Designation: E1875 − 13Standard Test Method forDynamic Young’s Modulus, Shear Modulus, and Poisson’sRatio by Sonic Resonance1This standard is issued under the fixed designation E1875; 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. Scope*1.1 This test method covers the determination of the dy-namic elastic properties of elastic materials. Specimens ofthese materials possess specific mechanical resonant frequen-cies that are determined by the modulus of elasticity, 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 is specifically appropriate for materialsthat are elastic, homogeneous, and isotropic (1).2Materials ofa composite character (particulate, whisker, or fiber reinforced)may be tested by this test method with the understanding thatthe character (volume fraction, size, morphology, distribution,orientation, elastic properties, and interfacial bonding) of thereinforcement in the test specimen will have a direct effect onthe elastic properties. These reinforcement effects must beconsidered in interpreting the test results for composites. Thistest method is not satisfactory for specimens that have cracksor voids that are major discontinuities in the specimen. Neitheris the test method satisfactory when these materials cannot befabricated in a uniform rectangular or circular cross section.1.3 A high-temperature furnace and cryogenic cabinet aredescribed for measuring the dynamic elastic moduli as afunction of temperature from –195 to 1200°C.1.4 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.1.5 There are material-specific ASTM standards that coverthe determination of resonance frequencies and elastic proper-ties of specific materials by sonic resonance or by impulseexcitation of vibration. Test Methods C215, C623, C747, C848,C1198, and C1259 may differ from this test method in severalareas (for example; sample size, dimensional tolerances,sample preparation). The testing of these materials shall bedone in compliance with these material specific standards.Where possible, the procedures, sample specifications, andcalculations are consistent with these test methods.1.6 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.7 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility 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:3C215 Test Method for Fundamental Transverse,Longitudinal, and Torsional Resonant Frequencies ofConcrete SpecimensC623 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 SonicResonance1This test method is under the jurisdiction of ASTM Committee E28 onMechanical Testing and is the direct responsibility of Subcommittee E28.04 onUniaxial Testing.Current edition approved Nov. 1, 2013. Published May 2014. Originallyapproved in 1997. Last previous edition approved in 2008 as E1875-08. DOI:10.1520/E1875-13.2The boldface numbers in parentheses refer to a list of references at the end ofthis standard.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.*A Summary of Changes section appears at the end of this standardCopyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States1C848 Test Method for Young’s Modulus, Shear Modulus,and Poisson’s Ratio For Ceramic Whitewares by Reso-nanceC1198 Test Method for Dynamic Young’s Modulus, ShearModulus, and Poisson’s Ratio for Advanced Ceramics bySonic ResonanceC1259 Test Method for Dynamic Young’s Modulus, ShearModulus, and Poisson’s Ratio for Advanced Ceramics byImpulse Excitation of VibrationE6 Terminology Relating to Methods of Mechanical TestingE177 Practice for Use of the Terms Precision and Bias inASTM Test MethodsE691 Practice for Conducting an Interlaboratory Study toDetermine the Precision of a Test Method3. Terminology3.1 Definitions:Terms common to mechanical testing.3.1.1 dynamic mechanical measurement, n— a technique inwhich either the modulus or damping, or both, of a substanceunder oscillatory applied force or displacement is measured asa function of temperature, frequency, or time, or a combinationthereof.3.1.2 elastic limit [FL–2],n—the greatest stress that amaterial is capable of sustaining without permanent strainremaining upon complete release of the stress.3.1.2.1 Discussion—Due to practical considerations in de-termining the elastic limit, measurements of strain using asmall force, rather than zero force, are usually taken as theinitial and final reference. (E6)3.1.3 modulus of elasticity [FL–2],n—the ratio of stress tocorresponding strain below the proportional limit.3.1.3.1 Discussion—The stress-strain relationships of manymaterials do not conform to Hooke’s law throughout the elasticrange, but deviate therefrom even at stresses well below theelastic limit. For such materials, the slope of either the tangentto the stress-strain curve at the origin or at a low stress, thesecant drawn from the origin to any specified point on thestress-strain curve, or the chord connecting any two specifiedpoints on the stress-strain curve is usually taken to be the“modulus of elasticity.” In these cases, the modulus should bedesignated as the “tangent modulus,” the “secant modulus,” orthe “chord modulus,” and the point or points on the stress-strain curve described. Thus, for materials where the stress-strain relationship is curvilinear rather than linear, one of thefour following terms may be used:(a) initial tangent modulus [FL–2], n—the slope of thestress-strain curve at the origin.(b) tangent modulus [FL–2], n—the slope of the stress-strain curve at any specified stress or strain.(c ) secant modulus [FL–2], n—the slope of the secantdrawn from the origin to any specified point on the stress-strain curve.(d) chord modulus [FL–2], n—the slope of the chorddrawn between any two specified points on the stress-straincurve below the elastic limit of the material.3.1.3.2 Discussion—Modulus of elasticity, like stress, isexpressed in force per unit of area (pounds per square inch,etc.).3.1.4 Poisson’s ratio, µ,n—the negative of the ratio oftransverse strain to the corresponding axial strain resultingfrom an axial stress below the proportional limit of thematerial.3.1.4.1 Discussion—Poisson’s ratio may be negative forsome materials, for example, a tensile transverse strain willresult from a tensile axial strain.3.1.4.2 Discussion—Poisson’s ratio will have more than onevalue if the material is not isotropic. (E6)3.1.5 proportional limit [FL–2] ,n—the greatest stress that amaterial is capable of sustaining without deviation fromproportionality of stress to strain (Hooke’s law).3.1.5.1 Discussion—Many experiments have shown thatvalues observed for the proportional limit vary greatly with thesensitivity and accuracy of the testing equipment, eccentricityof loading, the scale to which the stress-strain diagram isplotted, and other factors. When determination of proportionallimit is required, the procedure and the sensitivity of the testequipment should be specified. (E6)3.1.6 shear modulus (G) [FL–2],n—the ratio of shear stressto corresponding shear strain below the proportional limit, alsocalled torsional modulus and modulus of rigidity.3.1.6.1 Discussion—The value of the shear modulus maydepend on the direction in which it is measured if the materialis not isotropic. Wood, many plastics and certain metals aremarkedly anisotropic. Deviations from isotropy should besuspected if the shear modulus differs from that determined bysubstituting independently measured values of Young’smodulus, E, and Poisson’s ratio, µ, in the relation:G 5 E/@2~11µ!#3.1.6.2 Discussion—In general, it is advisable in reportingvalues of shear modulus to state the range of stress over whichit is measured.3.1.7 Young’s modulus (E) [FL–2] ,n—the ratio of tensile orcompressive stress to corresponding strain below the propor-tional limit of the material. (E6)3.2 Definitions of Terms Specific to This Standard:3.2.1 anti-nodes, n—two or more locations in an uncon-strained slender rod or bar in resonance that have localmaximum displacements.3.2.1.1 Discussion—For the fundamental flexure resonance,the anti-nodes are located at the two ends and the center of thespecimen.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.3.2.2.1 Discussion—Most elastic materials conform to thisdefinition well enough to make this resonance test valid.3.2.3 flexural vibrations, n—oscillations that occur in aslender rod or bar in a vertical plane normal to the lengthdimension.E1875 − 1323.2.4 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.3.2.4.1 Discussion—Practically, as long as the geometricaldimensions of the test specimen are large with respect to thesize of individual grains, crystals, or components, the body canbe considered homogeneous.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.3.2.5.1 Discussion—Materials are considered isotropic on amacroscopic scale, if they are homogeneous and there is arandom distribution and orientation of phases, crystallites, andcomponents.3.2.6 nodes, n—one or more locations of a slender rod or barin resonance that have a constant zero displacement.3.2.6.1 Discussion—For the fundamental flexuralresonance, the nodes are located at 0.224 L from each end,where L is the length of the specimen.3.2.7 resonance, n—state of slender rod or bar driven intoone of the modes of vibration described in 3.2.3 or 3.2.9 whenthe imposed frequency is such that the resultant displacementsfor a given amount of driving force are at a maximum.3.2.7.1 Discussion—The resonant frequencies are naturalvibration frequencies that are determined by the modulus ofelasticity, 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 rangefrom 20 to 25.3.2.9 torsional vibrations, n—oscillations that occur in eachcross-sectional plane of a slender rod or bar, such that the planetwists around the length dimension axis.4. 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 has advantages in certain respects overthe use of static loading systems for measuring moduli.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 of elasticity measure-ments calculated from static loading.5.2 This test method is suitable for detecting whether amaterial meets the specifications, if cognizance is given to oneimportant fact in materials are often sensitive to thermalhistory. Therefore, the thermal history of a test specimen mustbe considered in comparing experimental values of moduli toreference or standard values. Specimen descriptions shouldinclude any specific thermal treatments that the specimens havereceived.6. Apparatus6.1 The test apparatus is shown in Fig. 1. It consists of avariable-frequency audio oscillator, used to generate a sinusoi-dal voltage, and a power amplifier and suitable transducer toconvert the electrical signal to a mechanical driving vibration.A frequency meter (preferably digital) monitors the audiooscillator output to provide an accurate frequency determina-tion. A suitable suspension-coupling system supports the testspecimen. Another transducer acts to detect mechanical vibra-tion in the specimen and to convert it into an electrical signalthat is passed through an amplifier and displayed on anindicating meter. The meter may be a voltmeter,microammeter, or oscilloscope. An oscilloscope is recom-mended because it enables the operator to positively identifyresonances, including higher order harmonics, by Lissajousfigure analysis. If a Lissajous figure is desired, the output of theoscillator is also coupled to the horizontal plates of theoscilloscope. If temperature-depend