Dynamic properties of rubber.

Editors note -- this is the first installment of a series that will run in alternating months.

Rubber is unique in that its response to a mechanical deformation is quite different from other materials. It is the only known material that can be stretched up to 1,000% elongation and then return to its original length upon release. However, it not only has elastic properties like a metallic spring, but also has energy absorbing properties characteristic of a viscous liquid. It is this combination of properties that gives rubber its uniqueness. These viscoelastic properties allow a rubber product to maintain a constant shape after repeated deformation, while simultaneously absorbing mechanical energy.

In an engine mount, the elastic properties of the rubber store and return to the engine most of the input energy from the high frequency vibrations generated during operation. This reduces the transmission of these vibrations to the passenger compartment. On the other hand, the viscous properties dampen out the low frequency vibrations generated during idling of the engine. These vibrations have trequencies similar to the natural frequency of the system and because of resonance, increase in amplitude unless dampened out. The same is true for other automobile parts that function as vibration dampeners.

In a tire, the elastic properties of the rubber contribute to low heat build-up and improved mileage for the vehicle. At the same time, the viscous properties of the rubber help dissipate noise and shock vibrations; contribute to the frictional forces preventing slippage during cornering; and assist in transferring kinetic energy to the brakes when reducing the speed of the vehicle.

These viscoelastic properties, therefore, have a definite influence on the performance of any rubber product that is subjected to a cyclic deformation. The nature of this effect depends on the amplitude of the applied cyclic stress or strain, the frequency and the temperature.

Theory of viscoelasticity 

Elastic response

Although perfect elasticity can never be fully achieved, a high degree of elasticity is exhibited by metallic springs. If one end of a spring is fixed in a permanent position and a strain placed on the other end, a force is set up in the spring acting in a direction opposite to the strain. According to Hooke's Law, the force or stress is proportional to the strain, or:

F= SX where: F-, force (stress); S = .spring constant (modulus); X = displacement (strain).

In a perfectly elastic spring, the force develops instantaneously and does not relax or decay over a period of time. On the molecular scale, there is bending and stretching of the individual chemical bond angles and lengths of the metallic structure. In the process, the imposed mechanical energy is converted to potential energy and stored until the strain is removed. Once the strain is removed, the bond angles and lengths return to their original positions and the resisting force is reduced to zero. Since no energy is absorbed, the material is 100% elastic and thus behaves ideally. This high elasticity, however, is exhibited only at low strains (<1%). At higher strains, chemical bonds are broken, and the imposed mechanical energy is converted to heat. When the strain is reduced to zero, the force is not zero. Instead, a force in the opposite direction is generated in the spring. Note should be made that the construction of coiled metallic springs allows for a high elongation while only imposing a small flexural strain on individual segments. In any case, once a certain strain is exceeded and chemical bonds are broken, the spring does not return to its original length without the development of an opposing force.

The elasticity of a rubber is very different from the elasticity in a metal since it is still present at high elongations. The source of this elasticity must therefore be very different from the bending and stretching of bonds associated with metallic springs. According to the kinetic theory, rubber's elasticity is derived from the micro-Brownian motion of long chain molecules. These motions are caused by the intermolecular forces exerted by neighboring molecules and resemble the thermal motions of molecules in a liquid. The only difference is that the movement of molecules in a rubber is somewhat restricted by the long length and consequent entanglement of the chains. The movement of the molecules in a liquid and the chains in a rubber both tend toward positions of highest disorientation, or highest entropy. Because of this micro-Brownian movement, rubber chains can assume an extremely large number of conformations (ref. 1). The entropy of the entire system is the average entropy of all the conformations. The number of conformations that a given rubber molecule can assume has been mathematically shown to decrease when the distance between the ends of the molecule increases. When a strain is placed on a rubber, each individual chain is elongated and in the process becomes more oriented. Since the distance between the chain ends increases, the average chain assumes a reduced number of conformations and undergoes a reduction in entropy. In the process, the kinetic motion of the total chain tends toward conformations of highest entropy, and consequently exerts a counter force resisting the imposed strain. This counter force increases with increasing strain. This is analogous to two people swinging a jump rope. Here, the kinetic motion of the rope exerts a force pulling the ends together. Pulling the ends further apart increases the resisting force. Of course, the motion of a rubber chain is much more random in nature. When the strain on the rubber is released, the kinetic motion of the polymer chain pulls the ends together, and it assumes a more random conformation having a higher entropy. The sources of elasticity in a metallic spring and a rubber are therefore different. In the spring, individual bond angles and lengths are bent and stretched, and in the process set up forces resisting the imposed strain. In a rubber, the imposed strain decreases the entropy of the system, while the kinetic motion of the molecules set up an opposing force pulling the system to the higher entropy of the original length.

Viscous response

Viscosity is the internal friction of a liquid or polymer that resists the movement of one molecule past another. The viscosity of a liquid can be determined by measuring its flow rate through a tube; or the force resisting the rotation of a disk in the liquid. According to Newton's law:

F = [eta]dx/dt where: F = force; [eta] = viscosity; dx/dt = velocity (strain rate).

The resisting force, therefore, is proportional to the viscosity and to the imposed velocity or strain rate; but is independent of the actual strain. This can be exemplified by enclosing a liquid in a cylinder with a movable piston at one end and a small orifice at the other end. The force resisting the movement of the piston toward the orifice is proportional to the velocity of the piston times the viscosity of the liquid. At low velocity, the force approaches zero. When increasing the velocity, the resisting force increases and eventually reaches such magnitude that the cylinder fractures. The stress to strain relationship of this cylinder is very different from that of the spring. With the spring, the resultant stress is proportional to the strain and independent of the strain rate; while the stress on the cylinder piston is proportional to the strain rate and independent of the strain. Both the elastic response and the viscous response are time independent. If the strain on the spring is instantly reduced to zero, the stress instantly goes to zero. If the strain rate on the piston is instantly reduced to zero, the stress does the same.

The viscosities of a rubber and liquid are similar in nature. In a rubber, it is caused by the internal friction that resists the conformational changes resulting from an imposed strain. Like the viscosity of a liquid, it also is proportional to the strain rate, but independent of the strain.

Viscoelastic response

Viscoelastic materials have behavior characteristics that are both elastic and viscous. It is this combination that makes rubber unique. Take for example a strip of crosslinked rubber that is stretched to a given elongation (say 10%) and held there for a period of time. If an infinitely small strain rate is used, the force resisting the elongation is almost 100% elastic. The polymer chains reorient to the lower entropy conformation with almost no frictional resistance due to the viscous component. If the sample is held at this elongation for an infinite period of time, the stress remains fairly constant. This is shown in curve A of figure 1. If a higher strain rate is used (curve B), the results are quite different. The force resisting elongation now has two components. The viscous component imparts a frictional resistance to the reorientation of the polymer to the lower entropy conformation. Therefore, when reaching 10% elongation, the polymer is not yet at the low entropy conformation as in example A. However, over a period of time, the frictional resistance is overcome and the polymer reorients to the lower entropy conformation. The resisting force is then 100% elastic. The resultant stress therefore relaxes as a function of time. This time dependency is the exact opposite of what is found in the spring and cylinder examples of pure elastic and pure viscous materials, where the resultant stress is independent of time. If the strain rate of the example is increased even further, (curve C), the viscous component contributes an even higher amount to the resultant stress. If the strain rate is high enough, the frictional resistance reaches such magnitude that no conformational change occurs, and the only change is a bending and stretching of the chemical bonds. Under these conditions the polymer will fracture. This is similar to what happened when a high strain rate was applied to the cylinder. Here the frictional resistance of the liquid was so high that it could not flow out of the orifice, and the cylinder fractured.

Temperature has a large effect on viscoelastic properties. In the cylinder example, an increase in temperature reduces the viscosity or frictional resistance of the liquid. This results in a reduced resisting force at a given strain rate. With the spring, however, a change in temperature has little effect on the spring constant, and the resultant force remains fairly constant. In a viscoelastic material, temperature variations have the same influence on its viscous properties. An increase in temperature reduces the frictional resistance to movement of the polymer chain. Therefore at a given strain rate the viscous component of the resisting force is reduced. The effect of temperature on the elastic component of a rubber is different from that of a spring. The elasticity of a viscoelastic material is derived from the micro-Brownian motion of polymer chains. Based on the kinetic theory, an increase in temperature increases this motion and the resultant elastic force resisting a given strain. This is confirmed in the literature on studies of the effect of temperature on physical properties of gum rubber (ref. 2). Increasing the temperature of a strained rubber gives a proportional increase in the resulting stress.

References

[1.] L.R.G. Treloar, The physics of rubber elasticity, 3rd edition, 1975, Clarendon Press, Oxford, p. 45. [2.] M. Shen, D.A. McQuarrie and J.L. Jackson, J. Appl. Phys., 38, 791 (1967).

Ron Schaefer is a senior research and development chemist at Zeon Chemicals USA and has 29 years experience in the rubber industry. Before joining Zeon he worked for BFGoodrich in the Corporate, Tire and Chemical divisions.