The Maxwell Fluid

Kinematics, the analysis of the deformation of the element, require that 

Comparison with Equation 3.3 shows that for the Maxwell fluid, 

To find the constant of integration C note that as t — O, that the strain will be the initial elastic response since there has been no time elapsed for creep to occur. Thus, 

FIGU RE 3.5 Creep and relaxation of the Maxwell fluid, (a) Creep curve for a Maxwell fluid 

This equation represents the relaxation of a Maxwell fluid. As t °o, o — 0. The stress completely relaxes out, which is also characteristic of a fluid. 

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The simplest model that can show the most important aspects of viscoelastic behaviour is the Maxwell fluid. A mechanical model of the Maxwell fluid is a viscous element (a piston sliding in a cylinder of oil) in series with an elastic element (a spring). The total extension of this mechanical model is the sum of the extensions of the two elements and the rate of extension is the sum of the two rates of extension. It is assumed that the same form of combination can be applied to the shearing of the Maxwell fluid. 

In Chapter 1 it was pointed out that the Maxwell fluid is a very simple model of the first order effects observed with viscoelastic liquids. The equation of a Maxwell fluid is... 

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. 

Figure 22.9 Physical representation of the Maxwell fluid element.

Table 9.3. Components of the Maxwell fluid. Equation 9.16, for two-dimensional flows...

The models described in the preceding section are useful in developing mathematical relations between stress and strain in viscoelastic polymers and in giving insight to their response to creep, relaxation and other types of loading. Consider again the Maxwell fluid from Fig. 3.21,... 

On the other hand, the Maxwell fluid model explains the response of complex fluids to an oscillatory shear rate. The frequency-dependent behavior of this model, displayed into linear responses to applied shear rates has been found to be applicable to a variety of complex fluid systems. Although the linear viscoelasticity is useful for understanding the relationship between the microstructure and the rheological properties of complex fluids, it is important to bear in mind that the linear viscoelasticity theory is only valid when the total deformation is quite small. Therefore, its ability to distinguish complex fluids with similar micro- and nanostructure or molecular structures (e.g. linear or branched polymer topology) is limited. However, complex fluids with similar linear viscoelastic properties may show different non-linear viscoelastic properties [31]. 

The single spring element of the Maxwell fluid (Niven 2003) does not exhibit any time-related strain behavior and the response to stress (Tq at t = 0 can be expressed as follows ... 

Solution 3.12 is plotted in Figure 3.5a. The Maxwell fluid has an initial elastic modulus... 

Next, consider a relaxation test. Figure 2.4a, applied to the Maxwell fluid. It is more realistic to consider relaxation after some time tj, after some creep has occurred as shown in Figure 3.5b. Thus, for t > h, e = 0, e = Ej = constant, and from Equation 3.9,... 

In viscoelastic behavior of polymers there is another mechanical property which is characteristic. It is the relaxation time denoted as x. It is the time (arbitrarily defined for convenience) when the stress has relaxed to (1/e) of its original value. For the Maxwell fluid, the relaxation time is... 

The Maxwell fluid and Kelvin solid model with one spring and one dashpot actually fits very little experimental data well. Experimental results of creep and stress relaxation often flts with a Prony series that is a series of Maxwell fluid elements in parallel or a series of Kelvin solids in series. It is typical to utilize one element per decade of time. While this is not always necessary, it has historically been standard convention. The Prony series for the Maxwell fluid and Kelvin solid are... 

The differential equation for the Maxwell fluid (Equation 3.9) assuming true stress and true strain, is... 

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