Generalized Kelvin-Voigt model

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. 

Fig. 5. (a) The generalized Maxwell model, (b) The generalized Kelvin—Voigt model. 

Fig. 2.15 (a) GeneraUzed MaxweU model, (b) generalized Kelvin-Voigt model... 

Creep and the Retardation Spectra Generalized Kelvin-Voigt Model... 

In a similar way, for an infinite number of elements in the generalized Kelvin-Voigt model /(A) maybe used to express the probability density of retardation times and the creep function (f) for the spectrum in the case of the generalized Kelvin-Voigt model can be written as ... 

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)). 

Analyses of the results obtained depend on the shape of the specimen, whether or not the distribution of mass in the specimen is accounted for and the assumed model used to represent the linear viscoelastic properties of the material. The following terms relate to analyses which generally assume small deformations, specimens of uniform cross-section, non-distributed mass and a Voigt-Kelvin solid. These are the conventional assumptions. 

The Voigt-Kelvin model is a generalization of the Voigt element that results from connecting Voigt elements in series (Figure 3-9). Here the compliance functions are easily calculated, while the modulus functions are rather complicated. The results are summarized in Table 3-2 a sample calculation is provided below. 

Table 3-2. Behavior of Generalized Maxwell and Voigt-Kelvin Models in Various Experiments...

The generalized Voigt element or the Voigt-Kelvin model is a series arrangement of an arbitrary number of Voigt elements (Figure 14.12). Under creep, the creep response of each individual element is given by... 

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

More complex arrangements of elements are often used, especially if multiple relaxations are involved or if accurate representations of engineering data are required. The Maxwell-Weichert model consists of a very large (or infinite) number of Maxwell elements in parallel (2). The generalized Voigt-Kelvin model places a number of Kelvin elements in series. In each of these models, a spring or a dashpot may be placed alone, indicating elastic or viscous contributions. 

However, this model still has no instantaneous elasticity and a free spring is normally included in series with the generalized Kelvin solid with the result (sometimes referred to as the Voigt-Kelvin model),... 

The Voigt model is good for modeling viscoelastic solids in creep experiments. The more generalized version is the Voigt-Kelvin model, which is a series expansion of the Voigt model (Fig. 8). 

For creep tests, a generalized Voigt-Kelvin model is used (Figure 15.11). The creep response of an individual Voigt-Kelvin element is given by... 

FIGURE 15.11 Generalized Voigt-Kelvin model for an input shear stress, r. This model could also be used for modeling tensile stress, tr, with the corresponding moduli E,-. 

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 ... 

An analogous procedure based on (18.25) can be used to extract the and Ji = 1/Gj in a generalized Voigt-Kelvin model from creep data, (t). It requires a knowledge of the equilibrium elastic compliance, (oo). Its development will be left as an end-of-chapter exerdse. 

Describe how to apply Procedure X to recoverable elastic compliance data, to obtain the discrete parameters J/ and A.- in a generalized Voigt-Kelvin model... 

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