Polymer Viscoelasticity stress relaxation

In many applications, plastic parts carry reasonably constant mechanical loads over periods up to few years. The polymer will creep during the lifetime of the part. At moderate load levels, long-term prediction of creep from short-term tests is possible, because the viscoelastic response of polymers (creep, stress relaxation) measured at different temperatures superimpose when shifted along the time axis [24]. 

There are three fundamental test methods for characterization of the viscoelastic behavior of polymers creep, stress relaxation, and dynamic mechanical analysis. Although the primary focus for this chapter is DMA, it is useful first to discuss the fundamentals of creep and stress relaxation, not only because they are conceptually simpler but because most DMA instruments also are capable of operating in either a creep or stress relaxation mode. All three of the methods are related, and numerical techniques are available for calculating creep and stress relaxation data from dynamic mechanical data (Ferry 1980). 

Furthermore, the magnitude of the relaxation modulus is a function of temperature to more fuUy characterize the viscoelastic behavior of a polymer, isothermal stress relaxation measurements must be conducted over a range of temperatures. Figure 15.6 is a schematic log ,(f)-versus-log time plot for a polymer that exhibits viscoelastic behavior. Curves generated at a variety of temperatures are included. Key features of this plot are that (1) the magnitude of EXt) decreases with time (corresponding to the decay of stress. Equation 15.1), and (2) the curves are displaced to lower EXt) levels with increasing temperature. 

The EMT analysis indicated that the stress relaxes in proportion to the number of bonds removed. The initial linear decrease of E/Eq with is intuitively appealing and is the basis for many linear constitutive theories of polymers. An example is the Doi-Edwards theory of viscoelasticity of linear polymer melts [49] in which... 

There are two further related sets of tests that can be used to give information on the mechanical properties of viscoelastic polymers, namely creep and stress relaxation. In a creep test, a constant load is applied to the specimen and the elongation is measured as a function of time. In a stress relaxation test, the specimen is strained quickly to a fixed amount and the stress needed to maintain this strain is also measured as a function of time. 

Stress relaxation tests are alternative ways of measuring the same basic phenomenon in viscoelastic polymers as creep tests, Le. the time-dependent nature of their response to an applied stress. As such, they have also been of value in understanding the behaviour of these materials. The essence of stress relaxation tests is that strain increases with time for a given stress, so that if stress is decreased with time in a controlled manner ( relaxed ), a state... 

Creep and stress-relaxation tests measure the dimensional stability of a material, and because the tests can be of long duration, such tests are of great practical importance. Creep measurements, especially, are of interest to engineers in any application where the polymer must sustain loads for long periods. Creep and stress relaxation are also of major importance to anyone interested in the theory of or molecular origins of Viscoelasticity. 

In a further development of the continuous chain model it has been shown that the viscoelastic and plastic behaviour, as manifested by the yielding phenomenon, creep and stress relaxation, can be satisfactorily described by the Eyring reduced time (ERT) model [10]. Creep in polymer fibres is brought about by the time-dependent shear deformation, resulting in a mutual displacement of adjacent chains [7-10]. As will be shown in Sect. 4, this process can be described by activated shear transitions with a distribution of activation energies. The ERT model will be used to derive the relationship that describes the strength of a polymer fibre as a function of the time and the temperature. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

Because of equipment limitations in measuring stress and strain in polymers, the time-temperature superposition principle is used to develop the viscoelastic response curve for real polymers. For example, the time-dependent stress relaxation modulus as a function of time and temperature for a PMMA resin is shown in... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

The effects attributed to entangling interactions, e.g., the plateau region in stress relaxation, appear most prominently at high concentrations and in melts. It is important, however, to distinguish this interaction from other types which are present at lower polymer concentrations. To make the separation properly, it is necessary to examine viscoelastic behavior at all levels of concentration, beginning at infinite dilution. 

The sample must have reached steady state before cessation of the test or the application of a second step. Steady state in a creep test is seen as a constant slope in the strain curve. A constant slope in the stress curve may also be seen in a stress relaxation test, but often the signal is lost in the noise. A material that is liquid-like in real time will need a test period of 5 to 10 min. A stress relaxation test is likely to be somewhat shorter than a creep test since the signal inevitably decays into the noise at some point. A creep test will last indefinitely but will probably reach steady state within an hour. For a material that is a solid in real time, all experiments should be longer as molecular motion is, by definition, slower. Viscoelastic materials will lie in between these extremes. Polymer melts can take 1 hr or more to respond in a creep test, but somewhat less time in a stress relaxation test. 

In a stress relaxation test, a polymer test specimen is deformed by a fixed amount, eo, and the stress required to hold that amount of deformation is recorded over time. This test is very cumbersome to perform, so the design engineer and the material scientist have tended to ignore it. In fact, several years ago, the standard relaxation test ASTM D2991 was dropped by ASTM. Rheologists and scientists, however, have been consistently using the stress relaxation test to interpret the viscoelastic behavior of polymers. 

Stress relaxation master curve. For the poly-a-methylstyrene stress relaxation data in Fig. 1.33 [8], create a master creep curve at Tg (204°C). Identify the glassy, rubbery, viscous and viscoelastic regions of the master curve. Identify each region with a spring-dashpot diagram. Develop a plot of the shift factor, log (ax) versus T, used to create your master curve log (ot) is the horizontal distance that the curve at temperature T was slid to coincide with the master curve. What is the relaxation time of the polymer at the glass transition temperature ... 

According to the change of strain rate versus stress the response of the material can be categorized as linear, non-linear, or plastic. When linear response take place the material is categorized as a Newtonian. When the material is considered as Newtonian, the stress is linearly proportional to the strain rate. Then the material exhibits a non-linear response to the strain rate, it is categorized as Non Newtonian material. There is also an interesting case where the viscosity decreases as the shear/strain rate remains constant. This kind of materials are known as thixotropic deformation is observed when the stress is independent of the strain rate [2,3], In some cases viscoelastic materials behave as rubbers. In fact, in the case of many polymers specially those with crosslinking, rubber elasticity is observed. In these systems hysteresis, stress relaxation and creep take place. 

Very important phenomena in polymer behaviour, such as viscoelasticity, stress, strain, volume and enthalpy relaxation, ageing, etc., are characterised by time-dependence of the polymer properties. 

Three types of experiments are used in the study of viscoelasticity. These involve creep, stress relaxation, and dynamic techniques. In creep studies a body is subjected to a constant stress and the sample dimensions are monitored as a function of lime. When the polymer is lirst loaded an immediate deformation occurs, followed by progressively slower dimensional changes as the sample creeps towards a limiting shape. Figure 1-3 shows examples of the different behaviors observed in such experiments. 

Figure 9.2 Schematic phase diagram of a polymer/solvent mixture, where y is the Flory chi parameter, and xe = 1/2 is x at the theta temperature. The quantity Xe X along the ordinate is a reduced temperature, and is the polymer volume fraction. CP is the critical point, and BL is the binodal line. SSL and KSL are the static symmetry line and the kinetic symmetry line, respectively. These lines define the phase-inversion boundaries during quenches. In quenches that end at the right of such a line, the polymer-rich phase is the continuous phase, while to the left of the line the solvent-rich phase is the continuous one. SSL applies at long times, after viscoelastic stresses have relaxed, while KSL applies at shorter times before relaxation of viscoelas-...

---