Designation: E1876 − 15Standard Test Method forDynamic Young’s Modulus, Shear Modulus, and Poisson’sRatio by Impulse Excitation of Vibration1This standard is issued under the fixed designation E1876; 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 determination of the dynamicelastic properties of elastic materials at ambient temperatures.Specimens of these materials possess specific mechanicalresonant frequencies that are determined by the elasticmodulus, mass, and geometry of the test specimen. Thedynamic elastic properties of a material can therefore becomputed if the geometry, mass, and mechanical resonantfrequencies of a suitable (rectangular or cylindrical geometry)test specimen of that material can be measured. DynamicYoung’s modulus is determined using the resonant frequencyin either the flexural or longitudinal mode of vibration. Thedynamic shear modulus, or modulus of rigidity, is found usingtorsional resonant vibrations. Dynamic Young’s modulus anddynamic shear modulus are used to compute Poisson’s ratio.1.2 Although not specifically described herein, this testmethod can also be performed at cryogenic and high tempera-tures with suitable equipment modifications and appropriatemodifications to the calculations to compensate for thermalexpansion.1.3 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.1.4 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.5 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:2C215 Test Method for Fundamental Transverse,Longitudinal, and Torsional Resonant Frequencies ofConcrete SpecimensC372 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-nanceC1161 Test Method for Flexural Strength of AdvancedCeramics at Ambient TemperatureC1198 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 Methods3. Terminology3.1 Definitions:3.1.1 The definitions of terms relating to mechanical testingappearing in Terminology E6 and C1198 should be consideredas applying to the terms used in this test method.1This 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 Dec. 15, 2015. Published March 2016. Originallyapproved in 1997. Last previous edition approved in 2009 as E1876 – 09. DOI:10.1520/E1876-15.2For 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 States13.1.2 dynamic elastic modulus, n—the elastic modulus,either Young’s modulus or shear modulus, that is measured ina dynamic mechanical measurement.3.1.3 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 combinationthereof.3.1.4 elastic limit [FL–2],n—the greatest stress that amaterial is capable of sustaining without permanent strainremaining upon complete release of the stress. E63.1.5 modulus of elasticity [FL–2],n—the ratio of stress tocorresponding strain below the proportional limit.3.1.5.1 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-straincurve.(d) chord modulus [FL–2], n—the slope of the chord drawnbetween any two specified points on the stress-strain curvebelow the elastic limit of the material.3.1.5.2 Discussion—Modulus of elasticity, like stress, isexpressed in force per unit of area (pounds per square inch,etc.).3.1.6 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.3.1.6.1 Discussion—Poisson’s ratio may be negative forsome materials, for example, a tensile transverse strain willresult from a tensile axial strain.3.1.6.2 Discussion—Poisson’s ratio will have more than onevalue if the material is not isotropic. E63.1.7 proportional limit [FL–2] ,n—the greatest stress that amaterial is capable of sustaining without deviation fromproportionality of stress to strain (Hooke’s law). E63.1.7.1 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.3.1.8 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.3.1.8.1 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 5E2~11µ!3.1.8.2 Discussion—In general, it is advisable in reportingvalues of shear modulus to state the range of stress over whichit is measured. E63.1.9 Young’s modulus, E [FL–2],n—the ratio of tensile orcompressive stress to corresponding strain below the propor-tional limit of the material. E63.2 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, which will be eliminated uponremoval of the stress, with the body returning instantly to itsoriginal size and shape without energy loss. Most elasticmaterials conform to this definition well enough to make thisresonance test valid.3.2.3 flexural vibrations, n—the vibrations that occur whenthe oscillations in a slender rod or bar are in a plane normal tothe length dimension.3.2.4 homogeneous, adj—the condition of a specimen suchthat the composition and density are uniform, so 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, components, pores, ormicrocracks, the body can be considered homogeneous.3.2.5 in-plane flexure, n—for rectangular parallelepipedgeometries, a flexure mode in which the direction of displace-ment is in the major plane of the test specimen.3.2.6 isotropic, adj—the condition of a specimen such thatthe values of the elastic properties are the same in all directionsin the material.3.2.6.1 Discussion—Materials are considered isotropic on amacroscopic scale, if they are homogeneous and there is aE1876 − 152random distribution and orientation of phases, crystallites,components, pores, or microcracks.3.2.7 longitudinal vibrations, n—the vibrations that occurwhen the oscillations in a slender rod or bar are parallel to thelength of the rod or bar.3.2.8 nodes, n—one or more locations of a slender rod or barin resonance that have a constant zero displacement.3.2.8.1 Discussion—For the fundamental flexuralresonance, the nodes are located at 0.224 L from each end,where L is the length of the specimen.3.2.9 out-of-plane flexure, n—for rectangular parallelepipedgeometries, a flexure mode in which the direction of displace-ment is perpendicular to the major plane of the test specimen.3.2.10 resonant frequency, n—naturally occurring frequen-cies of a body driven into flexural, torsional, or longitudinalvibration that are determined by the elastic modulus, mass, anddimensions of the body.3.2.10.1 Discussion—The lowest resonant frequency in agiven vibrational mode is the fundamental resonant frequencyof that mode.3.2.11 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 25.3.2.12 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.3.3 Symbols:A = plate constant; used in Eq A1.1D = diameter of rod or diameter of diskDe= effective diameter of the bar; defined in Eq 10 andEq 11E = dynamic Young’s modulus; defined in Eq 1 and Eq 4,and Eq A1.4E1= first natural calculation of the dynamic Young’smodulus, used in Eq A1.2E2= second natural calculation of the dynamic Young’smodulus. used in Eq A1.3G = dynamic shear modulus, defined in Eq 12, Eq 14, andEq A1.5K = correction factor for the fundamental longitudinalmode to account for the finite diameter-to-length ratioand Poisson’s Ratio, defined in Eq 8Ki= geometric factor for the resonant frequency of order i,see Table A1.2 and Table A1.3L = specimen lengthMT= dynamic elastic modulus at temperature T (either thedynamic Young’s modulus E, or the dynamic shearmodulus G)M0= dynamic elastic modulus at room temperature (eitherthe dynamic Young’s modulus E or the dynamic shearmodulus G)R = correction factor the geometry of the bar, defined in Eq13T1= correction factor for fundamental flexural mode toaccount for finite thickness of bar and Poisson’s ratio;defined in Eq 2T1 = correction factor for fundamental flexural mode toaccount for finite diameter of rod, Poisson’s ratio;defined in Eq 4 and Eq 6b = specimen widthf = frequencyf0= resonant frequency at room temperature in furnace orcryogenic chamberf1= first natural resonant frequency; used in Eq A1.2f2= second natural frequency; used in Eq A1.3ff= fundamental resonant frequency of bar in flexure; usedin Eq 1fl= fundamental longitudinal resonant frequency of aslender bar; used in Eq 7 and Eq 9fT= resonant frequency measured in the furnace or cryo-genic chamber at temperature T, used in Eq 16ft= fundamental resonant frequency of bar in torsion; usedin Eq 12 and Eq 14m = specimen massn = the order of the resonance (n=1,2,3,.)r = radius of the disk, used in Eq A1.1t = specimen, disk or bar, thicknessT1= correction factor for fundamental flexural mode toaccount for finite thickness of the bar and Poisson’sratio; defined in Eq 2T’1= correction factor for fundamental flexural mode toaccount for finite thickness of the rod and Poisson’sratio; defined in Eq 4∆T = temperature difference between the test temperature Tand room temperature, used in Eq 16α = average linear thermal expansion coefficient(mm/mm/°C) from room temperature to test tempera-ture; used in Eq 16µ = Poisson’s ratioρ = density of the disk; used in Eq A1.14. Summary of Test Method4.1 This test method measures the fundamental resonantfrequency of test specimens of suitable geometry by excitingthem mechanically by a singular elastic strike with an impulsetool. A transducer (for example, contact accelerometer ornon-contacting microphone) senses the resulting mechanicalvibrations of the specimen and transforms them into electricsignals. Specimen supports, impulse locations, and signalpick-up points are selected to induce and measure specificmodes of the transient vibrations. The signals are analyzed, andthe fundamental resonant frequency is isolated and measuredby the signal analyzer, which provides a numerical reading thatis (or is proportional to) either the frequency or the period ofthe specimen vibration. The appropriate fundamental resonantfrequencies, dimensions, and mass of the specimen are used tocalculate dynamic Young’s modulus, dynamic shear modulus,and Poisson’s ratio.5. Significance a