Viscoelasticity Kelvin model

FIGURE 28.7 Viscoelastic Kelvin Model. (Redrawn from Bhattacharya, S.N., Rheology Fundamentals and Measurements, RMIT University, Melbourne, Australia, 2004.)... 

It is apparent therefore that the Superposition Principle is a convenient method of analysing complex stress systems. However, it should not be forgotten that the principle is based on the assumption of linear viscoelasticity which is quite inapplicable at the higher stress levels and the accuracy of the predictions will reflect the accuracy with which the equation for modulus (equation (2.33)) fits the experimental creep data for the material. In Examples (2.13) and (2.14) a simple equation for modulus was selected in order to illustrate the method of solution. More accurate predictions could have been made if the modulus equation for the combined Maxwell/Kelvin model or the Standard Linear Solid had been used. 

During a test on a polymer which is to have its viscoelastic behaviour described by the Kelvin model the following creep data was obtained when a stress of 2 MN/m was applied to it. 

We have developed the idea that we can describe linear viscoelastic materials by a sum of Maxwell models. These models are the most appropriate for describing the response of a body to an applied strain. The same ideas apply to a sum of Kelvin models, which are more appropriately applied to stress controlled experiments. A combination of these models enables us to predict the results of different experiments. If we were able to predict the form of the model from the chemical constituents of the system we could predict all the viscoelastic responses in shear. We know that when a strain is applied to a viscoelastic material the molecules and particles that form the system gradual diffuse to relax the applied strain. For example, consider a solution of polymer... 

Note 7 There are definitions of linear viscoelasticity which use integral equations instead of the differential equation in Definition 5.2. (See, for example, [11].) Such definitions have certain advantages regarding their mathematical generality. However, the approach in the present document, in terms of differential equations, has the advantage that the definitions and descriptions of various viscoelastic properties can be made in terms of commonly used mechano-mathematical models (e.g. the Maxwell and Voigt-Kelvin models). 

Note 4 Comparison with the general definition of linear viscoelastic behaviour shows that the polynomial /"(D) is of order zero, 0(D) is of order one, ago = a and a = p. Hence, a material described by the Voigt-Kelvin model is a solid (go > 0) without instantaneous elasticity (/"(D) is a polynomial of order one less than 0(D)). 

In the Voigt-Kelvin model for viscoelastic deformation, it is assumed that the total stress is equal to the sum of the viscous and elastic stress, 5 = + So, so that... 

The static tests considered in Chapter 8 treat the rubber as being essentially an elastic, or rather high elastic, material whereas it is in fact viscoelastic and, hence, its response to dynamic stressing is a combination of an elastic response and a viscous response and energy is lost in each cycle. This behaviour can be conveniently envisaged by a simple empirical model of a spring and dashpot in parallel (Voigt-Kelvin model). 

Figure 2 Graphical representation of the Voigt-Kelvin model (a) and the Maxwell model (b) of viscoelasticity. rd is the retardation time and

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... 

For a more complete discussion of Maxwell and Kelvin models and viscoelasticity (96-98,101). 

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]. 

The gel layer is treated as a Kelvin-Voight viscoelastic material where the gel shear is acting in parallel. This viscoelastic material stress has both an elastic and viscous component acting in parallel. This viscoelastic material model was substituted into the momentum equation and the resulting gel layer equation of motion is... 

Figure 5.14 Common viscoelastic models a) Voigt/Kelvin model b) Zener model/standard linear solid.

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 theory of non-isothermal viscoelastic behavior as developed by Hopkins [2] and Haugh [3] may be based on the representation of linear viscoelastic behavior by mechanical models. The linear viscoelastic behavior of polymers in simple shear at constant temperature and prescribed stress history may be expressed in terms of the deformation of a generalized Kelvin model. Spring constants and dashpot viscosity constants of the model have to be appropriately chosen the choice depends on temperature. For the non-isothermal treatment, the elasticity of the springs and the viscosities of the dashpots have to be inserted as functions of temperature. Due to the prescribed temperature history, they become functions of time. 

We have seen that the Maxwell model describes the stress relaxation of a viscoelastic solid to a first approximation, and the Kelvin model the creep behaviour, but that neither model is adequate for the general behaviour of a viscoelastic solid where it is necessary to describe both stress relaxation and creep. 

Figure 2.13 Three-element model for viscoelastic behavior of liquids (general relaxation model) as a combination of Maxwell and Voigt-Kelvin models.

For small stresses, we can use the approximation sinha w a in equation (8.3) so that the strain rate is proportional to the applied stress. In this case, the behaviour is linear and viscous. As stresses are small, the deformation is not plastic, but elastic, for there is a restoring force corresponding to the spring element in figure 8.7(a), whereas equation (8.3) describes the dashpot element of the Kelvin model. The behaviour is thus linear viscoelastic. At larger stresses, deviations from linearity occur, although the behaviour is still viscoelastic. 

FIGURE 2. Two-element models for linear viscoelasticity (a) Maxwell model (b) Voigt (Kelvin) model. 

Figure 5. Solutions to the Ting model for contact of a rigid probe with a viscoelastic substrate described by (a) the Maxwell model and (b) the Voigt/Kelvin model.

In a mechanical model, each spring or dashpot represents a mechanical analogue to the response of the material. However, the most complex mechanical model may not be able to describe polymer concrete. In the case of rPET polymer concrete, the Maxwell and Kelvin models have elements which allow representation of the viscoelastic response. A combination of these two models in series satisfactorily describes the creep response of rPET polymer concrete. This multiparameter model is shown in Figure 4.12. 

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