Maxwell-Wiechert model

In practice the stress relaxation behaviour has to be described expressed with N Maxwell elements connected in parallel, each with its own spring constant E and relaxation time t (the so-called Maxwell-Wiechert model) ... 

Although the Maxwell-Wiechert model and the extended Burgers element exhibit the chief characteristics of the viscoelastic behaviour of polymers and lead to a spectrum of relaxation and retardation times, they are nevertheless of restricted value it is valid for very small deformations only. In a qualitative way the models are useful. The flow of a polymer is in general non-Newtonian and its elastic response non-Hookean. 

In a Maxwell-Wiechert model both moduli become... 

FIG. 13.21 Double logarithmic plot of the reduced dynamic moduli of a Maxwell-Wiechert model with two relaxation times of 10,000 s and 1 s, vs. angular frequency. The corresponding spring constants are G and 100G, respectively. 

The reality, however, is not as simple as that. There are several possibilities to describe viscosity, 77, and first normal stress difference coefficient, P1. The first one originates from Lodge s rheological constitutive equation (Lodge 1964) for polymer melts and the second one from substitution of a sum of N Maxwell elements, the so-called Maxwell-Wiechert model (see Chap. 13), in this equation (see General references Te Nijenhuis, 2005). 

FIGURE 13-96 Schematic diagram of die Maxwell-Wiechert model. 

One obvious way of introducing a range of relaxation and retardation times into the problem is to construct mathematical models thai are equivalent to a number of Maxwell and/or Voigt models connected in parallel (and/or series). The Maxwell-Wiechert model (Figure 13-96), for example, consists of an arbitrary number of Maxwell elements connected in parallel. For simplicity let s see what you get with, say, three Maxwell elements and then extrapolate later to an arbitrary number, n. 

Transient loading patterns. Creep relaxation, stress relaxation, constant rate of strain and constant rate of stress are all examples of transient loading patterns. The Maxwell model (Fig. 7) is used to represent a viscoelastic liquid and is especially useful for stress relaxation experiments. This simple model can be generalized to give good approximations for real material systems by combining numerous elements in parallel. This is known as a Maxwell-Wiechert model. 

The mechanical models discussed above are based on single relaxation (or retardation) time. Real polymer fibers have a spectrum or distribution of relaxation and retardation times due to the existence of different types of conformational changes. One convenient way to introduce a range of relaxation times into the problem is to constmct models consisting of a number of Maxwell and/or Kelvin-Voigt sub-models connected in parallel and/or series. Figure 16.24 shows a Maxwell-Wiechert model, which is constmcted by connecting an aibitraiy number of... 

The total modulus of the Maxwell-Wiechert model also can be obtained ... 

The Maxwell-Wiechert model also can be used to describe the creep behavior of polymer fibers. However, for the creep behavior, it is mathematically more convenient to create a model involving a range of retardation times by connected a number of Kelvin-Voigt sub-models in series. 

Fig. 3 Common viscoelastic models (a) Maxwell, (b) Kelvin-Voigt, (c) Wiechert, (d) Kelvin, and (e) standard linear solid...

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