Viscoelasticity Kelvin element

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

FIGURE 40.32 Modeling the viscoelastic behavior of wood with the number of Kelvin elements in series. 

A few viscoelastic behaviors can be modeled adequately by a two-element model, but usually it is necessary to combine Maxwell and Kelvin elements. A series arrangement of the two elements, known as the four-element model, is the simplest model that exhibits all the features of viscoelasticity (Fig. 15). It is beyond the scope of this introductory chapter to derive the mathematical equations that describe the various models. Several excellent texts exist and can be consulted (66-68). 

The Four-Element Model While a few problems in viscoelasticity can be solved with the Maxwell or Kelvin elements alone, more often they are used together or in other combinations. Figure 10.5 illustrates the combination of the Maxwell element and the Kelvin element in series, known as the four-element model. It is the simplest model that exhibits all the essential features of viscoelasticity. 

The quantities E and t] of the models shown above are not, of course, simple values of modulus and viscosity. However, as shown below, they can be used in numerous calculations to provide excellent predictions or understanding of viscoelastic creep and stress relaxation. It must be emphasized that E and rj themselves can be governed by theoretical equations. For example, if the polymer is above T, the theory of rubber elasticity can be used. Likewise the WLF equation can be used to represent that portion of the deformation due to viscous flow, or for the viscous portion of the Kelvin element. 

FIGURE 15.1 Linear viscoelastic models (a) linear elastic (b) linear viscous (c) Maxwell element (d) Voigt-Kelvin element (e) three-parameter (f) four-parameter. 

Note that the Voigt-Kelvin element does not continue to deform as long as stress is applied, and it does not exhibit any permanent set (see Figure 15.7). It therefore represents a viscoelastic solid, and gives a fair qualitative picture of the creep response of some crosslinked polymers. 

The next step in the development of linear viscoelastic models is the so-called three-parameter model (Figure 15.le). By adding a dashpot in series with the Voigt-Kelvin element, we get a liquid. The differential equation for this model may be written in operator form as... 

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 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. 

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

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. ... 

In which element or model for a viscoelastic body will the elastic response be retarded by viscous resistance (a) Maxwell or (b) Voigt-Kelvin ... 

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. 

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. 

The simplest model that can be used for describing a single creep experiment is the Burgers element, consisting of a Maxwell model and a Voigt-Kelvin model in series. This element is able to describe qualitatively the creep behaviour of viscoelastic materials... 

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. 

The viscoelasticity properties are also important, because they can supply information directly related to the form of the macromolecules. The models of the linear viscoelasticity are developed from two elements a spring and a dashpot. Two of those elements in line constitute the Maxwell model and in parallel the Kelvin model (or Vogt).20 Normally, those models don t represent the behavior of complex materials satisfactorily. Other models such as the Burgers model, where the Maxwell and Kelvin models are connected in line, are used to determine the modulus of elasticity (Yj and Y2) and the coefficients of viscosity ( and t]2).21... 

This model indicates that the modulus of the polymer is the result of the individual moduli of each element, E, and the stress depends on the relaxation times, X, of each element. This equation is a better approximation to the behavior of polymers. To model the viscoelastic behavior of polymers, other models have been proposed, such as the Kelvin model [12]. 

In more detail, the flow of glass is more complex due to the combined elastic and viscous response to any type of applied stress, known as viscoelasticity. Several models have been proposed to describe viscoelasticity. Among them. Burger s model has been shown to characterize reasonably well the behavior of inorganic glasses [5]. In this version, illustrated in Fig. 3a, viscous (771) and elastic (El) elements are combined in series with a Kelvin solid, where two other elements (772, 2) are arranged in parallel and reflect the slow elastic properties. The rate of deformation under constant tensile stress a and zero initial deformation is made up from the rate of Newton s viscous deformation,... 

Fig. 6.10 Illustration of a series connection of the Maxwell model and the Kelvin model fw the four-element model to describe the viscoelastic creep behaviors of polymers...

A series crmnection of the Maxwell and Kelvin models makes the four-element model, known as the Burger s model (Burgers 1935), which can describe the viscoelastic creep behaviors of polymers, as given by... 

The Kelvin—Voigt Model. The other two-element mechanical model for viscoelasticity is the Kelvin-Voigt model in which the spring and dashpot are in parallel. In this model, the deformation or creep response to the imposition of a constant load is illustrated. In this instance a constant load is applied at t = 0 and the deformation is monitored. The Kelvin-Voigt model and its response are illustrated in Fignre 3. The material property of interest in this case is the creep compliance Jit) and it is written as... 

---