Loss compliance functions

Dynamic storage and loss compliance function for the cubic array are presented in Eqs. (T 11) and (T 12), respectively. 

When CO 0, the loss compliance function for viscoelastic solids is given by... 

Accordingly, the loss compliance function presents a maximum in the frequency domain at lower frequency than the loss relaxation modulus. This behavior is illustrated in Figure 8.18, where the complex relaxation modulus, the complex creep compliance function, and the loss tan 8 for a viscoelastic system with a single relaxation time are plotted. Similar arguments applied to a minimum in tan 8 lead to the inequalities... 

To obtain the frequency dependence of the loss compliance function for liquids, use must be made of Eq. (6.29b). Accordingly... 

Therefore, a first-order approximation for the retardation spectrum can be derived from the loss compliance function by means of the expression... 

Fig. 2. Reduced loss compliance or analogous dissipative dielectric permittivity for the linear array as a function of molecular weight...

In order to go further in the analysis, it is worth considering the loss compliance, /", shown at 1 Hz in Fig. 139. Indeed, the loss compliance allows one to perform quantitative comparisons based on additive contributions of the various units, as already mentioned (Sects. 6.3.1 and 7.1.1). Such an approach is illustrated as a function of temperature in Fig. 140 for MGIM36. 

Figure 9. Contour plot (70) of dynamic loss compliance as a function of frequency and temperature for poly(styrene-b-butadiene-styrene).

Vn>en the area under the Log (S") versus temperature curve Is plotted against epoxy moisture content, there Is a definite Increase with Increased moisture content. One method for presentation of the data Is to simply plot the area under the loss compliance transition as a function of sample moisture content. This could be done for each epoxy under consideration. 

Figure 3 Loss compliance (S") as a function of temperature for samples with different equilibrium moisture contents Key left, N-5208 samples and right, DGEBA-TETA.

The 25 PHR-DDS N-5208 epoxy sample covers a broader range of mole-moisture per mole-nitrogen values In Figure 4. However, at an abscissa value of 0.8, the differential loss compliance area does not appear saturated. The higher functionality of the N-5208 components result In a cured network with fewer tertiary amine links (6). This, In conjunction with the presence of the DDS sulfone groups, can be used to rationalize the lack of a plateau In the differential area curves of the 25 PHR-DDS epoxy data. 

Tan 5, storage compliance, and loss compliance values for these experiments are plotted as a function of time in Figure 8. This transient temperature cycle data illustrates interactions between the dynamic mechanical plasticization and blocking behavior just discussed as well as the epoxy s equilibrium moisture uptake behavior (3), and the temperature behavior of dynamic mechanical properties observed for this epoxy in Figure 2a. Perhaps the easiest comparison to consider involves the relationship between transient temperature cycling data of Figure 8 and the thermal behavior observed for N-5208 epoxy tan 6 data of Figure 2a. 

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Loss compliance quantifies the heat generated during dynamic loading in applications like tyres, wheels and rollers. Polyurethane elastomers with lower loss compliance values will experience less heat build up and, consequently, may suffer fewer field failures [14]. Figure 8.8 shows the graph of loss compliance as a function of temperature for the cast elastomers made with HER extended materials. The results are summarised in Table 8.5. 

In order to compare the Tg observed from the peak temperatures of loss modulus (G ), tan 8 and loss compliance (J ) curves, the peak temperatures are plotted as a function of the amorphous hard segment and shown in Figure 8.9. As can be seen in the figure, the amorphous hard segment produced by the high MW aromatic diol extenders has a pronounced effect on the soft segment Tg which, in turn, ultimately dictate the low temperature properties of the elastomers made from them. 

The energy dissipated per revolution is perhaps an indeterminate function of loss compliance, loss modulus, and loss tangent. For our studies, tan 6 determined at 40 Hz and at temperature of 60°C (a temperature fairly far above the Tg of rubbers of interest) proved to be the significant material property that relates to energy dissipated per cycle in a rolling tire, i.e. the tire s rolling resistance. 

Integration over equations 41 and 42 with appropriate limits by equations 19, 23, and 24 of Chapter 3 and addition of vRT/2 in the first two cases provides the viscoelastic functions G(t), G, and G" for the Chompff-Duiser theory. The corresponding curve for the loss compliance J" is included in Fig. 10-7. It extends farther to the low-frequency side than the others, as would be expected from the additional slow relaxation mechanisms. 

Oscillatory shear can also be performed by varying the stress sinusoidally and measuring the resulting strain as a function of time. The results can then be interpreted in terms of the real and imaginary components of the complex compliance, ] =J -i J". These components, the storage and loss compliances, are simply related to the storage and loss moduli as shown below. 

The storage and loss compliances are closely related to the retardation spectrum function I(T)as follows ... 

The idea of plotting one material function versus another has been applied to the viscoelastic properties of polymer. For example, one might plot the loss modulus versus the storage modulus or the loss compliance versus the storage compliance. We note that in this type of... 

Figure 11. Complex shear loss compliance of Galcit I as a function of reduced frequency. Tq=30 C.

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