Mechanical models Voigt model

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

Through the dashpot a viscous contribution was present in both the Maxwell and Voigt models and is essential to the entire picture of viscoelasticity. These have been the viscosities of mechanical units which produce equivalent behavior to that shown by polymers. While they help us understand and describe observed behavior, they do not give us the actual viscosity of the material itself. 

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. ... 

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.

Mechanical tests indicate that these blends do not behave like conventional blends and suggest that the polystyrene phase is continuous in the substrate. The moduli of the blends as a function of blend composition is plotted in Figure 10.6. The Voigt and Reuss models are provided for comparison (Nielsen, 1978) These are the theoretical upper and lower bounds, respectively, on composite modulus behavior our data follows the Voigt model, suggesting that both the polystyrene and polyethylene phases are continuous. In most conventional composites of polystyrene and HDPE, the moduli fall below the Voigt prediction indicating that the phases are discontinuous and dispersed (Barentsen and Heikens, 1973 Wycisk et al., 1990). 

This equation for the dielectric constant is the analogue of the compliance of a mechanical model, the so-called Jeffreys model, consisting of a Voigt-Kelvin element characterised by Gi and rp and t =t /Glr in series with a spring characterised by Gz- The creep of this model under the action of a constant stress aQ is (Bland, 1960)... 

Stoner et al. (1974) have proposed a mechanical model for postmortem striated muscle it is shown in Figure 8-28. The model is a combination of the Voigt model with a four-element viscoelastic model. The former includes a contractile element (CE), which is the force generator. The element SE is a spring that is passively elongated by the shortening of the CE and thus develops an... 

Viscoelasticity Models For characterization with viscoelasticity models, simulation models have been developed on the basis of Kelvin, Maxwell, and Voigt elements. These elements come from continuum mechanics and can be used to describe compression. 

Figure 12.10 shows the mechanical response as a funetion of time of two structures formed by combining a spring and a dashpot in series (Maxwell model) and in parallel (Voigt model). Creep is slow deformation of a viseoelastic material under constant stress (o), while relaxation is the time response of the stress after imposing a constant deformation (e). Thus, a simple mathematieal expression accounts for the relation between structure (combination of elements) and a property (creep or relaxation). Evidently, more complex responses ean be obtained by eombining several elements in series and in parallel. 

A mechanical model for such response would include a parallel arrangement of a spring for elastic behavior and a dashpot for the viscous component. (A dashpot is a piston inside a container filled with a viscous liquid.) This model, shown in Fig. 11-16a, is called a Kelvin or a Voigt element. When a force is applied across such a model, the stress is divided between the two components and the elongation of each is equal. 

Fig. 11-16. Simple mechanical models of viscoelastic behavior, (a) Voigt or Kelvin element and (b) Maxwell element.

If the creep experiment is extended to infinite times, the strain in this element does not grow indefinitely but approaches an asymptotic value equal to tq/G. This is almost the behavior of an ideal elastic solid as described in Eq. (11 -6) or (11 -27). The difference is that the strain does not assume its final value immediately on imposition of the stress but approaches its limiting value gradually. This mechanical model exhibits delayed elasticity and is sometimes known as a Kelvin solid. Similarly, in creep recovery the Maxwell body will retract instantaneously, but not completely, whereas the Voigt model recovery is gradual but complete. 

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

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. 

We shall now show that a series combination of a capacitor and a resistance in an electrical circuit leads to the same linear differential equation describing the time dependence of the charge as that for the Voigt model describing the time dependence of the stress in the mechanical case.1... 

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... 

FIGURE 13.12 Stress-strain (a-e) behavior of two simple mechanical models (a) the Maxwell model and (b) the Voigt-Kelvin model. 

Time-dependent hysteresis effects can also occur in crystalline materials and these lead to mechanical damping. Models, such as the SLS and the generalized Voigt model, have been used extensively to describe anelastic behavior of ceramics. It is, thus, useful to describe the sources of internal friction in these materials that lead to anelasticity. The models discussed in the last section are also capable of describing permanent deformation processes produced by creep or densification in crystalline materials. For polycrystalline ceramics, creep is usually considered from a different perspective and this will be discussed further in Chapter 7. 

The Kelvin—Voigt Model. The other two-element mechanical model for viscoelasticity is the Kelvin-Voigt model in which the spring and dashpot are in parallel. In this model, the deformation or creep response to the imposition of a constant load is illustrated. In this instance a constant load is applied at t = 0 and the deformation is monitored. The Kelvin-Voigt model and its response are illustrated in Fignre 3. The material property of interest in this case is the creep compliance Jit) and it is written as... 

The mechanics of materials approach is the simplest and least useful. It assumes that each phase is subjected either to the same strain (the Voigt model Equation 9.4) or the same stress (the Reuss model Equation 9.5). This yields the relationships ... 

We now illustrate these solutions for parameters typical of SFM contacts. We use mechanical models whose behavior brackets the range of response expected in real materials but are simple enough that analytical solutions can be obtained. In essentially all realistic cases, analytical solutions will not be possible. Specifically we chose the Maxwell and Voigt/Kelvin models. The Maxwell model is a series combination of an elastic spring with modulus E and dashpot with viscosity 7j. The relaxation functions are... 

According to the hterature the effects of relaxation and creep can be described by the mechanical Maxwell and Voigt model, respectively (Gent 1992). The corresponding meehanieal equivalent circuits toe related force-deformation characteristic and the equations describing toe motion of the network, and toe transient effect are shown in Table 1. 

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