Relaxation and Retardation Times

The various models were invented explicitly to provide a method of mathematical analysis of polymeric viscoelastic behavior. The Maxwell element expresses a combination of Hooke s and Newton s laws. For the spring, 

Since the Maxwell model has a spring and dashpot in series, the strain on the model is the sum of the strains of its components  

On a molecular scale the relaxation time of a polymer indicates the order of magnitude of time required for a certain proportion of the polymer chains to relax—that is, to respond to the external stress by thermal motion. It should be noted that the chains are in constant thermal motion whether there is an external stress or not. The stress tends to be relieved, however, when the chains happen to move in the right direction, degrade, and so on. 

Alternatively, Ti can be a measure of the time required for a chemical reaction to take place. Common reactions that can be measured in this way include bond interchange, degradation, hydrolysis, and oxidation. Combining equations (10.14) and (10.15) leads to 

The stress relaxation experiment requires that deldt = 0 that is, the length does not change, and the strain is constant. Integrating equation (10.16) under these conditions leads to 

The first thing you should remember is that the picture representation on the left assumes that the strain in each of the three Maxwell elements is identical, (e = = e2 = e3) while the stress, o, applied to the system as a whole must equal the sum of the stress experienced 

In a stress relaxation we can put de/dt = 0 and obtain for each of the Maxwell elements the equations shown in Equations 13-94. 

The result we have just obtained, that the relaxation modulus is the sum of the responses of the individual elements, is essentially a statement of the superposition 

As you might expect, you can also create a model using a bunch of Voigt models (in this case, in series) to obtain an equation for creep compliance (Equation 13-97)  

These models can be extended to the situation where there is a continuous distribu- - 

In considering systems where there are very many Maxwell elements employed in the model, that is, z in equation (3-34) is large, it is often convenient to replace the summation in the equation by an integration. Thus  

The various E s are replaced by a continuous function, ED(x), of the relaxation time where this function is called a distribution of relaxation times. Note that the dimensions of the variable Ed(t) and Et are not the same nor is E(r) a modulus instead it is akin to a probability density. The variable r has dimensions of time. The physical interpretation is as follows E x)dx is the fraction of the total relaxation modulus that has relaxation times between x and x + dr. In addition to the distribution E(x), one often encounters H(x), which is defined as 

Note that H(x) has the dimensions of modulus (e.g., units of Pa). In terms of H(x), equation (3-40) becomes 

In addition, dynamic modulus functions can be calculated via this distribution function E (cd), for example, is given as 

A similar method is used to consider compliance functions. Here, however, the distribution of retardation times L(t), defined as 

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When Z is sufficiently large, it is convenient to replace the sums involved in the above expressions by integrals and define continuous distributions of relaxation and retardation times. These spectral functions are usually defined on log time scales through the relations (9) ... 

The number N of retardation times needed depends on the required agreement between theory and experimental behaviour that is required. Instead of a description of viscoelastic behaviour with the aid of a discrete spectrum of relaxation and retardation times, also continuous relaxation or retardation time spectra can be used. In some cases these are easier to handle. 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

One obvious way of introducing a range of relaxation and retardation times into the problem is to construct mathematical models thai are equivalent to a number of Maxwell and/or Voigt models connected in parallel (and/or series). The Maxwell-Wiechert model (Figure 13-96), for example, consists of an arbitrary number of Maxwell elements connected in parallel. For simplicity let s see what you get with, say, three Maxwell elements and then extrapolate later to an arbitrary number, n. 

Baumgartel M and Winter HH (1989) Determination of discrete relaxation and retardation time spectra from dynamic mechanical data. Rheol Acta 28 511-9. 

In order to compare the relaxation and retardation times, let us consider first the Laplace transform of the relaxation modulus of a solid. According to Eq. (6.4),... 

The restraining influence of the crystallite alters the mechanical behavior by raising the relaxation time T and changing the distribution of relaxation and retardation times in the sample. Consequently, there is an effective loss of short T, causing both the modulus and yield point to increase. The creep behavior is also curtailed and stress relaxation takes place over much longer periods. Semicrystalline polymers are also observed to maintain a relatively higher modulus over a wider temperamre range than an amorphous sample. 

A material, which can be described as an SLS, was found to have unrelaxed and relaxed Young s modulus values of 70 and 50 GPa, respectively. Determine the relaxation and retardation times. Plot graphically the compliance of the material as a function of time, under the action of a constant uniaxial tensile stress. 

TABLE 26.1. WLF parameters characterizing temperature dependencies of shift factors for relaxation and retardation times in various polymer systems. 

Relaxation and retardation time spectra can be calculated exactly from stress relaxation, creep and dynamic mechanical measurements using Fourier or Laplace... 

The reader may use the Alfrey approximation (see Section 4.3.2) to derive relaxation and retardation time spectra from the data of Figure 6.1. These spectra can be approximated by a wedge and box distribution [3], shown by the dotted lines in Figure 6.2. 

To make an order-of-magnitude rule about the duration of a creep experiment, it may be noted that the terminal retardation time is very roughly equal to the product 7 qo cf. equations 10 and 11 of Chapter 10, recalling that the terminal relaxation and retardation times are similar in magnitude). Though 7 depends on polymer concentration, molecular weight, and molecular weight distribution, it is often of the order of 10 to 10" cm /dyne (10 to lO Pa )- This sets both... 

Having examined the nature of the temperature and pressure dependence of the relaxation and retardation times, we now turn attention to the details of the time and frequency dependence of the basic viscoelastic functions and their correlation with chemical structure. Each zone of time scale represents a separate problem. The one most characteristic of polymers is the subject of this chapter, the transition from rubberlike to glasslike consistency, where the moduli increase and the compliances decrease by several powers of 10 as a function of time or frequency, as illustrated in Chapter 2. 

An additional nonlinear effect which appears in extension under high stresses, attributable to the volume expansion associated with the fact that Poisson s ratio is less than i, will be discussed in Chapter 18 it is manifested by a decrease in all the relaxation and retardation times. This effect is especially prominent in more complicated stress patterns such as combined tension and torsion, as studied by Sternstein. ° Nonllnear creep behavior under combined tension and torsion with the additional complication of changing temperature during the experiment has been studied by Mark and Findley. ... 

For much higher pressure changes, a marked departure from linearity may be expected because of the diminution of free volume under pressure. The effects of such volume reduction on shear relaxation times (and dielectric and nuclear magnetic relaxation times) have been described in Sections D and E of Chapter 11 the bulk relaxation and retardation times are presumably similarly affected. The origin of this kind of nonlinearity can be understood qualitatively on the basis of a theory developed by Kovacs for a single retardation time. 

Mechanical Models, Relaxation and Retardation Time Spectra... 

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