Frequency dependent compliance

At constant stress (the constraint), the observed response is the change in strain and we call the phenomenon strain relaxation with stress retardation. 

We have seen that if the stress on a sample of rubber is removed instantaneously, the strain falls to zero in an exponential way, i.e. constant stress equal to zero. 

In the same way, if the strain is raised instantaneously from zero to a given value and held constant, the observed response is the change in stress and we call it stress relaxation with strain retardation, i.e. under constant strain, the stress appHed, equal to the internal force opposing strain, decays exponentially as the molecules take time to deform  

To understand the reason for the move, we have to appreciate that the time constant, T, in the compliance equation is the strain relaxation time. However, for modulus we obtain the stress relaxation time t. Surprisingly these two relaxation times are not exactly equal. 

Alternatively, we can take the reciprocal of the complex compliance, set the rubber modulus to zero, and change from T to t, when we get the result  

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Although several experimental studies [22, 62, 63] have been reported on p, G, and the compliance of polyelectrolyte solutions, systematic investigations on the frequency dependence is still lacking. 

Isotherms at several temperatures showing the frequency dependence of the real component / (co) of the complex compliance / (coi) of a viscoelastic material are plotted on a double logarithmic scale in Figure 8.9 (7). At high temperatures and low frequencies, J (co) decreases slightly with increas-... 

To obtain the frequency dependence of the loss compliance function for liquids, use must be made of Eq. (6.29b). Accordingly... 

Figure 3-6. Frequency dependence of the complex dynamic compliance for the Voigt model.

Equation 103 shows that—at this level of approximation—only the quantity J" (the viscous compliance) enters the frequency shift, whereas only the quantity J (the elastic compliance) enters the bandwidth. This has far-reaching consequences in the data analysis. Because the thickness and the viscous compHance additively enter the frequency shift, it is difficult to derive the viscous comphance without independent knowledge of the thickness. J cannot be inferred from the n-dependence of A/ because the n-dependence of / is unknown. The elastic comphance, on the other hand, can be derived with fair accuracy, because the mass only enters the bandwidth as a prefactor. Even its frequency dependence (Eq. 73) is obtained. 

The frequency-dependent viscous compliance J" co) can be determined for thin films in air. 

When two compartments have different time constants due to differences in resistance or compliance, gas can be flowing out of one at the same time as it is flowing into the other. This pendelluft phenomenon reduces effective tidal volume results in frequency dependence of dynamic compliance and resistance measurements, and again makes prediction of aerosol distribution in the presence of disease highly complex (269) (Fig. 18). 

In addition to the expressions mentioned for predicting moduli in the elastic state, blending equations developed by Ninomoya and Maekawa (1966) have been adapted to predict frequency-dependent moduli of filler-polymer systems. Compliances were considered to be additive, and the following relations for relative moduli (Dp/D were tested using a rubber-... 

Like electrical resistance and capacitance, airway resistance and lung compliance together impose a frequency-dependent impedance to ventilation. Thus, the normal lung emptying passively follows an exponential decay with a single time constant equal to the product of the resistance and the... 

It should be remembered that the moduli and compliances under discussion are functions of frequency. The quantities E, D etc. should thus be written E (a>), D (a>), and so forth. The frequency dependence of these quantities is governed by the same distribution of relaxation or retardation times as is stress relaxation, creep or other time-dependent mechanical phenomena. Single relaxation or retardation times cannot depict the frequency dependence of the dynamic mechanical behavior of polymers. 

The Viscoelastic Material Functions. In linear viscoelasticity, the moduli discussed for the elastic case can be recast as time- or fi equency-dependent functions. The same is true for the compliance functions that are discussed here. For simplicity, consider the shear modulus G which becomes G(t) or G (a>) in the case of the viscoelastic material. An important point here is that the viscoelastic modulus functions all exhibit time (frequency) dependence. Hence, one will have functions for K(t) and E(t) [or, eg, G t) and v i)] and these are required in the case of a three-dimensional strain or stress field. 

Although the Cole-Cole plot was first introduced in the context of a dielectric relaxation spectrum, it helped discover that the molecular mechanism underlying both dielectric relaxation and stress relaxation are substantially identical (44). Figure 8.13 provides an illustration, with temperature instead of frequency. Specifically, the same molecular motions that generate a frequency dependence for the dielectric spectrum are also responsible for the relaxation of orientation in polymers above Tg. Subsequently the Cole-Cole type of plot has been applied to the linear viscoelastic mechanical properties of polymers, especially in the vicinity of the glass transition, including the dynamic compliance and dynamic viscosity functions. 

Minimum frequencies for compliance monitoring should depend on the risk assessment made in the water risk management strategy and could be higher. 

In the ladder model treatment of Blizard, an alternative termination of the line of springs (Fig. 10-3) was considered in which each end, rather than being fixed, is attached to three other such lines, each of these to three more, and so on indefinitely, thus reproducing the connectivity of a tetrafunctional network. This.proyision increases the equilibrium compliance by a factor of 2 (corresponding-fo the factor of (/ — 2)//mentioned in Section I above), and it modifies the frequency dependence, which is now expressed by a rather complicated combination of hyperbolic functions. This frequency dependence of J" is also shown in Fig. 10-7 the maximum is slightly broader than for fixed cross-links (i.e., cross-links with affine deformation). 

It should be noted again that the proportionality of moduli to Tp/ToPo (and of compliances to Topo/Tp) and the applicability of a single friction coefficient fo are more general principles than the detailed predictions of any specific theory, and these principles can be applied without assuming that the form of the time or frequency dependence follows any prescribed function. There have been some otha-schemes for constructing composite curves in which it was assumed at the outset that log Ot was a linear function of reciprocal absolute temperature (by analogy with the theory of rate processes ), and it has sometimes been erroneously believed that the method of reduced variables implies a specific form such as this for the temperature dependence. On the contrary, the shape of the curve in Fig. 11-4 is determined empirically from the data, although it may be useful to fit it subsequently to a suitable analytical expression (Section B below). 

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. 

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