Shear creep

Many other techniques of measuring viscoelastic parameters, such as transient shear, creep and sinusoidally-varying shear, are available. A good description, together with the merits and demerits of each of these techniques, is available in Whorlow(19. ... 

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)-... 

Figure 11 Shear creep of polyethylene (density = 0.950) at different loads after 10 min, and as a function of applied stress. Deviation firm the value of 1.0 indicates a dependence of creep compliance on load.

The shear creep compliance, J(t), is related to the relaxation modulus through ... 

Nemoto.N, Moriwaki,M., Odani,H., Kurata,M. Shear creep studies of narrow-distribution poly(cis-isoprene). Macromolecules 4,215-219 (1971). 

Two types of measurements were made on these samples. In the region where moduli are higher than 109 dynes/sq. cm., a Clash-Berg torsional creep apparatus (7) was used. For moduli below 109 dynes/sq. cm., a modified Gehman apparatus (14) was employed. In both cases shear creep compliance, Je(t), was obtained. To convert this to relaxation modulus, Gr(t), the following equation was used ... 

Real world materials are not simple liquids or solids but are complex systems that can exhibit both liquid-like and solid-like behavior. This mixed response is known as viscoelasticity. Often the apparent dominance of elasticity or viscosity in a sample will be affected by the temperature or the time period of testing. Flow tests can derive viscosity values for complex fluids, but they shed light upon an elastic response only if a measure is made of normal stresses generated during shear. Creep tests can derive the contribution of elasticity in a sample response, and such tests are used in conjunction with dynamic testing to quantity viscoelastic behavior. 

In Chap. 13 the creep recovery of a Burgers element was discussed and from Fig. 13.18 it becomes clear that the recoverable shear creep strain is in the present terms equal to... 

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

FIG. 16.11 Shear creep compliance of poly(vinyl acetate), M = 240 kg/mol, and four solutions in diethyl phthalate with indicated values of the polymer volume fraction, John Wiley Sons, Inc. 

Fig. 9. Master shear creep curve of PVAc. A comparison between theory (solid [32] curve) and experiment (circles [33])...

Fig. 10. Comparison of calculated [32] and measured shift factor of PVAc. Circles are data from the shear creep measurement [33]...

We will first consider the parameters we are trying to model. Let us start with stress relaxation, where it is usual to describe properties in terms of a relaxation modulus, defined in Table 13-5 for tensile [ (r)] and shear [G(r)] experiments. The parameter used to describe the equivalent creep experiments are the tensile creep compliance [D(r)] and shear creep compliance [7(0]. It is important to realize that the modulus and the compliance are inversely related to one another for linear, tune-independent behavior, but this relationship no longer holds if the parameters depend on time. 

In creep measurements of polyacrylonitrile gels [78,82], the shear creep compliance J(t) behawd as JAt/tof/U + (t/to)"l where is the steady state compliance, the time constant to could be of the order of a minute, and n 0.75. This implies G co) (ko)" for coto 1 and S(q, t) for t to. We thus expect emergence of the power law (6.39) or more complicated transient decays in many cases. 

In a shear creep experiment of this type, the material undergoes a stress cr = shear strain with time is registered. As shown in Figure 5.4b, the shear strain for an ideal solid is instantaneous [e(0 = eoFT(0], remaining constant with time. However, the strain for ideal liquids is a linear function of time (Fig. 5.4c). As Eq. (4.135) suggests, the shear strain for an ideal liquid is given by... 

Figure 5.9 Excitations and responses in linear shear creep experiments.

Figure 5.10 Diagrams showing the transition from linear to nonlinear behavior in shear creep experiments. Note that the data are taken from creep experiments at different deformations.

Figure 5.11 Schematic representation of the double logarithmic plot of shear creep compliance in the time domain.

Figure 5.15 Form of the fading memory of Eq. (5.40) in a shear creep experiment.

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

The Poisson ratio, like the bulk, tensile, and shear creep compliance, is an increasing function of time because the lateral contraction cannot develop instantaneously in uniaxial tension but takes an infinite time to reach its ultimate value. In response to a sinusoidal uniaxial stretch, the complete Poison ratio is obtained ... 

Uppuluri, S. Morrison, E.A. Dvornic, P.R. Rheology of dendrimers. 2. Bulk polyamidoamine dendrimers under steady shear, creep, and dynamic oscillatory shear. Macromolecules 2000, 33, 2551-2560. 

G 3. Gibbs, D. A., and E. W. Merrill A shear creep viscometer for rheological studies of polymers. Proc. of Fourth Intern. Congr. on Rheology 2, 183—192 (1965). 

Gribb, T. T. Cooper, R. F. 2000. The effect of an equilibrated melt phase on the shear creep and attentuation behaviour of polycrystalline olivine. Geophysical Research Letters, 27, 2341-2344. 

The two experiments just described are, respectively, creep and stress relaxation, both in tension. Figure 2-9, shows a crude form of a creep experiment in shear. In such an experiment the sample is subjected to constant shear stress cr0 and its shear strain y is measured as a function of time. The shear creep compliance J(t) resulting from such an experiment is ... 

Figure 2-9. A simple apparatus to measure shear creep.

The Boltzmann superposition principle is one of the simplest but most powerful principles of polymer physics.2 We have previously defined the shear creep compliance as relating the stress and strain in a creep experiment. Solving equation (2-6) for strain gives... 

Show that the form of the Boltzmann principle given in equation (2-45) reverts to the defining equation for the shear creep compliance, equation (2-9), when a sample, initially at rest, is subjected to an instantaneous increment of stress at t = 0, which is thereafter held constant. 

Thus 8.2 x 103 kg (roughly 9 tons) placed on the sample pan in Figure 2-9 would cause the pointer to move down 0.40 cm in KT4 s. Clearly all the apparatus is assumed to have an infinite modulus and inertial effects are ignored. Clearly, this is not the preferred way to run a shear creep experiment on such a polymer. Torsional deformation of a rod would be a much better choice. 

In the above discussion, six functions Go(w), d(w), G (w), G"(w), /(w), and J"(oj) have been defined in terms of an idealized dynamic testing, while earlier we defined shear stress relaxation modulus G t) (see Equation 3.19) and shear creep compliance J(t) (see Equation 3.21) in terms of an idealized stress relaxation experiment and an idealized creep test, respectively. Mathematical relationships relating any one of these eight functions to any other can be derived. Such relationships for interconversion of viscoelastic function are described by Ferry [5], and interested readers are referred to this treatise for the same. 

The shear creep compliance of an HOPE at 29 °C vs. time on logarithmic scales. The dashed curve is for the multiple Voigt element model of Figure 7.1. The response of a single Voigt element having = 300 MPa and t = 1000 s is also shown. 

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