Designation: C1548 − 02 (Reapproved 2012)Standard Test Method forDynamic Young’s Modulus, Shear Modulus, and Poisson’sRatio of Refractory Materials by Impulse Excitation ofVibration1This standard is issued under the fixed designation C1548; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This test method covers the measurement of the funda-mental resonant frequencies for the purpose of calculating thedynamic Young’s modulus, the dynamic shear modulus (alsoknown as the modulus of rigidity), and the dynamic Poisson’sratio of refractory materials at ambient temperatures. Speci-mens of these materials possess specific mechanical resonantfrequencies, which are determined by the elastic modulus,mass, and geometry of the test specimen. Therefore, thedynamic elastic properties can be computed if the geometry,mass, and mechanical resonant frequencies of a suitablespecimen can be measured. The dynamic Young’s modulus isdetermined using the resonant frequency in the flexural modeof vibration and the dynamic shear modulus is determinedusing the resonant frequency in the torsional mode of vibration.Poisson’s ratio is computed from the dynamic Young’s modu-lus and the dynamic shear modulus.1.2 Although not specifically described herein, this methodcan also be performed at high temperatures with suitableequipment modifications and appropriate modifications to thecalculations to compensate for thermal expansion.1.3 The values are stated in SI units and are to be regardedas the standard.1.4 This standard may involve hazardous materials,operations, and equipment. This standard does not purport toaddress all of the safety concerns, if any, associated with itsuse. It is the responsibility of the user of this standard toestablish appropriate safety and health practices and deter-mine the applicability of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:2C71 Terminology Relating to RefractoriesC215 Test Method for Fundamental Transverse,Longitudinal, and Torsional Resonant Frequencies ofConcrete SpecimensC885 Test Method for Young’s Modulus of RefractoryShapes by Sonic ResonanceC1259 Test Method for Dynamic Young’s Modulus, ShearModulus, and Poisson’s Ratio for Advanced Ceramics byImpulse Excitation of Vibration3. Summary of Test Method3.1 The fundamental resonant frequencies are determinedby measuring the resonant frequency of specimens struck oncemechanically with an impacting tool. Frequencies are mea-sured with a transducer held lightly against the specimen usinga signal analyzer circuit. Impulse and transducer locations areselected to induce and measure one of two different modes ofvibration. The appropriate resonant frequencies, dimensions,and mass of each specimen may be used to calculate dynamicYoung’s modulus, dynamic shear modulus, and dynamic Pois-son’s ratio.4. Significance and Use4.1 This test method is non-destructive and is commonlyused for material characterization and development, designdata generation, and quality control purposes. The test assumesthat the properties of the specimen are perfectly isotropic,which may not be true for some refractory materials. The testalso assumes that the specimen is homogeneous and elastic.Specimens that are micro-cracked are difficult to test since theydo not yield consistent results. Specimens with low densities1This test method is under the jurisdiction of ASTM Committee C08 onRefractories and is the direct responsibility of Subcommittee C08.01 on Strength.Current edition approved March 1, 2012. Published April 2012. Originallyapproved in 2002. Last previous edition approved in 2007 as C1548 – 02 (2007).DOI: 10.1520/C1548-02R12.2For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1have a damping effect and are easily damaged locally at theimpact point. Insulating bricks can generally be tested with thistechnique, but fibrous insulating materials are generally tooweak and soft to test.4.2 For quality control use, the test method may be used formeasuring only resonant frequencies of any standard sizespecimen.An elastic modulus calculation may not be needed oreven feasible if the shape is non-standard, such as a slide gateplate containing a hole. Since specimens will vary in both sizeand mass, acceptable frequencies for each shape and materialmust be established from statistical data.4.3 Dimensional variations can have a significant effect onmodulus values calculated from the frequency measurements.Surface grinding may be required to bring some materials intothe specified tolerance range.4.4 Since cylindrical shapes are not commonly made fromrefractory materials they are not covered by this test method,but are covered in Test Method C215.5. Apparatus5.1 Electronic System—The electronic system in Fig. 1consists of a signal conditioner/amplifier, a signal analyzer, afrequency readout device, and a signal transducer for sensingthe vibrations. The system should have sufficient precision tomeasure frequencies to an accuracy of 0.1 %. Commercialinstrumentation is available which meets this requirement.35.1.1 Frequency Analyzer—This consists of a signalconditioner/amplifier to power the transducer and a digitalwaveform analyzer or frequency counter with storage capabil-ity to analyze the signal from the transducer. The waveformanalyzer shall have a sampling rate of at least 20 000 Hz. Thefrequency counter should have an accuracy of 0.1 %.5.1.2 Sensor—A piezeoelectric accelerometer contact trans-ducer is most commonly used, although non-contact transduc-ers based on acoustic, magnetic, or capacitance measurementsmay also be used. The transducer shall have a frequencyresponse in the range of 50 Hz to 10 000 Hz, and have aresonant frequency above 20 000 Hz. The sensor shall have amark identifying the maximum sensitivity direction so that itcan be properly oriented for each vibration mode.5.2 Impactor—Because refractory materials are tested withspecimens of various sizes, it is not feasible to specify animpactor with a specific size, weight, or construction method.However, hammer style impactors which have light weighthandles with the impacting mass concentrated near the end arepreferred to dropping vertical impactors. Steel hammer styleimpactors, with head weights between 0.3 and 3 % of thespecimen weight, are recommended. To avoid damaging thesurface of insulating bricks or other weak materials, plastic orrubber shapes should be substituted for the steel impactors.5.3 Specimen Support—The support shall permit the speci-men to vibrate freely without restricting the desired mode ofvibration. For room temperature measurements, soft rubber orplastic strips located at the nodal points are typically used.Alternately, the specimen can be placed on a thick soft rubberpad. For elevated temperature measurements, the specimenmay be suspended from support wires wrapped around thespecimen at nodal points and passing vertically out of the testchamber.6. Test Specimen6.1 Preparation—Test specimens shall be prepared to yielduniform rectangular shapes. Normally, brick sized specimensare used. Although smaller bars cut from bricks are easilytested for flexural resonant frequencies, it is more difficult toobtain torsional resonance in specimens of square cross-section. Some pressed brick shapes are dimensionally uniformenough to test without surface grinding, but specimens cutfrom larger shapes or prepared by casting or other means oftenrequire surface grinding of one or more surfaces to meet thedimensional criteria noted below.6.2 Heat Treatment—All specimens shall be prefired to thedesired temperature and oven dried before testing.6.3 Dimensional Ratios—Specimens having either verysmall or very large ratios of length to maximum transversedimensions are frequently difficult to excite in the fundamentalmodes of vibration. Best results are obtained when this ratio isbetween 3 and 5. For use of the equations in this method, theratio must be at least 2.3Equipment found suitable is available from J. W. Lemmons, Inc., 3466Bridgeland Drive, Suite 230, St. Louis, MO 63044-260.FIG. 1 Diagram of Test ApparatusC1548 − 02 (2012)26.4 Dimensional Uniformity—Rectangular specimens shallhave surfaces that are flat and parallel to within 60.5 % of thenominal measured value.6.5 Weight (or Mass) and Dimensions—Determine theweight (or mass) to the nearest 60.5 %. Measure each dimen-sion to within 60.5 %.7. Measurement of Impulse Resonant Frequencies7.1 Transverse Frequency:7.1.1 Support the specimen so that it may vibrate freely inthe fundamental transverse mode. In this mode the nodal points(where the displacement is zero) are located at 0.224L fromeach end, where L is the specimen length. Vibrational displace-ments are a maximum at the ends of the specimen and about3/5 maximum at the center. The nodal points are shown in Fig.2 along with recommended impact points and sensor locations.If the specimen does not have a square cross-section, supportthe specimen on its largest face such that it vibrates perpen-dicular to its thinnest dimension.7.1.2 Turn on the electronic system and warm it up accord-ing the manufacturers instructions.7.1.3 Position the sensor on the side face of the specimen atmid length, with the sensor oriented such that the mostsensitive pick-up direction coincides with the vibration direc-tion. In Fig. 2, the dot on the sensor indicates the most sensitivepickup direction of the sensor and it is pointed upward towarda top-center impact point. The sensor is typically held againstthe specimen with very light hand pressure, but some typescould be temporarily attached to large specimens.7.1.4 Select an impact hammer with a head weight 0.3 to3 % of the specimen weight and lightly tap the top center of thespecimen perpendicular to the surface. Note the readingdisplayed by the electronic system, allow a few seconds forexisting vibrations to dampen in the specimen, and repeat theprocedure at least 3 times until a consistent value is repro-duced. Record that value and calculate the resonant frequencyfrom it per the manufacturer’s instructions if frequency is notdisplayed directly. If a consistent value cannot be obtained,either the specimen is damaged or other modes of vibration areinterfering with the measurement.7.2 Torsional Frequency7.2.1 Support the specimen so that it may vibrate freely intorsion. In this mode there is a single nodal point at the centerand vibrations are a maximum at the ends. The impact andsensor pickup points are located at 0.224L from the ends. Thislocation is a nodal point for flexural vibration and minimizesinterference from flexural vibrations.7.2.2 Turn on the electronic system and warm it up accord-ing to the manufacturers instructions.7.2.3 Position the sensor on the side face of the specimen at0.224L, with the sensor oriented such that the sensitive pick-updirection coincides with the vibration direction. In Fig. 2, thedot on the sensor indicates the most sensitive pickup directionof the sensor and it is pointed upward toward a top impactpoint.7.2.4 Select an impact hammer with a head weight 0.3 to3 % of the specimen weight and lightly tap the top of thespecimen at a 0.224L location perpendicular to the surface.Note the reading displayed by the electronic system, allow afew seconds for existing vibrations to dissipate, and repeat theprocess at least 3 times until a consistent value is reproduced.Record that value and calculate the resonant frequency from itif frequency is not displayed directly.8. Calculations8.1 Dynamic Young’s Modulus4,5:8.1.1 From the fundamental flexural vibration of a rectan-gular bar:E 5 0.9465Smff2bDSL3t3 DT1(1)where:E = Young’s modulus, Pa,m = mass of the bar, g,b = width of the bar, mm,L = length of the bar, mm,t = thickness of the bar, mm,ff= fundamental resonant frequency of the bar in flexure,Hz, andT1= correction factor for fundamental flexural made toaccount for finite thickness of bar, Poisson’s ratio, etc.T15 116.585 ~110.0752µ10.8109µ2! StLD22 0.868StLD2HS8.340 ~110.2023µ12.173µ2! StLD4DJHS1.00016.338 ~110.1408µ11.536µ2! StLD2DJµ = Poisson’s ratio.8.1.1.1 If L / t ≥ 20, T1can be simplified to:T15S1.00016.585StLD2Dand E can be calculated directly.8.1.1.2 If L / t 20, then an initial Poisson’s ratio must beassumed to start the computations. An iterative process is then4Spinner, S., Reichard, T. W., and Tefft, W. E., “A Comparison of Experimentaland Theoretical Relations Between Young’s Modulus and the Flexural and Longi-tudinal Resonance Frequencies of Uniform Bars,” Journal of Research of theNational Bureau of Standards—A. Physics and Chemistry, Vol 64A, #2, March-April, 1960.5Spinner, S., and Tefft, W. E., “A Method for Determining MechanicalResonance Frequencies and for Calculating Elastic Moduli from these Frequencies,”Proceedings, ASTM, 1961, pp. 1221-1238.C1548 − 02 (2012)3used to determine a value of Poisson’s ratio, based on experi-mental Young’s modulus and shear modulus. This iterativeprocess is shown in Fig. 3 and described below.(1) Determine the fundamental flexural and torsional reso-nant frequencies of the rectangular test specimen. Using Eq 2,the dynamic shear modulus of the test specimen is calculatedfrom the fundamental torsional resonant frequency and thedimension and mass of the specimen.(2) Using Eq 1, the dynamic Young’s modulus of therectangular test specimen is calculated from the fundamentalflexural resonant frequency, the dimensions and mass of thespecimen, and the initial/iterative Poisson’s ratio.(3) The dynamic shear modulus and Young’s values modu-lus.calculated in steps (1) and (2) are substituted into Eq 3, forPoisson’s ratio. A new value for Poisson’s ratio is thencalculated for another iteration starting at step (2).FIG. 2 Impact Points and Transducer LocationsC1548 − 02 (2012)4(4) Steps (2) and (3) are repeated until no significantdifference (2 % or less) is observed between the last iterativevalue and the final computed value of the Poisson’s ratio.Self-consistent values for the moduli are thus obtained.8.2 Dynamic Shear Modulus6,7:8.2.1 From the fundamental torsional vibration of a rectan-gular bar:G 5H~4 Lmft2!~bt!JHB~11A!J(2)where:G = shear modulus, Pa, andft= fundamental resonant frequency of the bar in torsion,Hz.B 5HSbtD1StbDJH4StbD2 2.52StbD210.21StbD6JA 5H0.5062 2 0.8776SbtD10.3504SbtD22 0.0078SbtD3JH12.03SbtD19.892SbtD2J8.3 Poisson’s Ratio:8.3.1 From E and G:µ 5SE2GD2 1 (3)6Pickett, G., “Equations for Computing Elastic Constants