Creep Kelvin element

Note 5 The retardation spectrum (spectrum of retardation times) describing creep in polymers may be considered as arising from a group of Voigt-Kelvin elements in series. 

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)... 

In a creep experiment the Voigt-Kelvin element is instantaneously subjected to a stress [Pg.414]

The limiting value of the creep is equal to 00 = a0/E = Voigt-Kelvin element is only able to describe qualitatively the creep behaviour of rubberlike materials with a limited creep and not the creep of an elastic liquid. In general the creep compliance may be expressed as... 

Neither simple mechanical model approximates the behavior of real polymeric materials very well. The Kelvin element does not display stress relaxation under constant strain conditions and the Maxwell model does not exhibit full recovery of strain when the stress is removed. A combination of the two mechanical models can be used, however, to represent both the creep and stress relaxation behaviors... 

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... 

This relation describes a parallel arrangement of an elastic spring g and a kind of dashpot which is responsible for the stress relaxation according to do/d(ln t) = -X. This arrangement resembles the classical Kelvin element, as is demonstrated by comparing the creep response of this element... 

Figure 14 Creep behavior of the Maxwell and Kelvin elements. The Maxwell element exhibits viscous flow throughout the deformation, whereas the Kelvin element reaches an asymptotic limit to deformation.

As examples of the behavior of combinations of springs and dashpots, the Maxwell and Kelvin elements will be subjected to creep experiments. In such an experiment a stress, a, is applied to the ends of the elements, and the strain, e, is recorded as a function of time. The results are illustrated in Figure 10.4. 

The quantities E and t] of the models shown above are not, of course, simple values of modulus and viscosity. However, as shown below, they can be used in numerous calculations to provide excellent predictions or understanding of viscoelastic creep and stress relaxation. It must be emphasized that E and rj themselves can be governed by theoretical equations. For example, if the polymer is above T, the theory of rubber elasticity can be used. Likewise the WLF equation can be used to represent that portion of the deformation due to viscous flow, or for the viscous portion of the Kelvin element. 

It is of interest to compare the retardation time with the relaxation time. The retardation time is the time required for E2 and % in the Kelvin element to deform to 1 - lie, or 63.21% of the total expected creep. The relaxation time is the time required for 1 and 1/3 to stress relax to lie or 0.368 of ob, at constant strain. Both Ti and T2, to a first approximation, yield a measure of the time frame to complete about half of the indicated phenomenon, chemical or physical. A classroom demonstration experiment showing the determination of the constants in the four-element model is shown in Appendix 10.2. 

Draw the creep and creep recovery curves for the three-element model consisting of a Kelvin element and a spring in series. 

Derive an equation for the creep recovery of a Kelvin element, beginning after a creep experiment extending it to 6, a later time. 

It can be seen that the first 2 terms in the first square brackets correspond to the Maxwell element response and the other terms correspond to the Kelvin element response. The multiparameter model describes the response observed for rPET polymer concrete under a constant load. To obtain the creep strain using Equation 4.33 for a given stress, it is necessary to compute multiparameters, which are the 3 elastic constants ( i, 2 and 3) for the springs and the 3 viscous constants rp, % and 7/3) for the dashpots. Eigure 4.14 shows graphically how to obtain these values from the experimental results. 

A single Maxwell element is not realistic for characterizing a polymer as no transient response is shown in a creep test, i.e., the creep response is linear with time. A single Kelvin element is also not accurate as no instantaneous elastic response occurs in a creep test. A more realistic result for creep is obtained if a Kevin solid is combined with a Maxwell fluid to obtain the four-parameter fluid as in Fig. 3.22. 

The differential equation can be derived by following similar procedures as previously given for the Maxwell and Kelvin elements (see problem 3.4). The resulting equation can then be solved for the case of creep. However, the creep response can also be obtained by superposition by adding the creep response of Kelvin and Maxwell elements to obtain. 

The four-parameter fluid can also be evaluated in relaxation but typically, Maxwell elements in parallel are used for relaxation and Kelvin elements in series are used for creep. 

The creep compliance of a Kelvin element is D(t) = l-e j. Using the Boltzman superposition principle, find an equation for the strain vs. time in a constant stress rate test. Sketch your results, i.e., e vs. t. 

FIGURE 15.7 Creep response of a Voigt-Kelvin element to an applied shear stress. 

Note that the Voigt-Kelvin element does not continue to deform as long as stress is applied, and it does not exhibit any permanent set (see Figure 15.7). It therefore represents a viscoelastic solid, and gives a fair qualitative picture of the creep response of some crosslinked polymers. 

Its creep response is the sum of the creep responses of the Maxwell and Voigt-Kelvin elements ... 

Consider, for example, the creep response of the four-parameter model (Rgure 15.8). For this model, a logical choice for Xc would be the time constant for its Voigt-Kelvin component, r]2/G2- For De1 (t Xc), the Voigt-Kelvin element and dashpot 1 will be essentially immobile, and the response will be due almost entirely to spring 1, that is, almost purely elastic. For De 0 Xc), the instantaneous and retarded elastic... 

For creep tests, a generalized Voigt-Kelvin model is used (Figure 15.11). The creep response of an individual Voigt-Kelvin element is given by... 

The Voigt (or Kelvin) element resembles the Maxwell element except that the spring and dashpot are in parallel rather than in series. Show that the creep compliance of such an element is... 

The dashpot constant, rj2, for the Kelvin-Voigt element may be determined by selecting a time and corresponding strain from the creep curve in a region where the retarded elasticity dominates (i.e. the knee of the curve in Fig. 2.40) and substituting into equation (2.42). If this is done then r)2 = 3.7 X 10 MN.s/m ... 

The creep curve for polypropylene at 4.2 MN/m (Fig. 2.5) is to be represented for times up to 2 X 10 s by a 4-element model consisting of a Maxwell unit and a Kelvin-Voigt unit in series. Determine the constants for each of the elements and use the model to predict the strain in this material after a stress of 5.6 MN/m has been applied for 3 x 10 seconds. 

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.

A parallel array of E and h gives a Kelvin-Voigt element. This model does not allow an instantaneous deformation (the stress on the dashpot would be infinite), and it does not show stress relaxation. At a constant stress it exhibits creep at time t its strain is ( ) the stress in the spring then is ... 

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. 

In a similar way the creep can be represented by a number of Kelvin-Voigt elements in series ... 

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