Voigt spring-and-dashpot model

Figure 5.61 (a) Kelvin-Voigt spring and dashpot in parallel model of viscoelasticity and... 

The Kelvin model (also called as the Voigt model or the Kelvin-Voigt model) is a parallel connection of the spring and dashpot models (Kelvin, L. (Thompson, W.) 1875 Voigt 1892), representing the anealstic behavior, as given by... 

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]. 

Total strain e in a Voigt-Kelvin spring-and-dashpot model [11] is... 

Figure 2.35 Spring and dashpot models for the viscoelasticity of polymers. The spring has Young s modulus E and the dashpot an associated viscosity rj. (a) In series (Maxwell model), (b) in parallel (Voigt model)...

The Maxwell and Voigt models are of historical interest. They are also occasionally useful in theoretical analysis as examples for which particularly complex general results simplify drastically. They have some theoretical importance deriving from the fact that by combining Maxwell or Voigt elements in the sense of spring and dashpot models, one generates spectrum models [Ferry (1970)]. 

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.

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. 

Because of the assumption that linear relations exist between shear stress and shear rate (equation 3.4) and between distortion and stress (equation 3.128), both of these models, namely the Maxwell and Voigt models, and all other such models involving combinations of springs and dashpots, are restricted to small strains and small strain rates. Accordingly, the equations describing these models are known as line viscoelastic equations. Several theoretical and semi-theoretical approaches are available to account for non-linear viscoelastic effects, and reference should be made to specialist works 14-16 for further details. 

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... 

In contrast, in a model proposed by Voigt and Kelvin (Figure 5.5), in which the spring and dashpot are in parallel, the applied stress is shared, and each element is deformed equally. Thus the total stress S is equal to the sum of the viscous stress ij (dy/dt) plus the elastic stress Gy ... 

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). 

Figure H3.3.4 Mechanical models are often used to model the response of foods in creep or stress relaxation experiments. The models are combinations of elastic (spring) and viscous (dashpot) elements. The stiffness of each spring is represent by its compliance (J= strain/stress), and the viscosity of each dashpot is represent by a Newtonian viscosity (ri). The form of the arrangement is often named after the person who originally proposed the model. The model shown is called a Burgers model. Each element in the middle—i.e., a spring and dashpot arranged in parallel—is called a Kelvin-Voigt unit.

The parallel arrangement, also shown in Figure 2.49, is called the Voigt model. It is used to model the behavior of a cross-linked but sluggish polymer, such as one of the polyacrylates. Since the spring and dashpot must move in parallel, both the deformation and the recoverability are retarded. 

In 1874, Boltzmann formulated the theory of viscoelasticity, giving the foundation to the modem rheology. The concept of the relaxation spectmm was introduced by Thompson in 1888. The spring-and-dashpot analogy of the viscoelastic behavior (Maxwell and Voigt models) appeared in 1906. The statistical approach to polymer problems was introduced by Kuhn [1930]. 

In the Kelvin or Voigt model the spring and dashpot elements are connected in parallel, as shown in Figure 3.13a. This model roughly approximates the behavior of rubber. When the load is applied at zero time, the elastic deformation cannot occur immediately because the rate of flow is limited by the dashpot. Displacements continue until the strain equals the elastic deformation of the spring and it resists further movement. On removal of the load the spring recovers the displacement by reversing the... 

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... 

Fig. 7.8 The Kelvin or Voigt model spring and dashpot in parallel.

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