Viscoelasticity Kelvin-Voigt model

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... 

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

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

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.

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). 

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 ... 

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. 

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

In addition, a time-dependent viscoelastic component described by the so-called Voigt-Kelvin configuration, i.e. the combination of spring E and damper Yf comes into action. Further, a viscoplastic flow component may exist, modelled by the damper of viscosity... 

Spring-and-dashpot models are extended by the Voigt-Kelvin (V-K) model, which broadens linear viscoelastic concepts. The spring and dashpot are always in parallel. The V-K spring-and-dashpot models are useful for understanding creep behavior [11]. 

This makes the simplest possible assumption that the shear stresses related to strain and strain rate are additive. The equation represents one of the simple models for linear viscoelastic behaviour (the Voigt or Kelvin model) and will be discussed in detail in Section 5.2.5. 

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. 

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... 

Earlier in the theory of viscoelasticity many rheological models with combinations of the Maxwell and the Voigt - Kelvin bodies were considered (see Freudental and Geiringer [1]). These models have constitutive laws for stresses o j and strains eij which include time derivatives of arbitrary order. 

A variety of models have been employed to explain the viscoelastic behaviour of polymeric materials. The Maxwell unit, consisting of a spring and dashpot in series, and the Kelvin (or Voigt) unit, consisting of a spring and dashpot in parallel, are the simplest of these models (see Figures 7 and 8). The Maxwell and Kelvin models lead to analogous equations and have similar limitations. Here we consider only the Maxwell model. 

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. 

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