Maxwell-Weichert model

If one of the Maxwell elements in the Maxwell-Weichert model is replaced with a spring or a dashpot of infinite viscosity, then the stress in such a model would decay to a finite value rather than zero. This would approximate the behavior of a cross-linked polymer. 

As can be seen, the Maxwell-Weichert model possesses many relaxation times. For real materials we postulate the existence of a continuous spectrum of relaxation times (A,). A spectrum-skewed toward lower times would be characteristic of a viscoelastic fluid, whereas a spectrum skewed toward longer times would be characteristic of a viscoelastic solid. For a real system containing crosslinks the spectrum would be skewed heavily toward very long or infinite relaxation times. In generalizing, A may thus he allowed to range from zero to infinity. The concept that a continuous distribution of relaxation times should be required to represent the behavior of real systems would seem to follow naturally from the fact that real polymeric systems also exhibit distrihutions in conformational size, molecular weight, and distance between crosslinks. 

More complex arrangements of elements are often used, especially if multiple relaxations are involved or if accurate representations of engineering data are required. The Maxwell-Weichert model consists of a very large (or infinite) number of Maxwell elements in parallel (2). The generalized Voigt-Kelvin model places a number of Kelvin elements in series. In each of these models, a spring or a dashpot may be placed alone, indicating elastic or viscous contributions. 

The response to an applied strain of 200% (e = 2) was then studied. As shown, the logarithmic term of Eq. (6) may be neglected for large N, and the response of the model then becomes equivalent to that of a generalized Maxwell body (Weichert body) with 48 discrete relaxation times, in parallel with a spring. 

The behavior of a polymer system is so complicated that we cannot represent it with the response time of a single Maxwell element. In other words, the simple model described above cannot approach the behavior of a real system. In 1893, Weichert showed that stress-relaxation experiments could be represented as a generalization of Maxwell s equation. The mechanical model according to Weichert s formulation is shown in Figure 3.11 it consists of a large number of Maxwell elements coupled in parallel. 

FIGURE 3.11 Generalized Maxwell model (Weichert s formulation). 

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