ASTM C1259-21

Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics 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 modulus of advanced ceramics that are elastic, homogeneous, and isotropic (1).4  
5.3 This test method addresses the room temperature determination of dynamic moduli of elasticity of slender bars (rectangular cross section) and rods (cylindrical). Flat plates and discs may also be measured similarly, but the required equations for determining the moduli are not addressed herein.  
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. Deviations in specimen dimensions or mass and internal flaws (cracks, delaminations, inhomogeneities, porosity, etc.) will change the resonant frequency for that specimen. Any specimen with a resonant frequency falling outside the prescribed frequency range is rejected. The actual modulus of each specimen need not be determined as long as the limits of the selected frequency range are known to include the resonant frequency that the specimen must possess if its geometry and mass and in...
SCOPE
1.1 This test method covers determination of the dynamic elastic properties of advanced ceramics 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, cylindrical, or disc geometry) test specimen of that material can be measured. The resonant frequencies in flexure and torsion are measured by excitation of vibrations of the test specimen in a supported mode by a singular elastic strike with an impulse tool (Section 4 and Fig. 1, Fig. 3, and Fig. 4). Dynamic Young’s modulus is determined using the resonant frequency in the flexural 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.
FIG. 1 Block Diagram of Typical Test Apparatus  
1.2 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, in accordance with Subsections 9.2, 9.3, and 10.4 of Test Method C1198.  
1.3 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 C215, C623, C747, C848, C1198, E1875, and E1876 may differ from this test method in several areas (for example, sample size, dimensional tolerances, sample preparation, calculation details, etc.). The testing of those materials should be done in compliance with the appropriate material-specific standards. Where possible, the pro...

General Information

• Status: Published
• Publication Date: 31-Jul-2021
• ICS: 81.060.99 - Other standards related to ceramics
• Technical Committee: C28 - Advanced Ceramics
• Drafting Committee: C28.01 - Mechanical Properties and Performance

Relations

• Refers: ASTM C747-23 - Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• Effective Date: 01-Oct-2023
• Refers: ASTM C1198-20 - Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Sonic Resonance
• Effective Date: 01-Jan-2020
• Refers: ASTM C1145-19 - Standard Terminology of Advanced Ceramics
• Effective Date: 01-Jul-2019
• Refers: ASTM E2001-18 - Standard Guide for Resonant Ultrasound Spectroscopy for Defect Detection in Both Metallic and Non-metallic Parts
• Effective Date: 01-Nov-2018
• Refers: ASTM C747-16 - Standard Test Method for Moduli of Elasticity and Fundamental Frequencies of Carbon and Graphite Materials by Sonic Resonance
• Effective Date: 01-Oct-2016
• Refers: ASTM C215-14 - Standard Test Method for Fundamental Transverse, Longitudinal, and<brk/> Torsional Resonant Frequencies of Concrete Specimens
• Effective Date: 15-Dec-2014
• Refers: ASTM E177-14 - Standard Practice for Use of the Terms Precision and Bias in ASTM Test Methods
• Effective Date: 01-May-2014
• Refers: ASTM E2001-13 - Standard Guide for Resonant Ultrasound Spectroscopy for Defect Detection in Both Metallic and Non-metallic Parts
• Effective Date: 01-Dec-2013
• Refers: ASTM E1875-13 - Standard Test Method for Dynamic Young's Modulus, Shear Modulus, and Poisson's Ratio by Sonic Resonance
• Effective Date: 01-Nov-2013
• Refers: ASTM C1161-13 - Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
• Effective Date: 01-Aug-2013
• 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 E691-13 - Standard Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
• Effective Date: 01-May-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 C1145-06(2013)e1 - Standard Terminology of Advanced Ceramics
• Effective Date: 01-Feb-2013
• Refers: ASTM C1145-06(2013) - Standard Terminology of Advanced Ceramics
• Effective Date: 01-Feb-2013

Overview

ASTM C1259-21: Standard Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio for Advanced Ceramics by Impulse Excitation of Vibration specifies a non-destructive process for determining the dynamic elastic properties of advanced ceramics at ambient temperature. Utilizing impulse excitation of vibration, this standard evaluates fundamental resonant frequencies of specimens based on their mass and geometry, allowing computation of dynamic Young’s modulus, dynamic shear modulus, and Poisson’s ratio.

This test method is tailored for elastic, homogeneous, and isotropic advanced ceramics shaped as slender bars (rectangular cross sections), rods (cylindrical), and discs. The approach offers several advantages over static loading and continuous resonance techniques, providing meaningful characterization while minimizing specimen damage.

Key Topics

• Dynamic Elastic Properties: Determines Young’s modulus, shear modulus, and Poisson’s ratio by measuring mechanical resonant frequencies.
• Specimen Requirements: Applies to bars, cylindrical rods, and discs of advanced ceramics, emphasizing precise measurements of dimensions and mass.
• Testing Procedure: Specimens are excited using a single elastic strike (impulse tool); resulting vibrations are monitored by transducers and analyzed electronically to isolate fundamental frequencies.
• Non-Destructive Testing: Tests are performed with minute strains, ensuring minimal risk of fracture, and specimens remain available for further evaluation.
• Quality Control: Resonant frequencies serve as quality indicators, with deviations suggesting flaws such as cracks, delaminations, or inhomogeneities.
• Flexibility: While designed for ambient conditions, with equipment modifications, the method can also be adapted for cryogenic or high temperatures.

Applications

ASTM C1259-21 is widely utilized in various aspects of ceramic materials engineering, including:

• Material Development: Used to characterize and optimize advanced ceramics during research and product development.
• Design Data Generation: Provides reliable elastic property data essential for engineering calculations and finite element modeling.
• Quality Assurance: Resonant frequency measurements allow rapid screening of specimens for compliance with material specifications.
• Process Control: Facilitates monitoring and maintaining consistency in ceramic production processes.
• Non-Destructive Evaluation (NDE): Detects internal flaws and variations in composition or geometry without damaging the specimen.
• Suitability for Complex Shapes: Can accommodate both regular (bars, rods, discs) and complex geometries, making the method versatile for a broad range of applications.

By enabling accurate, non-destructive determination of moduli at or near the origin of the stress-strain curve, this method is of practical importance wherever the mechanical integrity of advanced ceramics is critical.

Related Standards

ASTM C1259-21 is part of a comprehensive family of ASTM and ISO standards addressing dynamic testing and characterizing of advanced ceramics and other rigid materials, including:

• ASTM C215: Resonant frequency and modulus determination for concrete specimens
• ASTM C623: Young’s modulus, shear modulus, and Poisson’s ratio for glass and glass-ceramics
• ASTM C747, C848, C1198: Sonic resonance methods for various ceramic and graphite materials
• ASTM E1875, E1876: Sonic resonance and impulse excitation for dynamic modulus testing
• ISO 14704: Flexural strength of monolithic ceramics at room temperature

Organizations involved in ceramics research, manufacturing, and certification rely on these and related standards to ensure accuracy, repeatability, and global comparability of mechanical property data.

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Keywords: ASTM C1259-21, dynamic elastic modulus, Young’s modulus, shear modulus, Poisson’s ratio, impulse excitation of vibration, advanced ceramics, non-destructive testing, quality control, resonant frequency, ceramic mechanical properties, ceramic material testing.

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: C1259 − 21
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio for Advanced Ceramics by Impulse Excitation of
Vibration
This standard is issued under the fixed designation C1259; 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* procedures, sample specifications, and calculations in this
standard are consistent with the other test methods.
1.1 This test method covers determination of the dynamic
elastic properties of advanced ceramics at ambient tempera- 1.4 This test method uses test specimens in bar, rod, and
tures. Specimens of these materials possess specific mechani- disc geometries. The rod and bar geometries are described in
cal resonant frequencies that are determined by the elastic the main body. The disc geometry is addressed in Annex A1.
modulus, mass, and geometry of the test specimen. The
1.5 A modification of this test method can be used for
dynamic elastic properties of a material can therefore be
quality control and nondestructive evaluation, using changes in
computed if the geometry, mass, and mechanical resonant
resonant frequency to detect variations in specimen geometry
frequencies of a suitable (rectangular, cylindrical, or disc
and mass and internal flaws in the specimen. (See 5.5.)
geometry) test specimen of that material can be measured. The
1.6 The values stated in SI units are to be regarded as
resonant frequencies in flexure and torsion are measured by
standard. The non-SI unit values given in parentheses are for
excitation of vibrations of the test specimen in a supported
information only and are not considered standard.
mode by a singular elastic strike with an impulse tool (Section
4 and Fig. 1, Fig. 3, and Fig. 4). Dynamic Young’s modulus is 1.7 This standard does not purport to address all of the
safety concerns, if any, associated with its use. It is the
determined using the resonant frequency in the flexural mode
of vibration. The dynamic shear modulus, or modulus of responsibility of the user of this standard to establish appro-
priate safety, health, and environmental practices and deter-
rigidity, is found using torsional resonant vibrations. Dynamic
Young’s modulus and dynamic shear modulus are used to mine the applicability of regulatory limitations prior to use.
1.8 This international standard was developed in accor-
compute Poisson’s ratio.
dance with internationally recognized principles on standard-
1.2 Although not specifically described herein, this test
ization established in the Decision on Principles for the
method can also be performed at cryogenic and high tempera-
Development of International Standards, Guides and Recom-
tures with suitable equipment modifications and appropriate
mendations issued by the World Trade Organization Technical
modifications to the calculations to compensate for thermal
Barriers to Trade (TBT) Committee.
expansion,inaccordancewithSubsections9.2,9.3,and10.4of
Test Method C1198.
2. Referenced Documents
1.3 There are material-specific ASTM standards that cover 2
2.1 ASTM Standards:
the determination of resonance frequencies and elastic proper-
C215 Test Method for Fundamental Transverse,
ties of specific materials by sonic resonance or by impulse
Longitudinal, and Torsional Resonant Frequencies of
excitationofvibration.TestMethodsC215,C623,C747,C848,
Concrete Specimens
C1198, E1875, and E1876 may differ from this test method in
C372 Test Method for Linear Thermal Expansion of Porce-
several areas (for example, sample size, dimensional
lainEnamelandGlazeFritsandFiredCeramicWhiteware
tolerances, sample preparation, calculation details, etc.). The
Products by Dilatometer Method
testing of those materials should be done in compliance with
C623 Test Method for Young’s Modulus, Shear Modulus,
theappropriatematerial-specificstandards.Wherepossible,the
and Poisson’s Ratio for Glass and Glass-Ceramics by
Resonance
This test method is under the jurisdiction of ASTM Committee C28 on
Advanced Ceramics and is the direct responsibility of Subcommittee C28.01 on
Mechanical Properties and Performance. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved Aug. 1, 2021. Published August 2021. Originally contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
approved in 1994. Last previous edition approved in 2015 as C1259 – 15. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/C1259-21. 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
C1259 − 21
3.1.4 dynamic mechanical measurement, n—a technique in
which either the modulus or damping, or both, of a substance
under oscillatory load or displacement is measured as a
function of temperature, frequency, or time, or combination
thereof. (E1876)
–2
3.1.5 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)
FIG. 1 Block Diagram of Typical Test Apparatus
–2
3.1.6 elastic modulus [FL ],n—the ratio of stress to strain
below the proportional limit. (E6)
C747 Test Method for Moduli of Elasticity and Fundamental
3.1.7 Poisson’s ratio (µ) [nd],n—the absolute value of the
Frequencies of Carbon and Graphite Materials by Sonic
ratio of transverse strain to the corresponding axial strain
Resonance
resulting from uniformly distributed axial stress below the
C848 Test Method for Young’s Modulus, Shear Modulus,
proportional limit of the material.
and Poisson’s Ratio For Ceramic Whitewares by Reso-
3.1.7.1 Discussion—In isotropic materials, Young’s Modu-
nance
lus (E), shear modulus (G), and Poisson’s ratio (µ) are related
C1145 Terminology of Advanced Ceramics
by the following equation:
C1161 Test Method for Flexural Strength of Advanced
µ 5 E/2G 21 (1)
~ !
Ceramics at Ambient Temperature
(E6)
C1198 Test Method for Dynamic Young’s Modulus, Shear
–2
Modulus, and Poisson’s Ratio for Advanced Ceramics by 3.1.8 proportional limit [FL ],n—the greatest stress that a
Sonic Resonance
material is capable of sustaining without deviation from
E6 Terminology Relating to Methods of Mechanical Testing proportionality of stress to strain (Hooke’s law). (E6)
E177 Practice for Use of the Terms Precision and Bias in
–2
3.1.9 shear modulus (G) [FL ],n—the elastic modulus in
ASTM Test Methods
shear or torsion. Also called modulus of rigidity or torsional
E691 Practice for Conducting an Interlaboratory Study to
modulus. (E6)
Determine the Precision of a Test Method
–2
3.1.10 Young’s modulus (E) [FL ],n—the elastic modulus
E1875 Test Method for Dynamic Young’s Modulus, Shear
in tension or compression. (E6)
Modulus, and Poisson’s Ratio by Sonic Resonance
E1876 Test Method for Dynamic Young’s Modulus, Shear
3.2 Definitions of Terms Specific to This Standard:
Modulus, and Poisson’s Ratio by Impulse Excitation of
3.2.1 antinodes, n—two or more locations that have local
Vibration
maximum displacements, called antinodes, in an unconstrained
E2001 Guide for Resonant Ultrasound Spectroscopy for
slender rod or bar in resonance. For the fundamental flexure
Defect Detection in Both Metallic and Non-metallic Parts
resonance, the antinodes are located at the two ends and the
2.2 ISO Standard:
center of the specimen.
ISO 14704 Test Method for Flexural Strength of Monolithic
3.2.2 elastic, adj—the property of a material such that an
Ceramics at Room Temperatures
application of stress within the elastic limit of that material
making up the body being stressed will cause an instantaneous
3. Terminology
and uniform deformation, which will be eliminated upon
3.1 Definitions:
removal of the stress, with the body returning instantly to its
3.1.1 The definitions of terms relating to mechanical testing
original size and shape without energy loss. Most advanced
appearing inTerminology E6 should be considered as applying
ceramics conform to this definition well enough to make this
to the terms used in this test method. The definitions of terms
resonance test valid.
relatingtoadvancedceramicsappearinginTerminologyC1145
3.2.3 flexural vibrations, n—the vibrations that occur when
should be considered as applying to the terms used in this test
the displacements in a slender rod or bar are in a plane normal
method. Directly pertinent definitions as listed in Terminolo-
to the length dimension.
gies E6 and C1145 are shown in the following paragraphs with
the appropriate source given in brackets.
3.2.4 homogeneous, adj—the condition of a specimen such
3.1.2 advanced ceramic, n—a highly engineered, high-
that the composition and density are uniform, so that any
performance, predominately nonmetallic, inorganic, ceramic
smaller specimen taken from the original is representative of
material having specific functional attributes. (C1145)
thewhole.Practically,aslongasthegeometricaldimensionsof
3.1.3 dynamic elastic modulus, n—the elastic modulus, thetestspecimenarelargewithrespecttothesizeofindividual
either Young’s modulus or shear modulus, that is measured in
grains, crystals, components, pores, or microcracks, the body
a dynamic mechanical measurement. (E1876) can be considered homogeneous.
3.2.5 in-plane flexure, n—for rectangular parallelepiped
geometries, a flexure mode in which the direction of displace-
Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
4th Floor, New York, NY 10036, http://www.ansi.org. ment is in the major plane of the test specimen.
C1259 − 21
3.2.6 isotropic, adj—the condition of a specimen such that 5.2 This test method is specifically appropriate for deter-
thevaluesoftheelasticpropertiesarethesameinalldirections mining the modulus of advanced ceramics that are elastic,
in the material.Advanced ceramics are considered isotropic on homogeneous, and isotropic (1).
a macroscopic scale, if they are homogeneous and there is a
5.3 This test method addresses the room temperature deter-
random distribution and orientation of phases, crystallites,
mination of dynamic moduli of elasticity of slender bars
components, pores, or microcracks.
(rectangular cross section) and rods (cylindrical). Flat plates
and discs may also be measured similarly, but the required
3.2.7 nodes, n—oneormorelocationsinaslenderrodorbar
equations for determining the moduli are not addressed herein.
in resonance having a constant zero displacement. For the
fundamental flexural resonance of such a rod or bar, the nodes
5.4 This dynamic test method has several advantages and
are located at 0.224 L from each end, where L is the length of
differences from static loading techniques and from resonant
the specimen.
techniques requiring continuous excitation.
5.4.1 The test method is nondestructive in nature and can be
3.2.8 out-of-plane flexure, n—for rectangular parallelepiped
used for specimens prepared for other tests. The specimens are
geometries, a flexure mode in which the direction of displace-
subjected to minute strains; hence, the moduli are measured at
ment is perpendicular to the major plane of the test specimen.
or near the origin of the stress-strain curve, with the minimum
3.2.9 resonant frequency, n—naturally occurring frequen-
possibility of fracture.
cies of a body driven into flexural, torsional, or longitudinal
5.4.2 The impulse excitation test uses an impact tool and
vibration that are determined by the elastic modulus, mass, and
simple supports for the test specimen. There is no requirement
dimensions of the body. The lowest resonant frequency in a
for complex support systems that require elaborate setup or
given vibrational mode is the fundamental resonant frequency
alignment.
of that mode.
5.5 Thistechniquecanbeusedtomeasureresonantfrequen-
3.2.10 slender rod or bar, n—in dynamic elastic property
cies alone for the purposes of quality control and acceptance of
testing, a specimen whose ratio of length to minimum cross-
test specimens of both regular and complex shapes.Arange of
sectional dimension is at least 5 and preferably in the range of
acceptable resonant frequencies is determined for a specimen
20 to 25.
with a particular geometry and mass. Deviations in specimen
dimensions or mass and internal flaws (cracks, delaminations,
3.2.11 torsional vibrations, n—the vibrations that occur
inhomogeneities, porosity, etc.) will change the resonant fre-
when the oscillations in each cross-sectional plane of a slender
quency for that specimen. Any specimen with a resonant
rod or bar are such that the plane twists around the length
frequency falling outside the prescribed frequency range is
dimension axis.
rejected. The actual modulus of each specimen need not be
determinedaslongasthelimitsoftheselectedfrequencyrange
4. Summary of Test Method
are known to include the resonant frequency that the specimen
4.1 This test method measures the fundamental resonant
mustpossessifitsgeometryandmassandinternalstructureare
frequency of test specimens of suitable geometry (bar, rod, or
within specified tolerances. The technique is particularly suit-
disc) by exciting them mechanically by a singular elastic strike ablefortestingspecimenswithcomplexgeometries(otherthan
with an impulse tool. A transducer (for example, contact parallelepipeds, cylinders/rods, or discs) that would not be
accelerometer or noncontacting microphone) senses the result- suitable for testing by other procedures. This is similar to the
ingmechanicalvibrationsofthespecimenandtransformsthem evaluation method described in Guide E2001.
intoelectricsignals.Specimensupports,impulselocations,and
5.6 If a thermal treatment or an environmental exposure
signal pickup points are selected to induce and measure
affects the elastic response of the test specimen, this test
specific modes of the transient vibrations. The signals are
methodmaybesuitableforthedeterminationofspecificeffects
analyzed, and the fundamental resonant frequency is isolated
of thermal history, environment exposure, etc. Specimen de-
and measured by the signal analyzer, which provides a numeri-
scriptions should include any specific thermal treatments or
cal reading that is (or is proportional to) either the frequency or
environmental exposures that the specimens have received.
the period of the specimen vibration. The appropriate funda-
mental resonant frequencies, dimensions, and mass of the
6. Interferences
specimen are used to calculate dynamic Young’s modulus,
6.1 The relationships between resonant frequency and dy-
dynamic shear modulus, and Poisson’s ratio.
namic modulus presented herein are specifically applicable to
homogeneous, elastic, isotropic materials.
5. Significance and Use
6.1.1 This method of determining the moduli is applicable
to composite ceramics and inhomogeneous materials only with
5.1 This test method may be used for material development,
careful consideration of the effect of inhomogeneities and
characterization, design data generation, and quality control
anisotropy. The character (volume fraction, size, morphology,
purposes.
distribution, orientation, elastic properties, and interfacial
The boldface numbers in parentheses refer to the list of references at the end of
this test method.
C1259 − 21
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, etc.) 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.
FIG. 2 Diagram of Typical Impulse Tool for Small Specimens
6.1.4 The test method is not satisfactory for specimens that
have major discontinuities, such as large cracks (internal or
mode.Thelocationsoftheimpulsepointandtransducershould
surface) or voids.
not be changed in multiple readings; changes in position may
6.2 This test method for determining moduli is limited to
develop and detect alternate vibration modes. In the same
specimens with regular geometries (rectangular parallelepiped,
manner, the force used in impacting should be consistent in
cylinders, and discs) for which analytical equations are avail-
multiple readings.
able to relate geometry, mass, and modulus to the resonant
6.5 If the frequency readings are not repeatable for a
vibration frequencies. The test method is not appropriate for
specific set of impulse and transducer locations on a specimen,
determining the elastic properties of materials that cannot be
it may be because several different modes of vibration are
fabricated into such geometries.
being developed and detected in the test. The geometry of the
6.2.1 The analytical equations assume parallel and concen-
test bar and desired vibration mode should be evaluated and
tric dimensions for the regular geometries of the specimen.
used to identify the nodes and antinodes of the desired
Deviations from the specified tolerances for the dimensions of
vibrations. More consistent measurements may be obtained if
the specimens will change the resonant frequencies and intro-
the impulse point and transducer locations are shifted to induce
duce error into the calculations.
and measure the single desired mode of vibration.
6.2.2 Edge treatments such as chamfers or radii are not
considered in the analytical equations. Edge chamfers on
7. Apparatus
flexure bars prepared according to Test Method C1161 will
7.1 Apparatus suitable for accurately detecting, analyzing,
change the resonant frequency of the test bars and introduce
andmeasuringthefundamentalresonantfrequencyorperiodof
error into the calculations of the dynamic modulus. It is
a vibrating free-free beam is used. The test apparatus is shown
recommended that specimens for this test method not have
in Fig. 1. It consists of an impulse tool, a suitable pickup
chamfered or rounded edges.Alternately, if narrow rectangular
transducer to convert the mechanical vibration into an electri-
specimens with chamfers or edge radii are tested, then the
cal signal, an electronic system (consisting of a signal
procedures in Annex A2 should be used to correct the
conditioner/amplifier, a signal analyzer, and a frequency read-
calculated Young’s modulus, E.
outdevice),andasupportsystem.Commercialinstrumentation
6.2.3 For specimens with as-fabricated and rough or uneven
is available that measures the frequency or period of the
surfaces, variations in dimensions can have a significant effect
vibrating specimen.
in the calculations. For example, in the calculation of dynamic
7.2 Impulse Tool—The exciting impulse is imparted by
modulus, the modulus value is inversely proportional to the
lightly striking the specimen with a suitable implement. This
cube of the thickness. Uniform specimen dimensions and
implement should have most of its mass concentrated at the
precise measurements are essential for accurate results.
point of impact and have mass sufficient to induce a measur-
6.3 The test method assumes that the specimen is vibrating
able mechanical vibration, but not so large as to displace or
freely, with no significant restraint or impediment. Specimen
damage the specimen physically. In practice, the size and
supports should be designed and located properly in accor-
geometryoftheimpulsetooldependsonthesizeandweightof
dance with 9.3.1, 9.4.1, and 9.5.1 so the specimen can vibrate
the specimen and the force needed to produce vibration. For
freely in the desired mode. In using direct contact transducers,
commonly tested geometries (small bars, rods, and discs) in
the transducer should be positioned away from antinodes and
advanced ceramics, an example of such an impulse tool would
with minimal force to avoid interference with free vibration.
be a steel sphere 0.5 cm in diameter glued to the end of a
With noncontacting transducers, the maximum sensitivity is
flexible 10 cm long polymer rod. (See Fig. 2.) An alternate
accomplished by placing the transducer at an antinode.
impulse tool would be a solid metal, ceramic, or polymer
6.4 Proper location of the impulse point and transducer is sphere (0.1 to 1.0 cm in diameter) dropped on the specimen
important in introducing and measuring the desired vibration through a guide tube to ensure proper impulse position.
C1259 − 21
7.3 Signal Pickup—Signal detection can be via transducers dimensions should therefore be selected with this relationship
in direct contact with the specimen or by noncontact transduc- in mind. The selection of size shall be made so that, for an
ers. Contact transducers are commonly accelerometers using estimated modulus, the resonant frequencies measured will fall
piezoelectric or strain gage methods to measure the vibration. within the range of frequency response of the transducers and
Noncontact transducers are commonly acoustic microphones, electronics used. For a slender rod, the ratio of length to
but they may also use laser, magnetic, or capacitance methods minimum cross-sectional dimension shall have a value of at
tomeasurethevibration.Thefrequencyrangeofthetransducer least10.However,aratioofapproximately20 ≈25ispreferred
shall be sufficient to measure the expected frequencies of the for ease in calculation. For shear modulus measurements of
specimens of interest. A suitable range would be 100 Hz to rectangular bars, a ratio of width to thickness of 5 or greater is
50 kHz for most advanced ceramic test specimens. (Smaller recommended for minimizing experimental difficulties.
and stiffer specimens vibrate at higher frequencies.) The
8.3 All surfaces on the rectangular specimen shall be flat.
frequency response of the transducer across the frequency
Opposite surfaces across the length and width shall be parallel
range of interest shall have a bandwidth of at least 10 % of the
within 0.01 mm or 60.1 %, whichever is greater. Opposite
maximum measured frequency before –3 dB power loss oc-
surfaces across the thickness shall be parallel within 0.002 mm
curs.
or 60.1 %, whichever is greater. The cylindrical specimen
7.4 Electronic System—The electronic system consists of a shall be round and constant in diameter within 0.002 mm or
signal conditioner/amplifier, signal analyzer, and a frequency 60.1 %, whichever is greater.
readoutdevice.Thesystemshouldhaveaccuracyandprecision
8.4 Test specimen mass shall be determined within 60.1 %
sufficient to measure the frequencies of interest to an accuracy
or 10 mg, whichever is greater.
of 0.1 %.The signal conditioner/amplifier should be suitable to
8.5 Test specimen length shall be measured to within
power the transducer and provide and appropriate amplified
60.1 %. Test specimen cross-sectional dimensions (thickness
signal to the signal analyzer. The signal analysis system
and width in rectangular beams; diameter in cylindrical rods)
consists of a frequency counting device and a readout device.
shall be measured within 60.1 % or 0.01 mm at three equally
Appropriate devices are frequency counter systems with stor-
spaced locations along the length and an average value
age capability or digital storage oscilloscopes with a frequency
determined.
counter module. With the digital storage oscilloscope, a Fast
Fourier Transform signal analysis system may be useful for
8.6 Porous materials and those susceptible to hydration
analyzing more complex waveforms and identifying the fun-
should be dried in air at 120 °C in a drying oven until the mass
damental resonant frequency.
is constant (less than 0.1 % or 10 mg difference in measured
mass with 30 min of additional drying).
7.5 Support System—The support shall serve to isolate the
specimen from extraneous vibration without restricting the
8.7 It is recommended that the laboratory obtain and main-
desired mode of specimen vibration. Appropriate materials
tain an internal reference specimen with known and recorded
should be stable at the test temperatures. Support materials can
fundamental resonant frequencies in flexure and torsion. The
be either soft or rigid for ambient conditions. Examples of soft
reference specimen should be used to check and confirm the
materials would be a compliant elastomeric material, such as
operation of the test system on a regular basis. It can also be
polyurethane foam strips. Such foam strips would have simple
used to train operators in the proper test setup and test
flat surfaces for the specimen to rest on. Rigid materials, such
procedure. The reference specimen can be a standard ceramic
as metal or ceramic, should have sharp knife edges or cylin-
material (alumina, silicon carbide, zirconia, etc.) or it may be
drical surfaces on which the specimen should rest. The rigid
of a similar size, composition, and microstructure to the types
supports should be resting on isolation pads to prevent ambient
of ceramic specimens commonly tested at the laboratory. The
vibrations from being picked up by the transducer. Wire
referencespecimenmustmeetthesize,dimensionaltolerances,
suspension can also be used. Specimens shall be supported
and surface finish requirements of Section 8.
along node lines appropriate for the desired vibration in the
locations described in Section 8.
9. Procedure
9.1 Activate all electrical equipment, and allow it to stabi-
8. Test Specimen
lize according to the manufacturer’s recommendations.
8.1 The specimens shall be prepared so that they are either
9.2 Use a test specimen established as a verification/
rectangular or circular in cross section. Either geometry can be
calibration standard to verify the equipment response and
used to measure both dynamic Young’s modulus and dynamic
accuracy.
shear modulus. Although the equations for computing shear
moduluswithacylindricalspecimenarebothsimplerandmore
9.3 Fundamental Flexural Resonant Frequency (Out-of-
accurate than those used with a rectangular bar, experimental
Plane Flexure):
difficulties in obtaining torsional resonant frequencies for a
9.3.1 Place the specimen on the supports located at the
cylindrical specimen usually preclude its use for determining
fundamental nodal points (0.224 L from each end; see Fig. 3).
shear modulus. (See Annex A1 for disc specimens.)
9.3.2 Determine the direction of maximum sensitivity for
8.2 Resonantfrequenciesforagivenspecimenarefunctions the transducer. Orient the transducer so that it will detect the
of the specimen dimensions as well as its mass and moduli; desired vibration.
C1259 − 21
FIG. 3 Rectangular Specimens Tested for In-Plane and Out-of-Plane Flexure
9.3.2.1 Direct Contact Transducers—Placethetransducerin 9.3.4 Record the resultant reading, and repeat the test until
contact with the test specimen to pick up the desired vibration. a recommended ten readings are obtained that lie within
Ifthetransducerisplacedatanantinode(locationofmaximum 610 % of the mean. The round-robin interlaboratory study
displacement), it may mass load the specimen and modify the
(12.2) showed that data points significantly (>10 %) out of
natural vibration. The transducer should preferably be placed
range were measurements of spurious vibration modes or
only as far from the nodal points as necessary to obtain a
secondary harmonics. If ten readings cannot be taken, a
reading (see Fig. 3). This location will minimize the damping
minimum of five readings that lie within 610 % of the mean
effect from the contacting transducer. The transducer contact
shall be required for estimating the mean. Use the mean of
force should be consistent, with good response and minimal
these readings to determine the fundamental resonant fre-
interference with the free vibration of the specimen.
quency in flexure.
9.3.2.2 Noncontact Transducers—Place the noncontact
9.4 Fundamental Flexural Resonant Frequency (In-Plane
transducer over an antinode point and close enough to the test
Flexure):
specimen to pick up the desired vibration, but not so close as
9.4.1 This procedure is the same as that above (9.3), except
to interfere with the free vibration (see Fig. 3).
9.3.3 Strikethespecimenlightlyandelastically,eitheratthe that the direction of vibration is in the major plane of the
specimen.This measurement can be performed in two ways. In
center of the specimen or at the opposite end of the specimen
from the detecting transducer (see Fig. 3). one case, move the transducer and impulse tool 90° around the
C1259 − 21
FIG. 4 Rectangular Specimen Tested for Torsional Vibration
longaxisofthetestspecimentointroduceanddetectvibrations 10.1.1 For the fundamental flexure frequency of a rectan-
in the major plane (see Fig. 3). In the alternate method, rotate gular bar (2),
the test bar 90° around its long axis and reposition it on the 2 3 3
E 5 0.9465 mf /b L /t T (2)
~ !~ !
f 1
specimen supports. Transpose the width and thickness dimen-
where:
sions in the calculations. For homogeneous, isotropic
materials, the calculated moduli should be the same as the E = Young’s modulus, Pa,
m = mass of the bar, g (see Note 2),
moduli calculated from the out-of-plane frequency. The com-
b = width of the bar, mm (see Notes 1 and 2),
parison of in-plane and out-of-plane frequency measurements
L = length of the bar, mm (see Note 2),
can thus be used as a cross check of experimental methods and
t = thickness of the bar, mm (see Notes 1 and 2),
calculations.
f = fundamental resonant frequency of bar in flexure, Hz,
f
9.5 Fundamental Torsional Resonant Frequency:
and
9.5.1 Support the specimen at the midpoint of its length and
T = correction factor for fundamental flexural mode to
width (the torsional nodal planes) (see Fig. 4).
account for finite thickness of bar, Poisson’s ratio, etc.
9.5.2 Locatethetransduceratonequadrantofthespecimen,
2 2 4
T 5 116.585 ~110.0752 µ10.8109 µ !~t/L! 2 0.868 ~t/L!
preferably at approximately 0.224 L from one end and toward
(3)
theedge.Thislocationisanodalpointofflexuralvibrationand
will minimize the possibility of detecting a spurious flexural
2 4
8.340 110.2023 µ12.173 µ t/L
~ !~ !
mode (see Fig. 4). 2
F G
2 2
1.00016.338 110.1408 µ11.536 µ t/L
~ !~ !
9.5.3 Strike the specimen on the quadrant diagonally oppo-
µ 5 Poisson’s ratio
site the transducer, again at 0.224 L from the end and near the
NOTE 1—The width (b) and thickness (t) values used in the modulus
edge. Striking at a flexural nodal point will minimize the
calculations (Eq 2 and 3) for the rectangular specimens will depend on the
possibility of exciting a flexural mode of vibration (see Fig. 4).
type of vibration (out-of-plane or in-plane) induced in the specimen. The
9.5.4 Record the resultant reading, and repeat the test until
cross-sectional dimension t will always be parallel to the vibrational
a recommended ten readings are obtained that lie within
motion. The dimension b will always be perpendicular to the vibrational
motion. In effect, the two different flexural modes will give two different
610 % of the mean. The round-robin interlaboratory study
fundamental resonant frequencies, but the calculations for the two modes
(12.2) showed that data points significantly (>10 %) out of
should give the same modulus value, because the values for b and t are
range were measurements of spurious vibration modes or
exchanged in the calculations for the two different flexure modes.
secondary harmonics. If ten readings cannot be taken, a
NOTE2—Inthemodulusequations,themassandlengthtermsaregiven
minimum of five readings that lie within 610 % of the mean
inunitsofgramsandmillimetres.However,thedefinedequationscanalso
shall be required for estimating the mean. Use the mean of be used with mass and length terms in units of kilograms and metres with
no changes in terms or exponents.
these readings to determine the fundamental resonant fre-
quency in torsion.
10.1.1.1 IfL/t ≥ 20, T can be simplified to the following:
T 5 1.00016.585 t/L (4)
@ ~ ! #
10. Calculation
10.1 Dynamic Young’s Modulus (2, 3): and E can be calculated directly.
C1259 − 21
FIG. 5 Process Flowchart for Iterative Determination of Poisson’s Ratio
10.1.1.2 IfL/t < 20 and Poisson’s ratio is known, then T 10.1.2 For the fundamental flexural frequency of a rod of
can be calculated directly from Eq 3 and then used to calculate circular cross section (2):
E.
3 4 2
E 5 1.6067 L /D mf T ' (5)
~ !~ !
f 1
10.1.1.3 IfL/t < 20 and Poisson’s ratio is not known, then
where:
an initial Poisson’s ratio must be assumed to begin the
computations. An iterative process is then used to determine a D = diameter of rod, mm (see Note 1), and
T ' = correction factor for fundamental flexural mode to
value of Poisson’s ratio, based on experimental Young’s
account for finite diameter of bar, Poisson’s ratio, etc.
modulus and shear modulus. The iterative process is flow-
charted in Fig. 5 and described in paragraphs (1) through (5) 2 2
T ' 5 114.939 110.0752 µ10.8109 µ D/L (6)
~ !~ !
below.
(1) Determine the fundamental flexural and torsional reso- 20.4883 D/L
~ !
nantfrequencyoftherectangulartestspecimen,asdescribedin
2 4
4.691 ~110.2023 µ12.173 µ !~D/L!
Section 9. Using Eq 8 and 9, calculate the dynamic shear
F G
2 2
1.00014.754 110.1408 µ11.536 µ D/L
~ !~ !
modulus of the test specimen for the fundamental torsional
resonant frequency.
10.1.2.1 If L/ D ≥ 20, then T ' can be simplified to the
(2) Using Eq 2 and 3, calculate the dynamic Young’s
following:
modulusoftherectangulartestspecimenfromthefundamental
T ' 5 @1.00014.939 ~D/L! # (7)
flexural resonant frequency, dimensions and mass of the
specimen, and initial/iterative Poisson’s ratio. Care shall be
10.1.2.2 IfL/D < 20 and Poisson’s ratio is known, then T '
exercised in using consistent units for all of the parameters
can be calculated directly from Eq 6 and then used to calculate
throughout the computations.
E.
(3) Substitute the dynamic shear modulus and Young’s
10.1.2.3 IfL/D < 20 and Poisson’s ratio is not known, then
modulus values calculated in steps (1) and (2) into Eq 11 for
an initial Poisson’s ratio must be assumed to start the compu-
Poisson’s ratio satisfying isotropic conditions. Calculate a new
tations. Final values for Poisson’s ratio, dynamic Young’s
value for Poisson’s ratio for another iteration beginning at Step
modulus, and dynamic shear modulus are determined, using
(2).
the same method shown in Fig. 5 and described in paragraphs
(4) Repeat Steps (2) and (3) until no significant difference
(1)through(5)in10.1.1.3,butusingthemodulusequationsfor
(2 % or less) is observed between the last iterative value and
circular bars (Eq 5, Eq 6, and Eq 10).
the final computed value of the Poisson’s ratio.
10.2 Dynamic Shear Modulus (3, 4):
(5) Self-consistent values for the moduli are thus obtained.
10.2.1 For the fundamental torsional frequency of a rectan-
10.1.1.4 If the rectangular specimen is narrow and the four
gular bar (3):
long edges of the rectangular bar have been chamfered or
4Lmf
rounded, then the calculated Young’s modulus, E, should be
t
G 5 R (8)
corrected in accordance with Annex A2. bt
C1259 − 21
where: sional variability, surface finish, edge conditions, observed
changes after cryogenic or high-temperature testing, etc.,
G = dynamic shear modulus, Pa, and
where pertinent.
f = fundamental resonant frequency of bar in torsion, Hz.
t
11.1.3 For each specimen tested: the specimen geometry,
b
dimensions and mass, specimen test temperature, the number
S D
t 0.00851 b
R 5 11 of measurements taken, the individual numerical values and
F G
t 1.991 L
3 4 the mean value obtained for the measured fundamental reso-
4 2 2.521 1 2
b
S D
π
b
e t 11
nant frequency for each test mode, and the calculated values
for dynamic Young’s modulus, dynamic shear modulus, and
b 2 b
2 0.060 2 1 (9)
S D S D
Poisson’s ratio.
L t
11.1.4 Date of test and name of the person performing the
Eq 9 should be accurate to within ~0.2 % forb/L ≤ 0.3 and
test.
b/t ≤ 10 in the fundamental mode of torsional vibration,
11.1.5 Laboratory notebook number and page on which test
otherwise the errors are estimated to be ≤1%.(3)
data are recorded or the computer data file name, or both, if
10.2.2 For the fundamental torsion frequency of a cylindri-
used.
cal rod (3):
2 2
12. Precision and Bias
G 5 16 mf L/πD (10)
~ !
t
12.1 An evaluation (5) was conducted and published in
10.3 Poisson’s Ratio:
1990 by Smith, Wyrick, and Poole, of three different methods
µ 5 ~E/2G! 21 (11)
of modulus measurement of mechanically alloyed materials.
As part of that evaluation, the impulse modulus measurement
where:
method, using a commercial instrument, was used. With that
µ = Poisson’s ratio,
instrument, the precision of the impulse method was measured
E = Young’s modulus, and
G = shear modulus. using a NIST Standard Reference Material 718 (alumina
reference bar No. C1) in flexural vibration. The NIST standard
10.4 If measurements are made at elevated or cryogenic
had a measured and specified fundamental flexural frequency
temperatures, the calculated moduli must be corrected for
of 2043.3 Hz. The fundamental flexural resonant frequency of
thermal expansion effects using Eq 12.
the NIST reference bar was measured by the impulse method
M 5 M f /f 1/ 11α∆T (12)
@ # @ ~ !#
T 0 T 0 and reported by Smith, Wyrick, and Poole as 2044.6 Hz. This
was a percentage error of +0.06 %, indicating the level of bias
where:
that is achievable with the impulse method.
M = modulus at temperature T (either Young’s modulus E
T
12.2 An intralaboratory round-robin test was conducted in
or shear modulus G),
1993 to measure the precision of frequency measurement on
M = modulus at room temperature (eitherYoung’s modulus
twomonolithicceramictestbars.Abiastestwasnotconducted
E or shear modulus G),
f = resonant frequency in furnace or cryogenic chamber at because suitable standard reference bars were not readily
T
available.
temperature T,
f = resonant frequency at room temperature in furnace or
12.2.1 The tests were conducted with an alumina test bar
cryogenic chamber, (10 g, 83.0 by 6.9 by 4.8 mm) and a silicon nitride bar (2.0 g,
α = average linear thermal expansion (mm/mm · °C) from
50 by 4.0 by 3.0 mm). The silicon nitride bar was machined to
room temperature to test temperature (Test Method
Test Method C1161 tolerances; the alumina bar was not
C372 is recommended), and
machined and varied from 4.5 to 4.8 mm in thickness along its
∆T = temperaturedifferentialin°Cbetweentesttemperature
length. The variations in the alumina bar thickness were
T and room temperature.
deliberate; it provided a test of the robustness of the frequency
measurement technique.
10.5 Use the following stress conversion factor for English
12.2.2 Torsional frequency measurements were not per-
units.
formed because the width-thickness ratio of the bars was not
1Pa 5 1.450 310 psi (13)
suitable for torsional frequency measurements.
12.2.3 The bars were tested in flexural vibration at eight
11. Report
laboratories using ten combinations of different frequency
11.1 Report the following information: analyzer test systems, impulse tools, contact and noncontact
transducers, and supports systems. For the alumina bar, the
11.1.1 Identification of specific tests performed, a detailed
mean measured flexural frequency for the ten tests was
description of apparatus used (impulse tool, transducer, elec-
trical system, and support system), and an explanation of any
deviations from the described test method.
Grindosonic instrument, available from J.W. Lemmens, 10801 PearTree Lane,
11.1.2 Complete description of material(s) tested stating
St. Louis, MO 63074.
composition, number of specimens, specimen geometry and
Supporting data have been filed at ASTM International Headquarters and may
mass, specimen history, and any treatments to which the
beobtainedbyrequestingResearchReportRR:C28-1000.ContactASTMCustomer
specimens have been subjected. Include comments on dimen- Service at [e-mail].
C1259 − 21
TABLE 2 Effects of Variable Error on Modulus Calculation
6581 Hz, with a standard deviation of 20 Hz. This corresponds
to a coefficient of variation of 0.3 %. For the silicon nitride bar, Variable Exponent in Calculation
Variable Measurement Error
Modulus Equation Error
the mean measured flexural frequency for the ten tests was
Frequency (f) 0.1 % f 0.2 %
11 598 Hz, with a standard deviation of 34 Hz. This corre-
Length (L) 0.1 % L 0.3 %
sponds to a coefficient of variation of 0.3 %.
Mass (m) 0.1 % m 0.1 %
–1
12.2.4 The intralaboratory study did show that individuals Width (b) 0.1 % b 0.1 %
–3
Thickness (t) 0.1 % t 0.3 %
with experience in using the impulse test method for a given
–4
Diameter (D) 0.1 % D 0.4 %
specimen geometry produced data sets with smaller standard
deviations. For example, with the alumina test bar, the coeffi-
cients of variation for individual laboratories ranged from
a range of errors in the calculation of the modulus based on the
0.001 to 0.6 % among the ten test sets. For the silicon nitride
variable exponent in the equations. Table 2 gives the calcula-
bar,therangeofcoefficientsofvariationwas0.001to1.0%for
tion error effects of errors in the different experimental vari-
the individual laboratories.
ables.
12.2.5 Based on this intralaboratory study of the impulse
12.4 It is expected that the major sources of experimental
test method, the repeatability and reproducibility coefficients at
variation in modulus values for this test method will be in two
the 95 % confidence level are listed in Table 1.
measurements: the fundamental frequency and the smallest
dimension (thickness/diameter) of the test bars. If a fundamen-
TABLE 1 Within- and Between-Laboratory Precision
talresonantfrequencyof6000Hzismeasurabletoanaccuracy
Test Bar No. and Type Al O Si N
2 3 3 4
of18Hz/(0.3%)anda3mmthickbarisparallelandmeasured
Measured fundamental flexural frequency (Hz) 6581 11.598
to an accuracy of 0.01 mm (0.3 %), the error in the thickness
A
95 % repeatability limit (within laboratory) 2.8 CV, % 0.8 % 1.1 %
r
A measurement will have the greater effect on the modulus
95 % reproducibility limit (between laboratories) 2.8 CV, % 1.2 % 1.4 %
R
A calculation (0.9 % for thickness error versus 0.6 % for
Calculated in accordance with Practice E691, Section 21, and reported in
frequency error).
accordance with Practice E177, Section 28.
13. Keywords
12.3 Apropagation of errors analysis of the equations for E 13.1 advanced ceramics; bar; beam; cylindrical rod; disc;
and G using the stated tolerances for dimensions, mass, and dynamic; elastic modulus; elastic properties; flexure; impulse;
frequency measurements in this test method has shown that a Poisson’s ratio; resonance; resonant beam; shear modulus;
0.1 % error in the measurement of the key variables produces torsion; Young’s modulus
ANNEXES
(Mandatory Information)
A1. DISC-SHAPED SPECIMENS FOR DYNAMIC YOUNG’S MODULUS, SHEAR MODULUS, AND POISSON’S RATIO FOR
ADVANCED CERAMICS BY IMPULSE EXCITATION OF VIBRATION
A1.1 Scope A1.2.1.2 first natural vibration, n—the vibrations that occur
when the displacements in the cross-sectional plane (the plane
A1.1.1 In testing advanced ceramic disc specimens for
thatisparalleltotheflatofthedisc)arenormaltotheplaneand
Young’s modulus, shear modulus, and Poisson’s ratio, the disc
symmetrical around two orthogonal diameters in the plane of
geometry requires a significantly different set of equations and
the disc, producing a twisting of the disc.This is an orthogonal
method of calculation and some minor changes in procedures.
This annex describes
...

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: C1259 − 15 C1259 − 21
Standard Test Method for
Dynamic Young’s Modulus, Shear Modulus, and Poisson’s
Ratio for Advanced Ceramics by Impulse Excitation of
Vibration
This standard is issued under the fixed designation C1259; 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 Scope*
1.1 This test method covers determination of the dynamic elastic properties of advanced ceramics 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, cylindrical, or disc geometry) test specimen of that material can
be measured. The resonant frequencies in flexure and torsion are measured by excitation of vibrations of the test specimen in a
supported mode by a singular elastic strike with an impulse tool (Section 4 and Fig. 1, Fig. 3, and Fig. 4). Dynamic Young’s
modulus is determined using the resonant frequency in the flexural 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 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. Specimen supports, impulse locations, and signal pick-up points are
selected to induce and measure specific modes of the transient vibrations. A transducer (for example, contact accelerometer or
non-contacting microphone) senses the resulting mechanical vibrations of the specimen and transforms them into electric signals.
(See Fig. 1.) The transient 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.
This test method is under the jurisdiction of ASTM Committee C28 on Advanced Ceramics and is the direct responsibility of Subcommittee C28.01 on Mechanical
Properties and Performance.
Current edition approved Feb. 1, 2015Aug. 1, 2021. Published April 2015August 2021. Originally approved in 1994. Last previous edition approved in 20142015 as
C1259 – 14.C1259 – 15. DOI: 10.1520/C1259-15.10.1520/C1259-21.
FIG. 1 Block Diagram of Typical Test Apparatus
*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
C1259 − 21
1.2 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, in
accordance with sectionsSubsections 9.2, 9.3, and 10.4 of Test Method C1198.
1.3 Where possible, the procedures, sample specifications, and calculations in this test method are consistent with 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 C215, C623, C747, C848, C1198and, C1198E1875., and
E1876 may differ from this test method in several areas (for example, sample size, dimensional tolerances, sample preparation,
calculation details, etc.). The testing of those materials should be done in compliance with the appropriate material-specific
standards. Where possible, the procedures, sample specifications, and calculations in this standard are consistent with the other test
methods.
1.4 This test method uses test specimens in bar, rod, and disc geometries. The rod and bar geometries are described in the main
body. The disc geometry is addressed in Annex A1.
1.5 A modification of this test method can be used for quality control and nondestructive evaluation, using changes in resonant
frequency to detect variations in specimen geometry and mass and internal flaws in the specimen. (See 5.5).)
1.6 The values stated in SI units are to be regarded as the standard. The non-SI unit values given in parentheses are for information
only and are not considered standard.
1.7 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 safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
1.8 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
C1145 Terminology of Advanced Ceramics
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
D4092 Terminology for Plastics: Dynamic Mechanical Properties
E6 Terminology Relating to Methods of Mechanical Testing
E177 Practice for Use of the Terms Precision and Bias in ASTM Test Methods
E691 Practice for Conducting an Interlaboratory Study to Determine the Precision of a Test Method
E1875 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Sonic Resonance
E1876 Test Method for Dynamic Young’s Modulus, Shear Modulus, and Poisson’s Ratio by Impulse Excitation of Vibration
E2001 Guide for Resonant Ultrasound Spectroscopy for Defect Detection in Both Metallic and Non-metallic Parts
2.2 ISO Standard:
ISO 14704 Test Method for Flexural Strength of Monolithic Ceramics at Room Temperatures
3. Terminology
3.1 Definitions:
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.
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
C1259 − 21
3.1.1 The definitions of terms relating to mechanical testing appearing in Terminology E6 should be considered as applying to the
terms used in this test method. The definitions of terms relating to advanced ceramics appearing in Terminology C1145 should be
considered as applying to the terms used in this test method. Directly pertinent definitions as listed in Terminologies E6, C1145,
andand C1145 D4092are shown in the following paragraphs with the appropriate source given in brackets.
3.1.2 advanced ceramic, n—a highly engineered, high-performance, predominately nonmetallic, inorganic, ceramic material
having specific functional attributes. (C1145)
3.1.3 dynamic elastic modulus, n—the elastic modulus, either Young’s modulus or shear modulus, that is measured in a dynamic
mechanical measurement. (E1876)
3.1.4 dynamic mechanical measurement, n—a technique in which either the modulus or damping, or both, of a substance under
oscillatory load or displacement is measured as a function of temperature, frequency, or time, or combination thereof.
(D4092E1876)
–2
3.1.5 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
3.1.6 elastic modulus [FL ] ], , n—the ratio of stress to strain below the proportional limit. (E6)
3.1.7 Poisson’s ratio (μ) [nd],n—the absolute value of the ratio of transverse strain to the corresponding axial strain resulting from
uniformly distributed axial stress below the proportional limit of the material.
3.1.7.1 Discussion—
In isotropic materials, Young’s Modulus (E), shear modulus (G), and Poisson’s ratio (μ) are related by the following equation:
µ5 E/2G 2 1 (1)
~ !
(E6)
–2
3.1.8 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)
–2
3.1.9 shear modulus (G) [FL ] ], , n—the elastic modulus in shear or torsion. Also called modulus of rigidity or torsional modulus.
(E6)
–2
3.1.10 Young’s modulus (E) [FL ] ], , n—the elastic modulus in tension or compression. (E6)
3.2 Definitions of Terms Specific to This Standard:
3.2.1 antinodes, n—two or more locations that have local maximum displacements, called antinodes, in an unconstrained slender
rod or bar in resonance. For the fundamental flexure resonance, the antinodes 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 advanced ceramics conform to this
definition well enough to make this resonance test valid.
3.2.3 flexural vibrations, n—the vibrations that occur when the displacements in a slender rod or bar are in a plane normal to the
length dimension.
3.2.4 homogeneous, adj—the condition of a specimen such that the composition and density are uniform, so that any smaller
specimen taken from the original is representative of the whole. 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, the body can be
considered homogeneous.
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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. Advanced ceramics 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 nodes, n—one or more locations in a slender rod or bar in resonance having a constant zero displacement. For the
fundamental flexural resonance of such a rod or bar, the nodes are located at 0.224 L from each end, where L is the length of the
specimen.
3.2.8 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.
3.2.9 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. The lowest resonant frequency in a given vibrational
mode is the fundamental resonant frequency of that mode.
3.2.10 slender rod or bar, n—in dynamic elastic property testing, a specimen whose ratio of length to minimum cross-sectional
dimension is at least 5 and preferably in the range of 20 to 25.
3.2.11 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.
4. Summary of Test Method
4.1 This test method measures the fundamental resonant frequency of test specimens of suitable geometry (bar, rod, or disc) by
exciting them mechanically by a singular elastic strike with an impulse tool. A transducer (for example, contact accelerometer or
non-contactingnoncontacting microphone) senses the resulting mechanical vibrations of the specimen and transforms them into
electric signals. Specimen supports, impulse locations, and signal pick-uppickup 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 modulus of advanced ceramics that are elastic, homogeneous,
and isotropic (1).
5.3 This test method addresses the room temperature determination of dynamic moduli of elasticity of slender bars (rectangular
cross-section) cross section) and rods (cylindrical). Flat plates and disksdiscs may also be measured similarly, but the required
equations for determining the moduli are not addressed herein.
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.
The boldface numbers in parentheses refer to the list of references at the end of this test method.
C1259 − 21
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. Deviations in specimen dimensions or mass and internal flaws (cracks, delaminations,
inhomogeneities, porosity, etc)etc.) will change the resonant frequency for that specimen. Any specimen with a resonant frequency
falling outside the prescribed frequency range is rejected. The actual modulus of each specimen need not be determined as long
as the limits of the selected frequency range are known to include the resonant frequency that the specimen must possess if its
geometry and mass and internal structure are within specified tolerances. The technique is particularly suitable for testing
specimens with complex geometries (other than parallelepipeds, cylinders/rods, or discs) that would not be suitable for testing by
other procedures. This is similar to the evaluation method described in Guide E2001.
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, etc. 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 modulus presented herein are specifically applicable to
homogeneous, elastic, isotropic materials.
6.1.1 This method of determining the moduli is applicable to composite ceramics 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, etc.) 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 The test method is not satisfactory for specimens that have major discontinuities, such as large cracks (internal or surface)
or voids.
6.2 This test method for determining moduli is limited to specimens with regular geometries (rectangular parallelepiped, cylinders,
and discs) for which analytical equations are available to relate geometry, mass, and modulus to the resonant vibration frequencies.
The 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.
6.2.2 Edge treatments such as chamfers or radii are not considered in the analytical equations. Edge chamfers on flexure bars
prepared according to Test Method C1161 will change the resonant frequency of the test bars and introduce error into the
calculations of the dynamic modulus. It is recommended that specimens for this test method not have chamfered or rounded edges.
Alternately, if narrow rectangular specimens with chamfers or edge radii are tested, then the procedures in Annex A2 should be
used to correct the calculated Young’s modulus, E.
6.2.3 For specimens with as-fabricated and rough or uneven surfaces, variations in dimensions can have a significant effect in the
calculations. For example, in the calculation of dynamic modulus, the modulus value is inversely proportional to the cube of the
thickness. Uniform specimen dimensions and precise measurements are essential for accurate results.
C1259 − 21
FIG. 2 Diagram of Typical Impulser Impulse Tool for Small Specimens
6.3 The 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 9.3.1, 9.4.1, and 9.5.1 so the specimen can vibrate freely in the desired
mode. In using direct contact transducers, the transducer should be positioned away from antinodes and with minimal force to
avoid interference with free vibration. With noncontacting transducers, the maximum sensitivity is accomplished by placing the
transducer at an antinode.
6.4 Proper location of 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 antinodes 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, impulse tool, a suitable pickup
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—Impulse Tool—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 impulse tool depends on the size and weight of the specimen and the force needed to produce vibration. For commonly
tested geometries (small bars, rods, and discs) in advanced ceramics, an example of such an impulser impulse tool would be a steel
sphere 0.5 cm in diameter glued to the end of a flexible 10–cm10 cm long polymer rod. (See Fig. 2.) An alternate impulser impulse
tool would be 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.
7.3 Signal Pickup—Signal detection can be via transducers in direct contact with the specimen or by non-contactnoncontact
transducers. Contact transducers are commonly accelerometers using piezoelectric or strain gage methods to measure the vibration.
Non-contactNoncontact transducers are commonly acoustic microphones, but they may also use laser, magnetic, or capacitance
methods to measure the vibration. The frequency range of the transducer shall be sufficient to measure the expected frequencies
of the specimens of interest. A suitable range would be 100 Hz to 50 kHz 50 kHz for most advanced ceramic test specimens.
C1259 − 21
(Smaller and stiffer specimens vibrate at higher frequencies.) The frequency response of the transducer across the frequency range
of interest shall have a bandwidth of at least 10 % of the maximum measured frequency before -3 dB –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 and 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.
7.5 Support System—The support shall serve to 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 can be either soft
or rigid for ambient conditions. Examples of soft materials would be a compliant elastomeric material, such as polyurethane foam
strips. Such foam strips would 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 be resting on
isolation pads to prevent ambient vibrations from being picked up by the transducer. Wire suspension can also be used. Specimens
shall be supported along node lines appropriate for the desired vibration in the locations described in Section 8.
8. Test Specimen
8.1 The specimens shall be prepared so that they are either rectangular or circular in cross section. Either geometry can be used
to measure both dynamic Young’s modulus and dynamic shear modulus. Although the equations for computing shear modulus with
a cylindrical specimen are both simpler and more accurate than those used with a rectangular bar, experimental difficulties in
obtaining torsional resonant frequencies for a cylindrical specimen usually preclude its use for determining shear modulus. (See
Annex A1 for disc specimens.)
8.2 Resonant frequencies for a given specimen are functions of the specimen dimensions as well as its mass and moduli;
dimensions should therefore be selected with this relationship in mind. The selection of size shall be made so that, for an estimated
modulus, the resonant frequencies measured will fall within the range of frequency response of the transducers and electronics
used. For a slender rod, the ratio of length to minimum cross-sectional dimension shall have a value of at least 10. However, a
ratio of approximately 20 ≈ 25 is preferred for ease in calculation. For shear modulus measurements of rectangular bars, a ratio
of width to thickness of 5 or greater is recommended for minimizing experimental difficulties.
8.3 All surfaces on the rectangular specimen shall be flat. Opposite surfaces across the length and width shall be parallel within
0.01 mm or 60.1 %, whichever is greater. Opposite surfaces across the thickness shall be parallel within 0.002 mm or 60.1 %,
whichever is greater. The cylindrical specimen shall be round and constant in diameter within 0.002 mm or 60.1 %, whichever
is greater.
8.4 Test specimen mass shall be determined within 60.1 % or 10 mg, whichever is greater.
8.5 Test specimen length shall be measured to within 60.1 %. 60.1 %. Test specimen cross sectional cross-sectional dimensions
(thickness and width in rectangular beams; diameter in cylindrical rods) shall be measured within 60.1 % or 0.01 mm at three
equally-spaced equally spaced locations along the length and an average value determined.
8.6 Porous materials and those susceptible to hydration should be dried in air at 120°C120 °C in a drying oven until the mass is
constant (less than 0.1%0.1 % or 10 mg difference in measured mass with 30 min of additional drying).
8.7 It is recommended that the laboratory obtain and maintain an internal reference specimen with known and recorded
fundamental resonant frequencies in flexure and torsion. The reference specimen should be used to check and confirm the operation
of the test system on a regular basis. It can also be used to train operators in the proper test setup and test procedure. The reference
specimen can be a standard ceramic material (alumina, silicon carbide, zirconia, etc.) or it may be of a similar size, composition,
and microstructure to the types of ceramic specimens commonly tested at the laboratory. The reference specimen must meet the
size, dimensional tolerances, and surface finish requirements of Section 8.
C1259 − 21
FIG. 3 Rectangular Specimens Tested for In-Plane and Out-of-Plane Flexure
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 (0.224 L from each end; see Fig. 3).
9.3.2 Determine the direction of maximum sensitivity for the transducer. Orient the transducer so that it will detect the desired
vibration.
9.3.2.1 Direct Contact Transducers—Place the transducer in contact with the test specimen to pick up the desired vibration. If the
transducer is placed at an antinode (location of maximum displacement), it may mass load the specimen and modify the natural
vibration. The transducer should preferably be placed only as far from the nodal points as necessary to obtain a reading (see Fig.
3). This location will minimize the damping effect from the contacting transducer. The transducer contact force should be
consistent, with good response and minimal interference with the free vibration of the specimen.
C1259 − 21
FIG. 4 Rectangular Specimen Tested for Torsional Vibration
9.3.2.2 Non-ContactNoncontact Transducers—Place the non-contactnoncontact transducer over an antinode point 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 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 a recommended ten readings are obtained that lie within 610 % 610 %
of the mean. The round-robin interlaboratory study (12.2) showed that data points significantly (>10 %) out of range were
measurements of spurious vibration modes or secondary harmonics. If ten readings cannot be taken, a minimum of five readings
that lie within 610 % of the mean shall be required for estimating the mean. Use the mean of these 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 that above (9.3), except that the direction of vibration is in the major plane of the specimen.
This measurement can be performed in two ways. In one case, move the transducer and impulser impulse tool 90° around the long
axis of the test specimen to introduce and detect vibrations in the major plane (see Fig. 3). In the alternate method, rotate the test
bar 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) (see Fig. 4).
9.5.2 Locate the transducer at one quadrant of the specimen, preferably at approximately 0.224 L from one end and toward the
edge. This location is a nodal point of flexural vibration and will minimize the possibility of detecting a spurious flexural mode
(see Fig. 4).
9.5.3 Strike the specimen on the quadrant diagonally opposite the transducer, again at 0.224 L from the end and near the edge.
Striking at a flexural nodal point will minimize the possibility of exciting a flexural mode of vibration (see Fig. 4).
9.5.4 Record the resultant reading, and repeat the test until a recommended ten readings are obtained that lie within 610 % 610 %
of the mean. The round-robin interlaboratory study (12.2) showed that data points significantly (>10%)(>10 %) out of range were
C1259 − 21
measurements of spurious vibration modes or secondary harmonics. If ten readings cannot be taken, a minimum of five readings
that lie within 610 % of the mean shall be required for estimating the mean. Use the mean of these readings to determine the
fundamental resonant frequency in torsion.
10. Calculation
10.1 Dynamic Young’s Modulus (2, 3):
10.1.1 For the fundamental flexure frequency of a rectangular bar (2),
2 3 3
E 5 0.9465 mf /b L /t T (2)
~ !~ !
f 1
where:
E = Young’s modulus, Pa,
m = mass of the bar, g (see Note 2),
b = width of the bar, mm (see Notes 1 and 2),
L = length of the bar, mm (see Note 2),
t = thickness of the bar, mm (see Notes 1 and 2),
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, etc.
2 2 4
T 5 116.585 ~110.0752 µ10.8109 µ ! ~t/L! 2 0.868 ~t/L! (3)
2 4
8.340 ~110.2023 µ12.173 µ !~t/L!
F G
2 2
1.00016.338 110.1408 µ11.536 µ t/L
~ ! ~ !
µ5 Poisson’s ratio
µ5 Poisson’s ratio
NOTE 1—The width (b) and thickness (t) values used in the modulus calculations (Eq 2Eq 2 and 3 and Eq 3) for the rectangular specimens will depend
on the type of vibration (out-of-plane or in-plane) induced in the specimen. The cross-sectional dimension t will always be parallel to the vibrational
motion. The dimension b will always be perpendicular to the vibrational motion. In effect, the two different flexural modes will give two different
fundamental resonant frequencies, but the calculations for the two modes should give the same modulus value, because the values for b and t are
exchanged in the calculations for the two different flexure modes.
NOTE 2—In the 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, ≥ 20, T can be simplified to the following:
T 5 @1.00016.585 ~t / L! # (4)
and E can be calculated directly.
10.1.1.2 If L / t <20 < 20 and Poisson’s ratio is known, then T can be calculated directly from Eq 3 and then used to calculate
E.
10.1.1.3 If L / t <20 < 20 and Poisson’s ratio is not known, then an initial Poisson’s ratio must be assumed to begin the
computations. An iterative process is then used to determine a value of Poisson’s ratio, based on experimental Young’s modulus
and shear modulus. The iterative process is flow-charted in Fig. 5 and described in paragraphs (1) through (5) below.
(1) Determine the fundamental flexural and torsional resonant frequency of the rectangular test specimen, as described in
Section 9. Using Eq 8 and Eq 9, calculate the dynamic shear modulus of the test specimen for the fundamental torsional resonant
frequency.
(1) Determine the fundamental flexural and torsional resonant frequency of the rectangular test specimen, as described in
Section 9. Using Eq 8 and 9, calculate the dynamic shear modulus of the test specimen for the fundamental torsional resonant
frequency.
(2) Using Eq 2 and 3, 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. Care shall be exercised 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 11 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.
C1259 − 21
FIG. 5 Process Flowchart for Iterative Determination of Poisson’s Ratio
(5) Self-consistent values for the moduli are thus obtained.
(2) Using Eq 2 and Eq 3, 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. Care shall be exercised 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 11 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.
10.1.1.4 If the rectangular specimen is narrow and the four long edges of the rectangular bar have been chamfered or rounded,
then the calculated Young’s modulus, E, should be corrected in accordance with Annex A2.
10.1.2 For the fundamental flexural frequency of a rod of circular cross section (2):
3 4 2
E 5 1.6067 ~L /D !~mf !T ' (5)
f 1
where:
D = diameter of rod, mm (see Note 1), and
T ' = correction factor for fundamental flexural mode to account for finite diameter of bar, Poisson’s ratio, etc.
2 2
T '5 114.939 ~110.0752 µ10.8109 µ ! ~D/L! (6)
2 0.4883 ~D/L!
2 4
4.691 110.2023 µ12.173 µ D/L
~ ! ~ !
F G
2 2
1.00014.754 ~110.1408 µ11.536 µ !~ D/L!
2 2
T '5 114.939 ~110.0752 µ10.8109 µ ! ~D/L! (6)
20.4883 D/L
~ !
2 4
4.691 110.2023 µ12.173 µ D/L
~ ! ~ !
F G
2 2
1.00014.754 ~110.1408 µ11.536 µ !~ D/L!
10.1.2.1 If L / D ≥20, ≥ 20, then T ' can be simplified to the following:
C1259 − 21
T '5 1.00014.939 D/L (7)
@ ~ ! #
10.1.2.2 If L / D <20 < 20 and Poisson’s ratio is known, then T ' can be calculated directly from Eq 6 and then used to calculate
E.
10.1.2.3 If L / D <20 < 20 and Poisson’s ratio is not known, then an initial Poisson’s ratio must be assumed to start the
computations. Final values for Poisson’s ratio, dynamic Young’s modulus, and dynamic shear modulus are determined, using the
same method shown in Fig. 5 and described in paragraphs (1) through (5) in 10.1.1.3, but using the modulus equations for circular
bars (Eq 5, Eq 6, and Eq 10).
10.2 Dynamic Shear Modulus (3, 4):
10.2.1 For the fundamental torsional frequency of a rectangular bar.bar (3):
4 L m f
t
G 5 R (8)
b t
where:
G = dynamic shear modulus, Pa,
f = fundamental resonant frequency of bar in torsion, Hz,
t
G = dynamic shear modulus, Pa, and
f = fundamental resonant frequency of bar in torsion, Hz.
t
b
S D
2 2
t 0.00851 b b 2 b
R 5 11 2 0.060 2 1 (9)
F G S D S D
t 1.991 L L t
342 2.521 12 4
b
S D
b π
e t 11
Eq 9 should be accurate to within ~ 0.2 % ~0.2 % for b/L ≤0.3 b / L ≤ 0.3 and b/t ≤10 b / t ≤ 10 in the fundamental mode of
torsional vibration, otherwise the errors are estimated to be ≤1 %. (3)
10.2.2 For the fundamental torsion frequency of a cylindrical rod (3):
2 2
G 5 16 mf L/πD (10)
~ !
t
10.3 Poisson’s Ratio:
µ5 ~E/2G! 2 1 (11)
where:
μ = Poisson’s ratio,
E = Young’s modulus, and
G = shear modulus.
10.4 If measurements are made at elevated or cryogenic temperatures, the calculated moduli must be corrected for thermal
expansion effects using Eq 12.
M 5 M f /f 1/ 11αΔT (12)
@ # @ ~ !#
T 0 T 0
where:
M = modulus at temperature T (either Young’s modulus E or shear modulus G),
T
M = modulus at room temperature (either Young’s modulus E or shear modulus G),
f = resonant frequency in furnace or cryogenic chamber at temperature T,
T
f = resonant frequency at room temperature in furnace or cryogenic chamber,
α = average linear thermal expansion (mm/mm · °C) from room temperature to test temperature (Test Method C372 is
recommended), and
ΔT = temperature differential in °C between test temperature T and room temperature.
C1259 − 21
10.5 Use the following stress conversion factor for English units.
1 Pa 5 1.450 310 psi (13)
11. Report
11.1 Report the following information:
11.1.1 Identification of specific tests performed, a detailed description of apparatus used (impulser, (impulse tool, transducer,
electrical system, and support system), and an explanation of any deviations from the described test method.
11.1.2 Complete description of material(s) tested stating composition, number of specimens, specimen geometry and mass,
specimen history, and any treatments to which the specimens have been subjected. Include comments on dimensional variability,
surface finish, edge conditions, observed changes after cryogenic or high-temperature testing, etc., where pertinent.
11.1.3 For each specimen tested – tested: the specimen geometry, dimensions and mass, specimen test temperature, the number
of measurements taken, the individual numerical values and the mean value obtained for the measured fundamental resonant
frequency for each test mode, and the calculated values for dynamic Young’s modulus, dynamic shear modulus, and Poisson’s ratio.
11.1.4 Date of test and name of the person performing the test.
11.1.5 Laboratory notebook number and page on which test data are recorded or the computer data file name, or both, if used.
12. Precision and Bias
12.1 An evaluation (5) was conducted and published in 1990,1990 by Smith, Wyrick, and Poole, of three different methods of
modulus measurement of mechanically alloyed materials. As part of that evaluation, the impulse modulus measurement method,
using a commercial instrument, was used. With that instrument, the precision of the impulse method was measured using a NIST
Standard Reference Material 718 (alumina reference bar No. C1) in flexural vibration. The NIST standard had a measured and
specified fundamental flexural frequency of 2043.3 Hz. The fundamental flexural resonant frequency of the NIST reference bar was
measured by the impulse method and reported by Smith, Wyrick, and Poole as 2044.6 Hz. This was a percentage error of +0.06 %,
indicating the level of bias that is achievable with the impulse method.
12.2 An intralaboratory round-robin test was conducted in 1993 to measure the precision of frequency measurement on two
monolithic ceramic test bars. A bias test was not conducted because suitable standard reference bars were not readily available.
12.2.1 The tests were conducted with an alumina test bar (10 g, 83.0 by 6.9 by 4.8 mm) and a silicon nitride bar (2.0 g, 50 by
4.0 by 3.0 mm). The silicon nitride bar was machined to Test Method C1161 tolerances; the alumina bar was not machined and
varied from 4.5 to 4.8 mm in thickness along its length. The variations in the alumina bar thickness were deliberate; it provided
a test of the robustness of the frequency measurement technique.
12.2.2 Torsional frequency measurements were not performed because the width-thickness ratio of the bars was not suitable for
torsional frequency measurements.
Grindosonic instrument, available from J. W. Lemmens, 10801 Pear Tree Lane, St. Louis, MO 63074.
Supporting data have been filed at ASTM International Headquarters and may be obtained by requesting Research Report RR:C28-1000. Contact ASTM Customer
Service at [e-mail].
C1259 − 21
TABLE 2 Effects of Variable Error on Modulus Calculation
Variable Exponent in Calculation
Variable Measurement Error
Modulus Equation Error
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
Thickn
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