Voigt/Kelvin model

In a creep test, a constant stress o is applied onto a fiber at / = 0. The governing equation can be rewritten as  

Solving this linear differential eqnation, the strain during creep can be described as a function of time  

r is called retardation time, instead of relaxation time, since this is a creep test. If the constant stress is released at t = / , the elastic component (i.e., the spring) would retard the fiber and the strain decreases according to the following 

Another disadvantage of Kelvin-Voigt model is that it cannot be used to describe the stress relaxation behavior of polymer fibers. Under a constant strain the governing equation of the Kelvin-Voigt model becomes  

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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. 

Maxwell and Kelvin-Voigt models are to be set up to simulate the creep behaviour of a plastic. The elastic and viscous constants for the Kelvin-Voigt models are 2 GN/m and 100 GNs/m respectively and the viscous constant for the Maxwell model is 200 GNs/m. Estimate a suitable value for the elastic constant for the Maxwell model if both models are to predict the same creep strain after 50 seconds. 

A plastic which behaves like a Kelvin-Voigt model is subjected to the stress history shown in Fig. 2.87. Use the Boltzmanns Superposition Principle to calculate the strain in the material after (a) 90 seconds (b) 150 seconds. The spring constant is 12 GN/m and the dashpot constant is 360 GNs/m. ... 

Here the time derivative of the strain is represented by Newton s dot. This is the response of a purely viscous fluid. Now suppose we consider a combination of these models. The two simplest arrangements that we can visualise is the models in series or parallel. When they are placed in series we have a Maxwell model and in parallel we have a Kelvin (or sometimes a Kelvin-Voigt) model. 

Figure 4.8 A multiple Maxwell model and a multiple Kelvin- Voigt model...

The Kelvin — Voigt Model. A similar development can be followed for the case of a spring and dashpot in parallel, as shown schematically in Figure 5.61a. In this model, referred to as the Kelvin-Voigt model of viscoelasticity, the stresses are additive... 

The Four-Element ModeF. The behavior of viscoelastic materials is complex and can be better represented by a model consisting of four elements, as shown in Figure 5.62. We will not go through the mathematical development as we did for the Maxwell and Kelvin-Voigt models, but it is worthwhile studying this model from a qualitative standpoint. 

According to the Kelvin (Voigt) model of viscoelasticity, what is the viscosity (in Pa-s) of a material that exhibits a shear stress of 9.32 x 10 Pa at a shear strain of 0.5 cm/cm over a duration of 100 seconds The shear modulus of this material is 5 x 10 Pa. 

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. 

The Kelvin-Voigt model, also known as the Voigt model, consists of a Newtonian damper and Hookean elastic spring connected in parallel, as shown in the picture. It is used to explain the stress relaxation behaviors of polymers. 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

The models described so far provide a qualitative illustration of the viscoelastic behaviour of polymers. In that respect the Maxwell element is the most suited to represent fluid polymers the permanent flow predominates on the longer term, while the short-term response is elastic. The Kelvin-Voigt element, with an added spring and, if necessary, a dashpot, is better suited to describe the nature of a solid polymer. With later analysis of the creep of polymers, we shall, therefore, meet the Kelvin-Voigt model again in more detailed descriptions of the fluid state the Maxwell model is being used. 

Summarizing The basic idea, mentioned in chapter 6, that creep of solid polymers could be represented by a simple four-parameter model (the Burgers model), composed of a Maxwell and a Kelvin-Voigt model in series, appears to be inadequate for three reasons ... 

The angle 5 measures the lag of strain behind stress and is known as the loss angle of the material and provides a measure of the internal damping of stress waves. A simpler model of viscoelasticity, the Kelvin-Voigt model places j8 =0, and tan 5 = wy. 

Here the Kelvin-Voigt model is assumed to adequately describe the viscoelastic properties of the elastomer and the Lame constants can be written to include the characteristic relaxatiog times of the material. Theg become the operators A... 

Figure 1-8 Maxwell Model (Left) and Kelvin-Voigt Model (Right) Illustrate Mechanical Analogs of Viscoelastic Behavior.

SO that the Maxwell and the Kelvin-Voigt models and their combinations were used in the interpretation of results of such studies (Chapters 3 and 5). 

Figure 10,4 Schematic representation of the Kelvin-Voigt model.

The Maxwell and Kelvin-Voigt models are unable to represent conveniently the material response of a viscoelastic system. A better approach to the actual behavior is achieved by using more complex models. 

We notice that the elements in series in the mechanical model are transformed in parallel in the electrical analogy. The converse is true for the Kelvin-Voigt model. The electrical analog of a ladder model is thus an electrical filter. 

Appropriate combinations of the coefficients of Eq. (16.20) can reproduce determined idealized behavior of viscoelastic materials such as those corresponding to the Maxwell and Kelvin-Voigt models. Thus, for the Maxwell model in shear,... 

A series combination of these elements corresponds to the Maxwell model, while their parallel combination corresponds to the Kelvin-Voigt model (Fig. 54). 

The main disadvantage of the Maxwell model is that the static shear modulus p0 vanishes in this model, while the drawback of the Kelvin-Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin-Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131]... 

Figure 54. Models of viscoelastic properties (a) Maxwell model (/ Kelvin-Voigt model.

FIG. 5. Schematic diagram of Maxwell and Kelvin-Voigt models. 

Higher density aerogels have elastic moduli well above 10 MPa, but for lower densities elastic moduli become small enough to allow the atmospheric air pressure to become noticeable. The gas inside the aerogel pores influences both the total (dynamic) modulus and, at very low aerogel densities, the total density of the system. According to a simple model (Kelvin-Voigt model), the sound velocity becomes... 

To eliminate the Newtonian simplification, a rheological constitutive equation is replaced in the equations that require it. Or, in the case where viscoelasticity effects are required, the simple Kelvin-Voigt model can be used. In this case, the stress is decomposed into its viscous and elastic components, as shown in the following equation ... 

The Kelvin-Voigt model has the advantage of not needing the derivative of the stress, which is difficult to obtain experimentally, as in the Maxwell model. 

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