Viscoelastic functions approximate

DETERMINATION OF SPECTRA FROM VISCOELASTIC FUNCTIONS USING FIRST-ORDER APPROXIMATIONS... 

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 calculation of viscoelastic functions by means of spectra calculated by first-order approximations may lead to values of these functions that are in error with respect to the true values. These errors are lower if correction... 

An alternative procedure for calculating the spectra involves fitting the experimental results for the viscoelastic functions by means of spline functions. The derivatives of Eqs. (9.81) and (9.82) are determined by means of these functions, and thus the spectra can be obtained. A summary of these and other approximations used to calculate retardation and relaxation spectra from the measured compliance and relaxation functions, respectively, can be found in Refs. 1 and 5. 

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... 

It has been remarked that time (frequency) - temperature reduced data on carbon black filled rubbers exhibit increased scatter compared to similar data on unfilled polymers. Payne (102) ascribes this to the effects of secondary aggregation. Possibly related to this are the recent observations of Adicoff and Lepie (174) who show that the WLF shift factors of filled rubbers giving the best fit are slightly different for the storage and loss moduli and that they are dependent on strain. Use of different shift factors for the various viscoelastic functions is not justified theoretically and choice of a single, mean ar-funetion is preferred as an approximation. The result, of course, is increased scatter of the experimental points of the master curve. This effect is small for carbon black... 

Initially, for characterisation by mechanical spectroscopy, the strain dependence of, for example, the complex shear modulus (G ) is established. Typical results are shown schematically in Figure 2.8. This experiment establishes the linear viscoelastic region of the system, within which the viscoelastic functions are independent of strain. In other words, the applied strain does not perturb the sample. For entanglement networks the linear viscoelastic region extends to approximately 25% strain. 

Approximate Interrelations among The Linear Viscoelastic Functions... 

Generally, the approximation methods have an analytical foundation based oh the properties of the integrands of the corresponding exact equations. Such an integrand is usually the product of the viscoelastic function initially known and an... 

In other cases, simple explicit solutions can be obtained for integrals in equations such as 21 and 22 of Chapter 3 if the viscoelastic function in the integrand is assumed to have a certain form and the intensity function is retained without approximation. Again, it may be sufficient for the form assumed for the viscoelastic function to be valid over perhaps two decades of logarithmic time scale. ... 

It is difficult to compare the shapes of the viscoelastic functions for single crystal mats and bulk crystalline polymer from the data of Figs. 16-1, 16-2, 16-S, and 16-6 without extensive recalculation under circumstances where approximation methods give poor precision because the functions vary so slowly with time or frequency cf. equation 40 of Chapter 4). Comparisons of the two types of samples will be shown for isochronal viscoelastic functions in Section B below. 

The exact formal relationships between the various viscoelastic functions are conveniently expressed using Fourier or Laplace transform methods (cf. Section 5.4.2). However, it is often adequate to use simple approximations due to Alfrey in which the exponential term for a single Kelvin or Maxwell unit is replaced by a step function, as shown schematically in Figure 5.18. 

On a log-log plot, many viscoelastic functions are approximately linear over significant time ranges. This suggests the use of a relaxation function of the form... 

This approximation assumes that T is much less than the decay times of the viscoelastic medium. It is the assumption underlying the work of Hunter (1960) and Graham (1978). The viscoelastic functions are expanded in a Taylor expansion about t = 0, and only linear terms are retained. Therefore, from (5.3.9) ... 

Viscoelastic properties have been discussed in relation to molar mass, concentration, solvent quality and shear rate. Considering the molecular models presented here, it is possible to describe the flow characteristics of dilute and semi-dilute solutions, as well as in simple shear flow, independent of the molar mass, concentration and thermodynamic quality of the solvent. The derivations can be extended to finite shear, i.e. it is possible to evaluate T) as a function of the shear rate. Furthermore it is now possible to approximate the critical conditions (critical shear rate, critical rate of elongation) at which the onset of mechanical degradation occurs. With these findings it is therefore possible to tune the flow features of a polymeric solution so that it exhibits the desired behaviour under the respective deposit conditions. 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

When a spring and a dash pot are connected in series the resulting structure is the simplest mechanical representation of a viscoelastic fluid or Maxwell fluid, as shown in Fig. 3.10(d). When this fluid is stressed due to a strain rate it will elongate as long as the stress is applied. Combining both the Maxwell fluid and Voigt solid models in series gives a better approximation for a polymeric fluid. This model is often referred to as the four-parameter viscoelastic model and is shown in Fig. 3.10(e). Atypical strain response as a function of time for an applied stress for the four-parameter model is found in Fig. 3.12. 

The time dependence of the displacement of a macromolecule, shown in Fig. 9 as a function of the ratio t/r, is typical for diffusion of Brownian particle in viscoelastic fluid (Zanten and Rufener 2000 Zanten et al. 2004). The function (5.5) for big values of B can be approximated as... 

One can notice that the dissipative terms in the dynamic equation (3.11) (taken for the case of zero velocity gradients, z/jj = 0) have the form of the resistance force (D.3) for a particle moving in a viscoelastic liquid, while the memory functions are (with approximation to the numerical factor) fading memory functions of the viscoelastic liquid. The macromolecule can be considered as moving in a viscoelastic continuum. In the case of choice of memory functions (3.15), the medium has a single relaxation time and is characterised by the dynamic modulus... 

Detailed analysis of the isothermal dynamic mechanical data obtained as a function of frequency on the Rheometrics apparatus lends strong support to the tentative conclusions outlined above. It is important to note that heterophase (21) polymer systems are now known to be thermo-rheologically complex (22,23,24,25), resulting in the inapplicability of traditional time-temperature superposition (26) to isothermal sets of viscoelastic data limitations on the time or frequency range of the data may lead to the appearance of successful superposition in some ranges of temperature (25), but the approximate shift factors (26) thus obtained show clearly the transfer viscoelastic response... 

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