Retardation time, Kelvin

In this case complete recovery occurs, as illustrated in Figure 20.8. The system is characterised by a Kelvin retardation time Tu that is also given by the ratio of r,(o)/G(0). 

Here the term ik is the retardation time. It is given by the product of the compliance of the spring and the viscosity of the dashpot. If we examine this function we see that as t -> 0 the compliance tends to zero and hence the elastic modulus tends to infinity. Whilst it is philosophically possible to simulate a material with an infinite elastic modulus, for most situations it is not a realistic model. We must conclude that we need an additional term in a single Kelvin model to represent a typical material. We can achieve this by connecting an additional spring in series to our model with a compliance Jg. This is known from the polymer literature as the standard linear solid and Jg is the glassy compliance ... 

As with the elastic solid we can see that as the stress is applied the strain increases up to a time t = t. Once the stress is removed we see partial recovery of the strain. Some of the strain has been dissipated in viscous flow. Laboratory measurements often show a high frequency oscillation at short times after a stress is applied or removed just as is observed with the stress relaxation experiment. We can replace a Kelvin model by a distribution of retardation times ... 

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. 

In a limited way, the Maxwell element describes a liquid. Similarly, the Kelvin (Voight) element describes a solid. As the relaxation time, Tj, is defined for the Maxwell element, the retardation time, T2, is defined for the Kelvin element. For the Kelvin element under stress. 

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. 

The ratio shear viscosity to shear modulus is often symbolised by the time x = ti / G. For the Maxwell model, x is called the stress relaxation time. In the Kelvin model x is a measure of the time required for the extension of the spring to its equilibrium length under a constant stress. X is called the retardation time. 

Kelvin model, with modulus of elasticity of 10 dyne/cm and retardation time of 10 sec. Polymer B follows a Maxwell model with modulus of elasticity of 10 ° dyne/cm and relaxation time of 2 seconds. In both cases the tensile strain rate was 5% per second. Polymer A broke after 40 seconds and B after 30 seconds. 

Let the number of individual units in multi-element models tend to infinity. For creep, an infinite number of Kelvin units gives an infinite number of retardation times this is called the spectrum of retardation times. The analagous development for stress relaxation leads to the spectrum of relaxation times. 

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. 

A distribution of retardation times based on a generalized Kelvin model leads to a retardation spectrum, L(x), defined by,... 

Hence, it is found that a Kelvin solid flows under constant stress once time is of the order of Tr (Figure 6.17a). Some authors refer to Kelvin Tr as retardation time. As discussed below, glass delayed elasticity and flow can be captured with a Burger solid that combines in series a Kelvin and a Maxwell solid. A Kelvin solid yields retardation while a Maxwell one yields relaxation. Relaxation time informs on the time scales at which a viscoelastic solid will behave elastically or relax. Let us consider glass transition the viscosity is 10 -Pas while shear modulus of most glasses scales with Pa so that relaxation time is of the... 

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 ... 

On fitting the creep data (for 900 s) to a model comprising of two Kelvin units (K1+K2), a decrease in the error norm is observed when compared with a single Kelvin unit. The two Kelvin units (K1+K2) have two retardation time constants, Z and X2. The shorter time constant, Ti, is in the range of 1-2 s while the longer one, Zz, is approximately 300 s. The second Kelvin unit (K2) has a compliance, D2, of 2.82E-06 Pa and a viscosity, ri2, of 1.06E+05 kPa.s. On performing similar experiments, Westman et al. (42) observed that the viscosity of the dashpot was in the range of 5-20 GPa.s. 

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. 

Use a Kelvin model for creep and show that the inflection point on the log scale plot corresponds to the creep retardation time t. Then the asymptotic time can be estimated at 3-5x even though the... 

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 ... 

If shear continued, more links between the structural units would break and re-form, but as weaker links do so at smaller time points, there is some retardation of this process. In Fig. 13A, this phase, which is called the retarded elastic region, is presented by the curved compliance-time profile between the points B and C. In the mechanical model (Fig. 13B), this region corresponds to a slow movement of spring 2 and dashpot 1, i.e., the Kelvin unit. The value of the retarded compliance can be obtained from ... 

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