Relaxed compliances

Figure 8.12 Creep compliance (inverse of modulus) as a function of log (time). The rate of transition from the unrelaxed compliance (higher modulus) to the relaxed compliance (lower limiting modulus) depends on the parameter m.

Here = Jra — Jua, where and are, respectively, the unrelaxed and relaxed compliance functions in the a relaxation process. The values of these quantities could in principle be obtained by extrapolation methods from complex plane plots of /"( ) versus / (co). If (j(t) = Gq Im exp(/o)r), then Eq. (12.1) can be written as... 

Note that the same parametersy(0),/(oo), li(0) and h(oo) appear in (22) as in (23). Figure 20 shows stress-relaxation data for the same oriented PET sheet plotted as unrelaxed and relaxed (t 0 and t-rco) stress-relaxation compliance v. the applied strain sq. The full lines on this figure are a least squares fit to the stress-relaxation data, and the dashed lines are predicted directly from the creep data. This excellent internal consistency is a... 

Fif). 20. The strmn dependence of the isochronal stress-relaxation compliance for oriented PET tested in tension perpendicular to the initial draw (O) t 114 s, ( ) t 7000 s. The full lines are a least squaresfit to these data, the darted lines predicted entirely from the creep data of Fig. 19 (after Brereton et al. j. 

Not here . Also of great interest in relation to shape-memory performance is the relaxed modulus Er. It determines the recovery stress available to overcome any resistance to shape recovery during heating. It is determined by entropic elasticity of the network and is therefore expected to be proportional to absolute temperature. To compare the various polymers, therefore, relaxed moduli were all reduced to a common temperature T = 92.3°C the mean of the reference temperatures. Thus values were obtained from relaxed compliances thus... 

Ferry and co-workers [7], on the basis of the molecular theory of viscoelasticity, proposed that superposition should incorporate a small vertical shift factor Fopo/TP, where p is the density at the experimental temperature T and po relates to the reference temperature Tq. Further corrections have been suggested by McCrum and Morris [8] to deal with the changes in unrelaxed and relaxed compliances with temperature. 

Fig. 4.1. Linear viscoelastic creep (a) constant stress a, applied at r = 0 leads to time-dependent strain y, (t) (b) a higher stress applied at r = 0 leads to time dependent strain Vj ( ) (c), from (a) and (b) the strains at tg, y (f,), and at time fb-y (fb), are linear in the stress (d) the observed dependence of J (r) (eqn (4,3)) on log t through one complete relaxation. and are the unrelaxed and relaxed compliances.

Before discussing tire complex mechanical behaviour of polymers, consider a simple system whose mechanical response is characterized by a single relaxation time x, due to tire transition between two states. For such a system, tire dynamical shear compliance is [42]... 

Figure C2.1.14. (a) Real part and (b) imaginary part of tire dynamic shear compliance of a system whose mechanical response results from tire transition between two different states characterized by a single relaxation time X.

In principle, the relaxation spectrum H(r) describes the distribution of relaxation times which characterizes a sample. If such a distribution function can be determined from one type of deformation experiment, it can be used to evaluate the modulus or compliance in experiments involving other modes of deformation. In this sense it embodies the key features of the viscoelastic response of a spectrum. Methods for finding a function H(r) which is compatible with experimental results are discussed in Ferry s Viscoelastic Properties of Polymers. In Sec. 3.12 we shall see how a molecular model for viscoelasticity can be used as a source of information concerning the relaxation spectrum. 

At longer times an increase in compliance marks the relaxation of the glassy state to the rubbery state. Again, an increase of temperature through Tg would produce the same effect. 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

Figure 3.16 Some experimental dynamic components, (a) Storage and loss compliance of crystalline polytetrafluoroethylene measured at different frequencies. [Data from E. R. Fitzgerald, J. Chem. Phys. 27 1 180 (1957).] (b) Storage modulus and loss tangent of poly(methyl acrylate) and poly(methyl methacrylate) measured at different temperatures. (Reprinted with permission from J. Heijboer in D. J. Meier (Ed.), Molecular Basis of Transitions and Relaxations, Gordon and Breach, New York, 1978.)...

Dynamic mechanical measurements were made on PTEE samples saturated with various halocarbons (88). The peaks in loss modulus associated with the amorphous relaxation near —90°C and the crystalline relaxation near room temperature were not affected by these additives. An additional loss peak appeared near —30° C, and the modulus was reduced at all higher temperatures. The amorphous relaxation that appears as a peak in the loss compliance at 134°C is shifted to 45—70°C in the swollen samples. 

Not only are the creep compliance and the stress relaxation shear modulus related but in turn the shear modulus is related to the tensile modulus which itself is related to the stress relaxation time 0. It is therefore in theory possible to predict creep-temperature relationships from WLF data although in practice these are still best determined by experiment. 

Graphite will creep imder neutron irradiation and stress at temperatures where thermal creep is normally negligible. The phenomenon of irradiation creep has been widely studied because of its significance to the operation of graphite moderated fission reactors. Indeed, if irradiation induced stresses in graphite moderators could not relax via radiation creep, rapid core disintegration would result. The observed creep strain has traditionally been separated into a primary reversible component ( ,) and a secondary irreversible component (Ej), both proportional to stress and to the appropriate unirradiated elastic compliance (inverse modulus) [69]. The total irradiation-induced creep strain (ej is thus ... 

The computer-optimized y values obtained for a number of conditions are given in Table VI. It can be seen that the first condition assumes simple power functions only and a value for B strictly in compliance with Eq. (18). The rms error achieved is good, but marked improvements are obtained by relaxing the equations for A and B in stages, as shown, the final result giving a much better rms error. It was not necessary in the analysis to separate the data into low- and high-velocity regimes, as was the case for round-tube data, since the lowest mass velocity is not so low as to cause difficulty. 

Sympathetic stimulation of veins. The smaller, more compliant veins that serve generally as blood reservoirs as well as specific blood reservoirs are densely innervated by the sympathetic system. Stimulation of the vascular smooth muscle in the walls of these vessels causes vasoconstriction and a decrease in venous compliance. Vasoconstriction increases venous pressure in the veins the blood is squeezed out of the veins and, due to the presence of one-way valves, moves toward the heart so that VR increases. A decrease in sympathetic stimulation allows the veins to relax and distend. The vessels become more compliant and capable of holding large volumes of blood at low pressures. In this case, VR decreases. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

Time-crosslink density superposition. Work of Plazek (6) and Chasset and Thirion (3, 4) on cured rubbers suggests that there is one universal relaxation function in the terminal region, independent of the crosslink density. Their results indicate that the molar mass between crosslinks might be considered as a reducing variable. However, these findings were obtained from compliance measurements on natural rubber vulcanizates,... 

Master curves are important since they give directly the response to be expected at other times at that temperature. In addition, such curves are required to calculate the distribution of relaxation times as discussed earlier. Master curves can be made from stress relaxation data, dynamic mechanical data, or creep data (and, though less straightforwardly, from constant-strain-rate data and from dielectric response data). Figure 9 shows master curves for the compliance of poly(n. v-isoprene) of different molecular weights. The master curves were constructed from creep curves such as those shown in Figure 10 (32). The reference temperature 7, for the... 

If the Boltzmann superposition principle holds, the creep strain is directly proportional to the stress at any given time, f Similarly, the stress at any given lime is directly proportional to the strain in stress relaxation. That is. the creep compliance and the stress relaxation modulus arc independent of the stress and slrai . respectively. This is generally true for small stresses or strains, but the principle is not exact. If large loads are applied in creep experiments or large strains in stress relaxation, as can occur in practical structural applications, nonlinear effects come into play. One result is that the response (0 l,r relaxation times can also change, and so can ar... 

Chain branching affects the viscosity, the longest relaxation time, and the steady-state compliance and therefore influences creep and stress relaxation (19,163- 167). The effect is difficult to quantify because the length... 

Star-shaped polymer molecules with long branches not only increase the viscosity in the molten state and the steady-state compliance, but the star polymers also decrease the rate of stress relaxation (and creep) compared to a linear polymer (169). The decrease in creep and relaxation rate of star-shaped molecules can be due to extra entanglements because of the many long branches, or the effect can be due to the suppression of reptation of the branches. Linear polymers can reptate, but the bulky center of the star and the different directions of the branch chains from the center make reptation difficult. 

The temperature dependence of the compliance and the stress relaxation modulus of crystalline polymers well above Tf is greater than that of cross-linked polymers, but in the glass-to-rubber transition region the temperature dependence is less than for an amorphous polymer. A factor in this large temperature dependence at T >> TK is the decrease in the degree of Crystallinity with temperature. Other factors arc the reciystallization of strained crystallites ipto unstrained ones and the rotation of crystallites to relieve the applied stress (38). All of these effects occur more rapidly as the temperature is raised. 

Even in cases where the rigid polymer forms the continuous phase, the elastic modulus is less than that of the pure matrix material. Thus two-phase systems have a greater creep compliance than does the pure rigid phase. Many of these materials craze badly near their yield points. When crazing occurs, the creep rate becomes much greater, and stress relaxes rapidly if the deformation is held constant. 

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