# 11. Behaviour of polymers under cyclic load

A common way to investigate the behaviour of polymers is to subject them to a small cyclic deformation while measuring the resulting stress. With a cyclic deformation we mean a kind of push and pull action that is done (for example) once every second. A graphical representation of such a cyclic deformation, including the resulting stress, is shown in below. The shape of the deformation with time is called sinusoidal.

The cyclic deformation test can be done at varying temperatures. Then it gives us information about the temperatures where phase transitions occur and about the stiffness in the glass phase and rubber phase. We will discuss this test in more detail in this chapter.

The cyclic deformation test can also be performed at a constant temperature for a prolonged time for fatigue testing. The applied deformation is chosen in such a way that the resulting stress is below the yield stress. The test is continued until the specimen breaks. This gives us information about the sensitivity of a polymer for brittle failure. Details about this test will be discussed in another chapter.

Dynamic mechanical analysis

Dynamic mechanical analysis (DMA) is a technique used to study and characterize materials. A sinusoidal deformation is applied and the stress in the material is measured. The temperature of the sample or the cycle time of the deformation can be varied. Often a cycle time of about 1 second is chosen.

DMA allows a quick comparison of material properties between two materials. The technique can be used to determine the linear viscoelastic region of viscoelastic materials.

In the glass phase, where the polymer is rigid, the stress will follow the applied strain per Hook’s law. The ratio of stress and strain is the same as the glass elasticity modulus. The deformation applied to the polymer results in deformation by bending of the chain parts of the molecules.

At the glass transition temperature the rotation time of the Kuhn segments is 1 second. That means that during a cycle of 1 second the stress relaxes from the glass stress to the rubber stress. The nett result is that the time of maximum stress starts to deviate from the time of maximum deformation. The stress-time curve is now shifted from the strain-time curve, as shown below.

In the rubber phase the Kuhn segment rotation time is much less than 1 second. The deformation applied to the polymer results in an immediate deformation by rotation of the Kuhn segments. The stress follows the applied strain without any shift. The ratio between stress and strain in the rubber phase is the rubber elasticity modulus.

In the melt phase the reptation time is less than 1 second. During a 1 second cycle of the test the rubber stress relaxes due to reptation of the molecules. As near the glass transition temperature this leads to a shift in time between stress and strain. The minimum and maximum stresses are now found at the moment that the deformation speed is maximum, which means zero deformation. The stress is zero when the deformation speed is zero, which means at minimum or maximum deformation.

The ratio between stress and deformation and the time shift enables us to calculate a storage modulus and a loss modulus. The storage modulus gives information about the elastic behaviour of the polymer; the loss modulus gives information about the viscous behaviour of the polymer.

For a perfectly elastic solid, the resulting strain and the stress will be in phase. For a purely viscous fluid, there will be a half a cycle time delay of stress with respect to strain. Viscoelastic polymers have the characteristics in between where some time shift will occur during DMA tests. A typical DMA curve is shown below. Example of DMA curve. Green = storage modulus, red = loss modulus, blue = tangent of loss angle. At the glass transition point Tg the storage modulus suddenly reduces and the loss modulus peaks.

Starting from high temperature and reducing to 0 K a number of transitions will be found. They are labelled per the Greek alphabet:

• First transition is called alpha-transition. This is usually the glass transition temperature in case of amorphous polymers or the crystalline melting point in case of crystalline polymers.
• Second is called beta-transition. This transition is usually attributed to the first occurrence of chain rotation in case of amorphous polymers.
• Third is called gamma-transition.
• Etcetera.

## Summary

• When a cyclic deformation is applied to a polymer then the resulting stress shows a cyclic behaviour too.
• By changing the temperature, transitions in the polymer can be found. The stress suddenly changes near a transition or shows a time shift.
• The transitions in the polymer are named per the Greek alphabet: the transition at the highest temperature is called alpha-transition, next lower temperature beta-transition, etcetera.
• In amorphous polymers the alpha-transition is always the glass-rubber transition.