Equilibrium compliance

As a result, we find for sols that the divergence of the above zero shear viscosity rj0 and of two other linear viscoelastic material functions, first normal stress coefficient and equilibrium compliance 7°, depends on the divergence... 

We can also calculate other viscoelastic properties in the limit of low shear rate (linear viscoelastic limit) near the LST. The above simple spectrum can be integrated to obtain the zero shear viscosity 0, the first normal stress coefficient if/1 at vanishing shear rate, and the equilibrium compliance J... 

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. 

An analytic expression for the reduced creep function, obtained from Eq. (3.11) with Eq. (3.6) for the linear array, is shown in Table 4 Eq. (T4). Je is the elastic equilibrium compliance and y (1/2, w) is the incomplete gamma function of order 1/2, extensively tabulated and available in most computer subroutine libraries. The last term accounts for contribution of plastic flow, where present. Reduced creep functions (recoverable and including flow contributions) are plotted in Fig. 4 as a function of Z. The... 

DETEHMINATION OF VISCOSITY AND STEADY-STATE (EQUILIBRIUM) COMPLIANCE FROM RELAXATION AND RETARDATION SPECTRA... 

Both the steady-state compliance function, and the equilibrium compliance, Jg, can readily be obtained from the retardation spectrum. Actually, by taking the limit of Eq. (9.20) in the limit oo 0, the following relationship for Jg and Jg is obtained ... 

The response of viscoelastic materials lies between these two extremes. When a constant stress is applied, there is an instantaneous rise in compliance. The compliance then increases with time to the equilibrium value. When the stress is released, there is an instantaneous drop in the compliance followed by a time-dependent decrease. For a viscoelastic solid, all the energy is stored, and hence there is a total energy release upon removal of the shear stress. As a result, the final equilibrium compliance is zero (Fig. 12d). For a viscoelastic liquid, however, viscoelastic flow takes place and there is only a partial recovery when the stress is removed (Fig. 12e). 

It can seen from Figure 5.4 that the equilibrium compliance Je decreases uniformly from the 1007/DDS to the 828/DDS and is expected on the basis of the kinetic theory of rubberlike elasticity, since the concentration of network chains increases and the molecular weight per crosslinked unit, Mx, decreases in the same order. The Mx values calculated as /o/JT/g are remarkably close to the molecular weight values of the starting epoxy resins. 

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... 

From these results, no relaxation at lower frequencies is perceptible. However, Ross-Murphy and Higgs [8,9] measured the creep behaviour of gelatin gels (2.4-15 wt%) at room temperature. The gels show a typical creep behaviour of polymer solutions with equilibrium compliances, J, of the order of 10 -10 m /N and with rather high viscosities of the order of 10 Ns/m. For a 4 wt% solution at room temperature a retardation time of approximately... 

For the viscoelastic liquids Jg is the steady-state recoverable compliance and for the viscoelastic solids, beyond the gel point it is the equilibrium compliance. 

J = Ji+Ji+ h- Js can be considered as zero for a crosslinked or highly crystalline polymers. Table 7.7 lists creep-related data of PU/C20A nanocomposites. The instantaneous compliance decreases with the addition of clay, and the nanocomposites observes the nearly same equilibrium compliance except 5wt% C20A, which is similar to the equilibrium stress during stress relaxation. It is also similarly found that both the creep rate and retardant time increases with the addition of clay, which should also be the result of enhanced phase microseparation. 

Fig. 2.25. The equilibrium compliance and high-frequency limiting compliance JoQ plotted as functions ofT — T. Jq values for various molecular weights (580 (V), 3500 (o), and 10200 (A)) are from cyclic shear data of Gray et al. [202]. /e for molecular weight 3400 (-h) is from creep-recovery data of Plazek and O Rourke [90]. Curves for the M = 3500 and 10200 samples are calculated from the equation given in table 5 of [90]. The dashed curve is an extrapolation outside the range of measurement. The dotted lines are the values predicted by the Rouse theory, 0.4M/(p 7). Values of J o for all polymers were obtained from measurements at temperatures from Tg to Tg -h 20 K and extrapolated to higher temperatures.

These simple numerical exercises illustrate the point that the total Hamiltonian of the system in the presence of the electric field must not only include the well-known electrostatic energy term but also the hitherto ignored strain energy term. The small strains that are encountered in the polarization orocess also justify the use of the electrostatic equations of the sphere. It is also quite clear that for very unsymmetrical species where equilibrium compliance approaches infinity, such as the case of bulky polar liquids, the strain energy term may also be zero and the Onsager-Kirkwood equations are once more applicable. It should be pointed out that the strain energy is also zero for finite values of Do but symmetrical species such as spherical dipoles. 

Consider a case where the relaxation time of point dipoles in the system is much less than 10. Then, in the experimental frequency range of 10 -10 Hz, equilibrium dielectric constants will be measured. However, let us assume that y /D is such that Cq 1 reduced in magnitude. Furthermore, the equilibrium compliance D is a function of time such that Do(t) = Du(l - exp(-t/r) where t is the relaxation time somewhere... 

If the long-time behaviour predicted by this model is extrapolated back to zero time, the intercept is yo. If this is divided by the applies stress, we get the so-called equilibrium compliance, ]e, which can be related to the elastic elements we... 

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