Designation: C1198 − 09 (Reapproved 2013)Standard Test Method forDynamic Young’s Modulus, Shear Modulus, and Poisson’sRatio for Advanced Ceramics by Sonic Resonance1This standard is issued under the fixed designation C1198; 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 determination of the dy-namic elastic properties of advanced ceramics. Specimens ofthese materials possess specific mechanical resonant frequen-cies that are determined by the elastic modulus, mass, andgeometry of the test specimen. Therefore, the dynamic elasticproperties of a material can be computed if the geometry, mass,and mechanical resonant frequencies of a suitable test speci-men of that material can be measured. Dynamic Young’smodulus is determined using the resonant frequency in theflexural mode of vibration. The dynamic shear modulus, ormodulus of rigidity, is found using torsional resonant vibra-tions. Dynamic Young’s modulus and dynamic shear modulusare used to compute Poisson’s ratio.1.2 This test method measures the resonant frequencies oftest specimens of suitable geometry by mechanically excitingthem at continuously variable frequencies. Mechanical excita-tion of the bars is provided through the use of a transducer thattransforms a cyclic electrical signal into a cyclic mechanicalforce on the specimen.Asecond transducer senses the resultingmechanical vibrations of the specimen and transforms theminto an electrical signal. The amplitude and frequency of thesignal are measured by an oscilloscope or other means to detectresonant vibration in the desired mode. The resonantfrequencies, dimensions, and mass of the specimen are used tocalculate dynamic Young’s modulus and dynamic shear modu-lus. (See Fig. 1)1.3 This test method is specifically appropriate for advancedceramics that are elastic, homogeneous, and isotropic (1).2Advanced ceramics of a composite character (particulate,whisker, or fiber reinforced) may be tested by this test methodwith the understanding that the character (volume fraction,size, morphology, distribution, orientation, elastic properties,and interfacial bonding) of the reinforcement in the testspecimen will have a direct effect on the elastic properties.These reinforcement effects must be considered in interpretingthe test results for composites. This test method is notsatisfactory for specimens that have cracks or voids that aremajor discontinuities in the specimen. Neither is the testmethod satisfactory when these materials cannot be fabricatedin a uniform rectangular or circular cross section.1.4 A high-temperature furnace and cryogenic cabinet aredescribed for measuring the dynamic elastic moduli as afunction of temperature from −195 to 1200°C.1.5 Modification of this test method for use in qualitycontrol is possible. A range of acceptable resonant frequenciesis determined for a specimen with a particular geometry andmass. Any specimen with a frequency response falling outsidethis frequency range is rejected. The actual modulus of eachspecimen need not be determined as long as the limits of theselected frequency range are known to include the resonantfrequency that the specimen must possess if its geometry andmass are within specified tolerances.1.6 The procedures in this test method are, where possible,consistent with the procedures of Test Methods C623, C747,and C848. The tables of these test methods have been replacedby the actual formulas from the original references. With theadvent of computers and sophisticated hand calculators, theactual formulas can be easily used and provide greater accu-racy than factor tables.1.7 The values stated in SI units are to be regarded as thestandard. The values given in parentheses are for informationonly.1.8 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is the1This test method is under the jurisdiction of ASTM Committee C28 onAdvanced Ceramics and is the direct responsibility of Subcommittee C28.01 onMechanical Properties and Performance.Current edition approved Aug. 1, 2013. Published September 2013. Originallyapproved in 1991. Last previous edition approved in 2009 as C1198 – 09. DOI:10.1520/C1198-09R13.2The boldface numbers given in parentheses refer to a list of references at theend of the text.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1responsibility of the user of this standard to establish appro-priate safety and health practices and determine the applica-bility of regulatory limitations prior to use.2. Referenced Documents2.1 ASTM Standards:3C372 Test Method for Linear Thermal Expansion of Porce-lain Enamel and Glaze Frits and Fired Ceramic WhitewareProducts by the Dilatometer MethodC623 Test Method for Young’s Modulus, Shear Modulus,and Poisson’s Ratio for Glass and Glass-Ceramics byResonanceC747 Test Method for Moduli of Elasticity and FundamentalFrequencies of Carbon and Graphite Materials by SonicResonanceC848 Test Method for Young’s Modulus, Shear Modulus,and Poisson’s Ratio For Ceramic Whitewares by Reso-nanceC1145 Terminology of Advanced CeramicsC1161 Test Method for Flexural Strength of AdvancedCeramics at Ambient TemperatureD4092 Terminology for Plastics: Dynamic MechanicalPropertiesE2001 Guide for Resonant Ultrasound Spectroscopy forDefect Detection in Both Metallic and Non-metallic Parts3. Terminology3.1 Definitions:3.1.1 advanced ceramic, n—a highly engineered, highperformance, predominately nonmetallic, inorganic, ceramicmaterial having specific functional attributes. C11453.1.1.1 dynamic mechanical measurement, n—a techniquein which either the modulus or damping, or both, of a substanceunder oscillatory load or displacement is measured as afunction of temperature, frequency, or time, or combinationthereof. D40923.1.2 elastic limit [FL−2],n—the greatest stress that amaterial is capable of sustaining without permanent strainremaining upon complete release of the stress.3.1.3 elastic modulus [FL−2],n—the ratio of stress to strainbelow the proportional limit.3.1.4 Poisson’s ratio (µ) [nd],n—the absolute value of theratio of transverse strain to the corresponding axial strainresulting from uniformly distributed axial stress below theproportional limit of the material.3.1.4.1 Discussion—In isotropic materials Young’s modulus(E), shear modulus (G), and Poisson’s ratio (µ) are related bythe following equation:µ 5 ~ E/2G ! 2 13.1.5 proportional limit [FL−2],n—the greatest stress that amaterial is capable of sustaining without deviation fromproportionality of stress to strain (Hooke’s law).3.1.6 shear modulus (G) [FL−2],n—the elastic modulus inshear or torsion. Also called modulus of rigidity or torsionalmodulus.3.1.7 Young’s modulus ( E) [FL−2],n—the elastic modulus intension or compression.3.2 Definitions of Terms Specific to This Standard:3.2.1 anti-nodes, n—an unconstrained slender rod or bar inresonance contains two or more locations that have localmaximum displacements, called anti-nodes. For the fundamen-tal flexure resonance, the anti-nodes are located at the two endsand the center of the specimen.3.2.2 elastic, adj—the property of a material such that anapplication of stress within the elastic limit of that materialmaking up the body being stressed will cause an instantaneousand uniform deformation, that will be eliminated upon removalof the stress, with the body returning instantly to its originalsize and shape without energy loss. Most advanced ceramicsconform to this definition well enough to make this resonancetest valid.3.2.3 flexural vibrations, n—the vibrations that occur whenthe oscillations in a slender rod or bar are in the plane normalto the length dimension.3.2.4 homogeneous, adj—the condition of a specimen suchthat the composition and density are uniform, such that anysmaller specimen taken from the original is representative ofthe whole. Practically, as long as the geometrical dimensions ofthe test specimen are large with respect to the size of individualgrains, crystals, or components, the body can be consideredhomogeneous.3.2.5 isotropic, adj—the condition of a specimen such thatthe values of the elastic properties are the same in all directionsin the material.Advanced ceramics are considered isotropic ona macroscopic scale, if they are homogeneous and there is arandom distribution and orientation of phases, crystallites, andcomponents.3.2.6 nodes, n—a slender rod or bar in resonance containsone or more locations having a constant zero displacement,called nodes. For the fundamental flexural resonance, the nodesare located at 0.224 L from each end, where L is the length ofthe specimen.3.2.7 resonance, n—a slender rod or bar driven into one ofthe modes of vibration described in 3.2.3 or 3.2.9 is said to bein resonance when the imposed frequency is such that theresultant displacements for a given amount of driving force areat a maximum. The resonant frequencies are natural vibrationfrequencies that are determined by the elastic modulus, mass,and dimensions of the test specimen.3.2.8 slender rod or bar, n—in dynamic elastic propertytesting, a specimen whose ratio of length to minimum cross-sectional dimension is at least five and preferably in the rangeof 20 to 25.3.2.9 torsional vibrations, n— the vibrations that occurwhen the oscillations in each cross-sectional plane of a slenderrod or bar are such that the plane twists around the lengthdimension axis.3For 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.C1198 − 09 (2013)24. Summary of Test Method4.1 This test method measures the resonant frequencies oftest specimens of suitable geometry by exciting them atcontinuously variable frequencies. Mechanical excitation ofthe bars is provided through the use of a transducer thattransforms a cyclic electrical signal into a cyclic mechanicalforce on the specimen.Asecond transducer senses the resultingmechanical vibrations of the specimen and transforms theminto an electrical signal. The amplitude and frequency of thesignal are measured by an oscilloscope or other means to detectresonance. The resonant frequencies, dimensions, and mass ofthe specimen are used to calculate dynamic Young’s modulusand dynamic shear modulus.5. Significance and Use5.1 This test method may be used for material development,characterization, design data generation, and quality controlpurposes. It is specifically appropriate for determining themodulus of advanced ceramics that are elastic, homogeneous,and isotropic.5.1.1 This test method is nondestructive in nature. Onlyminute stresses are applied to the specimen, thus minimizingthe possibility of fracture.5.1.2 The period of time during which measurement stressis applied and removed is of the order of hundreds ofmicroseconds. With this test method it is feasible to performmeasurements at high temperatures, where delayed elastic andcreep effects would invalidate modulus measurements calcu-lated from static loading.5.2 This test method has advantages in certain respects overthe use of static loading systems for measuring moduli inadvanced ceramics. It is nondestructive in nature and can beused for specimens prepared for other tests. Specimens aresubjected to minute strains; hence, the moduli are measured ator near the origin of the stress-strain curve with the minimumpossibility of fracture. The period of time during whichmeasurement stress is applied and removed is of the order ofhundreds of microseconds. With this test method it is feasibleto perform measurements at high temperatures, where delayedelastic and creep effects would invalidate modulus measure-ments calculated from static loading.5.3 The sonic resonant frequency technique can also be usedas a nondestructive evaluation tool for detecting and screeningdefects (cracks, voids, porosity, density variations) in ceramicparts. These defects may change the elastic response and theobserved resonant frequency of the test specimen. GuideE2001 describes a procedure for detecting such defects inmetallic and nonmetallic parts using the resonant frequencymethod.6. Interferences6.1 The relationships between resonant frequency and dy-namic modulus presented herein are specifically applicable tohomogeneous, elastic, isotropic materials.6.1.1 This test method of determining the moduli is appli-cable to composite ceramics and inhomogeneous materialsonly with careful consideration of the effect of inhomogeneitiesand anisotropy. The character (volume fraction, size,morphology, distribution, orientation, elastic properties, andinterfacial bonding) of the reinforcement/inhomogeneities inthe specimens will have a direct effect on the elastic propertiesof the specimen as a whole. These effects must be consideredin interpreting the test results for composites and inhomoge-neous materials.6.1.2 If specific surface treatments (coatings, machining,grinding, etching, etc.) change the elastic properties of thenear-surface material, there will be accentuated effects on theproperties measured by this flexural method, as compared tostatic/bulk measurements by tensile or compression testing.6.1.3 This test method is not satisfactory for specimens thathave major discontinuities, such as large cracks (internal orsurface) or voids.6.2 This test method for determining moduli is limited tospecimens with regular geometries (rectangular parallelepipedand cylinders) for which analytical equations are available torelate geometry, mass, and modulus to the resonant vibrationfrequencies. This test method is not appropriate for determin-ing the elastic properties of materials which cannot be fabri-cated into such geometries.6.2.1 The analytical equations assume parallel/concentricdimensions for the regular geometries of the specimen. Devia-tions from the specified tolerances for the dimensions of thespecimens will change the resonant frequencies and introduceerror into the calculations.6.2.2 Edge treatments such as chamfers or radii are notconsidered in the analytical equations. Edge chamfers onflexure bars prepared according to Test Method C1161 willchange the resonant frequency of the test bars and introduceerror into the calculations of the dynamic modulus. It isrecommended that specimens for this test not have chamferedor rounded edges. Alternately, if narrow rectangular specimenswith chamfers or edge radii are tested, then the procedures inAnnex A1 should be used to correct the calculated Young’smodulus, E.6.2.3 For specimens with as-fabricated/rough or unevensurfaces, variations in dimension can have