Retardation time, Voigt-Kelvin

Note 2 The retardation time of a Voigt-Kelvin element is r = 1/g o = pia = (dashpot constant)/(spring constant). 

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. 

The usual way in which the deformation changes with time, has been dealt with in 6.1. The best representation appeared to be a Maxwell element with a Kelvin-Voigt element in series the deformation is then composed of three components an immediate elastic strain, which recovers spontaneously after removal of the load, a delayed elastic strain which gradually recovers, and a permanent strain. Moreover, we noticed that a single retardation time (a single Kelvin-Voigt element) is not sufficient we need to introduce a spectrum ... 

Figure 2 Graphical representation of the Voigt-Kelvin model (a) and the Maxwell model (b) of viscoelasticity. rd is the retardation time and

The integral form of Eq. (4.18) (Kaelble, 1971) shows that e is an exponential decay function of t/( /G). The dimensions of r /G reduce to seconds (Appendix 4) and the equation reaches a limiting l/e (0.37emax) in t = ti/G seconds. The retardation time (/rd) is the time required for emax of a Voigt-Kelvin fluid (Fig. 2a) to be reduced to 37% of emax after t has been removed (Barnes et al., 1989 Seymour and Carraher, 1981). A long retardation time is characteristic of a more elastic than viscous fluid. 

The Kelvin-Voigt elements are used to describe data from a creep experiment and the retardation time (t2) is the time required for the spring and the dashpot to deform to (1 — 1 /e), or 63.21 % of the total creep. In contrast, the relaxation time is that required for the spring and dashpot to stress relax to 1 /e or 0.368 of a (0) at constant strain. To a first approximation, both z and Z2 indicate a measure of the time to complete about half of the physical or chemical phenomenon being investigated (Sperling, 1986). 

In Equation (3.85), Jm is the mean compliance of all the bonds and Tm is the mean retardation time Tm equals Jmt m where ijm is the mean viscosity associated with elasticity. One can replace the mean quantities with a spectrum of retarded elastic moduli (Gj) and the viscosities (iji), where, J-, = l/G,. Typically, one or two Kelvin-Voigt elements can be used to describe the retarded elastic region. 

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... 

In a similar way, for an infinite number of elements in the generalized Kelvin-Voigt model /(A) maybe used to express the probability density of retardation times and the creep function (f) for the spectrum in the case of the generalized Kelvin-Voigt model can be written as ... 

The mechanical models discussed above are based on single relaxation (or retardation) time. Real polymer fibers have a spectrum or distribution of relaxation and retardation times due to the existence of different types of conformational changes. One convenient way to introduce a range of relaxation times into the problem is to constmct models consisting of a number of Maxwell and/or Kelvin-Voigt sub-models connected in parallel and/or series. Figure 16.24 shows a Maxwell-Wiechert model, which is constmcted by connecting an aibitraiy number of... 

The Maxwell-Wiechert model also can be used to describe the creep behavior of polymer fibers. However, for the creep behavior, it is mathematically more convenient to create a model involving a range of retardation times by connected a number of Kelvin-Voigt sub-models in series. 

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 ... 

Time response of different rheological systems to applied forces. The Maxwell model gives steady creep with some post stress recovery, representative of a polymer with no cross-linking. The Kelvin-Voigt model gives a retarded viscoelastic behavior expected from a cross-linked polymer. 

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