Retardation time spectra

We assume that the above solution is valid in about the same time range as the self-similar relaxation time spectrum, Eq. 1-5. The retardation time spectrum is also self-similar. It is characterized by its positive exponent n which takes on the same value as in the relaxation time spectrum. 

Complementary relationships can be used to obtain the retardation time spectrum in terms of the complex compliances and the creep compliance. 

Applying time-temperature equivalence (as discussed in Section 6.2 above) gives the storage compliance as a function of frequency over a very wide range of frequencies, as shown in Figure 6.13. Thus it is now possible to calculate the retardation time spectrum, and compare this with any theoretical models that may be proposed. 

From the compliance data, J and J", the retardation time spectrum L(t) can be obtained as... 

The relaxation time spectrum can be calculated exactly from the measured stress relaxation modulus using Fourier or Laplace transform methods, and similar eonsiderations apply to the retardation time spectrum and the creep compliance. It is more convenient to consider these transformations at a later stage, when the final representation of linear viscoelasticity, that of the complex modulus and complex compliance, has been discussed. 

It is important to recognise that the relaxation time spectrum and the retardation time spectrum are only mathematical descriptions of the macroscopic behaviour and do not necessarily have a simple interpretation in molecular terms. It is a quite separate exercise to correlate observed patterns in the relaxation behaviour, such as a predominant relaxation time, with a specific molecular process. It should also be emphasised, as will be apparent from the further detailed discussion, that qualitative interpretations in general molecular terms can often be obtained from the experimental data directly, without recourse to calculation of the relaxation time spectrum or the retardation time spectrum. 

Similar relationships hold between the creep compliance J(t), the real and imaginary parts of the complex compliance Ji and J2 and the retardation time spectrum L(t). These... 

The characteristic function in this integral which specifies the relaxation properties of the system is Lj(logr), being called the retardation time spectrum of the shear compliance J ... 

The retardation time spectrum (G(t,, .)) is obtained from the parameters gi and T, r. It is assumed that gi is independent of T and d, whereas T, (at temperature T and structural parameter S) is given by eq. (5.22). A change in temperature from Z fo will thus shift the spectrum along the log T axis by og aja ). The time-dependent variation in d can be obtained for any thermal prehistory of the type... 

So far the results have shown that the viscoelastic behavior of para-aromatic polyamide fibers and of other well-oriented fibers can be described by a serial arrangement of an energy-elastic spring representing the chain modulus, and a viscoelastic element. The latter embodies the chain rotations due to shear deformation of the crystallites with a retardation time spectrum /(t) whose distribution in case of aramid fibers is described by the function This viscoelastic element may be... 

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