Viscoelasticity Voigt element

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. 

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. 

In the model most often used to represent QCM-D measurements of viscoelastic films, probed in either gas or liquid environments, the films are represented by a Voigt element, characterized by a complex shear modulus ... 

Viscoelastic models employing (a) a single Voigt element (b) multiple Voigt elements connected in series. The values of the moduli and retardation times are used to model the creep of HOPE in Figure 7.3 also. 

Viscoelastic characteristics of polymers may be measured by either static or dynamic mechanical tests. The most common static methods are by measurement of creep, the time-dependent deformation of a polymer sample under constant load, or stress relaxation, the time-dependent load required to maintain a polymer sample at a constant extent of deformation. The results of such tests are expressed as the time-dependent parameters, creep compliance J t) (instantaneous strain/stress) and stress relaxation modulus Git) (instantaneous stress/strain) respectively. The more important of these, from the point of view of adhesive joints, is creep compliance (see also Pressure-sensitive adhesives - adhesion properties). Typical curves of creep and creep recovery for an uncross-Unked rubber (approximated by a three-parameter model) and a cross-linked rubber (approximated by a Voigt element) are shown in Fig. 2. 

Retardation time n. Of a stressed viscoelastic material (specifically, one behaving like a Voigt element), the time required after release of stress for the strain to decrease to 1/e (=0.368) of its original value. 

Voigt element n. This is a Voight model which is a component, together with other Voight or Maxwell components, of a more complex viscoelastic model system, such as the standard linear solid. 

The viscoelastic functions exhibited by the Voigt element are as follows ... 

A group of Voigt elements in series represents a discrete spectrum of retardation times, each time t/ being associated with a spectral compliance magnitude Since in a series arrangement the strains are additive, it turns out that for the Voigt model. Fig. 1-10, the viscoelastic functions /(/), J ((x>), and are obtained by... 

This model is known as the standard linear solid and is usually attributed to Zener [4]. It provides an approximate representation to the observed behaviour of polymers in their main viscoelastic range. As has been discussed, it predicts an exponential response only. To describe the observed viscoelastic behaviour quantitatively would require the inclusion of many terms in the linear differential Equation (5.9). These more complicated equations are equivalent to either a large number of Maxwell elements in parallel or a large number of Voigt elements in series (Figures 5.13(a) and (b)). 

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. 

A Standard Model for the viscoelastic behaviour of plastics consists of a spring element in scries with a Voigt model as shown in Fig. 2.86. Derive the governing equation for this model and from this obtain the expression for creep strain. Show that the Unrelaxed Modulus for this model is and the Relaxed Modulus is fi 2/(fi + 2>. 

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...

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. ... 

In which element or model for a viscoelastic body will the elastic response be retarded by viscous resistance (a) Maxwell or (b) Voigt-Kelvin ... 

The Four-Element ModeF. The behavior of viscoelastic materials is complex and can be better represented by a model consisting of four elements, as shown in Figure 5.62. We will not go through the mathematical development as we did for the Maxwell and Kelvin-Voigt models, but it is worthwhile studying this model from a qualitative standpoint. 

The simplest model that can be used for describing a single creep experiment is the Burgers element, consisting of a Maxwell model and a Voigt-Kelvin model in series. This element is able to describe qualitatively the creep behaviour of viscoelastic materials... 

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... 

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

Fig. 8 Three-element network describing a viscoelastic solid. Leaving out the spring on the right-hand side leads to the Voigt model [92]. tfowever, this model predicts infinite stress at infinite frequency. Since the frequency of the QCM is high, the Voigt model misses an essential bit of the pictme...

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