Kelvin-Voigt solid

Figure 10.5 Response of a Kelvin-Voigt solid to a shear step stress input.

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

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. 

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. 

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 ... 

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... 

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). 

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... 

There are other models based on springs and dashpots such as the simple Kelvin-Voigt model for viscoelastic solid and the Burgers model. Reader is referred to Refs. [1-5] for details. Other elementary models are the dumbbell, bead-spring representations, network, and kinetic theories. However, the most notable limitation of all these models is their restriction to small strain and strain rates [2, 3]. 

Compliant solid Continuum Viscoelastic Kelvin-Voigt or higher... 

The two basic viscoelastic models include the Kelvin-Voigt (K-V) and the Maxwell elements. The K-V element behaves as a solid when sheared, since the deformed material regains its initial state after the applied stress is relaxed. The components of equivalent shear modulus and equivalent viscosity, respectively, are... 

If we connect the single elastic and viscous elements in parallel, we end up with the simplest representation of a viscoelastic solid, and this has been named after Voigt, the German physicist (1850 - 1919), while some workers call it the Kelvin or the Kelvin-Voigt model, linking it with Lord Kelvin (1824 - 1907), another Scottish physicist) to please as many as possible, we shall call it the Kelvin-Voigt model. 

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

Figure 3.10 Basic mechanical elements for solids and fluids a) dash pot for a viscous response, b) spring for an elastic response, c) Voigt or Kelvin solid, d) Maxwell fluid, and e) the four-parameter viscoelastic fluid...

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. 

Note 4 Damping curves are conventionally analysed in terms of the Voigt-Kelvin solid giving a decaying amplitude and a single frequency. 

Note 5 Given the properties of a Voigt-Kelvin solid, a damping curve is described by the equation... 

Note 2 The logarithmic decrement can be used to evaluate the decay constant, p. From the equation for the damping curve of a Voigt-Kelvin solid. 

Note 2 For a Voigt-Kelvin solid, with P(D)=1 and Q y)=a+pD, where a is the spring constant and P the dashpot constant, the equation describing the deformation becomes... 

Note 7 Notes 2 and 5 show that application of a sinusoidal uniaxial force to a Voigt-Kelvin solid of negligible mass, with or without added mass, results in an out-of-phase sinusoidal uniaxial extensional oscillation of the same frequency. 

Note 1 For a Voigt-Kelvin solid of negligible mass, the absolute modulus can be evaluated from the ratio of the flexural force (/o) and the amplitude of the flexural deflection (y) with... 

Note 3 A material specimen which behaves as a Voigt-Kelvin solid under forced oscillation , with a mass added at the point of application of the applied oscillatory force... 

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