SIST EN 843-2:2007

SLOVENSKI STANDARD
01-maj-2007
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Advanced technical ceramics - Mechanical properties of monolithic ceramics at room
temperature - Part 2: Determination of Young's modulus, shear modulus and Poisson's
ratio
Hochleistungskeramik - Mechanische Eigenschaften monolithischer Keramik bei
Raumtemperatur - Teil 2: Bestimmung des Elastizitätsmoduls, Schubmoduls und der
Poissonzahl
Céramiques techniques avancées - Propriétés mécaniques des céramiques
monolithiques a température ambiante - Partie 2: Détermination du module d'Young, du
module de cisaillement et du coefficient de Poisson
Ta slovenski standard je istoveten z: EN 843-2:2006
ICS:
81.060.30 Sodobna keramika Advanced ceramics
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EUROPEAN STANDARD
EN 843-2
NORME EUROPÉENNE
EUROPÄISCHE NORM
December 2006
ICS 81.060.30 Supersedes ENV 843-2:1995
English Version
Advanced technical ceramics - Mechanical properties of
monolithic ceramics at room temperature - Part 2: Determination
of Young's modulus, shear modulus and Poisson's ratio
Céramiques techniques avancées - Propriétés mécaniques Hochleistungskeramik - Mechanische Eigenschaften
des céramiques monolithiques à température ambiante - monolithischer Keramik bei Raumtemperatur - Teil 2:
Partie 2: Détermination du module d'Young, du module de Bestimmung des Elastizitätsmoduls, Schubmoduls und der
cisaillement et du coefficient de Poisson Poissonzahl
This European Standard was approved by CEN on 11 November 2006.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
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This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
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© 2006 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN 843-2:2006: E
worldwide for CEN national Members.

Contents Page
Foreword.4
1 Scope.5
2 Normative references.5
3 Terms and definitions .6
4 Method A: Static flexure method.6
4.1 Principle.6
4.2 Apparatus.7
4.3 Test pieces.8
4.4 Procedure.8
4.5 Calculations.11
4.6 Measurement uncertainty.13
5 Method B: Resonance method.13
5.1 Principle.13
5.2 Apparatus.13
5.3 Test pieces.14
5.4 Procedure.16
5.5 Calculations.18
5.6 Measurement uncertainty.20
6 Method C: Ultrasonic method.20
6.1 Principle.20
6.2 Apparatus.20
6.3 Test pieces.22
6.4 Test method.22
6.5 Calculations.23
6.6 Measurement uncertainty.23
7 Method D: Impulse excitation method.24
7.1 Principle.24
7.2 Apparatus.24
7.3 Test pieces.24
7.4 Procedure.24
7.5 Calculations.27
7.6 Measurement uncertainty.27
8 Report.28
8.1 General.28
8.2 Method A.28
8.3 Method B.28
8.4 Method C.29
8.5 Method D.29
Annex A (informative) Impact excitation method applied to disc test pieces.30
A.1 Scope.30
A.2 Apparatus.30
A.3 Test pieces.30
A.4 Principle.30
A.5 Method.31
A.6 Calculations.32
A.7 Interferences.32
A.8 Measurement uncertainty.33
A.9 Report.33
Annex B (informative) Round-robin validation of test methods .37
B.1 Objectives.37
B.2 Materials.37
B.3 Test facilities.37
B.4 Results.37
B.5 Conclusions.38
Bibliography.39

Foreword
This document (EN 843-2:2006) has been prepared by Technical Committee CEN/TC 184 “Advanced
technical ceramics”, the secretariat of which is held by BSI.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by June 2007, and conflicting national standards shall be withdrawn at
the latest by June 2007.
This document supersedes ENV 843-2:1995.
EN 843 Advanced technical ceramics — Mechanical properties of monolithic ceramics at room temperature
comprises six parts:
Part 1: Determination of flexural strength
Part 2: Determination of Young’s modulus, shear modulus and Poisson’s ratio
Part 3: Determination of subcritical crack growth parameters from constant stressing rate flexural strength
tests
Part 4: Vickers, Knoop and Rockwell superficial hardness
Part 5: Statistical analysis
Part 6: Guidance for fractographic investigation
At the time of publication of this Revision of Part 2, Part 6 was available as a Technical Specification.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Cyprus, Czech Republic,
Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania,
Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden,
Switzerland and United Kingdom.
1 Scope
This part of EN 843 specifies methods for determining the elastic moduli, specifically Young’s modulus, shear
modulus and Poisson’s ratio, of advanced monolithic technical ceramics at room temperature. This European
Standard prescribes four alternative methods for determining some or all of these three parameters:
A The determination of Young’s modulus by static flexure of a thin beam in three- or four-point flexure.
B The determination of Young’s modulus by forced longitudinal resonance, or Young’s modulus, shear
modulus and Poisson’s ratio by forced flexural and torsional resonance, of a thin beam.
C The determination of Young’s modulus, shear modulus and Poisson’s ratio from the time-of-flight of an
ultrasonic pulse.
D The determination of Young’s modulus from the fundamental natural frequency of a struck bar (impulse
excitation method).
All the test methods assume the use of homogeneous test pieces of linear elastic materials.
NOTE 1 Not all ceramic materials are equally and linearly elastic in tension and compression, such as some porous
materials and some piezoelectric materials.
With the exception of Method C, the test assumes that the test piece has isotropic elastic properties. Method C
may be used to determine the degree of anisotropy by testing in different orientations.
NOTE 2 An ultrasonic method for dealing with anisotropic materials (ceramic matrix composites) can be found in
ENV 14186 [1]. An alternative to Method D for isotropic materials using disc test pieces is given in
Annex A.
NOTE 3 At high porosity levels all of the methods except Method C may become inappropriate. The methods are only
suitable for a maximum grain size (see EN 623-3), excluding deliberately added whiskers, of less than 10 % of the minimum
dimension of the test piece.
NOTE 4 The different methods given in this European Standard can produce slightly different results on the same
material owing to differences between quasi-isothermal quasi-static and quasi-adiabatic dynamic conditions. In addition,
the calculation routines for different methods have different origins and different potential uncertainties which have not
been rigorously evaluated in producing this European Standard. Some information is given in Annex B (see also reference
[2]).
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated references,
only the edition cited applies. For undated references, the latest edition of the referenced document (including any
amendments) applies.
EN 623-2, Advanced technical ceramics — Monolithic ceramics — General and textural properties — Part 2:
Determination of density and porosity
EN 623-3, Advanced technical ceramics — Monolithic ceramics — General and textural properties — Part 3:
Determination of grain size and size distribution (characterized by the Linear Intercept Method)
EN 623-4, Advanced technical ceramics — Monolithic ceramics — General and textural properties — Part 4:
Determination of surface roughness
EN 843-1:2006, Advanced technical ceramics — Mechanical properties of monolithic ceramics at room
temperature — Part 1: Determination of flexural strength
EN ISO 463, Geometrical Product Specifications (GPS) — Dimensional measuring equipment — Design and
metrological characteristics of mechanical dial gauges (ISO 463:2006)
EN ISO 7500-1, Metallic materials — Verification of static uniaxial testing machines — Part 1:
Tension/compression testing machines — Verification and calibration of the force-measuring system (ISO
7500-1:2004)
EN ISO/IEC 17025, General requirements for the competence of testing and calibration laboratories (ISO/IEC
17025:2005)
ISO 3611, Micrometer callipers for external measurement
ISO 6906, Vernier callipers reading to 0,02 mm
3 Terms and definitions
For the purposes of this document, the terms and definitions given in EN 843-1:2006 and the following apply.
3.1
Young’s modulus
stress required in a material to produce unit strain in uniaxial extension or compression
3.2
shear modulus
shear stress required in a material to produce unit angular distortion
3.3
Poisson’s ratio
negative value of the ratio of lateral strain to longitudinal strain in an elastic body stressed longitudinally
3.4
static elastic moduli
elastic moduli determined in a quasi-isothermal condition by stressing statically or quasistatically
3.5
dynamic elastic moduli
elastic moduli determined non-quasistatically, i.e. under quasi-adiabatic conditions, such as in the resonant,
ultrasonic pulse or impulse excitation methods
4 Method A: Static flexure method
4.1 Principle
Using three- or four-point flexure of a thin beam test piece, the elastic distortion is measured, from which Young’s
modulus may be calculated according to thin-beam equations.
4.2 Apparatus
4.2.1 Test jig
capable of three-point or four-point flexure.
The test jig shall be in accordance with that described in EN 843-1 in terms of its function, i.e. the support and
loading rollers shall be free to roll, and to articulate to ensure axial and even loading.
NOTE 1 Articulation is not essential for carefully machined flat and parallel-faced test pieces.
The outer span of the test jig shall be 40 mm or greater.
NOTE 2 If the availability of test material allows, a span of at least 100 mm is recommended to obtain large
displacements and to ensure that the compliance of the machine is a small correction if displacement is recorded as a
machine cross-head movement.
The test jig shall be for four-point flexure, if displacement is determined by strain gauges or differential transducer.
4.2.2 Test machine
capable of applying a force to the test jig at a constant displacement rate. The test machine shall be equipped
for recording the load applied to the test jig at any point in time. The accuracy of the test machine shall be in
accordance with EN ISO 7500-1, Grade 1 (1 % of indicated load), and shall be capable of recording to a
sensitivity of ≤ 0,1 % of the maximum load employed. The calibration shall have been checked within the
previous year.
4.2.3 Displacement or strain measuring device
4.2.3.1 General
Equipment shall be installed to measure the displacement or strain of the loaded test piece by one of three
methods, in accordance with 4.2.3.2, 4.2.3.3 or 4.2.3.4.
4.2.3.2 Method A.1
A facility is designed to measure the apparent displacements of the test machine with the test piece (Figure 1 a)),
and with the test piece replaced by a steel or ceramic bar at least 15 mm thick. The difference between these
displacements is equivalent to the displacement of the test piece in the test jig. The displacement recording device
shall be calibrated by comparing machine cross-head displacement with the movement indicated on a dial gauge
or other displacement measuring device (see 4.2.5) contacting the cross-head.
4.2.3.3 Method A.2
A facility is designed to measure the displacement of the test piece directly using transducers contacting two
defined points on the surface of the test piece between the support loading rollers in three-point or four-point
bending (Figure 1 b)). The defined points shall be the centre of the span and one or both loading rollers in four-
point bending, or the centre of the span and one or both support rollers in three-point bending. The transducer
shall be capable of detecting movements with an accuracy of 0,001 mm, shall have output linear to 0,01 % and
shall be calibrated to an accuracy of 0,1 %.
4.2.3.4 Method A.3
A facility is designed to record the strain on the surface of the test piece by using a strain gauge placed on the
surface of the test piece between the central loading rollers in four-point bending (Figure 1 c)). The strain gauge
and its associated bridge circuit shall have an accuracy of better than 0,1 % and shall be capable of resolving a
-5
strain of less than 10 .
NOTE It is recommended that the strain gauge should only be applied by experienced personnel in order to ensure it
performs accurately. It is also recommended that two or more gauges are fitted and their outputs recorded simultaneously
in order to provide a check on reproducibility.
4.2.4 Micrometer
in accordance with ISO 3611, but capable of recording to 0,002 mm, or other device of equivalent accuracy,
for measuring the dimensions of the test piece.
4.2.5 Dial gauge
in accordance with EN ISO 463 or other calibrated displacement measuring device, capable of recording to
0,01 mm.
4.3 Test pieces
Test pieces shall be rectangular section bars selected and prepared by agreement between parties. They may be
directly prepared close to final dimensions or machined from larger blocks. This test measures Young’s modulus
parallel to the length of the test piece. If the test material is likely to be elastically anisotropic, care shall be taken in
selection of the test piece orientation and in the interpretation of the test results. The maximum grain size (see
EN 623-3), excluding deliberately added whiskers, shall be less than 10 % of the minimum dimension of the test
piece.
The length of the test pieces shall be at least 10 mm longer than the test-jig span. The width of the test piece shall
be in the range 4 mm to 10 mm. For Method A.1 (4.2.3.2), the thickness of the test piece shall be in the range 0,8
mm to 1,5 mm. For methods A.2 (4.2.3.3) and A.3 (4.2.3.4), the test piece may be up to 3 mm thick. The test
pieces shall be machined to final dimensions. They shall be flat and parallel-faced to better than ± 0,5 % of
thickness on the faces to be placed on the loading rollers of the test-jig. They shall similarly be machined flat and
parallel-faced to better than ± 0,5 % of width on the side faces. For Method A.1 they shall not be chamfered.
NOTE For methods 2 and 3 they can be chamfered as specified in EN 843-1.
At least three test pieces shall be prepared.
4.4 Procedure
Measure the width and thickness of the test pieces at several places and record the average values.
Insert a test piece in the test-jig and centralize it in accordance with the requirements of EN 843-1. Select a
maximum force to be applied to the test piece which will avoid fracture.
NOTE 1 The upper level of force can be estimated by employing the strength calculation in EN 843-1:2006, Clause 8
and inserting a stress level of no more than 0,5σf, where σf is the mean fracture stress.
Apply a steadily increasing force to the test jig at a constant test machine cross-head displacement rate in the
range 0,001 mm/min to 0,5 mm/min. Record the load and displacement (either cross-head displacement (Method
A.1, 4.2.3.2), transducer displacement (Method A.2, 4.2.3.3), or strain gauge output (Method A.3, 4.2.3.4))
continuously. When the maximum selected force is achieved, reverse the direction of the machine and reduce the
load to zero. Repeat the cycle at least twice more to the same peak load, or until repeatable results are obtained.
Repeat the test on each test piece. If the machine displacement is to be employed (Method A.1) or if the
transducer method is employed using a support roller as one of the defined points (Method A.2), replace the test
piece with the thick parallel-sided steel or ceramic bar and repeat the loading cycles to the same peak load,
recording load and displacement.
NOTE 2 The use of both loading and unloading cycles is required in order to take into account machine hysteresis in
Method A.1, transducer hysteresis in Method A.2 and to test strain gauge adhesion in Method A.3.

a) Method A.1, using machine displacement

b) Method A.2, using a displacement transducer

c) Method A.3, using a strain gauge

Key
1 push-rod or top platen   8 rods detecting deflection
2 metallic half-sphere   9 support frame
3 metallic loading block   10 adjusting screw
4 loading rollers (freely rolling)  11 suspension springs
5 test piece    12 displacement transducer
6 support rollers (freely rolling)  13 load cell
7 support block    14 strain gauge

Figure 1 — Methods of measuring displacement or strain in quasi-statically loaded flexural test pieces,
a) Method A.1 using machine displacement, b) Method A.2 using a displacement transducer and c)
Method A.3 using a strain gauge
4.5 Calculations
4.5.1 From cross-head displacement (Method A.1)
Inspect the recordings of load and displacement for the test piece and the thick steel or ceramic bar for uniformity
and linearity. Select a region of the recordings from a minimum load of not less than 10 % of peak load or 0,2 N,
whichever is the greater, to a maximum load of not more than 90 % of the peak load applied. The same load
range shall be selected for each loading cycle on the test piece and the thick bar.
NOTE 1 The region of the recordings selected should avoid strong non-linearities at low load which may include
irreproducible effects of machine movement and test piece alignment and also the effects of cross-head reversal near
peak load.
Calculate or measure the displacement recorded over the selected load range for each loading and unloading
cycle for the test piece and for the thick bar. Calculate the average displacement in each direction. If the
displacement of the first cycle is more than 2 % different from that of the second or subsequent cycle, ignore the
first cycle when computing the average.
NOTE 2 The first cycle may show a different response to subsequent cycles as the test piece beds down into the test
jig and the machine movement stabilises.
Calculate Young’s modulus according to the following equations:
For displacement of loading points in three-point bending:
()F −F l
2 1
E = (1)
4bh()d −d
c s
For displacement of loading points in four-point bending:
2()F −F d(d + 3d)
2 1 1 1 2
E = (2)
bh()d −d
c s
where
-2
E is the Young’s modulus expressed in newtons per square metre (N m ) or pascals (Pa);
F is the lower load level selected from recordings, expressed in newtons (N);
F is the upper load level selected from recordings, expressed in newtons (N);
l is the test jig outer span in three-point or four-point bending, expressed in metres (m);
d is the test jig inner roller to outer roller spacing in four-point bending, expressed in metres (m);
d is the one half of the test jig inner span in four-point bending, expressed in metres (m);
b is the test piece width, expressed in metres (m);
h is the test piece thickness, expressed in metres (m);
d is the displacement recorded for the test piece in the jig over load interval F to F , expressed in metres
c 1 2
(m);
d is the displacement recorded for the thick bar in the jig over load interval F to F , expressed in metres.
s 1 2
NOTE 3 For the case of quarter-point bending, d = d , and Equation (2) reduces to:
1 2
()F −F l
2 1
E = (3)
8bh()d −d
c s
Calculate the average Young’s modulus figures for the loading and unloading curves. If these values differ by
more than 2 %, repeat the tests. If they differ by less than 2 %, take the overall average as the determined value
from the test.
4.5.2 From transducer displacement measurements (Method A.2)
Use the procedure specified in 4.5.1 to obtain displacements for a defined load range. If one of the defined points
for the transducer contact in three-point bending is the support roller, calculate the displacement recorded for the
thick bar. Subtract the mean value of the thick bar displacement from the mean specimen displacement over the
same load range for both loading and unloading.
For three-point bending using defined points at the span centre and under one or both support rollers, calculate
Young’s modulus using Equation (1).
For four-point bending using defined points at the span centre and under one or both loading rollers, calculate
Young’s modulus from the following equation:
3()F −F d d
2 1 1 2
E = (4)
bh d
t
where
d is the transducer displacement recorded between the test piece centre and the inner loading point in four-
t
point bending over the selected load range, expressed in metres (m).
NOTE For the case of quarter-point bending, d = d , and Equation (4) reduces to:
1 2
()
3 F −F l
2 1
E = (5)
64bh d
t
Calculate the average Young’s modulus figures for the loading and unloading parts of the cycles. If these values
differ by more than 2 %, repeat the tests. If they differ by less than 2 %, take the overall average as the
determined value from the test.
4.5.3 From strain gauges (Method A.3)
Use the procedure defined in 4.5.1 to obtain strain gauge outputs for a defined load range. Calculate the strain
change over the load range for each loading and unloading of the test piece.
Calculate Young’s modulus from the following equation:
3()F −F d
2 1 1
E = (6)
bh ε
where
ε  is the strain change over the defined load range, expressed as a fraction without units.
Calculate the average Young’s modulus figures for the loading and unloading parts of the cycles. If these values
differ by more than 2 %, repeat the tests. If they differ by less than 2 %, take the overall average as the
determined value from the test.
4.6 Measurement uncertainty
The uncertainty of Young’s modulus determined in accordance with this method derives primarily from the
parallelism of the test piece faces and the accuracy of measurement of thickness in the direction of flexure.
Additional factors are the alignment in the loading jig and the repeatability of measurement of deflections or strain.
For example, using test pieces 1 mm in thickness and with a span of 100 mm, a mechanically reliable fixture of
the piece typically permits a repeatability in load cycling of ± 2 % in flexural displacement or strain measurement.
Overall, an uncertainty of typically less than ± 5 % should be achievable.
5 Method B: Resonance method
5.1 Principle
A beam test piece is excited mechanically or electromechanically to vibrate at a given frequency and the
magnitude of the vibration is determined by a detector. The peak response is obtained at the resonant frequency,
either the fundamental or an overtone. The test is performed to excite either longitudinal or flexural and torsional
vibration. Young’s modulus may be determined from longitudinal resonance and Young’s modulus, shear
modulus and Poisson’s ratio may be determined from the flexural and torsional resonant frequencies, together
with the test piece dimensions and mass.
5.2 Apparatus
5.2.1 General
There are various techniques that may be used to determine the resonant frequency of the test piece. The test
piece may be excited by direct mechanical contact of a vibrator (especially appropriate for longitudinal
vibration) such as a piezoelectric transducer, or it may be suspended by a wire from a vibrator (appropriate for
flexural and torsional vibration), such as a record player cartridge or loudspeaker cone. Alternatively, it may be
driven electromagnetically by attaching thin foils of magnetic material to one surface, or electrostatically by
attaching an electrode to, or painting a conducting film of metal or graphite on, one surface.
5.2.2 Driving electronics
The driving electronics shall consist of a variable frequency oscillator and a record player cartridge assembly,
loudspeaker cone, or other suitable transducer. It is recommended that the oscillator is equipped with a digital
frequency display. It shall have sufficient power to drive high-modulus ceramic test pieces through the
transducer in the frequency range 100 Hz to 100 kHz, with a flat response curve (i.e. no resonances of its
own). The stability and accuracy of the digital display shall be checked against a standard frequency,
preferably from a transfer standard source.
5.2.3 Detecting electronics
The detecting electronics shall consist of a record player cartridge assembly or other suitable transducer, a
linear amplifier, and a voltmeter, ammeter or oscilloscope. The detector shall generate a voltage proportional
to the amplitude of vibration, the velocity, or the acceleration of the test piece.
NOTE The oscilloscope is recommended for identifying resonant conditions.
5.2.4 Test piece support
The test piece support shall permit the test piece to vibrate in the desired mode without significant restriction.
If the test piece is to be supported from beneath, the supports shall be made of rubber, cork, or similar
material and shall have a minimum contact area with the test piece.
NOTE 1 For the electrostatic method it may be necessary to make the support electrically conducting.
Alternatively, if the test piece is to be suspended from the driving and detecting transducers, fine thread or
metal wires shall be used. The supports shall be placed at or close to the ends of the vibrating test piece (see
5.4.1). The vibrating mass of the suspension system shall be negligible compared with the mass of the test
piece. For the electromagnetic or electrostatic method, the mass of any magnetic foil or electrode attached to
the test piece shall be negligible compared with the mass of the test piece.
NOTE 2 For the electrostatic method, a thin evaporated coating of a suitable metal will usually suffice. For the
electromagnetic method, the magnetic foil should be of nickel or iron, typically less than 0,05 mm thick and should be
attached to the test piece near the centre with a minimum of adhesive.
5.2.5 Laboratory balance
capable of weighing the test piece to the nearest 1 mg.
5.2.6 Micrometer
conforming to ISO 3611, but capable of recording to the nearest 0,002 mm or similar device of equivalent
accuracy for measuring the dimensions of the test piece.
5.2.7 Vernier callipers
conforming to ISO 6906, or other suitable device for measuring the length of the test piece to the nearest
0,05 mm.
5.2.8 Oven
for drying test pieces at 120 °C ± 10 °C, or other suitable device.
5.2.9 Desiccator
for storage of dried test pieces.
5.3 Test pieces
5.3.1 General
If the test material is likely to be elastically anisotropic, care shall be taken in the selection of test piece
orientations and in the interpretation of the test results.
NOTE This test method measures Young’s modulus parallel to the length of the test piece and shear modulus as an
aggregate of different directions.
The maximum grain size (see EN 623-3), excluding deliberately added whiskers, shall be less than 10 % of the
minimum dimension of the test piece.
5.3.2 Flexural resonance
5.3.2.1 General
The test piece shall be either a rectangular prism in accordance with 5.3.2.2 or a circular cross-section rod in
accordance with 5.3.2.3.
5.3.2.2 Rectangular prism
The test piece shall have a ratio l/h > 10 where h is the thickness of the test piece in the direction of flexural
vibration and l is the overall length (Figure 2a). The ratio l/b shall be > 10 where b is the width of the test piece.
The dimensions of the test piece shall be such as to have a fundamental flexural resonant frequency in the range
100 Hz to 20 kHz.
NOTE 1 For convenience, a flexural test piece as defined in EN 843-1 can be used, provided that the ends of the bar
are machined square and parallel, subject to the allowable frequency range above.
-2
NOTE 2 If the moduli of the test material are high (E > 200 GN m ), or when the available oscillator power is marginal,
it is recommended that l/h >> 10 and that 10 > b/h > 2,5. It is also recommended that b/h is > 1,1 or < 0,9 to avoid
confusion of different vibration modes.
The parallelism of the upper and lower surfaces perpendicular to the direction of flexural vibration shall be better
than h/100, of the sides parallel to the direction of vibration, better than b/100, and of the ends of the test piece,
better than l/200.
5.3.2.3 Circular rod
The test-piece shall be a circular cross-section rod with l /d >10 where d is the diameter of the test piece. The
diameter of the test piece shall be constant to within l /100, and the ends shall be flat and parallel to better than
l /200.
5.3.3 Torsional resonance
The test piece shall be a rectangular prism l /h > 20. The ratio b/h shall be greater than 3 and may be with
advantage as high as 10 (Figure 2 b)). The parallelism of the surfaces shall be better than h/100 and b/100, and of
the ends of the test piece, better than l /200
5.3.4 Longitudinal resonance
The test piece shall be either
 a rod with a length/diameter ratio of at least 20 (Figure 2 c)) and with the diameter uniform to d/100 over
its length, or
 a rectangular section bar with 1 < b/h < 2 and with b and h uniform to b/100 and h/100 respectively.
The ends of the rod or bar shall be square and parallel to better than l /200.
5.3.5 Number of test pieces
The minimum number of test pieces prepared shall be three.
5.4 Procedure
5.4.1 General
Dry the test pieces in the oven at 120 °C ± 10 °C to constant mass and cool in the desiccator.
Assemble the test apparatus, energize and allow it to reach equilibrium before testing.
Weigh a test piece to the nearest 1 mg. Measure its cross-sectional dimensions at least three positions along its
length to the nearest 0,002 mm with the micrometer and take the average figure. Measure the length to the
nearest 0,05 mm with the vernier callipers. Place appropriately on the test-piece support within the apparatus (see
5.2.4).
l
0,224 l
b
a)
b)
d
c)
Key
1 driver  2 detector 3 nodal positions
Figure 2 — Resonance method, a) flexural, b) torsional, and c) longitudinal resonance modes with
geometrical dimensions and support, driving and detecting positions indicated
h
5.4.2 Flexural resonance
Position the test piece in the apparatus so as to obtain flexural vibration in the desired direction (parallel to
dimension h). If the test piece is to be supported from below (e.g. in the magnetic or electrostatic methods),
position the supports at the nodal points (positions of minimum amplitude) of the fundamental (first order) vibration,
i.e. at a distance of 0,224 of the total length from each end (Figure 2 a)). If the test piece is to be suspended by
threads or wires which will be used to drive and detect vibration, position the threads outside the nodal points near
the ends and along the centre line of the test piece. If a microphone is used as the pick-up, position this on the
centre line of the test piece very close to the centre of the test piece. If the test piece is to be driven
electrostatically, position the driving electrode under the centre of the test piece. If the test piece is to be driven
electromagnetically, position the driving coil opposite the pick-up foil attached to the centre of the test piece.
NOTE 1 In some cases using mechanical driving it may be desirable to place the driving transducer at the centre point
of the test piece to encourage vibration in the fundamental mode.
NOTE 2 It is recommended that external noise is excluded as far as is practical to avoid extraneous excitations of the
test piece. For very small test pieces, the use of a vacuum envelope for the apparatus is recommended.
Force the test piece to vibrate at a low frequency, and note the amplitude of the vibration. Increase the frequency
steadily and detect any frequencies at which the amplitude shows well-defined maxima. Check that the greatest
maximum is the fundamental frequency of vibration. Record the fundamental mode frequency to an accuracy of
better than 1 Hz or 0,1 % whichever is the greater.
NOTE 3 There are various methods of checking that the fundamental frequency has been detected. One is to employ a
small directional microphone to detect the amplitude of vibration and to move this along the length of the test piece to
ensure that there are only two nodes. A second is to spread fine, free-flowing powder (<< 1 µm particle size, e.g. fine
sawdust) on the top surface of the test piece and observe its alignment at the nodal points. A third is to determine the
relationship between the frequencies of the resonances. The first overtone should occur at approximately 2,64 times the
fundamental frequency.
5.4.3 Torsional resonance
If the test piece is to be supported from below, position the support at the centre of the test piece. Position the
exciting and pick-up transducers or their connecting systems to diagonally opposite corners of the test piece to
excite torsional vibration (Figure 2 b)). If the test piece is to be supported on threads, position the supports at
diagonally opposite corners of the test piece. Follow the procedure described for flexural resonance (5.4.2) to
determine the resonant frequencies and check that the fundamental has been detected. Ensure that torsional and
not flexural resonance has occurred by moving the driving position to the centre-line of the test piece; the
resonance should disappear if it was torsional. Record the fundamental mode frequency to an accuracy of 0,1 %
or better.
5.4.4 Longitudinal resonance
Position the test piece in the apparatus and place the vibrator in contact with one end and the pick-up in contact
with the other (Figure 2 c)). Force the test piece to vibrate at a low frequency and note the amplitude of the
detected vibration. Increase the frequency steadily and detect any frequencies at which the amplitude shows well-
defined maxima. The greatest maximum can generally be taken as being at the fundamental frequency of
vibration. Record the fundamental mode frequency to an accuracy of at least 1 Hz or 0,1% whichever is the
smaller.
5.5 Calculations
5.5.1 For flexural resonance of a rectangular section test piece calculate the dynamic elastic modulus of
the test piece from the flexural mode equation:
 
mf
l
f  
 
E = 0,946 A (7)
 
f


b h
 
 
where
-2
E is the Young’s modulus, expressed in newtons per square metre (N m ) or pascals (Pa);
f is the resonance frequency for fundamental mode flexural vibration, expressed in hertz (Hz);
f
b is the width of test piece, expressed in metres (m);
h is the thickness of test piece, expressed in metres (m);
m is t
...