Dynamic creep compliance functions

The response of a material to a sinusoidal shear stress a = ao sin cot is delayed an angle 6 with regard to the perturbation, and the relaxation between the shear deformation (response) and the shear stress is given by 

By comparing the relations given in Eq. (6.18) with those of Eq. (6.5), one obtains 

According to Eqs. (6.19), the relationships between the components of the complex compliance function and those of the complex relaxation modulus 

7 TRANSFORMATION OF COMPLIANCE FUNCTIONS FROM THE FREQUENCY DOMAIN TO THE TIME DOMAIN AND VICE VERSA FOR VISCOELASTIC SOLIDS 

According to the Boltzmann superposition principle, the shear strain of a solid viscoelastic material under the action of a harmonic shear stress can be written as (2) 

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This is because although 0 = (10), in general, cr(10) oQ (it will usually be less). In principle, the quantities we have defined, E(t), Dit), Gif), and J(i), provide a complete description of tensile and shear properties in creep and stress relaxation (and equivalent functions can be used to describe dynamic mechanical behavior). Obviously, we could fit individual sets of data to mathematical functions of various types, but what we would really like to do is develop a universal model that not only provides a good description of individual creep, stress relaxation and DMA experiments, but also allows us to relate modulus and compliance functions. It would also be nice to be able formulate this model in terms of parameters that could be related to molecular relaxation processes, to provide a link to molecular theories. 

For viscoelastic fluids, the formalism of a viscous fluid and an elastic solid are mixed [31]. The equations for the effective viscosity, dynamic viscosity, and the creep compliance are given in Table 12.4 for a viscous fluid, an elastic solid, and a visco-elastic solid and fluid. For the viscoelastic fluid model the dynamic viscosity, >j (tu), and the elastic contribution, G (ti)), are plotted as a function of (w) in Figure 12.31. With one relaxation time, X, the breaks in the two curves occur at co. 

As mentioned above, it is very difficult, for experimental reasons, to measure the relaxation modulus or the creep compliance at times below 1 s. In this time scale region, dynamic mechanical viscoelastic functions are widely employed (5,6). However, in these methods the measured forces and displacements are not simply related to the stress and strain in the samples. Moreover, in the case of dynamic experiments, inertial effects are frequently important, and this fact must be taken into account in the theoretical methods developed to calculate complex viscoelastic functions from experimental results. 

In practice, viscoelastic properties can be determined by static and dynamic tests. The typical static test procedure is the creep test. Here, a constant shear stress is applied to the sample over a defined length of time and then removed. The shear strain is monitored as a function of time. The level of stress employed should be high enough to cause sample deformation, but should not result in the destruction of any internal structure present. A typical creep curve is illustrated in Fig. 13A together with the four-element mechanical model that can be used to explain the observations. The creep compliance represents the ratio between shear strain rate and constant stress at any time t. 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

In the above discussion, six functions Go(w), d(w), G (w), G"(w), /(w), and J"(oj) have been defined in terms of an idealized dynamic testing, while earlier we defined shear stress relaxation modulus G t) (see Equation 3.19) and shear creep compliance J(t) (see Equation 3.21) in terms of an idealized stress relaxation experiment and an idealized creep test, respectively. Mathematical relationships relating any one of these eight functions to any other can be derived. Such relationships for interconversion of viscoelastic function are described by Ferry [5], and interested readers are referred to this treatise for the same. 

Viscoelastic behavior is a time-dependent mechanical response and usually is characterized with creep compliance, stress-relaxation, or dynamic mechanical measurements. Since time is an additional variable to deformation and force, to obtain unique characterizing functions in these measurements one of the usual variables is held constant. 

The viscoelastic behavior is evaluated by means of two types of methods static tests and dynamic tests. In the first calegtuy a step change of stress or strain is applied and the stress or strain response is recorded as a function of time. Stress relaxation, creep compliance, and creep recovery are static methods. The dynamic tests involve the imposition of an oscillatory strain or stress. Every technique is described in the following sections. 

Another experimental quantity that also reflects the full dynamics of the solution is the creep compliance, t). A fixed stress is applied to the sample, and the strain is followed as a function of time. It is customary to represent the creep compliance in the form ... 

The storage and loss shear moduli, G and G", vs. oscillation frequency ta, and the creep compliance J vs. time t, measured at each concentration and tanperature, were temperature shifted with respect to frequency or time. These temperature master curves at each concentration were then shifted to overlap one another along the frequency or time axis. The dynamic shear moduli master curves as a function of reduced frequency (oa ac are shown in fig. 4.4, and the shear creep compliance master curves as a function of reduced time tlajUc are shown in fig. 4.5. Master curves... 

This principle applies to all of the viscoelastic functions defined previously for stress relaxation, creep, and dynamic mechanical experiments. A most comprehensive discussion of TTS is provided in the text by Ferry (1980). An example of TTS is shown in Fig. 5.14 for the modulus data of Fig. 5.15, which were determined at various temperatures (Mercier et al. 1965) by both stress relaxation and creep measurements. The creep compliance data were converted to modulus by the relationship E(t) = VJ t). The single curve formed by superposition of the various curves shifted on the time axis to the given reference temperature (Tr) is referred to as a master curve. In this case... 

Fig. 2.28. A plot of tan 3 as a function of actual frequencies at several temperatures for NBS-PIB. The data were obtained by using several instruments spanning the frequency range as shown in the abscissa. The high-frequency data at -35.8 °C (open circles) are from Fitzgerald et al, J. Appl. Phys. 24 (1953), 640. The rest of the data were obtained by a combination of creep-compliance and dynamic-modulus measurements [209]. From Plazek et al. by permission [209].

The linear viscoelastic properties G(t)md J t) are closely related. Both the stress-relaxation modulus and the creep compliance are manifestations of the same dynamic processes at the molecular level in the liquid at equilibrium, and they are closely related. It is not the simple reciprocal relationship G t) = 1/J t) that applies to Newtonian liquids and Hookean solids. They are related through an integral equation obtained by means of the Boltzmann superposition principle [1], a link between such linear response functions. An example of such a relationship is given below. 

So far, we have discussed the response of systems only for forces with special time dependencies. The creep compliance describes the reaction on a force which is switched on at zero time and then remains constant, the dynamic compliance specifies the response on a sinusoidally varying stress. What happens in the general case, when an arbitrary time dependent force ip(t) is applied There is a specific function which enables us to deal with this general situation, sometimes called the primary response function . It is introduced by considering the effect of an infinitely short pulse, as represented by... 

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. 

In this report we present a model for the primary creep of well-oriented aramid fibres. It has been shown for well-oriented fibres of PET, cellulose and poly-(p-phenylene terephthalamide), abbreviated here as PpPTA, that the dynamic compliance, S, is a linear function of the second moment of the orientation distribution of the chains / / /2/ /3/. By measuring S during creep and relaxation of a fibre, the changes in the orientation distribution can be followed. As shown here, such an experiment offers a valuable tool for the investigation of the viscoelasticity in polymer fibres. 

Fig. 2.31. The logarithm of the retardation spectrum L of poly(methyl methacrylate) as a function of the logarithm of the reduced retardation time r/ar. The solid curve was calculated from the reduced Jr(t) curve obtained from creep data taken at lower temperatures (14.4-34.7 °C) and longer times (10° s < r < 10 s) and shifted to 13.1° C. The dashed line was calculated from the dynamic compliances obtained by Williams and Ferry at higher temperatures and frequencies To was chosen to be 10.8 °C. From [217] by permission.

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