Spectra retardation

We determine the long time end of the retardation spectrum by approximating Eq. 1-5 with... 

The relaxation spectrum H is independent of the experimental time t and is a fundamental description of the system. The exponential function depends upon both the experimental time and the relaxation time. Such a function in the context of this integral is called the kernel. In order to describe different experiments in terms of a relaxation spectrum H or retardation spectrum L it is the kernel that changes. The integral can be formed in time or frequency depending upon the experiment being modelled. The inclusion of elastic properties at all frequencies and times can be achieved by including an additional process in the relaxation... 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

The contribution of this component is shown in Figures 4.16 and 4.17. One of the important features to recognise about the retardation spectrum is that it only has an indirect relationship to both the zero shear rate viscosity and the high frequency shear modulus. Both these properties are contained in the relaxation spectra. We shall see in Section 4.5.7 that, whilst a relationship exists between H and L it is somewhat complex. 

In the same manner as the modulus can be related to the relaxation spectrum so the compliance can be related to the retardation spectrum ... 

This result is very interesting because whilst we have shown that G(0) has been excluded from the relaxation spectrum H at all finite times (Section 4.4.5), it is intrinsically related to the retardation spectrum L through Jc. Thus the retardation spectrum is a convenient description of the temporal processes of a viscoelastic solid. Conversely it has little to say about the viscous processes in a viscoelastic liquid. In the high frequency limit where co->oo the relationship becomes... 

The high frequency elastic modulus does not appear in the retardation spectrum but is an intrinsic part of the relaxation spectrum. These features are reinforced when the interrelationship between the spectra are considered. 

Exact Inversions from the Relaxation or Retardation Spectrum... 

Note 5 The retardation spectrum (spectrum of retardation times) describing creep in polymers may be considered as arising from a group of Voigt-Kelvin elements in series. 

An additional step — application of the Gross integral relation, Eq. (3.5) — will generate the corresponding retardation spectrum, La(x). 

The spectmm L(t) is called the retardation spectrum. With the integration from t= 0 to t = °° the immediate elastic response and the irreversible flow are, respectively, automatically taken into account. 

Both the steady-state compliance function, and the equilibrium compliance, Jg, can readily be obtained from the retardation spectrum. Actually, by taking the limit of Eq. (9.20) in the limit oo 0, the following relationship for Jg and Jg is obtained ... 

In the transient compliance function, J(t), the retardation spectrum L(x) is modulated by the function 1 — exp(—t/x) [see Eq. (9.15)]. Plotting this function against In t/x gives the sigmoidal curve shown in Figure 9.6. We should note that the time of observation ( ) in the first quadrant is greater than the retardation times, and as a result x varies between zero and t. Then the creep compliance function for viscoelastic liquids is approximately given by (1,2)... 

The approximate value of the retardation spectrum is obtained from this equation by means of the relationship... 

Therefore, a first-order approximation for the retardation spectrum can be derived from the loss compliance function by means of the expression... 

This method can be used to obtain the retardation spectrum from the compliance function J(t). Equation (9.15) can be written as... 

Since J t) is a monotonous increasing function of time, the relation L(x) = Tx" can be assumed to be a good approximation for the retardation spectrum. Then the integral of Eq. (9.73) is give by... 

On the other hand, the approximations of first, second, and third order for the retardation spectrum are... 

Figure 9.8 Double logarithmic plot of the retardation spectrum versus the retardation time for a polystyrene fraction of molecular weight 3400 with a narrow molecular weight distribution. (From Ref. 8.)...

Once a relaxation (retardation) spectrum is obtained from a relaxation (creep compliance) viscoelastic function, any other function can be obtained. Alternatively, approximate methods have been developed to calculate viscoelastic functions from one another (10). By taking into account... 

Show that as a consequence of the retardation spectrum being positive,... 

Give an approximate expression for the retardation spectrum of a viscoelastic solid whose creep compliance is given by... 

The friction coefficient is customarily obtained from either the relaxation or retardation spectrum, H x) or L x), respectively. At short times, i.e., on the transition from the glassy-like to the rubbery plateau, the viscoelastic processes obey Rouse dynamics, and the relaxation modulus is given by Eq. (11.45). Since H x) = —dG/d nx t, one obtains... 

The coefficient of friction can alternatively be obtained from the retardation spectrum. In this case,... 

Resistivity, thermal Retardation spectrum Reynolds number. Re Rheological equation of state Rheological measurements... 

A curve of the logarithm of the modulus against time and temperature is shown in Figure 13.25. This provides a particularly useful description of the behavior of a polymer and allows one to estimate, among other things, either the relaxation or retardation spectrum. 

FIGURE 5.14 The logarithm of the reduced retardation spectrum, Lp, shown as function of the logarithm of the reduced retardation time, A./uj , for the three urethane-end linked polybutadiene elastomers, (o) TB-1, ( ) TB-2, ( )TB-3. The response has been reduced to corresponding state temperatures of 74°C, 0°C, and 17°C, respectively, for the primary softening transition. 

The retardation spectrum L X), where X is the retardation time, characterizes the contribution to the creep compliance J i), the dynamic storage compliance and the dynamic loss compliance J co) according to the following relations (for a viscoelastic liquid for a viscoelastic solid r is operationally infinite in the following equations),... 

Since the distribution function of retardation times (the retardation spectrum L) is a measure of the population of viscoelastic mechanisms along the time scale [17], one of the simplest propositions for obtaining the tear energy T C) from a material s viscoelastic behavior would be... 

We hasten to point out that in the literature [1] Co is commonly associated with the friction coefficient of and the relaxation times of all other longer time viscoelastic mechanisms (e.g., the entire relaxation or retardation spectrum). This common association of Co with is not used in the coupling model (CM/ which instead identifies Co as the friction coefficient of Tb [18, 59-61, 65, 68, 69], although the dependences of C> on T (/) and/or S (Sc) are retained. 

It was assumed that the results were reasonably close to linear behavior and the more extensive curve obtained from the cooling steps was analyzed to obtain a normalized retardation spectrum L(r). 

The Lo(t) that was obtained is shown in Fig. 12.22 with the retardation spectrum obtained from the shear creep compliance function, J(t). The levels of the short time behavior are matched. Three feamres should be noted. The functionality at short times is the same within experimental uncertainty. The 1/3 slope of log L(t) at short time indicates that the response is dominated by motions that contribute to Andrade creep. [66,85-87,141,142]... 

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