1、AMERICAN NATIONAL STANDARDGRAPHICAL PRESENTATION OF THECOMPLEX MODULUS OFVISCOELASTIC MATERIALSAccredited Standards Committee S2, Mechanical Vibration and ShockStandards SecretariatAcoustical Society of America35 Pinelawn Road, Suite 114EMelville, NY 11747-3177ANSI S2.24-2001ANSIS2.24-2001The Americ
2、an National Standards Institute, Inc. (ANSI) is the na-tional coordinator of voluntary standards development and the clear-inghouse in the U.S. for information on national and internationalstandards.The Acoustical Society of America (ASA) is an organization of sci-entists and engineers formed in 192
3、9 to increase and diffuse theknowledge of acoustics and to promote its practical applications.American National StandardGraphical Presentation of the Complex Modulusof Viscoelastic MaterialsSecretariatAcoustical Society of AmericaApproved 3 July 2001American National Standards Institute, Inc.Abstrac
4、tThis Standard specifies the procedure for generating a graphical presentation of the frequency andtemperature dependence of the complex modulus of viscoelastic materials. This Standard is the Nationalcounterpart of ISO 10112, Damping materials - Graphical presentation of the complex modulus.ANSI S2
5、.24-2001AMERICAN NATIONAL STANDARDS ON ACOUSTICSThe Acoustical Society of America (ASA) provides the Secretariat for AccreditedStandards Committees S1 on Acoustics, S2 on Mechanical Vibration and Shock,S3 on Bioacoustics, and S12 on Noise. These committees have wide represen-tation from the technica
6、l community (manufacturers, consumers, and general-interest representatives). The standards are published by the Acoustical Society ofAmerica through the American Institute of Physics as American National Stan-dards after approval by their respective Standards Committees and the AmericanNational Sta
7、ndards Institute.These standards are developed and published as a public service to providestandards useful to the public, industry, and consumers, and to Federal, State,and local governments.Each of the Accredited Standards Committees (operating in accordance with pro-cedures approved by ANSI) is r
8、esponsible for developing, voting upon, and main-taining or revising its own Standards. The ASA Standards Secretariat administersCommittee organization and activity and provides liaison between the AccreditedStandards Committees and ANSI. After the Standards have been produced andadopted by the Accr
9、edited Standards Committees, and approved as AmericanNational Standards by ANSI, the ASA Standards Secretariat arranges for theirpublication and distribution.An American National Standard implies a consensus of those substantially con-cerned with its scope and provisions. Consensus is established wh
10、en, in thejudgment of the ANSI Board of Standards Review, substantial agreement hasbeen reached by directly and materially affected interests. Substantial agreementmeans much more than a simple majority, but not necessarily unanimity. Consen-sus requires that all views and objections be considered a
11、nd that a concertedeffort be made towards their resolution.The use of American National Standards are completely voluntary. Their existencedoes not in any respect preclude anyone, whether he or she has approved theStandards or not, from manufacturing, marketing, purchasing, or using products,process
12、es, or procedures not conforming to the Standards.NOTICE: This American National Standard may be revised or withdrawn at anytime. The procedures of the American National Standards Institute require thataction be taken periodically to reaffirm, revise, or withdraw this Standard.Standards SecretariatA
13、coustical Society of America35 Pinelawn Rd. Suite 114EMelville, New York 11747USATelephone: 11 631 390-0215Telefax: 11 631 390-0217E-mail: asastdsaip.orgInternet: http:/asa.aip.org 2001 by Acoustical Society of America. This Standard may not be reproduced in whole orin part in any form for sale, pro
14、motion, or any commercial purpose, or any purpose notfalling within the provisions of the Copyright Act of 1976, without prior written permission ofthe publisher. For permission, address a written request to the Standards Secretariat of theAcoustical Society of America. ContentsPageForeword iii0 Int
15、roduction 11 Scope, purpose, and applications 11.1 Scope . 11.2 Purpose . 11.3 Applications . 12 Informative references 13 Definitions . 13.1 Shear modulus . 13.2 Loss factor 13.3 Time-temperature superposition 23.4 Shift factor . 23.5 Glass transition temperature . 23.6 Thermorheologically simple m
16、aterial 24 Data check using wicket plot . 25 Reduced frequency concept . 26 Graphical presentation 56.1 Data points 56.2 Analytical representation 56.3 Diagonal constant temperature lines 57 The frequency-temperature nomogram and its use . 6Figures1 Loss factor versus shear modulus for standard mate
17、rial . 42 Shift factor versus temperature for standard material . 53 Nomogram for standard material . 6Table1 Shear modulus, loss factor, and shift factor of standard material asfunctions of temperature and frequency 34iForewordThis Foreword is for information only, and is not a part of ANSI S2.24.2
18、001 AmericanNational Standard Method of Graphical Presentation of the Complex Modulus of Viscoelas-tic Materials.This Standard was developed under the jurisdiction of Accredited Standards Com-mittee S2, Mechanical Vibration and Shock. This Standard is the National coun-terpart of ISO 10112, Damping
19、materials - Graphical presentation of the complexmodulus.Accredited Standards Committee S2, Mechanical Vibration and Shock, has thefollowing scope:Standards, specifications, methods of measurement and test, and terminologyin the fields of mechanical vibration and shock and condition monitoring anddi
20、agnostics of machines, but excluding those aspects which pertain to biologi-cal safety, tolerance, and comfort.At the time this Standard was submitted to Accredited Standards Committee S2,Mechanical Vibration and Shock, for approval, the membership was as follows:R. J. Peppin, ChairD. J. Evans, Vice
21、 ChairS. B. Blaeser, SecretaryAcoustical Society of America S.I.HayekB. E. Douglas (Alt.)American Industrial Hygiene Association L.HRoysterD. Driscoll (Alt.)Bruel hence it is not an intrinsic material property.3.6 thermorheologically simple material. A ma-terial for which the time-temperature superp
22、ositionprinciple is valid. For such a material, the complexmodulus is expressed as a complex valued func-tion of one independent variable, namely theshifted frequency, commonly referred to as the re-duced frequency, to represent its variation with fre-quency and temperature.4 Data check using wicket
23、 plotFor purposes of illustration in this Standard, a validset of complex modulus data for the standard ma-terial described in ANSI S2.21-1998 1 was ob-tained using the resonance method in accordancewith ANSI S2.22-1998 2. The measured data arelisted in Table 1.To check the consistency and scatter o
24、f the data,plot all data, regardless of frequency or tempera-ture, on a plot of log (loss factor) vs log (modulus,G8), as shown in Figure 1. This plot is commonlyreferred to as a wicket plot. If the data represent athermorheologically simple material and if the datahave no scatter, the data will plo
25、t as a single,smooth curve. As unshifted data are plotted in thewicket plot, no part of any scatter in this plot can beattributed to the shifting procedure.While not specifically used in the graphical presen-tation of data, the wicket plot shall be used as aqualitative indication of the scatter of t
26、he experi-mental data. The width of the band of data, as wellas the departure of individual points from the cen-ter of the band, are indicative of scatter. Nothing isrevealed about the accuracy of the temperatureand frequency measurements or about any sys-tematic error.5 Reduced frequency conceptThe
27、 significance of time-temperature superposi-tion is demonstrated through the concept of re-duced frequency. In general, the complex shearmodulus of a viscoelastic material is a function offrequency and temperature,G*5G*f,T!. (2)In a thermorheologically simple material, thesevariables appear only as
28、the product of frequencyand a function of temperature known as the relax-ation time,G*5G*ft T!. (3)Hence a change in frequency is equivalent to achange in temperature. Consequently, the shiftfactor can be expressed as the ratio of the relax-ation time at temperature T to the relaxation timeat a refe
29、rence temperature T0,aTT!5t T!/t T0!. (4)The complex modulus can be written asG*5G*faTT!t T0! (5)and the reduced frequency is defined asfR5faTT!. (6)Complex modulus can be expressed in twoequivalent waysG*5G*faTT!t T0!5G*fRt T0!, (7)so that a modulus value measured at frequency fand temperature T is
30、 equivalent to a value at re-ANSI S2.24-20012 2001 Acoustical Society of AmericaTable 1 Shear modulus, loss factor, and shift factor of standard material as functions of temperatureand frequencyT/K f/Hz GMPa tan d log aTT/Kf/Hz G/MPa tan d log aT212.9 2416.1 940.00 0.033 17.21 262.4 18917.9 319.27 0
31、.201 3.49212.9 8884.0 952.33 0.043 17.21 262.4 23598.7 325.03 0.158 3.49212.9 16366.2 947.33 0.049 17.21 267.4 995.2 142.13 0.381 2.85217.7 2360.3 897.00 0.030 14.72 267.4 4229.0 196.93 0.297 2.85217.7 8690.5 911.67 0.039 14.72 267.4 8188.8 220.77 0.262 2.85217.7 15969.1 902.33 0.046 14.72 267.4 124
32、46.2 236.80 0.252 2.85222.5 2294.8 848.00 0.031 12.66 267.4 16554.6 239.10 0.256 2.85222.5 8349.0 839.00 0.064 12.66 268.0 959.5 130.53 0.403 2.78222.5 15545.8 856.00 0.035 12.66 268.0 4118.8 185.17 0.312 2.78222.5 22890.5 854.00 0.044 12.66 268.0 7994.3 208.90 0.276 2.78227.8 2221.5 794.33 0.037 10
33、.76 268.0 12201.2 225.90 0.267 2.78227.8 15071.0 804.33 0.037 10.76 268.0 16275.0 231.37 0.253 2.78227.8 22171.5 801.33 0.041 10.76 272.9 760.7 74.73 0.546 2.22232.5 2132.7 731.67 0.048 9.33 272.9 3503.6 124.07 0.427 2.22232.5 7798.4 735.00 0.025 9.33 272.9 6938.4 148.63 0.374 2.22232.5 14584.6 753.
34、00 0.040 9.33 272.9 10676.5 164.47 0.362 2.22232.5 21442.4 749.33 0.043 9.33 277.8 564.1 36.33 0.710 1.71237.5 2031.1 663.33 0.052 8.01 277.8 2848.3 75.30 0.538 1.71237.5 7525.2 682.00 0.059 8.01 277.8 5730.4 92.63 0.507 1.71237.5 13968.1 690.33 0.047 8.01 282.7 386.9 16.05 0.791 1.24237.5 20554.5 6
35、88.00 0.049 8.01 282.7 2227.6 39.33 0.727 1.24242.5 1907.9 584.33 0.070 6.87 282.7 4652.5 55.10 0.646 1.24242.5 7098.8 606.33 0.068 6.87 287.6 267.3 7.60 0.800 0.81242.5 13256.8 620.67 0.061 6.87 287.6 1659.7 18.84 0.903 0.81242.5 19617.9 626.33 0.056 6.87 287.6 3772.1 30.43 0.870 0.81247.5 1761.7 4
36、96.67 0.091 5.86 292.6 199.2 4.73 0.655 0.40247.5 6633.7 527.67 0.087 5.86 292.6 1190.2 9.56 0.919 0.40247.5 12399.7 541.33 0.079 5.86 292.6 2708.6 15.00 0.927 0.40247.5 18309.4 543.33 0.081 5.86 296.1 173.9 3.80 0.583 0.14252.4 1591.8 402.67 0.129 4.99 296.1 963.2 6.75 0.828 0.14252.4 6088.7 441.33
37、 0.118 4.99 296.2 170.4 3.69 0.568 0.13252.4 11461.4 459.67 0.106 4.99 296.2 968.2 6.82 0.829 0.13252.4 16970.0 463.67 0.111 4.99 296.2 2253.3 10.53 0.910 0.13252.4 22628.2 470.67 0.105 4.99 296.3 170.4 3.70 0.563 0.12257.5 1405.3 309.40 0.183 4.19 296.3 964.8 6.78 0.828 0.12257.5 5489.6 355.00 0.15
38、3 4.19 297.5 155.9 3.33 0.457 0.04257.5 10421.0 376.00 0.143 4.19 297.5 847.1 5.55 0.756 0.04257.5 15528.7 384.67 0.143 4.19 302.4 139.4 2.87 0.326 -0.30257.5 20749.9 392.00 0.141 4.19 302.4 670.1 4.08 0.563 -0.30262.4 1204.4 220.33 0.267 3.49 302.4 1490.0 5.26 0.738 -0.30262.4 4881.6 273.53 0.218 3
39、.49 307.2 130.8 2.62 0.245 -0.62262.4 9326.5 295.77 0.193 3.49 307.2 576.5 3.36 0.428 -0.62262.4 14075.1 311.50 0.185 3.49 307.2 1216.2 4.00 0.566 -0.62312.0 125.9 2.48 0.194 -0.91 331.2 450.9 2.42 0.121 -1.90312.0 521.4 2.95 0.320 -0.91 331.2 861.1 2.55 0.163 -1.90ANSI S2.24-20013 2001 Acoustical S
40、ociety of Americaduced frequency fRand temperature T0. The re-duced frequency can be much greater than themeasured frequency (by a factor aT) since mea-surements made as a function of temperature areequivalent to measurements made over a widerfrequency range than measured.The shift factor has been p
41、resented here in a for-mal mathematical manner. The significance of theshift factor function, and the origin of its name, canbe illustrated graphically. Consider a log-log plot ofexperimental measurements of G versus fre-quency plotted as a series of isotherms. Pick oneisotherm temperature as the re
42、ference tempera-ture. The next highest temperature isotherm canbe shifted along the log frequency axis until it par-tially overlaps the reference isotherm. This pro-cess is continued with all the isotherms, in se-quence, both higher and lower than the reference.The result is a plot of log G over a w
43、ide range oflog reduced frequency values. This plot is knownas a master plot. The amount of shift required toproduce overlap is plotted as a function of tem-perature. Because this function was generated byshifting data, it is known as the shift factor function.A plot of the shift factor for the stan
44、dard material isshown in Figure 2.For uniformity in comparing data on different ma-terials, a standard reference temperature of 298 Kshall be used for this plot. The aTvalues from Fig-ure 2 are listed in Table 1. Additional referencetemperatures may also be useful for specific appli-cations.To obtai
45、n the complete nomogram, it is required tofit the aTdata to an analytical equation. One con-venient equation is the well established WLF (Wil-liams-Landel-Ferry) equation 4:logaT52C1T2T0!/C21T2T0!. (8)Table 1 ContinuedT/K f/Hz GMPa tan d log aTT/Kf/Hz G/MPa tan d log aT312.0 1068.1 3.40 0.430 -0.91
46、331.2 1290.3 2.60 0.199 -1.90316.8 123.2 2.40 0.156 -1.18 331.2 1754.2 2.73 0.216 -1.90316.8 489.4 2.72 0.244 -1.18 337.5 118.2 2.23 0.089 -2.18316.8 976.9 3.02 0.332 -1.18 337.5 445.2 2.37 0.100 -2.18316.8 1498.6 3.15 0.413 -1.18 337.5 844.2 2.48 0.130 -2.18321.6 121.0 2.33 0.130 -1.43 337.5 1257.0
47、 2.51 0.158 -2.18321.6 470.2 2.57 0.189 -1.43 337.5 1704.3 2.61 0.178 -2.18321.6 919.6 2.79 0.257 -1.43 337.5 2103.0 2.54 0.206 -2.18321.6 1394.6 2.88 0.318 -1.43 342.5 117.6 2.22 0.083 -2.38326.4 119.9 2.29 0.112 -1.67 342.5 441.4 2.34 0.084 -2.38326.4 458.3 2.48 0.149 -1.67 342.5 833.1 2.43 0.107
48、-2.38326.4 884.1 2.65 0.203 -1.67 342.5 1236.0 2.45 0.128 -2.38326.4 1331.3 2.71 0.248 -1.67 342.5 1668.3 2.53 0.149 -2.38326.4 1818.1 2.87 0.269 -1.67 342.5 2054.2 2.46 0.167 -2.38331.2 118.9 2.26 0.101 -1.90Figure 1 Loss factor versus shear modu-lus for standard material.ANSI S2.24-20014 2001 Acou
49、stical Society of AmericaThe solid line in Figure 2 is a fit to the WLF equa-tion, where C15 9.43, C25 131.7 K, and T05298 K.6 Graphical presentationThe graphical presentation of complex modulusshall be plotted using logarithmic scales as shownin Figure 3. The plot shall contain the real andimaginary parts of the complex modulus and theloss factor as a function of reduced frequency. Therequired reference temperature of 298 K is usedfor this plot.This figure will be exp
