Voigt element

The more gradual approach to equilibrium than the model predicts can be taken into account by imagining that the rise consists of a series of n smaller (and unresolved) steps. This is equivalent to expanding the model so that it consists of n Voigt elements as shown in Fig. 3.10b. Each of these Voigt elements is characterized by its own value for G, 77, and r. 

In addition to the set of Voigt elements, a Maxwell element could also be included in the model. The effect is to include a contribution given by Eq. (3.69) to the calculated compliance. This long time flow contribution to the compliance is exactly what we observe for non-cross-linked polymers in Fig. 3.12. 

Next suppose we consider the effect of a periodically oscillating stress on a Voigt element of modulus G and viscosity 77. Remember from the last section that for a Voigt element the appUed stress equals the sum of the elastic and viscous responses of the model. Therefore, for a stress which varies periodically, Eq. (3.64) becomes... 

Figure 3.16a shows the storage and loss components of the compliance of crystalline polytetrafluoroethylene at 22.6°C. While not identical to the theoretical curve based on a single Voigt element, the general features are readily recognizable. Note that the range of frequencies over which the feature in Fig. 3.16a develops is much narrower than suggested by the scale in Fig. 3.13. This is because the sample under investigation is crystalline. For amorphous polymers, the observed loss peaks are actually broader than predicted by a...

The spring constant, 2, for the Kelvin-Voigt element is obtained from the maximum retarded strain, 2, in Fig. 2.40. 

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

Note 1 The Voigt-Kelvin model is also known as the Voigt model or Voigt element. 

Voigt-Kelvin element Voigt-Kelvin model Voigt element Voigt model volume compression vorticity tensor width of the resonance curve Young s modulus zero-shear viscosity... 

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. 

Without the superposition principle we find the same result After taking away the stress the spring Ei is unloaded and we keep the deformed Kelvin-Voigt element with a strain e = a/Eiy - exp(-ii/T)) = e t ). 

The spring Ei now pulls back the Kelvin-Voigt element from t=ti with a stress (7, proportional to the remaining deformation e a = i e( -1 ). 

A parallel array of E and h gives a Kelvin-Voigt element. This model does not allow an instantaneous deformation (the stress on the dashpot would be infinite), and it does not show stress relaxation. At a constant stress it exhibits creep at time t its strain is ( ) the stress in the spring then is ... 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

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. 

In a similar way the creep can be represented by a number of Kelvin-Voigt elements in series ... 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

These three complications are schematically shown in Figure 7.11, using creep isochrones as a reference (a) a Kelvin-Voigt element has been chosen with a spring in series (a Burgers model without irreversible flow). 

Viscoelasticity Models For characterization with viscoelasticity models, simulation models have been developed on the basis of Kelvin, Maxwell, and Voigt elements. These elements come from continuum mechanics and can be used to describe compression. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

Surface crack length Fitted constant Fibre diameter Die exit diameter Young s modulus Crystal modulus Modulus of Maxwell element Modulus of Voigt element Strain energy release rate Boltzmann s constant... 

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