Viscoelasticity Voigt model

Keywords AT-cut quartz crystal Sauerbrey equation Thickness shear mode Oscillation Eigenfrequency Dissipation factor Viscoelasticity Voigt model... 

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... 

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. 

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>. 

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. 

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

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. ... 

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. 

There are several models to describe the viscoelastic behavior of different materials. Maxwell model, Kelvin-Voigt model, Standard Linear Solid model and Generalized Maxwell models are the most frequently applied. 

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. 

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... 

Recently, we have used QCM-D to study the adsorption of pectin on BSA, a model globular protein with a well-known structure (Foster 1977 Peters 1996). The adsorption amount of pectin onto BSA surface and the thickness of adlayers at various polysaccharide concentrations have been determined by the Voigt model, which can also be used to obtain viscoelastic properties of pectin layers on the BSA surface. 

QCM-D measurements that include dissipation allow a more accurate estimate of mass changes through application of Voigt model that takes into account the viscoelastic properties of the system. Modeling software QTools supphed by Q-Sense uses the full thick layer expressions to model the response. Here, this program has been used to estimate the mass, thickness, viscosity, and shear elastic modulus of the adsorbed pectin layer on BSA surface, with a best fit between the experimental and model/and D values. 

One unique capability of the QCM-D technique is its successful extraction of quantitative information about a film s viscoelasticity. Figure 8.9 shows the variation of viscosity and shear elastic moduli of a pectin layer during pectin adsorption on the BSA surface from the Voigt model. It is noted that the shear elastic moduli is much... 

In summary, the QCM-D technique has successfully demonstrated the adsorption of pectin on the BSA surface as well as determined the viscoelastic properties of the pectin layer. As pectin concentrations increase, the adsorbed mass of pectin estimated from the Voigt model show higher values than those estimated from the Sau-erbrey equation because the former takes into account the hydrated layer. But the similar increase of thickness of pectin suggests that the pectin chains form a multilayer structure. In agreement with our previous rheology results, the main elastic character of the pectin layer in terms of Q-tool software tells us the network structure of the pectin layer on the BSA surface. In summary, QCM-D cannot only help to better understand the polysaccharide/protein interactions at the interface, but also to gain information of the nanoscale structure of polysaccharide multilayers on protein surface. 

The angle 5 measures the lag of strain behind stress and is known as the loss angle of the material and provides a measure of the internal damping of stress waves. A simpler model of viscoelasticity, the Kelvin-Voigt model places j8 =0, and tan 5 = wy. 

Here the Kelvin-Voigt model is assumed to adequately describe the viscoelastic properties of the elastomer and the Lame constants can be written to include the characteristic relaxatiog times of the material. Theg become the operators A... 

Figure 1-8 Maxwell Model (Left) and Kelvin-Voigt Model (Right) Illustrate Mechanical Analogs of Viscoelastic Behavior.

The Maxwell and Kelvin-Voigt models are unable to represent conveniently the material response of a viscoelastic system. A better approach to the actual behavior is achieved by using more complex models. 

Appropriate combinations of the coefficients of Eq. (16.20) can reproduce determined idealized behavior of viscoelastic materials such as those corresponding to the Maxwell and Kelvin-Voigt models. Thus, for the Maxwell model in shear,... 

Figure 54. Models of viscoelastic properties (a) Maxwell model (/ Kelvin-Voigt model.

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...

In flow situations where the elastic properties play a role, viscoelastic fluid models are generally needed. Such models may be linear (e.g., Voigt, Maxwell) or nonlinear (e.g., Oldroyd). In general they are quite complex and will not be treated in this chapter. For further details, interested readers are referred to the textbooks by Bird et al. [6] and Barnes et al. [25],... 

In 1874, Boltzmann formulated the theory of viscoelasticity, giving the foundation to the modem rheology. The concept of the relaxation spectmm was introduced by Thompson in 1888. The spring-and-dashpot analogy of the viscoelastic behavior (Maxwell and Voigt models) appeared in 1906. The statistical approach to polymer problems was introduced by Kuhn [1930]. 

To eliminate the Newtonian simplification, a rheological constitutive equation is replaced in the equations that require it. Or, in the case where viscoelasticity effects are required, the simple Kelvin-Voigt model can be used. In this case, the stress is decomposed into its viscous and elastic components, as shown in the following equation ... 

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