Dynamic properties of rubber.

Mechanical models

Spring and dashpots are frequently used to make theoretical models which illustrate the interaction of the elastic and viscous components of rubber. The springs and dashpots can be combined in series or in parallel, representing the Maxwell or Voigt elements respectively (figure 2). The stress strain curves in figure 3 illustrate the effect of varying strains and strain rates on four simple models (ref. 3). The curves were calculated at two different strain rates; where strain rate [K.sub.2] is two times strain rate [K.sub.1]. The springs have moduli (E) of [10.sup.8] dynes/[cm.sup.2], and the dashpots have viscosities ([eta]) of [10.sup.8] poise. In curve A, which was calculated from a single spring, the stress and strain are related by the Hookean equation:

[sigma] = E[epsilon] where: [sigma] = stress; E = modulus; [epsilon] = strain

Since the stress is proportional to the strain and independent of the strain rate, a single straight line represents [K.sub.1] and K[sub.2]. In curve B, which is a single dashpot, stress and strain are related by the Newtonian equation:

[sigma] = K[eta] where: [sigma] = stress; K = strain rate; [eta] = viscosity

Since the stress is proportional to the strain rate but independent of the actual strain, two horizontal lines are obtained, representing K[sub.1], and K[sub.2]. In curve C, for the Voigt element, the spring and dashpot being connected in parallel are both forced to move at a constant strain rate. The resultant stress is the sum of the two, or:

[sigma] = K[eta] + E[epsilon]

In this case, the contribution from the dashpot immediately jumps to a constant value while the contribution from the spring increases with increased strain. When the strain rate is doubled, the initial constant value doubles but the slope remains constant. In curve D, for the Maxwell element, the stress and strain are related by the equation:

[sigma] = K[eta] ([1-e.sup.[-E[epsilon]K[eta])

At low strain levels, all the stress is generated by the stretching of the spring. Different strain rates have no effect and the curves are asymptotic at low strains. As the strain increases, the strain rate of the dashpot also increases; and the dashpot increases its contribution to the total stress. Eventually, the strain and resultant stress of the spring reach constant values. At this point, all movement is in the dashpot. Since its strain rate is now a constant ([K.sub.1] or [K.sub.2]), the slope of the curve becomes zero. Of course the curve with the larger strain rate ([K.sub.2]) reaches a higher stress before leveling off. This Maxwell model gives the most realistic stress strain curves. However, in this particular model, the stress is independent of strain rate over a range of low strains. In real polymer systems this is not quite true.

The curves in figure 4 show the response of these models to the application of a sudden stress and the resultant creep; and the application of a sudden strain and the resultant stress relaxation. In the creep experiments, a given weight or stress is placed on the element and the strain plotted as a function of time. With the spring (curve 4A), the addition of the stress immediately increases the strain to a constant value. At a later time, when the stress is removed, it immediately returns to zero. The addition of a stress to the dashpot (curve 4B) causes the strain to increase at a constant rate until the dashpot is at its maximum strain level. The removal of the stress has no effect on the strain. The Maxwell element (curve 4C) has two components. First the spring is stretched to a constant strain and then the dashpot moves at a constant strain rate. When the weight is removed, the spring immediately retracts while the dashpot remains stationary. The response of rubber is best represented by the Voigt element in curve 4D. Here the dashpot acts as a viscous resistance to the instantaneous movement of the spring. The equation for the deformation curve is:

[epsilon] = [sigma]/E ([1-e.sup.-t/[tau]) where: t = time; [tau] = relaxation time; [epsilon] = the strain at any given time

When the weight is removed, the equation for the element returning to its original shape is:

[epsilon] = [[epsilon].sub.o e.sup.-t/[tau]] where: [epsilon = initial strain; c = strain at any given time

In both cases, the strain time relationship is in the form of an exponential curve.

A different situation is found when a sudden strain is applied and the stress relaxation mea-sured as a function of time. Here, the dashpot (curve 4F) and the Voigt element (curve 4H) are not applicable since an instant strain cannot be applied to either of them. With the spring (curve 4E), the stress immediately increases to a constant value and immediately returns to zero when the strain is removed. With the Maxwell element (curve 4G), the spring responds immediately to the strain, but the movement of the dashpot allows the spring to relax. The stress then decays exponentially as a function of time according to the equation:

[sigma] = [[sigma].sub.o e.sup.-t/[tau]]

From these simple models it is evident that the Voigt element best approximates creep properties, while the Maxwell element best approximates stress relaxation properties.

Both of these elements can be combined to give a generalized mechanical model which describes both creep and stress relaxation (ref. 5). This model is shown in figure 5, along with the strain time curve for a typical creep experiment. The curve is interpreted by the generalized mechanical model. Although much over simplified, this model is a good representation of the viscoelastic properties of rubber. Rubber actually consists of an infinite number of such models, with a wide spectrum of spring constants and viscosities. Many of the components also deviate from both Hookean and Newtonian behavior. Nevertheless, this model gives a good groundwork for describing viscoelastic behavior.

Time-temperature superposition 

Temperature has a large effect on the viscoelastic properties of polymers. A typical modulus vs. temperature curve for a non crystalline (amorphous) polymer is shown in figure 6 (ref. 6). From the curve, it is readily observed that there are four distinct regions:

* Glassy region;

* Transition region;

* Rubbery region;

* Flow region.

Each region is associated with different molecular motions of the polymer chains. The glassy region exists at low temperatures and is characterized by a high modulus. On the molecular scale, there is not enough energy to allow for rotational or translational motion. The chain segments are essentially frozen in place and the only motion that can take place is due to bending and stretching of chemical bonds. The polymer therefore is highly elastic in this region. Since the only motion allowed is the bending and stretching of chemical bonds, only small strains can be applied to the polymer without fracturing it.

As the temperature is increased, the polymer goes through the glass transition (Tg) region, which extends over a temperature range of about 5[degrees] to 20[degrees] Centigrade. Over this temperature range, the modulus decreases by a factor of about 1,000. On the molecular scale, rotational motion increases and a leathery, behavior is observed. The frictional resistance to this motion, however, is very high and the polymer exhibits highly viscous properties in this region.

Once the temperature is sufficiently high to allow free rotational motion, the rubbery region is reached. Here, the modulus reaches a plateau. Since the frictional resistance to rotation is reduced, the polymer is again highly elastic. However, since the elasticity is associated with the kinetic motion of long chain segments, instead of the bending and stretching of chemical bonds, a much higher elongation is permitted before the rubber fractures. In spite of the low resistance to rotational motion, there is still a high resistance to translational motion. or the complete slippage of one polymer chain past another. This resistance to chain slippage is due to the effect of entanglements (ref. 7). Although of a less permanent nature, these entanglements function as crosslinks. If, for example, a strain is placed on a rubber, the entanglements eventually slip loose, and the resultant stress relaxes to zero. The entanglement network, however, is never really lost. Because of Brownian motion, new entanglements will reform as fast as others slip apart. Eventually, when the temperature is high enough, the entanglements come completely apart, and the polymer begins to flow. Since the effect of the entanglements is more pronounced with increasing molecular weight, the temperature at which flow begins, increases with increasing molecular weight (ref. 8). If the molecular weight is low enough, the rubbery region does not even exist and the polymer flows immediately after going through the Tg. At an even lower molecular weight the Tg is lowered.

In a vulcanized rubber, since the chemical crosslinks are stable through high temperatures, the flow region is absent. The rubbery region therefore extends beyond the normal flow region and there is little change in the modulus up to the temperature at which actual scission of the crosslinks takes place. The absence of the flow region imparts dimensional stability to vulcanized rubber.

Increasing the temperature of an elastomer accelerates the molecular and segmental motion. This causes the reduction in modulus shown in the previous modulus-temperature curve (figure 6). The same type of curve can be generated by measuring the modulus as a function of time (stress/relaxation). Such a stress relaxation curve would require a time interval of over [10.sup.10] units of time. However, by using the time-temperature superposition principle, a complete master curve can be plotted by running a series of stress relaxation