Creep compliance shear

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

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 ... 

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. 

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. 

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

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 ... 

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 ... 

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. 

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. 

When the strains or the strain rates are sufficiently small, the creep response is Unear. In this case, when the time-dependent strain is divided by the fixed stress, a unique creep compUance curve results that is, at each time there is only one value for this ratio, which is the compliance—y(t)lao = J t). The unique shear creep compliance function J t) (Pa or cm /dyne, 1 Pa = 0.1 cm /dyne) obtained for an amorphous polymer has the usual contributions... 

FIGURE 5.2 Reduced shear creep compliance curves Jp(t) cm /dyn, determined on Epon 1007/DDS at seven temperatures, as indicated, presented logarithmically as a function of logarithmic time t. 

FIGURE 5.3 Reduced shear creep compliance curves Jp(f) of Epon 1007/DDS shifted to superimpose with the curve at the reference temperature 100.7°C shown logarithmically versus the logarithm of the reduced time r/aj-. 

FIGURE 5.4 Comparison of reduced shear creep compliance curves of Epon 828, 1001, 1002, 1004, and 1007/DDS plotted logarithmically against time at the reference temperatures indicated that are close to the respective TgS. 

Torsional creep measurements were made on the urethane-crosslinked polybutadiene elastomers at temperatures between —68 C and 25 C. The average molecular weight of a networks chain. Me, is 3400, 5200, and 8300 for TB-1, TB-2, and TB-3, respectively. The reduced shear creep compliance Jpit/ax) curves obtained for the three samples are shown in Figure 5.13. The reference temperatures are chosen to be 7.4°C, 0.0 C, and 17.0°C for TB-1, TB-2, and TB-3, respectively, so that superposition is achieved at shorter times... 

FIGURE 5.13 The logarithm of the reduced shear creep compliance curves, Jp t) (in Pa ), for the three urethane-end Unked polybutadiene elastomers displayed as a function of the logarithm of the reduced time t/ap (in seconds). The reference temperatures of reduction are chosen so that superposition is achieved at short times in the primary softening dispersion, (o) TB-1, 74 C, ( ) TB-2, 0°C, (e) TB-3, 17°C. 

Use the shear creep data in Figure 4.4, together with the method of time-temperature superposition, to estimate the shear creep compliance for linear polyethylene at 20°C and a creep time 10 s. Ust the assumptions that you make in this long extrapx>lation of the creep data. 

The Epons 828,1001,1002,1004, and 1007 fully cured with stoichiometeric amounts of DDS are examples of well-characterized networks. Therefore, mechanical measurements on them offer insight into the viscoelastic properties of rubber networks. The shear creep compliance J t) of these Epons were measured above their glass temperatures [11, 12, 14]. From the statistical theory of rubber elasticity [1-5, 29-33] the equilibrium modulus Ge is proportional to the product Tp, where p is the density at temperature T, and hence the equilibrium compliance is proportional to (Tpy Thus J t) is expected to be proportional to and J(t)Tp is the quantity which should be compared at different temperatures. Actually the reduced creep compliance... 

FIGURE 6 Reduced shear creep compliance curves cmVdyne, determined on Epon... 

FIGURE 11 Logarithmic plot of the reduced shear creep compliance curves Jp t) (in cmVdyne = lOPa ) against the reduced time I/oy (in seconds).The reference temperature of reduction. To, is -20.0 C for all curves. Data points are shown only for Sample 11-A. 

Torsional creep measurements were made on the urethane-crosslinked polybutadiene elastomers at temperatures between -68 and 25°C.The average molecular weight of a networks chain. Me, is 3400, 5200, and 8300 for TB-1, TB-2, and TB-3 respectively. The reduced shear creep compliance Jp(t/ar) curves obtained for the three samples are shown in Fig. 18. The reference temperatures are chosen to be 7.4,0.0, and 17.0°C for TB-1, TB-2, and TB-3 respectively so that superposition is achieved at shorter times in the primary softening dispersion. The most loosely urethane-crosslinked TB-3 has the largest 7e = 2.5 x lO Pa There is a plateau intermediate between the glassy compliance 7g (not reached in these measurements) and /e, and its level is about 2.0 X 10 Pa in all three samples. The network chain density does not affect the form of the time-dependent response up to and including the intermediate plateau in the Jp(t) curves. Only the terminal dispersion, i.e., the approach to /e, is influenced. The shift factors, Uj, that were used to obtain the... 

---