Spring and dashpot in parallel

A spring and dashpot in parallel is called a Voigt model. 

Figure 3.10 Voigt models consisting of a spring and dashpot in parallel (a) simple Voigt unit and (b) set of units arranged in series.

Voigt-Kelvin model or element Model consisting of an ideal spring and dashpot in parallel in which the elastic response is retarded by viscous resistance of the fluid in the dashpot. 

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 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). 

The Kelvin body is a combination of the spring and dashpot in parallel. When elements are in parallel the stress of each element is added. 

The elements can be combined in series or parallel as shown in Figs 7.1 and 5.5. The convention for these models is that elements in parallel undergo the same extension. It is obvious that elements in series experience the same force. Thus, in the Maxwell model, the spring and dashpot in series experience the same force, while in Voigt model the spring and dashpot in parallel experience the same extension x. The total force f across the Voigt model can be written as the differential equation... 

Voigt-Kelvin model. A second simple mechanical model can be constructed from the ideal elements by placing a spring and dashpot in parallel. This is known as a Voigt-Kelvin model. Any applied stress is now shared between the elements, and each is subjected to the same deformation. The corresponding expression for strain is... 

An alternative model is obtained if one places the spring and dashpot in parallel and this is known as the Voigt or the Kelvin model (Fig. 5.14(a)). For this model, the strains on the two components are the same and the overall stress is the sum of the stresses on the dashpot and spring. The constitutive equation becomes... 

The experimental creep function is often analyzed as a nnm-ber of Kelvin elements in series (Figure 40.32), each having the property of a spring and dashpot in parallel (Genevaux, 1989 Martensson, 1992 Mohager and Toratti, 1993 Hanhijarvi, 1999 Passard and Perre, 2005). In the case of uniaxial load, this leads to... 

An appropriate physical model should be used to relate the specimen parameters (storage modulus, loss modulus, and damping factor), obtained in the DMA, to the effective properties (S-modulus, viscosity) of the material. On the basis of the Voigt model [8], consisting of the association of a spring and dashpot in parallel, the equation of motion can be expressed as ... 

Viscoelasticity can be expressed by either Maxwell or Voigt models, which shown in Fig. 2. The Maxwell model is composed of a spring and a dashpot in series, whereas the Voigt model is a spring and dashpot in parallel. The Maxwell model is used to understand stress relaxation behavior, and the Voigt model is used for creep behavior. 

A quite different response than the single Maxwell model may be obtained from the so-called single element Voigt model which now consists of a spring and dashpot in parallel. When this element is pulled, its components share a common strain, y, which leads to different stresses and aj in the spring and dashpot, respectively. Now the total stress is... 

We will now consider the viscoelastic properties of silicone gels with a mechanical model. It is possible to express complex mechanical behaviors by properly connecting a spring and dashpot, which are the mechanical models for modulus and viscosity, respectively. The Maxwell model, which is the model to connect a spring and dashpot in series, continues to deform upon application of external force. It is therefore liquidlike and is convenient to express the mechanical properties of a sol, which has elastic properties. In contrast, the Voigt model, which is the model to connect a spring and dashpot in parallel, reaches a finite deformation and exhibits equilibrium. Hence, it is solidlike and is convenient to express a gel that shows loss in mechanical energy [201]. 

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. 

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