Relaxed compliance/modulus

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. 

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.

For experiments performed in shear, there is a rather complicated relation between the time-dependent stress relaxation shear modulus G(t) defined by Equation 3.19 and the time-dependent creep compliance J t) defined by Equation 3.21. But if the slope of log G(r) versus log r is — m, then, to a good approximation. 

At intermediate times it will be seen that, in creep, the compliance passes from /u to /r with time constant r . In stress relaxation the modulus passes from G to Gr with time constant r. Thus, at very short and very long times the stress and strain are Hookean, but at intermediate times when the time t is of the order of the relaxation times this k not true and it in this region that we see viscoelastic effects. The relationship between theory (Figure 4.15) and experiment (Figures 4.4 and 4.7, for e mple) will be explored later the reader may well however compare these figmres now and see in outline how theory is in broad agreement with eqieriment... 

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... 

At intermediate times it will be seen that, in creep, the compliance passes from to Jr constant x . In stress relaxation the modulus passes from... 

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. 

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. 

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 ... 

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. 

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... 

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. 

In the same manner as the modulus can be related to the relaxation spectrum so the compliance can be related to the retardation spectrum ... 

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 ... 

According to the theory of linear elastico-viscous behaviour (47) the steady-state shear viscosity t] and the steady-state shear compliance Je depend in the following way on the shear relaxation modulus G (t), where t is here the time of the relaxation experiment ... 

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