Voigt model/element

Since the strain is the same in both elements in the Voigt model, the applied stress (subscript 0) must equal the sum of the opposing forces arising from the elastic and viscous response of the model ... 

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

The viscoelastic behaviour of a certain plastic is to be represented by spring and dashpot elements having constants of 2 GN/m and 90 GNs/m respectively. If a stress of 12 MN/m is applied for 100 seconds and then completely removed, compare the values of strain predicted by the Maxwell and Kelvin-Voigt models after (a) 50 seconds (b) 150 seconds. 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

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. ... 

Voigt-Kelvin model Voigt-Kelvin 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... 

The Four-Element ModeF. The behavior of viscoelastic materials is complex and can be better represented by a model consisting of four elements, as shown in Figure 5.62. We will not go through the mathematical development as we did for the Maxwell and Kelvin-Voigt models, but it is worthwhile studying this model from a qualitative standpoint. 

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. 

This equation for the dielectric constant is the analogue of the compliance of a mechanical model, the so-called Jeffreys model, consisting of a Voigt-Kelvin element characterised by Gi and rp and t =t /Glr in series with a spring characterised by Gz- The creep of this model under the action of a constant stress aQ is (Bland, 1960)... 

The Voigt-Kelvin element (retarded elastic response), represented by a spring and a dashpot in parallel. The elastic response is not instantaneous but retarded by a viscous resistance. The two contributions to the stress are additive in this model whereas the strains are equal ... 

Stoner et al. (1974) have proposed a mechanical model for postmortem striated muscle it is shown in Figure 8-28. The model is a combination of the Voigt model with a four-element viscoelastic model. The former includes a contractile element (CE), which is the force generator. The element SE is a spring that is passively elongated by the shortening of the CE and thus develops an... 

Figure 12.10 shows the mechanical response as a funetion of time of two structures formed by combining a spring and a dashpot in series (Maxwell model) and in parallel (Voigt model). Creep is slow deformation of a viseoelastic material under constant stress (o), while relaxation is the time response of the stress after imposing a constant deformation (e). Thus, a simple mathematieal expression accounts for the relation between structure (combination of elements) and a property (creep or relaxation). Evidently, more complex responses ean be obtained by eombining several elements in series and in parallel. 

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... 

The simplest flaws of the Maxwell and Voigt models, the fact that one cannot model creep while the other cannot model stress relaxation, can easily be fixed by combining our basic linear elements in different ways. One such is the so-called four-parameter model (Figure 13-94), which combines a Maxwell model in series with a Voigt model. The four parameters are the Maxwell modulus and viscosity, Eu and and the Voigt modulus and viscosity Ev and r v... 

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. 

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

We notice that the elements in series in the mechanical model are transformed in parallel in the electrical analogy. The converse is true for the Kelvin-Voigt model. The electrical analog of a ladder model is thus an electrical filter. 

In some cases, the models used for impedance are strictly defined. Others, such as the Voigt model, allow use of an arbitrary number of parameters. The fit of a Voigt model can be improved by sequentially adding RC elements, and the best model is achieved when the x statistic reaches a minimum value. 

A series combination of these elements corresponds to the Maxwell model, while their parallel combination corresponds to the Kelvin-Voigt model (Fig. 54). 

The Maxwell model conforms to the series connection of these elements, and the Voigt model conforms to the parallel connection. 

Fig. 8 Three-element network describing a viscoelastic solid. Leaving out the spring on the right-hand side leads to the Voigt model [92]. tfowever, this model predicts infinite stress at infinite frequency. Since the frequency of the QCM is high, the Voigt model misses an essential bit of the pictme...

We note that the Voigt model predicts that strain is not a continuous function of stress that is, the element does not deform continuously with the sustained application of a constant stress. The strain approaches an asymptomatic value given by (Oq/E). The strain of the element at equilibrium is simply that of an ideal elastic solid. The only difference is that the element does not assume this strain instantaneously, but approaches it gradually. The element is shown to exhibit retarded elasticity. In creep recovery, the Maxwell element retracts instantaneously but not completely, whereas the Voigt element exhibits retarded elastic recovery, but there is no permanent set. 

In the Kelvin or Voigt model the spring and dashpot elements are connected in parallel, as shown in Figure 3.13a. This model roughly approximates the behavior of rubber. When the load is applied at zero time, the elastic deformation cannot occur immediately because the rate of flow is limited by the dashpot. Displacements continue until the strain equals the elastic deformation of the spring and it resists further movement. On removal of the load the spring recovers the displacement by reversing the... 

The elements can be combined in series or parallel as shown in Figs 7.1 and 5.5. The convention for these models is that elements in parallel undergo the same extension. It is obvious that elements in series experience the same force. Thus, in the Maxwell model, the spring and dashpot in series experience the same force, while in Voigt model the spring and dashpot in parallel experience the same extension x. The total force f across the Voigt model can be written as the differential equation... 

The standard linear solid (SLS) is a more complicated model than the two previously considered. It combines series and parallel elements, as shown in fig. 7.9, and can describe both stress-relaxation and creep. For stress-relaxation the spring a remains at the original strain and only E), nd rj are involved in the relaxation. Hence r = r]/E, but the stress relaxes to eE, not to zero. For creep it can be shown that t = (l/ a + 1 Unlike the Voigt model, the SLS exhibits an immediate response, e = a/ E + E, because the two springs in parallel can extend immediately. Thus the SLS is a much better model than either of the simpler models. 

Two models that can duplicate viscoelastic behavior are the Voigt model with a spring and a dashpot coupled in parallel, and the Maxwell model with a spring and a dashpot coupled in series. Both are shown schematically in Fig. 4.158. The ratio of viscosity to modulus of one element of the Voigt model is called the retardation time... 

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