Voigt-Kelvin element

The spring constant, 2, for the Kelvin-Voigt element is obtained from the maximum retarded strain, 2, in Fig. 2.40. 

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

Evidently a fluid polymer cannot be considered in the model the deformation approaches to a limit. For a solid polymer the model seems more appropriate, though is represents neither a spontaneous elastic deformation nor permanent flow. Therefore a combination of a Kelvin-Voigt element with a spring and with a dashpot in series is, in principle, more appropriate. 

Without the superposition principle we find the same result After taking away the stress the spring Ei is unloaded and we keep the deformed Kelvin-Voigt element with a strain e = a/Eiy - exp(-ii/T)) = e t ). 

The spring Ei now pulls back the Kelvin-Voigt element from t=ti with a stress (7, proportional to the remaining deformation e a = i e( -1 ). 

A parallel array of E and h gives a Kelvin-Voigt element. This model does not allow an instantaneous deformation (the stress on the dashpot would be infinite), and it does not show stress relaxation. At a constant stress it exhibits creep at time t its strain is ( ) the stress in the spring then is ... 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

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. 

In a similar way the creep can be represented by a number of Kelvin-Voigt elements in series ... 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

These three complications are schematically shown in Figure 7.11, using creep isochrones as a reference (a) a Kelvin-Voigt element has been chosen with a spring in series (a Burgers model without irreversible flow). 

From data in a stress relaxation experiment (Chapter 3), where the strain is constant and stress is measured as a function oftime, a If), the relaxation time may be estimated from the time necessary for [cr(r)/cr(0)] to become (1/e) = 0.368. Typically, several Maxwell elements are used to fit experimental data, a t). For the Kelvin-Voigt element (Figure 1-8, right) under stress, the equation is ... 

The Kelvin-Voigt elements are used to describe data from a creep experiment and the retardation time (t2) is the time required for the spring and the dashpot to deform to (1 — 1 /e), or 63.21 % of the total creep. In contrast, the relaxation time is that required for the spring and dashpot to stress relax to 1 /e or 0.368 of a (0) at constant strain. To a first approximation, both z and Z2 indicate a measure of the time to complete about half of the physical or chemical phenomenon being investigated (Sperling, 1986). 

In Equation (3.85), Jm is the mean compliance of all the bonds and Tm is the mean retardation time Tm equals Jmt m where ijm is the mean viscosity associated with elasticity. One can replace the mean quantities with a spectrum of retarded elastic moduli (Gj) and the viscosities (iji), where, J-, = l/G,. Typically, one or two Kelvin-Voigt elements can be used to describe the retarded elastic region. 

An alternative model with a degree of complexity similar to that of the Maxwell model is the Kelvin-Voigt element. This model, consisting of a spring in parallel with a dashpot (see Fig. 10.4), is adequate to describe creep behavior. The total stress in the model is the sum of the stress in the spring and that in the dashpot,... 

The response to the stress input of the Kelvin-Voigt element is schematically represented in Figure 10.5. From Eq. (10.22), the creep compliance function is easily obtained as... 

What is commonly called the three-element standard, or simply the standard solid (or Zener s solid), is a combination of either a Kelvin-Voigt element in series with a spring or, alternatively, a Maxwell element in parallel with a spring (see Fig. 10.6). The strain response of the first model to the stress input CT = cjoH(t) can be written as... 

The Burgers model, also called a linear liquid of four elements, is a combination of the Maxwell model with a Kelvin-Voigt element (see Fig. 10.9). For a stress input, the strains are additive,... 

Figure 10.10 (a) Maxwell s elements in parallel (b) Kelvin-Voigt elements in... 

A standard solid (Kelvin-Voigt element in series with a spring) (Figure 10.2.1) is under a stress ct for a long time. Calculate the response of the solid after the load is eliminated (creep recovery experiment). 

Let us calculate sJ(s) for Kelvin-Voigt element. The constitutive equation is... 

Fit this behavior to that of a standard solid (spring in series with Kelvin-Voigt element). 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

To model the flexural dynamics of the test specimen, two masses and a Kelvin-Voigt element are used. The first mass, Wjc, represents the inertia of the central part of the specimen and it is also the mass first involved in the local interaction at the contact point. The second mass, Wj,., represents the inertia of the wings of the specimen. 

Retarded elasticity (Kelvin-Voigt element G2 is retarded by rj2)... 

The parameters of the Kelvin-Voigt model and the internal are fourth order tensors, while the strain and the stress are second order tensors. An imdetermined number of Kelvin-Voigt elements give flexibility to the model without increasing the complexity of the constitutive law as it will be discussed later in this section. Assuming a virgin material, having no p>ermanent strain due to earlier where J is the instantaneous elastic strain, where... 

Other quantities are also needed by finite element software, such as the tangent matrix for the use of Newton-Raphson methods, and are also function of the number N of Kelvin-Voigt elements. In the case of limiting the rheological model to N = 2 the tangent matrix is ... 

If we now stress the Burgers model in a creep test, it is easy to see what will happen, see figure 3 first, the unrestrained spring Gi will instantaneously deform to its expected extent, while the isolated dashpot will start to deform at its expected rate. However, the spring in the Kelvin-Voigt element cannot immediately respond, being hindered (i.e. retarded) by its dashpot. Nevertheless, it does begin to deform, and eventually comes to its expected steady-state deformation. It is possible to show that the overall deformation can be written down as ... 

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