Polymers Voigt-Kelvin model

If the generalized Voigt-Kelvin model is to represent a viscoelastic liquid such as a linear polymer, the modulus of one of the springs must be zero (infinite compliance), leaving a simple dashpot in series with all the other Voigt-Kelvin elements. Sometimes, the steady-flow response of this lone dashpot, ydashpot= (To/rjo)t, is subtracted from the overall response, leaving the compliances to represent only the elastic contributions to the overall response ... 

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 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. 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Evidently a fluid polymer cannot be considered in the model the deformation approaches to a limit. For a solid polymer the model seems more appropriate, though is represents neither a spontaneous elastic deformation nor permanent flow. Therefore a combination of a Kelvin-Voigt element with a spring and with a dashpot in series is, in principle, more appropriate. 

The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the stress relaxation behaviors of polymers. 

The models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

Time response of different rheological systems to applied forces. The Maxwell model gives steady creep with some post stress recovery, representative of a polymer with no cross-linking. The Kelvin-Voigt model gives a retarded viscoelastic behavior expected from a cross-linked polymer. 

A reasonable model for the behaviour of some high polymers is the Voigt or Kelvin substance, which may be represented by a spring in parallel with a dashpot. Show that, for such a substance (a) if at time zero when the strain is zero, a constant stress is applied... 

The Burgers Model. As a preview to the viscoelastic behavior of polymers, we next consider the four-element Burgers model that captures a minimum set of behaviors that is seen in polymeric materials and as discussed here (13). The insert in Figure 4 shows the Burgers model as a Maxwell model in series with a Kelvin-Voigt model. As shown in Figure 4, upon application of a constant stress Ti2 for a time ti followed by its removal, the model captures the folloAving aspects of polymer viscoelasticity ... 

Another disadvantage of Kelvin-Voigt model is that it cannot be used to describe the stress relaxation behavior of polymer fibers. Under a constant strain the governing equation of the Kelvin-Voigt model becomes ... 

This means in the Kelvin-Voigt model, the stress is constant during the stress relaxation test (Figure 16.22B), which is not trae for real polymer fibers. 

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