The Kelvin-Voigt Model

Note 1 The Voigt-Kelvin model is also known as the Voigt model or Voigt element. 

Note 3 The relationship defining the Voigt-Kelvin model may be written... 

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)). 

Note 5 The Voigt-Kelvin model may be represented by a spring and a dashpot filled with a Newtonian liquid in parallel, in which case a is the spring constant (force = a extension) and P is the dashpot constant (force = pxdAQ of extension). 

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... 

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

A second approximation of the tensile creep function is the Voigt-Kelvin model... 

The Voigt-Kelvin model may be mathematically described as follows ... 

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. 

Here we derive expressions for D and D" of a Voigt-Kelvin model consisting of z elements assuming a sinusoidal strain application. Applying equation (3-22) to the Voigt-Kelvin model experiencing a strain in the yth element given by... 

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 13.12 Stress-strain (a-e) behavior of two simple mechanical models (a) the Maxwell model and (b) the Voigt-Kelvin model. 

Figure 5 shows the major features of the Ting solution, for the Voigt/Kelvin model (Figure 5a) and the Maxwell model (Figure 5b). The elastic element has E = 2 MPa and the viscosity of the dashpot is selected to yield a relaxation time of r = 50 s. These values are similar to those of the poly(vinylethylene) sample freshly cast from...

Consider first the Voigt/Kelvin model (Figure 5a). The contact radius increases more slowly than in the elastic case and its maximum value is always smaller. As tcontact goes to values much shorter than T, tpeak shifts closer and closer to the end of the contact (e.g., tpeak tcomacd and farther and farther away from the time of maxi-... 

In the case of the Voigt/Kelvin model, not only does the contact radius increase, it shifts slightly toward shorter times as illustrated in Figure 7. This shift is at most about ten percent. Thus, the results shown in Figure 6 are only weakly affected by adhesion. 

Figure 7. Eff(K t of adhesion of the contact response of the Voigt/Kelvin model.

As a first example, consider the three-parameter model (sometimes known as the Voigt-Kelvin model) shown in Fig. 5.1. This model is best approached as a combination of a spring and a Kelvin model acting in series. The three sets of equations then become, where the subscripts 0 or s indicate the value of quantities in the free spring, the subscripts 1 or k indicate the value of quantities in the Kelvin element, and unsubscripted a and E are the remote values of stress and strain (the total stress and strain carried by the three-parameter solid). 

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). 

The Voigt-Kelvin model consists of an elastic element and a viscous element coupled in parallel (Fig. 5.23). The following constitutive equations are obtained by solving the differential equation ... 

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