Dynamic properties of rubber.

Theory of dynamic properties

Sinusoidal deformation

The initial study of viscoelasticity concerned itself with the response of a polymer to a non-cyclic deformation imposed at either a constant strain or a constant strain rate. It was found that the elastic response is proportional to the strain level and the viscous response is proportional to the strain rate. Rubber products, however, function under conditions of cyclically varying stress or strain rates and amplitudes. To correlate the viscoelastic properties of a rubber to its performance in a product, these properties must be measured under dynamic conditions. These dynamic properties are frequently determined by measuring the response of a rubber compound to a sinusoidally varying strain. In this manner, both the strain amplitude and the strain rate vary during a complete cycle. Under these vibratory conditions, a rubber absorbs energy which is partially stored, as in a spring; and partially dissiptated in overcoming internal friction, as in a dashpot. The ratio of the energy dissipated to the energy stored is a function of the viscoelastic properties of the rubber, the temperature of the rubber, the degree of deformation and the rate of deformation. Although most rubber products do not experience a true sinusoidal deformation, it is still possible under conditions of low deformation, to relate the response of a rubber to a sinusoidal vs. a non-sinusoidal deformation. The difficulty arises when trying to relate the response at low sinusoidal amplitude to the response at high amplitudes. Here the relationship is no longer linear and considerable empirical data are necessary to establish a good correlation.

Dynamic properties are also useful in determining the structure of polymers. They can be used to determine glass transition temperatures, crystallinity, degree of crosslinking and phase separations. In rubber compounding, dynamic properties are extremely useful in correlating the effect of elastomers, fillers, oils, vulcanization and processing, on final performance properties. They are also useful for determining the effects of temperature, degree of deformation and cyclic frequency on a rubber's mechanical properties.

In a simple apparatus for measuring dynamic properties a sinusoidally varying strain is applied via a motor driven eccentric. The resultant force is measured at the opposite end of the sample with a dynamometer ring or load cell. The angular difference between the input strain and the resultant stress is usually measured by mechanical or electronic methods. A graph of the sinusoidal strain and resultant stress, both plotted as a function of time or angle, is shown in figure 13. The measured maximum stress amplitude precedes the maximum strain amplitude by the phase angle [delta]. The stress amplitude ([F.sub.0]) is composed of contributions from both the elastic stress ([F.sub.1]) and the viscous stress ([F.sub.2]). The amount contributed by each is a function of the phase angle. Following Hooke's Law, the resultant stress due to the elastic portion of the rubber is in phase with and proportional to the strain. When the imposed strain reaches a peak value, the resultant elastic stress will also reach a peak value. The resultant stress due to the viscous portion of the rubber is governed by Newton's law and is 90[degrees] out of phase with the imposed strain. When the strain is at a maximum value, the strain rate (slope of the strain curve) is zero. Consequently, the resultant viscous stress is zero. At zero strain, the strain rate is at a maximum and the resultant viscous stress is at a peak value.

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The only values measured are the stress amplitude ([F.sub.0]) and the phase angle [delta]. The complex modulus (E*) is calculated by dividing the resultant maximum stress amplitude ([F.sub.0]) by the maximum imposed strain amplitude. Both the maximum elastic stress amplitude ([F.sub.1]) and the maximum viscous stress amplitude ([F.sub.2]) are calculated from the measured stress amplitude ([F.sub.0]) and the phase angle [delta] using simple trigonometric functions. Dividing these stress values by the strain gives the elastic or storage modulus (E') and the loss modulus (E").

E* = complex modulus = measured stress amplitude/strain amplitude

E' = elastic modulus = elastic stress amplitude/strain amplitude

E" = loss modulus = viscous stress amplitude/strain amplitude

E' = E* cos [delta]

E" = E* sine [delta] from the Pythagorean Theorem,

[(E*).sup.2] = [(E').sup.2] + [(E").sup.2]

and E* = [[[(E').sup.2] + [(E").sup.2]].sup.1/2]

also tan [delta] = E"/E'

The tan [delta] value, being the ratio of the viscous response to the elastic response is a measurement of hysteresis. If the experiment is run in shear rather than compression, the letter G is substituted for E.

The same type of experiment can also be run by applying a sinusoidally varying stress and measuring the resultant strain and phase angle [delta]. From this data, the following values are determined:

D* = complex compliance = measured strain amplitude/stress amplitude

D' = elastic compliance = elastic strain amplitude/stress amplitude

D" = loss compliance = viscous strain amplitude/stress amplitude

D' = D* cos [delta]

D" = D* sine [delta]

D* = [[[(D').sup.2] + [(D").sup.2]].sup.1/2]

Tan = D"/D'

If the experiment is run in shear rather than compression, the letter J is substituted for D.

The following relationships also exist between the compliance values and the modulus values:

D* = 1/E* or J* = 1/G* However:

D' [not equeal to] 1/E or D" [not equeal to] 1/E" but rather,

D' = E'/[(E').sup.2] + [(E").sup.2] = E'/[(E*).sup.2] and,

D" = E"/[(E').sup.2] + [(E").sup.2] = E"/[(E*).sup.2]

Hysteresis

Hysteresis is a measure of the energy dissipated by a rubber during deformation. When placing a sinusoidal strain on a sample, a portion of the energy input to the rubber during increasing strain is not returned during decreasing strain. The amount of energy dissipated is a function of the phase angle. This can be seen by taking the stress strain data from the sinusoidal curve and plotting it on an x y coordinate. The result is the hysteresis loop shown in figure 14. The stress strain curve for an increasing sinusoidal strain is the curve EFGHA. The area between this curve and the x axis is the total input energy. The corresponding stress strain curve for a decreasing sinusoidal strain is the curve ABCDE. The area between this curve and the x axis is the total energy returned. The area of the elliptical loop is the energy dissipated, or the hysteresis. This hysteresis loop is generated at a constant angular velocity, as is the sinusoidal curve. The strain rate therefore reaches a peak value at zero strain and becomes zero at the maximum and minimum strain. The effect of varying strains on the elastic stress and the viscous stress can be seen by drawing a straight line between the maximum and minimum strains (points A and E). The vertical distance between line AE and the x axis is the elastic stress. It is directly proportional to the strain. The vertical distance between line AE and the curve EFGHA is the viscous stress. It increases with increasing strain rate. The elastic stress reaches a maximum at the maximum strain (point A); and the viscous stress reaches a maximum at zero strain (point G). The phase angle ([delta]) is the angle between the maximum stress and the maximum strain.

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The area of the hysteresis loop, or the total energy lost per cycle, is given by the equation (ref. 16):

energy loss = [pi] FS sine [delta] where: F = maximum stress; S = maximum strain; sine [delta] = E"/E*. The hysteresis should therefore reach a maximum value at 90[degrees] phase angle and become zero at a 0[degrees] phase angle. The hysteresis loops generated by a 100% elastic and a 100% viscous material are shown in figures 15 and 16, respectively. With the 100% elastic material, the strain and resultant stress are in phase; and the phase angle is zero. The area of the hysteresis loop is also zero (sine 0[degrees] = 0). With the 100% viscous material, the strain and resultant stress are 90[degrees] out of phase. The area of the hysteresis loop is now at a maximum value, which in this case, happens to be a circle.

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As stated earlier, a rubber can be sinusoidally deformed in either the strain cycling mode or the stress cycling mode. Compound modifications can have different effects on hysteresis properties depending on the cycling mode. It is therefore necessary to determine hysteresis properties under conditions that relate to the type of deformation a rubber experiences during a particular service. Energy losses in either cycling mode can be calculated from dynamic properties by using the equation for the area of the hysteresis loop. In the constant strain cycling mode, the energy loss is calculated as follows:

energy loss = [pi] FS sine [delta]

              = [pi] SS E* sine [delta] (F = SE*)
              = [pi] [S.sup.2]E* (E"/E*) (sine [delta] = E"/E*)
              = [pi] [S.sup.2]E'
              [alpha]E" ([S.sup.2] is a constant)

Therefore under constant strain cycling conditions the hysteresis is directly proportional to the loss modulus.

The energy loss can similarly be calculated for the constant stress cycling mode:

energy loss = [pi] FS sine [delta]

              = [pi] FF D* sine [delta] (F = SD*)
              = [pi] [F.sup.2] D* (D"/D*) (sine [delta] = D"/D*)
              = [pi] [F.sup.2] D"
              = [pi] [F.sup.2] E"/[(E*).sup.2]
              [alpha] D" or E"/[(E*).sup.2] ([F.sup.2] is a constant)

Therefore in the constant stress cycling mode, the hysteresis is directly proportional to the loss compliance D", or E"/[(E*).sup.2].

Tan [delta] was previously defined as the ratio of the loss modulus to the elastic modulus (E"/E'). This is a dimensionless quantity that is proportional to the energy loss generated under conditions of constant energy input. The stress and strain cycling energy losses are both related to tan [delta]. For constant strain cycling:

energy loss = E" = E' tan [delta], (tan [delta] = E"/E') For constant stress cycling,

energy loss = E"/[(E*).sup.2] at typical tan [delta] values exhibited by rubber,

E* [alpha] E' Therefore,

energy loss [alpha] E"/[(E').sup.2]

[alpha] (tan [delta])/E' (tan [delta] = E"/E')

Both the stress and strain cycling energy losses are directly related to tan [delta] and oppositely related to the elastic or complex modulus. Therefore any compounding modification that changes a rubber's tan [delta] or modulus will also affect its hysteresis. The effect of modulus depends on the cycling mode.

The effect of varying the tan [delta] while maintaining a constant modulus is shown in figure 17. In going from curve A to curve B, tan was reduced by a factor of two, while the modulus was kept constant. Since the modulus is constant, the energy loss will show the same change whether in the stress or strain cycling mode. Curve B therefore is representative of both cycling modes. Its viscous stress (F2) is reduced by a factor of two; while its elastic and total stresses remain fairly constant. In going from A to B, the area of the hysteresis loop will decrease by a factor of about two. A different effect is found when varying the modulus while maintaining a constant tan [delta] (figure 18). The modulus of the compound in curves B and C is 1.5 times the modulus of the compound in curve A. Curve B was generated in the same stress cycling mode as curve A. Curve C was generated in the same strain cycling mode as curve A. In all three cases, [F.sub.2]/[F.sub.1] ([approximate] tan [delta]) remains constant. In the strain cycling mode, the area of the hysteresis loop is proportional to E' tan [delta] and therefore increases with increased modulus. For stress cycling, the area of the hysteresis loop is proportional to (tan [delta])/E' and therefore decreases with increased modulus. Both of these effects can be seen by the area changes of the hysteresis loops in figure 18. The effect of modulus on hysteresis generated under stress or strain cycling modes is extremely important and should be kept in mind when analyzing dynamic property data.

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Temperature-frequency superposition

The testing temperature and frequency must be specified when defining the dynamic properties of a rubber. The effect of these variables is seen in figure 19. At high temperatures or in the rubbery region, the complex, elastic and loss moduli are all low. Here, there is little frictional resistance impeding molecular rotational motion and the polymer is highly elastic. As the temperature is reduced to the transition region, the frictional resistance to rotational motion increases. This increases the loss modulus until it reaches a peak at the Tg. This frictional resistance also contributes to the force necessary to deform the polymer; and the complex and elastic moduli both increase with decreasing temperature. Below the Tg the frictional resistance increases to the point where rotational motion is very difficult and there is much bending and stretching of chemical bond angles and lengths. Since this type of motion is highly elastic, the loss modulus decreases below the Tg. Once all the motion is in the chemical bonds, the complex and elastic modulus plateau out at a value approximately 1,000 times greater than in the rubbery region.

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The effect of the frictional factor is to increase the relaxation time. Relaxation time is defined as the time necessary for a given stress to overcome the resistance to rotational motion. Since the frictional resistance increases with decreasing temperature, the relaxation time also increases. When subjected to a sinusoidal deformation of a given frequency, a rubber has one half the time of a complete cycle to relax (only one half of cycle is in compression). Above the Tg the frictional resistance is small and the relaxation time is much less than the time of a half cycle. Very little energy is therefore dissipated in overcoming the frictional resistance and the loss modulus is small. At the Tg, the relaxation time equals one half the time of a complete cycle and the total energy absorbed is at a maximum. Below the Tg, the relaxation time continues to increase. Because it is larger than the time of a half cycle, complete relaxation does not occur and the energy absorbed decreases.

The exact same effect can be obtained by holding the temperature constant and varying the frequency. As the frequency is increased, the time of a half cycle decreases. Once the time of a half cycle equals the relaxation time, the total energy is again at a maximum. At even higher frequencies, complete relaxation does not occur. Again the energy absorbed decreases. Increasing the frequency therefore has the same effect as decreasing the temperature. The same techniques described earlier to superposition temperature and time on a stress relaxation curve can be utilized to superposition temperature and frequency effects on dynamic properties. The elastic modulus curves in figure 20 were generated at a series of different temperatures and frequencies. By shifting each of the individual curves a given amount on the frequency axis, a master curve covering more than ten decades of frequency can be constructed. Usually about a 5[degrees]C to 10[degrees]C change in temperature is equivalent to a one decade change in frequency. Since most testing methods cover only a narrow range of frequencies, the temperature frequency superposition technique is extremely useful for calculating dynamic properties over a wide frequency range.

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All polymer systems have a wide distribution of relaxation times. The relaxation time of a total system at a particular temperature and frequency is the average of all the relaxation times of the polymer molecules. The superposition theory, however, assumes that all the different relaxation times have the same temperature dependence. If not used properly, it can therefore become prone to some significant errors. The following limitations are very helpful in using the technique:

* The polymer must behave in an amorphous manner.

* It should not be used below the Tg; where deformation involves bond bending and stretching.

* There must be no effects of crystallinity.

The addition of reinforcing fillers like carbon black can also introduce errors, so care must be exercised when using the superposition technique in carbon black filled rubbers.

Reference

16. A.G. Bushwell, E. Gee and E.R. Thornley, Journal of the IRI, 1, p. 43 (1967).