Kelvin-Voigt model linear viscoelasticity

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. 

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

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. 

One must note that the balance equations are not dependent on either the type of material or the type of action the material undergoes. In fact, the balance equations are consequences of the laws of conservation of both linear and angular momenta and, eventually, of the first law of thermodynamics. In contrast, the constitutive equations are intrinsic to the material. As will be shown later, the incorporation of memory effects into constitutive equations either through the superposition principle of Boltzmann, in differential form, or by means of viscoelastic models based on the Kelvin-Voigt or Maxwell models, causes solution of viscoelastic problems to be more complex than the solution of problems in the purely elastic case. Nevertheless, in many situations it is possible to convert the viscoelastic problem into an elastic one through the employment of Laplace transforms. This type of strategy is accomplished by means of the correspondence principle. 

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]. 

The equation represents one of the simple models for linear viscoelastic behaviour, the Kelvin or Voigt model, and is discussed in detail in Section 4.2.3 below. 

Fig. 3 Common viscoelastic models (a) Maxwell, (b) Kelvin-Voigt, (c) Wiechert, (d) Kelvin, and (e) standard linear solid...

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... 

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

The origin of the theory of viscoelasticity may be traced to various isolated researchers in the last decades of the nineteenth Century. This early stage of development is essentially due to the work of Maxwell, Kelvin and Voigt who independently studied the one dimensional response of such materials. The linear constitutive relationships introduced therein are the base of rheological models which are still used in many applications [121]. Their works led to Boltzmann s [122] first formulation of three dimensional theory for the isotropic medium, which... 

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 ... 

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