Viscoelastic functions dynamic function

In [343] it was shown, with regard to the hydrodynamic effect of fillers on the viscoelastic properties of composites, that the dynamic functions must obey the following equations ... 

A summary of analytic expressions obtained in this manner for all the viscoelastic functions is presented in Table 4 and 5 for the linear and cubic arrays. The well-known phenomenological analogy (8) between dynamic compliance and dielectric permittivity allows the formal use of Eqs. (T 5), (T 6), and (T 11), (T 12) for the dielectric constant, e (co), and loss, e"(co), of the linear and cubic arrays, respectively (see Table 6). The derivations of these equations are elaborated in the next section and certain molecular weight trends are discussed. 

An extensive study by Koppelmann (1958) of viscoelastic functions through one of the secondary mechanisms in Poly(methyl methacrylate) is shown in Fig. 13.15. Dynamic storage moduli and tan <5 s near 25 °C are plotted vs. angular frequency. It shows first, that a secondary mechanism is present, secondly that E and G are not completely parallel, because the Poisson constant is not a real constant, but also dependent on frequency (the numbers in between both moduli are the actual, calculated Poisson constants) and thirdly that tan <5E is practically equal to tan <5G. 

FIG. 13.59 Accuracy of interrelationships between static and dynamic viscoelastic functions. 

The appropriate viscoelastic functions are the dynamic rheological properties (storage modulus G and the loss modulus G", and the dynamic viscosities f and T ") extrapolated to infinite dilution and are called the intrinsic dynamic rheological properties ... 

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. 

The calculation from dynamic flexural experiments of elastic or viscoelastic functions is subject to errors arising from clamping (Ref. 6, p. 23). In the case of test samples whose section has dimensions of the order of magnitude of the free length, such a free length must be replaced by an effective length that represents that parameter in a more realistic way (see Fig. 7.7). 

A final comment seems to be pertinent. In most cases actual measurements are not made at the frequencies of interest. However, one can estimate the corresponding property at the desired frequency by using the time (fre-quency)-temperature superposition techniques of extrapolation. When different apparatuses are used to measure dynamic mechanical properties, we note that the final comparison depends not only on the instrument but also on how the data are analyzed. This implies that shifting procedures must be carried out in a consistent manner to avoid inaccuracies in the master curves. In particular, the shape of the adjacent curves at different frequencies must match exactly, and the shift factor must be the same for all the viscoelastic functions. Kramers-Kronig relationships provide a useful tool for checking the consistency of the results obtained. 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function... 

The crystalline phase affects the viscoelastic dynamic functions describing the glass-rubber relaxation. For example, the location of this absorption in the relaxation spectrum is displaced with respect to that of the amorphous polymer and greatly broadened. Consequently, the perturbing effects of crystal entities in dynamic experiments propagate throughout the amorphous fraction. The empirical Boyer-Beaman law (32)... 

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. 

More recently, a new, viscoelastic-plastic model for suspension of small particles in polymer melts was proposed [Sobhanie et al., 1997]. The basic assumption is that the total stress is divided into that in the matrix and immersed in it network of interacting particles. Consequently, the model leads to non-linear viscoelastic relations with yield function. The latter is defined in terms of structure rupture and restoration. Derived steady state and dynamic functions were compared with the experimental data. 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

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. 

In a rheometric dynamic spectrometer such as RDS7700 manufactured by Rheometrics, Inc., the torque and normal force generated in response to an imposed motion are measured by a transducer. A microcomputer determines stresses from these values with measured sample motion to calculate strains and viscoelastic functions such as G, G", and tan d [12]. 

The viscoelastic response of equilibrium rubber networks can be obtained by measuring the shear and tensile moduli or compliances as a function of time, or the corresponding dynamic moduli and compliances as a function of frequency. As discussed in Section 5.2, the measurements of any viscoelastic function can be converted to another viscoelastic function. 

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