Voigt model,

A spring and dashpot in parallel is called a Voigt model. 

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

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 ... 

Figure 3.10 Voigt models consisting of a spring and dashpot in parallel (a) simple Voigt unit and (b) set of units arranged in series.

It is interesting to note that the Voigt model is useless to describe a relaxation experiment. In the latter a constant strain was introduced instantaneously. Only an infinite force could deform the viscous component of the Voigt model instantaneously. By constrast, the Maxwell model can be used to describe a creep experiment. Equation (3.56) is the fundamental differential equation of the Maxwell model. Applied to a creep experiment, da/dt = 0 and the equation becomes... 

As we did in the case of relaxation, we now compare the behavior predicted by the Voigt model—and, for that matter, the Maxwell model—with the behavior of actual polymer samples in a creep experiment. Figure 3.12 shows plots of such experiments for two polymers. The graph is on log-log coordinates and should therefore be compared with Fig. 3.11b. The polymers are polystyrene of molecular weight 6.0 X 10 at a reduced temperature of 100°C and cis-poly-isoprene of molecular weight 6.2 X 10 at a reduced temperature of -30°C. 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

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 ... 

Through the dashpot a viscous contribution was present in both the Maxwell and Voigt models and is essential to the entire picture of viscoelasticity. These have been the viscosities of mechanical units which produce equivalent behavior to that shown by polymers. While they help us understand and describe observed behavior, they do not give us the actual viscosity of the material itself. 

Fig. 7. Voigt model analysis of (a) lateral contact stiffness and (b) the response time, t, for a silicon nitride tip vs. poly(vinylethylene) as a function of frequency and polymer aging times. Reprinted with permission from ref [71].

This is the governing equation for the Kelvin (or Voigt) Model and it is interesting to consider its predictions for the common time dependent deformations. 

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. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 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>. 

A plastic which behaves like a Kelvin-Voigt model is subjected to the stress history shown in Fig. 2.87. Use the Boltzmanns Superposition Principle to calculate the strain in the material after (a) 90 seconds (b) 150 seconds. The spring constant is 12 GN/m and the dashpot constant is 360 GNs/m. ... 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

In the parallel coupling of the Kelvin or Voigt model, the applied stress, (Tq, is simply the sum of the stresses of the individual components ... 

FIGURE 21.7 A double-Voigt model depicting measurement of dynamic nanofishing. 

FIGURE 21.8 The behavior of (a) chain stiffness, kz and (b) viscosity of the chain, tj2 against the extension. The values of 2 and -172 were calculated by a double-Voigt model. 

Figure 5. Representation of stress at advancing crack tip, and below, the state of viscoelastic strain indicated by Voigt models (35).

Here the time derivative of the strain is represented by Newton s dot. This is the response of a purely viscous fluid. Now suppose we consider a combination of these models. The two simplest arrangements that we can visualise is the models in series or parallel. When they are placed in series we have a Maxwell model and in parallel we have a Kelvin (or sometimes a Kelvin-Voigt) model. 

Figure 4.8 A multiple Maxwell model and a multiple Kelvin- Voigt model...

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. ... 

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... 

Describe the differences between the Kelvin and Voigt models for viscoelasticity and identify their corresponding equations. 

The Kelvin — Voigt Model. A similar development can be followed for the case of a spring and dashpot in parallel, as shown schematically in Figure 5.61a. In this model, referred to as the Kelvin-Voigt model of viscoelasticity, the stresses are additive... 

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. 

According to the Kelvin (Voigt) model of viscoelasticity, what is the viscosity (in Pa-s) of a material that exhibits a shear stress of 9.32 x 10 Pa at a shear strain of 0.5 cm/cm over a duration of 100 seconds The shear modulus of this material is 5 x 10 Pa. 

The parallel arrangement, also shown in Figure 2.49, is called the Voigt model. It is used to model the behavior of a cross-linked but sluggish polymer, such as one of the polyacrylates. Since the spring and dashpot must move in parallel, both the deformation and the recoverability are retarded. 

The use of the Voigt model to characterize cross-linked but sluggish polymers is illustrated in Figure 2.54. As shown by the corresponding equation in this case,... 

Mechanical tests indicate that these blends do not behave like conventional blends and suggest that the polystyrene phase is continuous in the substrate. The moduli of the blends as a function of blend composition is plotted in Figure 10.6. The Voigt and Reuss models are provided for comparison (Nielsen, 1978) These are the theoretical upper and lower bounds, respectively, on composite modulus behavior our data follows the Voigt model, suggesting that both the polystyrene and polyethylene phases are continuous. In most conventional composites of polystyrene and HDPE, the moduli fall below the Voigt prediction indicating that the phases are discontinuous and dispersed (Barentsen and Heikens, 1973 Wycisk et al., 1990). 

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