example; sample size, dimensional tolerances, sample
preparation). The testing of these materials shall be done in compliance with these material specific standards. Where possible, the
procedures, sample specifications and calculations are consistent with these test methods.test method shall not be used for
establishing accurate dynamic Young’s modulus, dynamic shear modulus or Poisson’s ratio for materials that cannot be fabricated
in uniform rectangular or cylindrical cross section.
1.8 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.
1.9 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.
1.10 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
C372 Test Method for Linear Thermal Expansion of Porcelain Enamel and Glaze Frits and Fired Ceramic Whiteware Products
by Dilatometer Method
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
C1161 Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
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
E6 Terminology Relating to Methods of Mechanical Testing
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
3. Terminology
3.1 Definitions:
3.1.1 The definitions of terms relating to mechanical testing appearing in Terminology E6 and C1198 should be considered as
applying to the terms used in this test method.
3.1.2 dynamic elastic modulus, n—the elastic modulus, either Young’s modulus or shear modulus, that is measured in a dynamic
mechanical measurement.
3.1.3 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 combination thereof.
–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. 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
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.
E1876 − 22
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 may 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). E6
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.
–2
3.1.8 shear 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 may 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:
E
G 5
2~11μ!
3.1.8.2 Discussion—
In general, it is advisable in reporting values of shear modulus to state the range of stress over which it is measured. E6
–2
3.1.9 Young’s 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 rod or bar in resonance that have local maximum
displacements.
3.2.1.1 Discussion—
For the fundamental flexure 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, which will be eliminated upon removal of the stress,
with the body returning instantly to its original size and shape without energy loss. Most elastic materials conform to this definition
well enough to make this resonance test valid.
3.2.3 flexural vibrations, n—the vibrations that occur when the oscillations in a slender rod or bar are in a plane normal to the
length dimension.
E1876 − 22
3.2.4 homogeneous, adj—the condition of a specimen such that the composition and density are uniform, values of the elastic
properties are uniform throughout, so 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, components, pores, or microcracks, micro-constituents, the body can be considered homogeneous. Interferences and
guidelines pertaining to micro-constituent sizing are discussed in 6.1.5.
3.2.5 in-plane flexure, n—for rectangular parallelepiped geometries, a flexure mode in which the direction of displacement is in
the major plane of the test specimen.
3.2.6 isotropic, adj—the condition of a specimen such that the values of the elastic properties are the same in all directions in the
material.
3.2.6.1 Discussion—
Materials are considered isotropic on a macroscopic scale, if they are homogeneous and there is a random distribution and
orientation of phases, crystallites, components, pores, or microcracks.
3.2.7 longitudinal vibrations, n—the vibrations that occur when the oscillations in a slender rod or bar are parallel to the length
of the rod or bar.
3.2.8 micro-constituents, n—Grains, crystals, components, pores, microcracks, or aggregates.
3.2.9 nodes, n—one or more locations of a slender rod or bar in resonance that have a constant zero displacement.
3.2.9.1 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.
3.2.10 out-of-plane flexure, n—for rectangular parallelepiped geometries, a flexure mode in which the direction of displacement
is perpendicular to the major plane of the test specimen.
3.2.11 resonant frequency, n—naturally occurring frequencies of a body driven into flexural, torsional, or longitudinal vibration
that are determined by the elastic modulus, mass, and dimensions of the body.
3.2.11.1 Discussion—
The lowest resonant frequency in a given vibrational mode is the fundamental resonant frequency of that mode.
3.2.12 slender rod or bar, n—in dynamic elastic property testing, a specimen whose ratio of length to minimum cross-sectional
dimension is at least five and preferably in the range from 20 to 25.
3.2.13 torsional vibrations, n—the vibrations that occur when the oscillations in each cross-sectional plane of a slender rod or bar
are such that the plane twists around the length dimension axis.
3.2.14 RVE, n—The representative volume element for effective physical properties characterization.
3.2.14.1 Discussion—
The RVE must be large enough to contain a sufficient number of micro-constituents in order to be macroscopically representative
of the material (2).
3.3 Symbols:
A = plate constant; used in Eq A1.1
D = diameter of rod or diameter of disk
D = effective diameter of the bar; defined in Eq 10 and
e
Eq 11
E = dynamic Young’s modulus; defined in Eq 1 and Eq 4, and Eq A1.4
E = first natural calculation of the dynamic Young’s modulus, used in Eq A1.2
E = second natural calculation of the dynamic Young’s modulus. used in Eq A1.3
G = dynamic shear modulus, defined in Eq 12, Eq 14, and Eq A1.5
E1876 − 22
K = correction factor for the fundamental longitudinal mode to account for the finite diameter-to-length ratio and Poisson’s
Ratio, defined in Eq 8
K = geometric factor for the resonant frequency of order i, see Table A1.2 and Table A1.3
i
L = specimen length
M = dynamic elastic modulus at temperature T (either the dynamic Young’s modulus E, or the dynamic shear modulus G)
T
M = dynamic elastic modulus at room temperature (either the dynamic Young’s modulus E or the dynamic shear modulus G)
O
R = correction factor the geometry of the bar, defined in Eq 13
T = correction factor for fundamental flexural mode to account for finite thickness of bar and Poisson’s ratio; defined in Eq
2.
'
T = correction factor for fundamental flexural mode to account for finite diameter of rod, Poisson’s ratio; defined in Eq 5.
b = specimen width
f = frequency
f = resonant frequency at room temperature in furnace or cryogenic chamber
f = first natural resonant frequency; used in Eq A1.2
f = second natural frequency; used in Eq A1.3
f = fundamental resonant frequency of bar in flexure; used in Eq 1
f
f = fundamental longitudinal resonant frequency of a slender bar; used in Eq 7 and Eq 9
l
f = resonant frequency measured in the furnace or cryogenic chamber at temperature T, used in Eq 16
T
f = fundamental resonant frequency of bar in torsion; used in Eq 12 and Eq 14
t
m = specimen mass
n = the order of the resonance (n=1,2,3,.)
r = radius of the disk, used in Eq A1.1
t = specimen, disk or bar, thickness
ΔT = temperature difference between the test temperature T and room temperature Eq 16
α = average linear thermal expansion coefficient
(mm/mm/°C) from room temperature to test temperature; used in Eq 16
μ = Poisson’s ratio
ρ = density of the disk; used in Eq A1.1
4. Summary of Test Method
4.1 This test method measures the fundamental resonant frequency of test specimens of suitable geometry by exciting them
mechanically by a singular elastic strike with an impulse tool. A transducer (for example, contact accelerometer or non-contacting
microphone) senses the resulting mechanical vibrations of the specimen and transforms them into electric signals. Specimen
supports, impulse locations, and signal pick-up points are selected to induce and measure specific modes of the transient vibrations.
The signals are analyzed, and the fundamental resonant frequency is isolated and measured by the signal analyzer, which provides
a numerical reading that is (or is proportional to) either the frequency or the period of the specimen vibration. The appropriate
fundamental resonant frequencies, dimensions, and mass of the specimen are used to calculate dynamic Young’s modulus, dynamic
shear modulus, and Poisson’s ratio.
5. Significance and Use
5.1 This test method may be used for material development, characterization, design data generation, and quality control purposes.
5.2 This test method is specifically appropriate for determining the dynamic elastic modulus of materials that are elastic,
homogeneous, and isotropic (13).
5.3 This test method addresses the room temperature determination of dynamic elastic moduli of elasticity of slender bars
(rectangular cross section) rods (cylindrical), and flat disks. Flat plates may also be measured similarly, but the required equations
for determining the moduli are not presented.
5.4 This dynamic test method has several advantages and differences from static loading techniques and from resonant techniques
requiring continuous excitation.
5.4.1 The test method is nondestructive in nature and can be used for specimens prepared for other tests. The specimens are
subjected to minute strains; hence, the moduli are measured at or near the origin of the stress-strain curve, with the minimum
possibility of fracture.
E1876 − 22
5.4.2 The impulse excitation test uses an impact tool and simple supports for the test specimen. There is no requirement for
complex support systems that require elaborate setup or alignment.
5.5 This technique can be used to measure resonant frequencies alone for the purposes of quality control and acceptance of test
specimens of both regular and complex shapes. A range of acceptable resonant frequencies is determined for a specimen with a
particular geometry and mass. The technique is particularly suitable for testing specimens with complex geometries (other than
parallelepipeds, cylinders/rods, or disks) that would not be suitable for testing by other procedures. Any specimen with a frequency
response falling outside the prescribed frequency range is rejected. The actual dynamic elastic 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.
5.6 If a thermal treatment or an environmental exposure affects the elastic response of the test specimen, this test method may be
suitable for the determination of specific effects of thermal history, environment exposure, and so forth. Specimen descriptions
should include any specific thermal treatments or environmental exposures that the specimens have received.
6. Interferences
6.1 The relationships between resonant frequency and dynamic elastic modulus presented herein are specifically applicable to
homogeneous, elastic, isotropic materials.
NOTE 2—Appendix X1 provides discussion on anisotropic materials and the reasons why this test method will not provide correct results.
6.1.1 This method of determining the moduli is applicable to composite and inhomogeneous materials only with careful
consideration of the effect of inhomogeneities and anisotropy. The character (volume fraction, size, morphology, distribution,
orientation, elastic properties, and interfacial bonding) of the reinforcement and inhomogeneities in the specimens will have a
direct effect on the elastic properties of the specimen as a whole. These effects must be considered in interpreting the test results
for composites and inhomogeneous materials.
6.1.2 The procedure involves measuring transient elastic vibrations. Materials with very high damping capacity may be difficult
to measure with this technique if the vibration damps out before the frequency counter can measure the signal (commonly within
three to five cycles).
6.1.3 If specific surface treatments (coatings, machining, grinding, etching, and so forth) change the elastic properties of the
near-surface material, there will be accentuated effects on the properties measured by this flexural method, as compared to
static/bulk measurements by tensile or compression testing.
6.1.4 This test method is not satisfactory for specimens that have major discontinuities, such as large cracks (internal or surface)
or voids.
6.1.5 Although inhomogeneity is also dependent on material selection, a minimum of 10 microconstituents per smallest edge, or
the maximum micro-constituents’ size less than or equal to one tenth of the smallest edge, is generally recommended to consider
a material homogenous. Otherwise, the elastic properties are best represented by an appropriate statistical distribution where the
number of samples to be tested must be increased according to the user’s acceptable error.
NOTE 3—Reference (2) results show that 1000 grains for representative volume element (RVE) of isotropic copper polycrystals give a maximum relative
error of 1.5%. The RVE for an Impulse Excitation Technique specimen may be considered as a cube with edges equal to the specimen smallest edge.
For example, a cubic RVE with 1000 micro-constituents has 10 micro-constituents per edge.
6.2 This test method for determining moduli is limited to specimens with regular geometries (rectangular parallelepiped, cylinders,
and disks) for which analytical equations are available to relate geometry, mass, and modulus to the resonant vibration frequencies.
This test method is not appropriate for determining the elastic properties of materials that cannot be fabricated into such
geometries.
6.2.1 The analytical equations assume parallel and concentric dimensions for the regular geometries of the specimen. Deviations
from the specified tolerances for the dimensions of the specimens will change the resonant frequencies and introduce error into
the calculations.
E1876 − 22
6.2.2 Edge treatments such as chamfers or radii are not considered in the analytical equations. Edge chamfers change the resonant
frequency of the test bars and introduce error into the calculations of the dynamic elastic modulus. It is recommended that
specimens for this test method not have chamfered or rounded edges.
6.2.3 For specimens with as-fabricated and rough or uneven surfaces, variations in dimension can have a significant effect in the
calculations. For example, in the calculation of dynamic elastic modulus, the modulus value is inversely proportional to the cube
of the thickness. Uniform specimen dimensions and precise measurements are essential for accurate results.
6.3 This test method assumes that the specimen is vibrating freely, with no significant restraint or impediment. Specimen supports
should be designed and located properly in accordance with the instructions so the specimen can vibrate freely in the desired mode.
In using direct contact direct-contact transducers, the transducer should be positioned away from anti-nodes and with minimal force
to avoid interference with free vibration.
6.4 Proper location to the impulse point and transducer is important in introducing and measuring the desired vibration mode. The
locations of the impulse point and transducer should not be changed in multiple readings; changes in position may develop and
detect alternate vibration modes. In the same manner, the force used in impacting should be consistent in multiple readings.
6.5 If the frequency readings are not repeatable for a specific set of impulse and transducer locations on a specimen, it may be
because several different modes of vibration are being developed and detected in the test. The geometry of the test bar and desired
vibration mode should be evaluated and used to identify the nodes and anti-nodes of the desired vibrations. More consistent
measurements may be obtained if the impulse point and transducer locations are shifted to induce and measure the single desired
mode of vibration.
7. Apparatus
7.1 Apparatus suitable for accurately detecting, analyzing, and measuring the fundamental resonant frequency or period of a
vibrating free-free beam is used. The test apparatus is shown in Fig. 1. It consists of an impulser, a suitable pickupdetecting
transducer to convert the mechanical vibration into an electrical signal, an electronic system (consisting of a signal
conditioner/amplifier, a signal analyzer, and a frequency readout device), and a support system. Commercial instrumentation is
available that measures the frequency or period of the vibrating specimen.
7.2 Impulser—The exciting impulse is imparted by lightly striking the specimen with a suitable implement. This implement should
have most of its mass concentrated at the point of impact and have mass sufficient to induce a measurable mechanical vibration,
but not so large as to displace or damage the specimen physically. In practice, the size and geometry of the impulser depends on
the size and weight of the specimen and the force needed to produce vibration. For commonly tested geometries (small bars, rods,
and disks) an example of such an impulser is a steel sphere 0.5 cm 0.5 cm in diameter glued to the end of a flexible 10-cm long
polymer rod. (See Fig. 2.) An alternate impulser is a solid metal, ceramic, or polymer sphere (0.1 to 1.0 cm in diameter) dropped
on the specimen through a guide tube to ensure proper impulse position.
7.2.1 An automatic electromagnetic impulser may be used instead of the implements described listed in 7.2, where a solid metal,
ceramic, or polymer projectile (0.1 cm to 1.0 cm in diameter and 1 cm to 10 cm in length) is shot against the test specimen. The
impulser shall not displace or damage the specimen and shall be at free flight at moment of impact.
7.3 Signal Pickup—Detection—Signal detection may be by means of transducers in direct contact with the specimen or by
noncontact transducers. Contact direct-contact detecting transducers or by non-contact detecting transducers. Direct contact
FIG. 1 Block Diagram of Typical Test Apparatus
E1876 − 22
FIG. 2 Diagram of Typical Impulser for Small Specimens
detecting transducers are commonly accelerometers using piezoelectric or strain gage methods to measure the vibration. Non
contactNon-contact detecting transducers are commonly acoustic microphones, but they may also use laser, magnetic, or
capacitance methods to measure the vibration. The frequency range of the detecting transducer shall be sufficient to measure the
expected frequencies of the specimens of interest. A suitable range would be from 100 Hz to 50 kHz for most advanced ceramic
test specimens. (Smaller and stiffer specimens vibrate at higher frequencies.) The frequency response of the detecting transducer
across the frequency range of interest shall have a bandwidth of at least 10 % of the maximum measured frequency before –3 dB
power loss occurs.
7.4 Electronic System—The electronic system consists of a signal conditioner/amplifier, signal analyzer, and a frequency readout
device. The system should have accuracy and precision sufficient to measure the frequencies of interest to an accuracy of 0.1 %.
The signal conditioner/amplifier should be suitable to power the transducer and provide an appropriate amplified signal to the
signal analyzer. The signal analysis system consists of a frequency counting device and a readout device. Appropriate devices are
frequency counter systems with storage capability or digital storage oscilloscopes with a frequency counter module. With the
digital storage oscilloscope, a Fast Fourier Transform signal analysis system may be useful for analyzing more complex waveforms
and identifying the fundamental resonant frequency.meter.
7.4.1 A computer with dedicated software and signal acquisition board may be used instead of the discrete devices listed in 7.4.
7.4.2 The signal conditioner/amplifier, signal analyzer, and frequency meter may be combined into a single device that displays
the fundamental frequency.
7.4.3 The accuracy and precision of the electronic system or computer with dedicated software shall be sufficient to measure the
frequencies of interest to an accuracy of 0.1 %.
7.4.4 The signal conditioner/amplifier or acquisition board shall be suitable to power the detecting transducer and shall provide
an appropriate amplified signal to the signal analyzer.
7.4.5 The signal analyzer may be a frequency meter with storage capability, a digital storage oscilloscope with a frequency meter,
or a computer with dedicated software.
7.4.5.1 With the digital storage oscilloscope or with dedicated software, a Fast Fourier Transform signal analysis system or other
dedicated frequency analyzing algorithm may be used for analyzing more complex waveforms and identifying the fundamental
resonant frequency.
7.5 Support System— The support shall isolate the specimen from extraneous vibration without restricting the desired mode of
specimen vibration. Appropriate materials should be stable at the test temperatures. Support materials may be either soft or rigid
for ambient conditions. An example of a soft material is a compliant elastomeric material, such as a polyurethane foam strip. Such
foam strips should have simple flat surfaces for the specimen to rest on. Rigid materials, such as metal or ceramic, should have
sharp knife edges or cylindrical surfaces on which the specimen should rest. The rigid supports should rest on isolation pads to
prevent ambient vibrations from being picked up by the detecting transducer. Wire suspension may also be used. Specimens shall
be supported along node lines appropriate for the desired vibration in the locations described in Section 8.
E1876 − 22
8. Test Specimen
8.1 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 4—Although the equations for computing shear modulus with a cylindrical specimen are both simpler and more accurate than those used with a
rectangular bar, experimental difficulties in obtaining torsional resonant frequencies for a cylindrical specimen usually preclude its use for determining
dynamic shear modulus.
8.2 Select the size so that, for an estimated dynamic elastic modulus, the resonant frequencies measured will fall within the range
of frequency response of the transducers and electronics used. For a slender test specimen, the ratio of length to minimum
cross-sectional dimension shall have a value of at least five (5). However, a ratio of approximately 20 to 25 should be used for
ease in calculation. For dynamic shear modulus measurements of rectangular test specimens, a ratio of width to thickness should
be five (5) or greater to minimize experimental difficulties.
NOTE 5—Resonant frequencies for a given specimen are functions of the specimen dimensions as well as its mass and moduli.
8.3 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 %.
8.4 Measure the specimen mass to within 0.1 %.
8.5 Measure the specimen length to within 0.1 %. Measure the thickness and width of the rectangular specimen to within 0.1 %
at three locations and determine an average. Measure the diameter of the cylindrical specimen to within 0.1 % at three locations
and determine an average. Take special care when measuring specimen dimensions that are less than 3 mm.
NOTE 6— The accuracy requirements of 8.4 and 8.5 set practical requirements for the minimum specimen dimensions. For example, the accuracy of a
typical laboratory micrometer is a = 0.003 mm; meeting the requirement of 8.5 requires that the minimum width, thickness, or diameter of the test
m
specimen be 3 mm. The accuracy of a typical precision laboratory balance is 0.001 g; meeting the requirement of 8.4 requires that the minimum mass
of the test specimen be 1 g. Given these minimum dimensions and mass, meeting the requirements of 8.4 and 8.5 requires that the minimum length of
a metal alloy test specimen with density of 0.008 g/mm be 13.9 mm. A test specimen of a typical ceramic, with lower density, will be correspondingly
longer
NOTE 7—Table 1 illustrates how uncertainties in the measured parameters influence the calculated dynamic elastic modulus. It shows that calculations
are most sensitive to uncertainty in the measurement of the thickness, diameter, and length.
9. Procedure
9.1 Activate all electrical equipment, and allow it to stabilize according to the manufacturer’s recommendations.
9.2 Use a test specimen established as a verification/calibration standard to verify the equipment response and accuracy.
9.3 Fundamental Flexural Resonant Frequency (Out-of-Plane Flexure):
9.3.1 Place the specimen on the supports located at the fundamental nodal points nodes (0.224 L from each end; see Fig. 3).
TABLE 1 Effects of Variable Uncertainty on Dynamic Elastic
Modulus Uncertainty
Variable Exponent in
Measurement Calculation
Variable Dynamic Elastic
Uncertainty Uncertainty
Modulus Equation
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 %
–3
Thickness (t) 0.1 % t 0.3 %
–4
Diameter (D) 0.1 % D 0.4 %
E1876 − 22
FIG. 3 Rectangular Specimens Tested for In-Plane and Out-of-Plane Flexure
9.3.2 Determine the direction of maximum sensitivity for the detecting transducer. Orient the detecting transducer so that it will
detect the desired vibration.
9.3.2.1 Direct-Contact Transducers—Place the detecting transducer in contact with the test specimen test specimen at point P1
in Fig. 3 to pick up the desired vibration. If the transducer is placed at an anti-node (location of maximum displacement), it may
mass load the specimen and modify the natural vibration. The transducer should be placed only as far from the nodal points node
as necessary to obtain a reading (see Fig. 3). This location will minimize the damping effect from the contacting direct-contact
detecting transducer. The transducer contact force should be consistent, with good response and minimal interference with the free
vibration of the specimen.
9.3.2.2 Non-Contact Transducers—Place the non-contact detecting transducer over an anti-node point at point M1.1 or M1.2 in
Fig. 3 and close enough to the test specimen to pick up the desired vibration, but not so close as to interfere with the free vibration
(see Fig. 3).
9.3.3 Strike the specimen lightly and elastically, either at the center of the specimen specimen, point X1 in Fig. 3, or at the opposite
end of the specimen from the detecting transducer (see Fig. 3).
9.3.4 Record the resultant reading, and repeat the test until five consecutive readings are obtained that lie within 1 % of each other.
Use the average of these five readings to determine the fundamental resonant frequency in flexure.
9.4 Fundamental Flexural Resonant Frequency (In-Plane Flexure):
9.4.1 This procedure is the same as 9.3, except that the direction of vibration is in the major plane of the specimen. This
measurement may be performed in two ways. In one case, move the detecting transducer and impulser 90° around the long axis
of the test specimen to points X2 and M2.1 and M2.2 to introduce and detect vibrations in the major plane (see Fig. 3). In the
alternate method, rotate the test barspecimen 90° around its long axis and reposition it on the specimen supports. Transpose the
width and thickness dimensions in the calculations. For homogeneous, isotropic materials, the calculated moduli should be the
same as the moduli calculated from the out-of-plane frequency. The comparison of in-plane and out-of-plane frequency
measurements can thus be used as a cross check of experimental methods and calculations.
9.5 Fundamental Torsional Resonant Frequency:
9.5.1 Support the specimen at the midpoint of its length and width (the torsional nodal planes) nodes) (see Fig. 4).
E1876 − 22
FIG. 4 Rectangular Specimen Tested for Torsional Vibration
9.5.2 Locate the detecting transducer at one quadrant of the specimen, preferably at approximately 0.224 L from one end and
toward the edge. edge at point P3 (direct-contact detecting transducer) or M3 (non-contact detecting transducer) in Fig. 4. This
location is a nodalnode point of flexural vibration and will minimize the possibility of detecting a spurious flexural vibration mode
(see Fig. 4).
9.5.3 Strike the specimen on the quadrant diagonally opposite the detecting transducer, again at 0.224 L from the end and near
the edge. edge at point X3 in Fig. 4. Striking at a flexural nodal pointvibration node will minimize the possibility of exciting a
flexural vibration mode of vibration (see Fig. 4).
9.5.4 Record the resultant reading, and repeat the test until five consecutive readings are obtained that lie within 1 % of each other.
Use the average of these five readings to determine the fundamental resonant frequency in torsion.
9.6 Fundamental Longitudinal Resonant Frequency:
9.6.1 Support the specimen at the midpoint of its length and width (the same as for torsion), or brace the specimen at its mid
length, the fundamental longitudinal nodal position. node.
9.6.2 Locate the detecting transducer at the center of one of the end faces of the specimen.specimen at point P4 or M4 in Fig. 5.
9.6.3 Strike the end face of the specimen at point X4 in Fig. 5, opposite to the face where the detecting transducer is located.
9.6.4 Record the resultant reading, and repeat the test, until five consecutive readings are obtained that lie within 1 % of each other.
Use the average of these five readings to determine the fundamental longitudinal resonant frequency.
FIG. 5 Rectangular Specimen Tested for Longitudinal Vibration
E1876 − 22
10. Calculation
10.1 Dynamic Young’s Modulus (13,24):
10.1.1 For the fundamental flexure resonant frequency of a rectangular bar (24),
2 3
mf L
f
E 5 0.9465 T (1)
S DS D
3 1
b t
where:
E = Dynamic Young’s modulus, Pa,
m = mass of the bar, g (see Note 5),
m = mass of the bar, g (see Note 8),
b = width of the bar, mm (see Note 5),
b = width of the bar, mm (see Note 8),
L = length of the bar, mm (see Note 5),
L = length of the bar, mm (see Note 8),
t = thickness of the bar, mm (see Note 5),
t = thickness of the bar, mm (see Note 8),
f = fundamental resonant frequency of bar in flexure, Hz, and
f
T = correction factor for fundamental flexural mode to account for finite thickness of bar, Poisson’s ratio, and so forth.
2 4
t t
T 5 116.585 110.0752 μ10.8109 μ 2 0.868 (2)
~ !S D S D
L L
t
8.340 ~110.2023 μ12.173 μ !
S D
L
t
3 2 4
1.00016.338 ~110.1408 μ11.536 μ !
S D
L
where:
μ = Poisson’s ratio.
NOTE 8—In the dynamic elastic modulus equations, the mass and length terms are given in units of grams and millimetres. However, the defined equations
can also be used with mass and length terms in units of kilograms and metres with no changes in terms or exponents.
10.1.1.1 If L/t ≥ 20, T can be simplified to the following:
t
T 5 1.00016.585 (3)
F S D G
L
and E can be calculated directly.
and E can be calculated directly.
10.1.1.2 If L/t < 20 and Poisson’s ratio is known, then T can be calculated directly from Eq 2 and then used to calculate E.
10.1.1.3 If L/t < 20 and Poisson’s ratio is not known, assume an initial Poisson’s ratio to begin the computations. Use an iterative
process to determine a value of Poisson’s ratio, based on experimental dynamic Young’s modulus and dynamic shear modulus. The
iterative process is flowcharted in Fig. 56 and described in (1) through (5),
(1) Determine the fundamental flexural and torsional resonant frequency of the rectangular test specimen, as described in
Section 9. Using Eq 12, calculate the dynamic shear modulus of the test specimen for the fundamental torsional resonant frequency.
(2) Using Eq 1 and Eq 2, calculate the dynamic Young’s modulus of the rectangular test specimen from the fundamental
flexural resonant frequency, dimensions and mass of the specimen, and initial/iterative Poisson’s ratio. Exercise care in using
consistent units for all of the parameters throughout the computations.
(3) Substitute the dynamic shear modulus and Young’s modulus values calculated in steps (1) and (2) into Eq 15 for Poisson’s
ratio satisfying isotropic conditions. Calculate a new value for Poisson’s ratio for another iteration beginning at Step (2).
(4) Repeat Steps (2) and (3) until no significant difference (2 % or less) is observed between the last iterative value and the
final computed value of the Poisson’s ratio.
(5) Self-consistent values for the moduli are thus obtained.
E1876 − 22
FIG. 56 Process Flow Chart for Iterative Determination of Poisson’s Ratio for Isotropic Materials
10.1.2 For the fundamental flexural resonant frequency of a rod of circular cross section (24) :
L
2 '
E 5 1.6067 ~mf ! T (4)
S D
4 f 1
D
where:
D = diameter of rod, mm (see Note 5), and
D = diameter of rod, mm (see Note 8), and
'
T = correction factor for fundamental flexural mode to account for finite diameter of rod, Poisson’s ratio, and so forth.
'
T =
1 2
D
114.939 110.0752 μ10.8109 μ
~ !S D
L
D
2 0.4883
S D
L
D
4.691 ~110.2023 μ12.173 μ !
S D
L
2 (5)
D
3 4
1.00014.754 110.1408 μ11.536 μ
~ !S D
L
'
10.1.2.1 If L/D ≥ 20, then T can be simplified to the following:
D
'
T 5 1.00014.939 (6)
F S D G
L
'
10.1.2.2 If L/D < 20 and Poisson’s ratio is known, then T can be calculated directly from Eq 5 and then used to calculate E.
10.1.2.3 If L/D < 20 and Poisson’s ratio is not known, assume an initial Poisson’s ratio to start the computations. Determine final
values for Poisson’s ratio, dynamic Young’s modulus, and dynamic shear modulus using the same method shown in Fig. 56 and
described in (1) through (5) in 10.1.1.3, but using the dynamic modulus equations for circular bars (Eq 4, and Eq 14).
10.1.3 For the fundamental longitudinal resonant frequency of a slender bar with circular cross-section:
L
E 5 16 m f (7)
F G
l 2
π D K
E1876 − 22
where:
f = fundamental longitudinal resonant frequency of bar, Hz
l
D = the diameter of the bar, mm
K = correction factor for the fundamental longitudinal mode to account for the finite diamet
...