Times retardation

A frequently used example of Oldroyd-type constitutive equations is the Oldroyd-B model. The Oldroyd-B model can be thought of as a description of the constitutive behaviour of a fluid made by the dissolution of a (UCM) fluid in a Newtonian solvent . Here, the parameter A, called the retardation time is de.fined as A = A (r s/(ri + s), where 7]s is the viscosity of the solvent. Hence the extra stress tensor in the Oldroyd-B model is made up of Maxwell and solvent contributions. The Oldroyd-B constitutive equation is written as... 

The load or stress has another effect on the creep behavior of most plastics. The volume of isotropic or amorphous plastic increases as it is stretched unless it has a Poisson ratio of 0.50. At least part of this increase in volume manifests itself as an increase in free volume and a simultaneous decrease in viscosity. This decrease in turn shifts the retardation times to being shorter. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

We assume that the above solution is valid in about the same time range as the self-similar relaxation time spectrum, Eq. 1-5. The retardation time spectrum is also self-similar. It is characterized by its positive exponent n which takes on the same value as in the relaxation time spectrum. 

T]he Origin of Species proposed a radically new idea, conceiving of time not as a power but as a factor whose effect could be perceived directly in distinct but complementary forms fossils, embryos, and rudimentary organs. The fossil was petrified time the embryo, operative time the rudimentary organ, retarded time. Together these bits of evidence constituted the archives of biological history —... 

Figures 5 and h show how the shape of the creep curve is modified by changes in the constants of the model. The values of the constants are given in Table I. Curve I is the same as shown in Figure 4, curve II shows onlv a small amount of viscous creep, and in curve 111, viscous flow is a prominent part of the total creep. The same data were used in Figures 5 and 6, but notice the dramatic, change in the shapes of the curves when a linear time scale is replaced by a logarithmic time scale. In the model, most of the recoverable creep occurs "Within about one decade of the retardation time.

In Section II, models were discussed that had only a single relaxation or retardation time. Actual polymers have a large number of relaxation or retardation times distributed over many decades of time. E(t) is then the sum of individual contributions, so equation (5) becomes... 

A distribution obtained by the use of equation (13) is only a first approximation to the real distribution. The corresponding distribution of retardation times is designated as L(T). It may be estimated from the slope of a compliance curve D(0 or J(t), for tensile or shear creep, respectively, plotted on a logarithmic time scale according to the equation (for shear creep)-... 

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

The distribution of relaxation or retardation times is much broader for cystallinc than for amorphous polymers, the Boltzmann superposition... 

A horizontal cantilever beam is made of an idealized material that has only two retardation times 10 and 1000 s. The beam is bent downward for 100 s. Then it is bent upward for 1 s and released without any vibrations taking place. Describe the motion of the beam for the next 10,000 s. 

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 ... 

The Lienard-Wiechert potentials (12) can also be derived from a rotation-free Lorentz transformation (boost) of the four potential of a static charge (13) to the moving frame at retarded time. For a charge moving at constant velocity the potentials can also be expressed in terms of the current position giving [21]... 

For example, Melville [26] studied the ultrasonically induced polymerisation of monomers such as styrene, methyl methacrylate and vinyl acetate in the presence and absence of polymethyl methacrylate and found that the polymerisation rates ( 1 % conversion/h) were not substantially increased by the presence of polymer. He concluded, in contrast to Driscoll, that the degradation of polymer was not the major source of radical production. Using hydroquinone as an inhibitor, he was able to deduce, from retardation times, that the rate of radical production was 2 X 10 mol dm s. A typical value for radical production using as an example the thermal initiation of AZBN (10 mol dm ) at 60 °C is 2 x 10" mol dm s" ... 

Does anyone know where to get an electronic easily-adjustable ignition system for a single cylinder lawn-mower type engine It would be great if you could advance or retard timing just by turning a knob. 

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

Note 3 The retardation time of a standard linear viscoelastic solid is t- 1/g o = Pilai. 

Note 4 Generally, a linear viscoelastic material has a spectrum of retardation times, which are reciprocals of = 0, 1in the polynomial Q D). 

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 retardation time t is the time for the strain to decrease to 1 — (1/e) or 1 — (1/2.7) = 0.63 of the original value. The viscoelastic flow of polymers is explained by approximate combinations of the dashpot and spring. The plots of the real data are compared with those predicted by various models. The relative importance of the various components of the model that fits the experimental data, dashpot and spring combinations, indicates the importance that the types of chain movement represented by the dashpot and spring have for that particular polymer under the particular experimental conditions. 

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