Viscoelasticity material functions

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

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. 

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. 

Interrelationships among the Viscoelastic Material Functions. There is a continuing disagreement within the molecular viscoelasticity commimity about which of the above methods should be used to characterize a material (20). In fact, if one can obtain the zero shear rate viscosity and any of the other functions, these methods are all equivalent. The issue, however, revolves aroimd the fact that some features that appear in the dynamic modulus disappear if the compliance is used as the function to represent the data and vice versa. Also, some measurements are more or less dominated by the viscosity contribution. As a result, some problems of misinterpretation of data could be averted if workers who prefer modulus representations would calculate the compliances. In addition, those who measure the compliance should calculate the moduli in order to provide the data in the format that is more common in the field because of the large number of commercial instruments that obtain dynamic moduli. The advent of modern software packages that make the interrelationships easily calculated makes this dispute seem to go away. The pathways to determine the different material functions, one from the other, are shown in Figure 6. 

Fig. 6. Chart showing the paths to interrelate the linear viscoelastic material functions. Equation numbers refer to Chapter 3 of Ferry s book (9) unless otherwise indicated. Determination of G (co) from G"(co) and J (o ) from J"(a>) and vice versa comes from the Kramer-Kronig relation and is discussed in Tschoegl (10).

S. W. Park and R. A. Schapery, Methods of Interconversion between Linear Viscoelastic material Functions. Part. I.-A Numerical Method Based on Prony Series Int. J. Solids and Struct. 36, 1653-1675 (1999). 

The introduction emphasized that in general one cannot use linear viscoelastic data to predict a nonlinear viscoelastic material function, nor as a rule can one use one nonlinear material function to predict another. Nevertheless, in shearing flows a few usefiil in-terrelaficms between material functions have often been observed to hold, at least approximately, for polymer melts and solutions. 

It is easier to load and unload viscous or soft solid samples with the parallel disk geometry than with cone and plate or concentric cylinders. Thus parallel disks are usually preferred for measuring viscoelastic material functions like G(r, y), G (), or 7(r, t) on polymer melts. To evaluate moduli or compliance, we use the strain and stress at the edge of the disk (eqs. 5.5.5 and 5.5.8), but now the stress must be corrected by d In A//d In y (Soskey and Winter, 1984). In the linear viscoelastic region G(r, y) = G(r) anddln Af/dlny = 1. 

The most commonly measured viscoelastic material function is G (a), T). It is so popular because sinusoidal oscillations can be used to follow viscoelastic changes with time, such as during curing and ciystallization. As discussed below, cross-correlation analysis of the signal can provide accurate G and G" values over a wide range of frequency and signal levels. 

Marrucci G, Maffettone PL (1993) Liquid crystalline polymers. Pergamon, New York Mead DW (1994) Determination of molecular weight distributions of linear flexible polymers from linear viscoelastic material functions. J Rheol 38 1797-1827 Moldenaers P, Yanase H, Mewis J (1990) Effect of shear history on the theological behavior of lyotropic liquid crystals. Liq Cryst Polym 24(1990) 370-380 Muir MC, Porter RS (1989) Processing rheology of liquid taystal polymers a review. Mol Cryst Liq Cryst 169 83-95... 

D. W. Honerkamp, Numerical interconversion of linear viscoelastic material functions, J. Rheol (1994) 38, pp. 1769-1795... 

D. W. Mead, Determinations of Molecular Weight Distributions of Linear Flexible Polymers from Linear Viscoelastic Material Functions J. Rheol. 38, 1797-1827 (1994). 

Master curves of the viscoelastic material functions G , G , and q were obtained by horizontally shifting these material function measured at three temperatures along their fi-equency axis. The data shifted very well with a common shift factors proving that all measurements were made in the LVE region, and that no thermal degradation had occurred. These master curves are shown in Figures 2 and 3 for the ABS neat resin and glass filled ABS, respectively. No shift factors are reported here, but it is noteworthy that they are different between these two samples. 

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