Dynamic Properties
MEASUREMENT OF DYNAMIC PROPERTIES OF RUBBER ROBERT S. MARVIN National Bureau of Standards, Washington 25, D. C.
T
HE dynamic mechanilimiting slope greatly exagT h e relation of the dynamic modulus to other rheologigerated. The recovery after cal properties of macal measurements on rubberlike materials is discussed. removal of the load is also terials are generally underThe experimental methods used in determining dynamic shown. As indicated by stood t o be those that govern properties of rubber are classified into three groups dethe response of matter to broken lines, this creep pending on the relative values of the sample dimensions stresses alternating with curve may be broken up into and the wave length of sound in the sample. Examples of time. They are usually exthree additive parts: an ineach group, with a short discussion of advantages and pressed in a form which asstantaneouselastic response, disadvantages of each, are given. An appendix includes mmes that both stress and a delayed elastic response, derivations, based on an infinite, linear differential equastrain have a sinusoidal time and a viscous flow. Of these tion, for forced and free vibration, and relates some of the only the, first two are revariation but can be defined more commonly used methods of expressing the results coverable. This type of beunder other conditions, as of dynamic measurements. will be discussed below. havior has been predicted We shall assume throughfor linear amorphous polyout that at a fixed frequency the strain is directly proportional to mers generally, and the fact that many materials actually do follow such curves, with the recovery curve representing the mirror the strese amplitude; this is equivalent to assuming that stress and strain can be related by a linear differential equation. illimage of the instantaneous elastic and delayed elastic portions of the deformation curve, has been confirmed experimentally though the total stress-strain curve for rubber is decidedly nonlinear, the approximation introduced here is reasonably accurate (9, 24, 27, 28). Parenthetically, it is often very difficult experifor the small strains that are encountered in dynamic testing mentally to differentiate between a true viscous flow and a de(usually under 1% and almost never greater than 10%). layed elastic response with a very long relaxation time (9, SO). Since rubber is a material of the type generally termed viscoTo the extent that the creep function, E ( ~ ) / (where u u repreelastic, the dynamic modulus must express both the effective sents stress and E strain), is independent of the magnitude of the stress, the behavior of the material is linear. This is equivalent “spring constant,” and the damping properties or energy loss to saying that the superposition principle applies-that is, the during cyclic deformation. These properties are of great pracstrain due to the combined stresses u1 u2 is equal to the sum of tical importance in the design of those rubber articles whose use the strain due to u1 alone plus that due to U Z alone. It is this same involves alternating rather than static deformations, and are also linearity of behavior, of course, that permits breaking of the of considerable theoretical interest in that they must provide ultimately a measure of the forces existing between the tangled creep function into three components as in Figure 1. Another specification of the properties of such a material chains that make up the rubber. employs a general linear differential equation, known as the The detailed theoretical analysis which would predict the operator equation ( 3 ) : rheological behavior of a rubber sample in terms of these forces is Bot yet available, hence a phenomenological treatment in which the general strese-strain relations are given in terms of certain macroscopic material constants will be used. n = O m = O
+
TYPES OF RHEOLOGICAL SPECIFICATIONS
There are several ways in which such a general relation (a rheological equation of state) may be expressed. The constants occurring in a general differential equation relating stress and strain may be specified, or the response of the material to some specified stress or strain may be given. I n principle, all such specifications are equivalent, and methods for converting from one type to any of the others have been developed (9,3, 8, 18, 27, 48, 68). These conversions involve mathematical difficulties and necessitate a knowledge of the functions over a complete range of time or frequency. Since such information is generally not available, in practice various approximation methods are required (2,3, IS, 49, 6463,67). Perhaps the simplest of these rheological relations to discuss is the creep function, which gives the response of the material to a constant stress a9 B function of time. Figure 1 shows such a curve for a material like an unvulcanized rubber exhibiting irreversible flow, with the initial instantaneous response and the 696
where D“ =
an/a
tn
The coefficients CY,, and pmspecify the rheological properties of a material, just as does the creep function, and may be derived from it, at least in principle. Equation 1 (and the creep function also) is perfectly general and if solved subject to the appropriate boundary conditions would give the response of the material to any arbitrary stress or strain. Such solutions, however, may be rather difficult t o obtain, and in any event it is often convenient to have a “model” of this equation from Fhich a good deal of qualitative information may be deduced without attempting a formal solution. One type of model which can represent Equation 1 is shown in Figure 2. Such models are made up of combinations of “springs,” across which the force is proportional to the strain, and “dashpots,” across which the force is proportional to the rate of strain. When arranged in the form of Figure 2, a spring-dashpot pair is
INDUSTRIAL AND ENGINEERING CHEMISTRY
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-ELASTOMERS-Dynamic known as a Voigt element, and the whole array as a Voigt Model
(66). It should perhaps be stressed that such models need not imply any physical assumptions regarding the actual molecular processes occurring in the material. They can be regarded as merely an alternate rheological specification of material properties. Here, the direct experimental functions are considered as the fundamental specifications and the others as derived.
real and imaginary parts of the ratio of stress to strain when the .time dependence of each is expressed as exp (id), where i = 2/--i. They may be combined into a complex modulus defined by G* = G' iG",which is the ratio of complex stress to complex strain (the actual stress and strain are the real parts of the complex expressions). The ratio G"/G' is the tangent of the angle, 8, between stress and strain, called the loss tangent. Alternately, the reciprocal of the loss tangent, designated Q, is used by some authors. The whole system of complex notation is exactly like that used in alternating current theory. A set of definitions which hold for any type of periodic stress and strain, and reduce to those given here for sinusoids, can be made in terms of energy concepts. If D is defined as the maximum energy stored in the sample per unit volume during a cycle, w as the energy dissipated per cycle per unit volume, and 60 as the amplitude of the strain, the components of the complex modulus may be defined as:
+
G' I
I
' -
I
- \
tl
t0
TIME -t
Figure 1.
Creep Function and Its Components
- ---
Stress imposed at h: released at ti Instantaneous elastic response Delayed elastic response * - - viscous flow = -Total response -
-
- -
Properties-
= 2v/eoa
(2)
G' = w/regZ
(3)
and the complex modulus, G*, as before. Although the quantities w and 0 would be rather difficult t o measure directly, they are concepts which can at least be visualized for a generalized periodic deformation. Several other methods' of expressing the results of dynamic experiments have been used by various workers, and a summary of some of these with equations relating them for the case of sinusoidal stress and strain is given in the appendix. 0-
The creep function, already considered, expresses directly certain experimental results. If it-could be determined over the complete range of times from zero t o infinity, it would provide a complete rheological epecification of a material. However, experimental limitations deny a direct measurement of this function a t very short times, say less than 1 second. Speaking in terms of the model of Figure 2, the elements whpse relaxation times, 7, are less than about 1 second will appear during a creep experiment as pure springs, and their delayed elastic character can only be evidenced by experiments conducted on a time scale of the order of magnitude of their relaxation times. Dynamic experiments provide, of course, the obvious means of obtaining such information, since the time scale of a steady-state sinusoidal vibration is equal to the reciprocal of the frequency. Hence, the definition of the dynamic modulus is discussed and then the various experimental methods which have been used in its measurement. DYNAMIC MODULUS FUNCTION
If a stress having a sinusoidal time variation is applied to a linear viscoelantic material, the steady-state strain will also have ti sinusoidal time dependence but will in general be out of phase with the stress as shown in Figure 3A or, since both quantities are sinusoids with the same period, by the rotating vector scheme of Figure 3B. o is the radian frequency, equal to 2r times the frequency, Y , and t is the time in seconds. Figure 3 shows that two quantities are required to specify the response to a sinusoidal stress. These may be taken as the ratio of the amplitudes of stress and strain, and the phase angle between them, or an two equivalent quantities. The two used here PS basic quantities for this report can be defined as follows:
G' = G,,
Component of stress in phase with strain Strain
Component of stress 90' out of phase with strain Strain
These quantities are chosen because they are, respectively, the
April 1952
b
0-
Figure 2.
Voigt Model
The dynamic modulus function is another specification of the rheological properties of a material, and is of course equivalent in principle to all the others. It can be determined directly in the range of short times but ia rather difficult to evaluate experimentally a t long times or very low frequencies. It happens that the transformation from a Voigt model of the type of Figure 2 to a dynamic modulus function is fairly simple, and it is interesting to make such a transformation for a model which would yield a creep function qualitatively like that of Figure 1. The results, for an arbitrary choice of parameters, are shown in Figure 4, where the real and imaginary parts of the modulus are shown logarithmically and the loss tangent semilogarithmically. At very low frequencies both G' and G" go t o zero, since the model selected exhibits irreversible flow; they approach zero in such a way that their ratio, tan 6, goes to infinity. As the frequency is increased, the response of the purely viscous ele-
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-ELASTOMERS-Dynamic
Propertie-
ment decreases (since velocity is proportional to frequency) and that of the delayed elastic element predominates. I n this region G’ remains nearly constant, while G” and tan 6 increase with increasing frequency until the response of the pure spring is appreciable compared to that of the damped spring. As this frequency is approached tan 6 goes through a rather sharp maximum corresponding to a maximum energy loss, G” also rises to a maximum, and G’ approaches asymptotically the value of the undamped spring, GI. At very high frequencies, the velocity of motion is so large that the dashpots represent essentially zero compliances, and the whole motion is associated with the undamped spring.
Figure 3. Sinusoidal Stress and.Strain
The model used here is idealized chiefly in that its dynamic modulus function shows much more abrupt changes with frequency than do actual rubberlike materials; this explains why so many of the earlier workers in this field found, for a narrow frequency range, dynamic moduli independent of frequency. Two groups of workers have fairly recently investigated a wide enough frequency range to cover the sharp rise in G’ as it approaches its high-frequency limiting value: Guth and his coworkers ($3) for Butyl and natural rubber and Nolle (39) for natural, Buna-N, Butyl, neoprene, and GR-S. Results now available from a large scale cooperative program initiated by the National Bureau of Standards in which over 20 laboratories throughout the world have made measurements of mamples from a single lot of polyisobutylene covering the f r e quency range of 0.01 to 7 x 105 cycles per second, with stress relaxation measurements extending the equivalent frequency range to about 10-8 cycle per second, demonstrate essentially all of the behavior shown in Figure 4. As expected, the functions change much more gradually with frequency than indicated in Figure 4, the final rise in the G‘ curve extending over a t least six decades. EffECT Of TEMPERATURE
It has been recognized for a long time that decreasing the temperature has an effect on the dynamic modulus similar to that due to increasing the frequency. This has been explained qualitatively by the common assumption that the effects of individual viscous mechanisms are proportional to exp (AI?“),A a constant. It has also been generally assumed, based on the kinetic theory of rubberlike elasticity, that individual spring mechanisms have moduli directly proportional t o the absolute temperature. On these conceptions, Ferry (12)has recently based a reduction scheme by which dynamic measurements made at various temperatures and frequencies may be compared, and has shown experimentally that such measurements do form single curves when so reduced. For the purposes of this discussion, it will be sufficient to ,698
note that increasing frequency at constant temperature corre sponds to decreasing temperature at constant frequency. TYPES OF MEASUREMENTS
No attempt will be made here to give a complete account of all the methods which have been developed for measuring dynamic properties of rubber. Rather a classification of the various types of methods will be set up and certain specific examples will be noted for each type of measurement. Some recent papers review this subject from various points of view: A discussion of many of these methods, particularly those used as test methods by the rubber industry, was given by Dillon and Gehman (10). Ferry, Sawyer, and Ashworth ( 1 4 )mention a few methods and classify and discuss the various possibilities, and Nolle (58) in a report of his outstanding developmental work in this field of instrumentation has given discussions of five different methods and has included derivations of the operating equations for each. Leaderman ($9) has presented a discussion of the response of viscoelastic systems to various types of strain and proposed a systematic nomenclature. It must be evident that in order to measure the whole range of frequency of interest in dynamic measurements, extending roughly from 1 to lo7 cycles per second, several widely different types of measurement will be necessary. One way to clamify such methods is based on the relative values of the sample dimensions and the wave length of sound in the material a t the frequency under consideration (14). If the dimension of the sample in the direction of the imposed strain is much smaller than the wave length of sound in the sample a t the test frequency the inertia of the sample may be neglected, and the analysis proceeds as though the sample were a single damped spring with frequency-dependent spring and damping constants. If this dimension is much larger than the wave length, the strain will be damped out, and it becomes possible to measure the attenuation of waves in the material. If the two quantities are of the same order of magnitude standing waves can be set up. Methods corresponding to all three of these cases are being used experimentally for dynamic measurements on rubber. By far the most common, those applicable at low frequencies (up to a few kilocycles per second) are the first type where wave motion in the sample can be neglected. Usually the rubber sample is used to provide the restoring force in a mass-spring system which is set into either forced or free vibration. In the case of forced vibration the “spring constant” or G’ is equal to a geometric constant depending on the dimensions of the sample, times the mass, times the square of the radian frequency of resonance. Strictly speaking the frequency a t which the velocity of motion of the mass (for a constant force) is a maximum should be used, but the use of an amplitude maximum rather than a velocity maximum leads to only a small error (which can be corrected for if desired) if the damping is not extreme. The methods used by Kosten and Zwikker ($6),Gehman (16), n’aunton and Waring (37), Moyal and Fletcher (%), Oberto and Palandri (41), Gehman, Woodford, and Stambaugh (17, 51), and Dillon, Prettyman, and Hall (11)are examples of this type. In general such devices, or a t least the more recent ones, have employed electromagnetic drives which are convenient to use and permit the design of a relatively compact apparatus. A slightly different type of resonance vibrator, which still must be considered under this classification, however, has been developed recently by Rorden and Grieco (44) and described by Hopkins ( l d ) . They measure the change in resonance frequency and breadth of the resonance curve of an electrically driven tuning fork caused by the introduction of rubber samples strained in shear by the motion of the fork, covering a frequency range of from 100 to 14,000 cycles per second. Nonresonance forced vibrators have been applied to rubberlike
INDUSTRIAL AND ENGINEERING CHEMISTRY
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-ELASTOMERS-Dynamic materials somewhat less frequently. I n such methods the real and imaginary parts of the dynamic modulus can be measured directly in terms of the definitions given here by a measurement of the force and velocity (or Bmplitude) and the phase angle between them. Electromagnetic drives are usually employed, and electromagnetic pickups can conveniently be used to measure the velocity (SI),although condenser pickups or strain gages could be employed. The frequencies covered by nonresonance methods may be varied continuously, and very small strains (10-6 or less) can be employed, thus practically ensuring that the material will respond in a linear manner. They are thus considerably more flexible than rfisonance vibrators. Blizard (6) has described measurements with a device of this type, using a special generator, over the frequency range of 0.0125 t o 750 cycles per second. A nonresonance method employing a mechanical drive and optical recording, but in which only the real part of the modulus was measured, was used by Aleksandrov and Lazurkin (1). A somewhat different nonresonance method was developed by Roelig (43)and has been used fairly extensively. I n this method the force applied to the mass in the mass-spring system is not measured, but rather the force transmitted through the sample. Roelig uses an optical method to trace out a stress-strain ellipse which can be analyzed to yield G' and G". The advantage of this method over those above is that the inertia of the apparatus does not enter into the equations for calculating the modulus, at least u p to frequencies of the order of 100 cycles per second. The machine as designed by Roelig uses rotating eccentric weights to apply the driving force. This type of driving system, plus the fact that the machine is designed to accept large test pieces, leads to a rather large piece of apparatus. Most of the very low frequency measurements (10 cycles per second and below) have been made utilizing free vibration devices (7, 19, 26, 55, 56). The directly measured quantities are the period of oscillation and the logarithmic decrement. The conversion from these quantities to the dynamic modulus as used here is given in the appendix. Since the strain ik not sinusoidal, but a damped sinusoid, there is, strictly speaking, a n approximation involved in calculating an equivalent dynamic modulus from these results, but the few measurements available t o date by which various methods can be compared indicate that any error involved is actually quite small. Probably the most familiar example of this class of measurements is the Yerzley oscillograph in which a rubber sample strained in compression (modifications have been used employing shear) supports an oscillating beam carrying adjustable weights and a pen which makes a record of the decaying oscillations oa a rotating drum. Many other devices employing free vibrations in compression, shear, and torsion have been developed, and a great deal of information has been obtained with them. They share with the resonance vibrators the disadvantage of requiring that a change in either the sample size or the inertia member must be made to change the frequency of measurement. The most common example of a standing wave method for use with solid rubbers is the vibrating reed technique. In this method a thin reed of material is vibrated either as a cantilever beam (38) or longitudinally (47). The amplitude of vibration of the free end of the beam may be observed directly with the aid of a low power microscope, or the modulated intensity of a light beam partially interrupted by the vibrating reed may be measured with a photocell. The frequency and band width of the amplitude resonance may be measured, the amplitude of free vibrations may be followed as they decay, or the amplitude-force ellipse may be displayed on an oscilloscope screen and a quantity proportional to G" calculated from the area of this ellipse divided by the square of the strain amplitude. Methods in which the decay with distance of the wave motion
April 1952
Propertie-
in a sample is measured are the longitudinal wave technique (4, 20, 38),which has been applied more extensively to textile fibers, and the compressional wave technique (6'8,40, 46). Another method which has been used at high frequencies (35) involves measuring the loading introduced on an electromechanical transducer by the presence of a sample. This type of measurement has also been used by Smith, Ferry, and Schremp (60) at low frequencies for measurements on polymer solutions. COMPARISON OF nqETHODS
The selection of the best of these several methods for any particular application or the comparison of the results obtained by these various methods is somewhat complicated. The first factor is a consideration of the frequency range to be covered, as indicated in the previous discussion. Changes in the design of the apparatus or size of the sample can allow a certain extension of the range of applicability, but even so there is a limitation of the order of three decades of frequency on the applicability of any one type of measurement.
1 ' " " ' ' 1 " 1 " ' ' " '
Figure 4.
Dynamic Response of Four-Parameter Model
Most of the measurements made a t low frequencies use either free vibration or forced resonance methods. Certain approximations are inherent in the analysis of such results as usually carried out. In the free vibration method the amplitude is constantly decreasing during the course of a measurement, so the dependence of modulus on amplitude cannot be checked too easily. In the resonance method, if resonance is detected by changing the frequency as is quite common, the assumption must be made that the components of the modulus are frequencyindependent over a limited frequency range. The experimental comparisons so far available between resonance and nonresonance vibrations seem to indicate that these approximations are valid within the precision of most of the methods now in use. This leaves, then, as the primary objections to the forced resonance method, the fact that it is relatively inconvenient to make measurements a t a large number of fre-
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-ELASTOMERS-Dynamic
Properties-
quencies within the range of the apparatus and also that this total range itself is quite limited; in practice most such devices are operated in the range 30 to 150 cycles per second, and many are used a t 60 cycles per second only. The free damped vibration measurements are quite convenient to set up and operate, but the analysis of their results is not too precise with the instrumentation ordinarily used. They are used most widely (Yerzley oscillograph) as a fairly routine method for comparing various stocks, but the instrumentation can be refined, and this method & the most convenient for measurements a t frequencies below the range of ordinary electronic equipment. Forced, nonresonance methods have the advantage, aside from yielding a direct measurement of the complex modulus, of being extremely flexible in the frequency and amplitude of strain employed. With electromagnetic driving and pickup units the sensitivity can be made very high. Such instruments do, of course, have certain inherent drawbacks. Although their total frequency range can be extended to perhaps 1500 or 2000 cycles per second, they require extraordinarily careful design and construction in order to avoid extraneous resonances which can lead to false results a t frequencies as low as 200 cycles per second unless care is taken. This work, and the assembly of the associated electronic equipment, require a good deal of time, and to ensure reliable results experienced operators are required. I n view of these factors and the rather high costs of such equipment, they have been used so far only as research instruments. The vibrating reed technique, in its simpler form where the amplitude is measured with the aid of a low powered microscope, requires relatively simple and inexpensive equipment and appears to be a rather convenient experimental technique. With a given sample, measurements can be obtained a t only one frequency for each temperature, and the determination of absolute modulus values is difficult because of the uncertainties in measuring the dimensions of the reed accurately. The measurements employed at higher frequencies all require the use of somewhat involved electronic circuits and require personnel with some electronic experience for their successful operation. Such equipment has been principally in the class of specially built research equipment up to the present time. TYPES OF STRAIN AND AMPLITUDE EPPECTS
Most of the low frequency test methods can be designed to employ any of the ordinary types of strain: shear (including torsion), compression' or, for a vulcanized material, extension. Of these, shear or extension appear8 to be the most satisfactory if an absolute modulus value is desired, as the effects of sample shape and size seem to have a great effect on the results of compression measurements. The use of a shape factor defined by the ratio area of one bonded surface/area of free surface ( 4 5 ) has been used with considerable success in static measurements but has not been tested as completely for dynamic measurements. Dynamic measurements in compression seem to yield values that depend not only on the shape factor but on the amplitude of strain and the static strain superposed during the test. It has been assumed throughout this discussion that dynamic moduli are independent of amplitude, but this does not hold true for strains greater than 1%, and there seems to be an effect present even below 1% strain. For both shear and compression the dependence is such that G' and G" decrease with increasing amplitude, and i t is apparently not a temperature effect ( 1 6 , 32, 4 1 ) . The effect is very pronounced with tread stocks but can be observed with gum stocks and even with pure polymer. No really satisfactory explanation seems to exist as yet, and a good deal more careful experimental work on this effect is needed. The high frequency measurements that involve the propagation of compressional waves present an additional complica-
700
tion, since the modulus which governs such waves is given by K* 4/3 G*, where K* is a complex bulk modulus. The calculation of the components of the shear modulus from such measurements, even assuming that the imaginary part of K* is zero, requires some independent determination of K', the real part of K*. Nolle ( 4 0 ) and Guth (25) have made the assumption that the bulk modulus of their polymers could be taken as equal to that of water. While they obtained values which fitted a reasonable extrapolation of lower frequency measurements of the shear modulus by the use of this estimate in connection with data taken a t about room temperature, there seems to be no real basis for expecting this assumption to hold true in general. At lower frequencies, up to a t least several kilocycles, there is evidence based on comparison of measured shear and Young's moduli (Young's moduli measured by extension methods) that Poisson's ratio is very close to one half as it is in the static case
+
(31). PRECISION OF MEASUREMENTS
The precision and accuracy of all these methods is a subject on which the information now available is not too satisfactory. The variability between duplicate results on the same sample seems to be of the order of *5oJ, for the best resonance and nonresonance forced vibration measurements and of the order of f 1 0 to 15% for some of the other methods which involve somewhat less precise observations. The variability between results obtained on duplicate samples ranges from about &5% to *20%. This is not a particularly satisfactory state of affairs, since the differences between many types of stocks can often fall within this range. A number of experimenters are now working on the problem of improving such measurements, and the outlook for attaining a precision of 1 or 2% seems, in some cases, rather good. One factor which is extremely important and which has received far too little attention in most of the work to date is that of close temperature control. Based on the polyisobutylene results from the National Bureau of Standards' cooperative program, it appears that for this material a t 25' C. and 60 cycles per second, the temperature coefficient of G' is about 4% per ' C., whereas a t 25' C. and 4000 cycles the coefficient is about five times. It should be noted, that for many rubbers (natural for instance) the variation of G' at room temperature and 60 cycles is much less, since in this case the measurements are on a relatively Rat part of the curve. As a general rule, then, it seems safe to say that in certain frequency and temperature regions, control to 0.1 or 0.2" C. should be sought. Also, temperatures within the samples should be checked by means of thermocouples, although this should probably be done in a dummy experiment the results of which are not used, since the presence of a thermocouple would very likely affect the stress concentration in some samples. NONSINUSOIDAL STRESS
So far the discussion has been limited to methods in which steady-state sinusoidal stresses were employed and from which both the real and imaginary parts of a modulus could be calculated. There have been used, and are used today, a number of methods which do not fulfill one or both of these conditions, but which should still be classified as dynamic methods. Some of the earliest measurements of "dynamic" properties were made with various types of flexometers (IO). These devices, in general, apply alternating stress or strain to a rubber sample but measure only the temperature rise as a result of the vibration. Although such a measurement can in principle be made the basis for a determination of G", since temperature rise is an indication of energy loss, the complications involved io calculating the actual energy loss are such that in practice this is not done,
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
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No. 4
-ELASTOMERS-Dynamic and they are used only to compare various stocks. The various devices are made to operate a t either constant amplitude or constant force, and, as Dillon and Gehman and many others have pointed out, care must be used in the interpretation of such results since actual apfilications of rubber articles may involve neither constant amplitude or force but some rather complicated intermediate case. Such devices seem definitely unsuited to any fundamental investigation. Mooney and Black (34) have developed an instrument in which the energy loss per cycle is measured for samples undergoing approximately sinusoidal, forced vibration. The various pendulum and rebound methods, in which the ratio of recovered to initial energy is measured by observing (essentially) the height of fall and recovery of a weight impinging on a sample, have been used also for many years with apparently rather satisfactory results in the comparison of different stocks. Such methods yield results from which the loss tangent can be calculated directly, but the frequency with which each measurement should be associated is somewhat difficult to determine accurately, and this frequency will of course depend on the modulus of the material under test. It is interesting to note, however, that the results of Powell and Aston using the Dunlop Pendulum and the Tripsometer on the “standard” polyisobutylene already referred to, agree surprisingly well with other more direct measurements. This result is termed surprising not only because of the difficulties in deciding on the frequency to associate with the measurements but also because such measurements approximate only a half cycle of sinusoidal vibration. -4t first glance one might be tempted t o assume that a steady,state condition would not be set up in such an experiment, and a total lack of agreement with steady-state measurements would hardly have been surprising. Finally, some rather recent methods which involved steadystate, nonsinusoidal vibrations should be mentioned. Breazeale (6) has developed an apparatus for measurements on textile fibers (which can also be used with some success on rubbers) employing triangular or saw-tooth longitudinal vibrations. Tolstoy and Feofilov (64) have discussed the development of a method employing square-wave pulses and observing the shape of the resulting strain. While such methods have not been fully evaluated (so far as this writer is aware) they offer interesting possibilities for a t least qualitative evaluations of dynamic properties. Since a square wave can be thought of, thrpugh a Fourier analysis, as representing the simultaneous application of sinusoidal waves consisting of the fundamental and all its harmonics, there seems a t least a speculative possibility here of obtaining some sort of modulus function which would be an integrated result of the dynamic moduli over the whole frequency range on the basis of a single measurement. No doubt such results would be difficult t o compare with moduli based on sinusoidal stress measurements, but the possibilities are intriguing.
APPENDIX RELATION OF DYNAMIC MODULUS TO OPERATOR EQUATION AND TO EXPERIMENTAL RESULTS
There follows a derivation relating the fundamental ( f r e quency-independent) parameters in the linear differential equation for a viscoelastic material to the quantities measured in free and forced vibration (where wave motion in the sample can be neglected) (49, and a summary of some of the other methods that have been used to express dynamic results, The operator equation (1) may be written as [SI
P ( D ) u = Q ( D )t where P =
2 n - 0
April 1952
m
anDn, Q =
&Dm m = O
Propertie-
For the forced vibration case, express the time dependence as exp ( i d )and there results
P(iw) u
= Q(iw)e
or U/€
= &(iw)/P(iw)
(5)
which is, by definition, the complex modulus. The ratio Q / P in Equation 5 contains both real and imaginary parts which are functions of the frequency and of the material constants, an, &,,. Therefore,
G’ = Re [ Q ( i w ) / P ( i w ) ] G” = Im [ & ( i w ) / P ( i w ) ]
(6)
For sinusoidal stress and strain only, the use of a simple differential e uation containing frequency-dependent “constants” in place of%quation 4 can be justified. If 7’ = G ” / w , Equation 6 shows that, for any given frequency, Equation 4 may be written u = G‘
+ 7’ De
e
(7)
which is the starting point for the analysis of dynamic experiments generally used. For free vibration experiments, the stress is due to an inertial force and may be expressed as - ( M / a ) D % ,and the strain must be written €0 exp ( i w - X ) t , the factor, exp ( - A t ) , expressing the decay of strain with time. Substituting these quantities into Equation 4 there results
- ( M / u ) (iw
- X)2P ( i w - X)e
= Q(iw
- X)e
(8)
Our aim here is to find some way of calculating G‘ and G” as defined in Equation 6 from the experimental results of a free vibration experiment in which W , or the period, and X, the attenuation of the vibration with time, are measured, To do this a first ap roximation may be made by assuming that X is so much smaser than w that Equation 8 may be written
+ 2iwX) = & ( i w ) / P ( i w ) (9) The right-hand side of Equation 9 is simply G’ + iG”from Equation 6. Thus ( M / a ) (w2
G’
= w2M/a
G” = 2wXM/a
(10)
In Equations $, 9, and 10 a is a constant depending on the sample dimensions such that: force/displacement = a stress/strain. A number of different methods of expressing experimental results have been used by workers in this field, all of which can be related (exactly or t o a close approximation) if sinusoidal stress and strain are assumed, For forced vibrations, the energy loss per cycle, per unit volume, is w = S c d r = r e o 2 G” (11) It should be noted that in evaluating this integral the actual
stress and strain-that is, the real parts of the complex expressions-must be used. The specific damping capacity is defined as the ratio of the energy loss, w, to the maximum energy stored In the sample during a cycle, Y $ = w / v = 2 a tan 6
c
(12)
Roelig defines a specific damping capacity in a slightly different manner, as the area of his stress-strain ellipse (w) divided by ‘/e the product of maximum stress and maximum strain (which he also measures directly from his ellipse). This gives a specific d p p i n g capacity equal to:
In his vibrating reed measurements, Sack uses the quantity a measure of energy loss
W / E O ~as
($1
W/E$
= wG”
(14)
The more common analysis of such measurements, wherein the frequency of amplitude resonance, YO, and the half-power width of the resonance curve, AV (the difference between frequencies a t
INDUSTRIAL A N D ENGINEERING CHEMISTRY
701
ELASTOMERS-Dynamic
Properties-
which the amplitude is 1/.\/2 of the maximum), are measured, yields A v / v ~= t a n 6 (15) The results of free damped vibrations are often expressed in terms of the period of vibration, T = 2 K / W , and the logarithmic decrement, which is defined as the logarithm of the ratio of the amplitudes of two successive vibrations.
Dynamic resilience is defined as the ratio of energy stored in two successive cycles of free vibrations. Since this energy is proportional t o the displacement squared, the dynamic resilience 1s
For pendulum or rebound tests only half a cycle of vibration is involved, so the resilience as measured in such tests is the square root of t h a t defined by Equation 17
R,
=
exp ( -
tan 6)
(18) R, is also generally used to express the results obtained with the Yerzley oscillograph. R
LITERATURE CITED
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ACCEPTE~D January 31, 1952.
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