Recoverable creep compliance

Figure 8.17 Isotherms showing the recoverable creep compliance for a narrow polystyrene distribution with molecular weight 3400. (From Ref. 16.)...

FIGURE 6.9 Loss tangent of polyisobutylene measured by dynamic mechanical spectroscopy and calculated from the recoverable creep compliance. These are not master curves the abscissa is the actual frequency (Plazek et al., 1995). 

PS (Commercial sample Hostyrene N-7000, mol. wt. not given.) 378 9.4 39 363.5 (from dilatometry at a cooling rate of 3K/h) [56] aj s of the entire softening dispersion from creep, Jr t), data with 10 < t < 10 s and temperature ranging from -90 to 130 °C. Its temperature dependence Is considerably weaker than that of ar,s found by Plazek and O Rourke [25] by recoverable creep compliance, Jr t), data with 10 ° < f < 10 s. If the difference In Tg of about 7.5° Is accounted for (Schwarzl s sample has a lower Tg) then there Is reasonable agreement between the the two sets of data. 

In Figure 24 (90) the lowest molecular weight, narrow distribution PS depicted as A25 (M = 47,000) has only two contributions, (i) at short times a common linear portion between log A,r of — 1 and 3 with a slope of 1/3 and it is followed by (2) a S5mimetrical peak centered at log A. = 6. The short-time region reflects the presence of Andrade creep in which the recoverable creep compliance is a linear fimction of An alternative description by a generalized Andrade creep with, with Pa not necessarily equals to 1/3 and varies from one polymer to another, is possible. Actually such a generalized Andrade creep is the short-time part of the... 

Fig. 30. The local segmental relaxation correlation time Tc of high molecular weight PS as a function of temperature obtained by two-dimensional exchange nmr (154,155) up to long times exceeding 100 s, and compared with the shift factor ot,s from time-temperature superposition of recoverable creep compliance Jr( ) curves of another high molecular weight...

Some empirical interrelations for obtaining J and J" from the recoverable creep compliance, / .(/) = J t) — t/t]o, have been given by Plazek and Raghupathi.34... 

FIG. 10-9. Normalized logarithmic recoverable creep compliance plot for Rouse theory (I) and for polystyrene of high molecular weight (II) (Example III of Fig. 2-1). 

FIG.l 1-15. Recoverable creep compliance from data of Fig. 11-14 reduced to 100° by shift factors chosen empirically in the transition zone and Fitted to equation 21. (The small rise to the right of 8 on the abscissa scale is attributed to a very small proportion of species of much higher molecular... 

FIG. 12-25. Recoverable creep compliance of narrow-distribution polystyrene with molecular weight 16,400, measured at various temperatures as indicated and reduced to 100°C with reduction factors appropriate in the transition zone. (Plazek and O Rourke. )... 

When dash pot and spring elements are connected in parallel they simulate the simplest mechanical representation of a viscoelastic solid. The element is referred to as a Voigt or Kelvin solid, and it is shown in Fig. 3.10(c). The strain as a function of time for an applied force for this element is shown in Fig. 3.11. After a force (or stress) elongates or compresses a Voigt solid, releasing the force causes a delay in the recovery due to the viscous drag represented by the dash pot. Due to this time-dependent response the Voigt model is often used to model recoverable creep in solid polymers. Creep is a constant stress phenomenon where the strain is monitored as a function of time. The function that is usually calculated is the creep compliance/(f) /(f) is the instantaneous time-dependent strain e(t) divided by the initial and constant stress o. ... 

Note 5 Creep is sometimes described in terms of non-linear viscoelastic behaviour, leading, for example, to evaluation of recoverable shear and steady-state recoverable shear compliance. The definitions of such terms are outside the scope of this document. 

After the stress has been removed (point D in Fig. 13A), the recovery phase follows a pattern mirroring the creep compliance curve to some degree First, there is some instantaneous elastic recovery (D-E return of spring 1 into its original shape Fig. 13A, B). Second, there is a retarded elastic recovery phase (E-F slow movement of the Kelvin unit into its original state Fig. 13A, B). However, during the Newtonian phase, links between the individual structural elements had been destroyed, and viscous deformation is non-recoverable. Hence, some deformation of the sample will remain this is in the mechanical model reflected in dash-pot 2, which remains extended (Fig. 13B). 

It was originally assumed that Gj should be proportional to T, and hence have a temperature dependence small compared to that for Xi, and further that all the Xi have a common dependence on temperatxure (7). Thus, it is assumed here that the temperature dependence of rj reflects that of only the T(. Recent investigators have suggested that for simple liquids, the Gf can have an appreciable temperatxure dependence xmder some conditions, so that it may be unreasonable to expect a simple fimction to correlate iq T) over extended temperature intervals 142a, 143b). Similar considerations may be necessary for a complete analysis of C as defined by Eq. (2.1) with temperature. Recent results of Plazek (777) showed that the temperature dependences of the viscous and recoverable contributions to the creep compliance of polystyrene are different from each other (after allowing for an expected proportionality of G,- to T). This can be interpreted as an indication of a significant temp>erature dependence of the G. Since our primary concern here is the temperature dependence of C for polymeric materials, for which the requisite data on deviation of Gj from proportionality to T do not exist, we will henceforth assume that there is only one contribution to the temperatiue dependence of (or of rj for simple liquids). Careful comparison of the temperature dependence of rj and of viscoelastic properties such as the recoverable compliance, may eventually provide an assessment of this assumption. 

Two steady states are recognized for the long-time creep compliance of materials. Either the sample is a solid and the compliance becomes time independent or the sample is a liquid and the compliance becomes linear in -time. Once steady state has been achieved in creep, the stress can be removed (a = 0) and the elastic recoil, called creep recovery, can be measured. Recovery strain is defined as 7r(0 s 7(0) — 7(0 for t > 0, where t is defined to be zero at the start of recovery. The recoverable compliance is defined as the ratio of the time-dependent recovery strain 7r(0 and the initially applied stress a, where both 7r and t are now defined to be zero at the start of recovery ... 

Boltzmann superposition relates the recoverable compliance after steady state has been achieved in creep to the creep compliance ... 

Find the relation between creep compliance J(t) and recoverable compliance /R(f) using the Boltzmann superposition principle. Dielectric spectroscopy indicates that water molecules respond to an oscillating electric field at a frequency of 17 GHz at room temperature. Is water still a Newtonian liquid at this high a frequency or is it viscoelastic If... 

The stress relaxation modulus and the creep compliance are both manifestations of the same dynamic process at the molecular level and are closely related. This relationship, however, is not a simple reciprocal relations that would be expressed as G t) = but rather in an integral equation that is derived from the Boltzman superposition principle. It relates recoverable compliance, to r/o, zero shear viscosity [22]. 

The positive curvature and the following maximum of the creep compliance L(r) indicate the contribution of polymeric molecular modes of motion to the recoverable compliance. Since no positive curvature is seen in the volume contraction Lq t), no polymeric modes are present and no long range coordinated motions of any kind are detected. 

Two quantities that play important roles in flow behavior of polymeric liquids are the steady-state viscosity at zero shear rate, rto, and the steady-state recoverable shear compliance, J°. Both are obtained quite directly from creep results, from (To and the shear rate fss in the steady-state region of the creep phase and J° from the total recoil strain (yr) in the recovery phase ... 

It has already been noted in Section A5 that the temperature reduction factors in the transition and terminal zones may be somewhat different. An example of a very complete study by Plazek on a polystyrene with almost uniform molecular weight 46,900 is shown in Fig. 11-14. The creep compliance Jp t) reduced to a reference temperature of 100° by shift factors ar calculated from the viscosity—i.e., from equation 13—provides satisfactory superposition of data in the terminal zone, but in the transition zone the reduced data diverge. Alternatively, the recoverable compliance Jp(t) — t/rjo can be satisfactorily reduced in the transition zone with a slightly different set of reduction factors these, however, appear to give a slight divergence in the terminal zone (Fig. 11-15). Both sets of shift factors follow the WLF equation, equation 21, but with slightly different coefficients in Fig. 11-14,... 

FIG. 15-3. Recoverable shear creep compliance and retardation spectrum of l 3 5-tri-a-naphthyl benzene, reduced to 64.2°C from measurements at that temperature and above (open circles) and at four other temperatures as indicated. (Points for spectrum calculated in several different ways.) (Plazek and Magill. )... 

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