Loss relaxation modulus

This equation indicates that the dissipated energy is proportional to both the square of the amplitude of the deformation and the loss relaxation modulus. 

Accordingly, the phenomenological theory of linear viscoelasticity predicts the same frequency dependence for the loss relaxation modulus of solids and liquids in the terminal region. 

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... 

The relationship between the relaxation modulus and the loss relaxation modulus established by Eqs. (6.8) leads to the expression... 

From this equation the relaxation spectrum is approximately obtained in terms of the loss relaxation modulus,... 

Figure 12.13 Shear loss relaxation modulus G as a function of temperature for three amorphous polyesters in the glassy state. (From Ref. 11.)...

The storage and loss moduli, G and G", are obtained from the relaxation spectrum in the usual way—that is, using G = Gi[co rl/(l + co zl)] G — G,[mT /(l -P The longest relaxation mode of the relaxation modulus in Eq. (3-67) is the dominant one it accounts for 96% of the zero-shear viscosity. Thus, the reptation model predicts that for a nearly monodisperse melt, the relaxation spectrum is dominated by a single relaxation time, T = Ta. This is in reasonable accord with experimental data at low and moderate frequencies (see the dashed line in Fig. 3-29). As the frequency increases, however, there... 

The storage relaxation modulus for liquids is also given by Eq. (9.8) with Ge = 0. Proceeding in the same way for the loss modulus, one obtains... 

Figure 12.31 Loss tensile relaxation modulus plotted versus temperature for various polyethylenes (From Ref. 38).

Use the Boltzmann superposition integral to derive the storage modulus of a viscoelastic liquid as a sine transform of the stress relaxation modulus G(t) [Eq. (7.149) with 6 eq = 0)]. Also derive the loss modulus as a cosine transform of G(t) [Eq. (7.150) with Ggq = 0] for a viscoelastic liquid. 

For high frequencies uj > 1 /tq, there are no relaxation modes in the Rouse model. The storage modulus becomes independent of frequency, and equal to the short time stress relaxation modulus, which is kT per monomer G uj) (pkTjb. This high-frequency saturation is not included in Eqs (8.49) and (8.50). At low frequencies a < 1/tr, the storage modulus is proportional to the square of frequency and the loss modulus is pro-portional to frequency, as is the case for the terminal response of any viscoelastic liquid. 

Using Eqs (7.149) and (7.150) with the approximate Zimm model prediction for the stress relaxation modulus [Eq. (8.64)] provides predictions of the storage and loss moduli that are valid for dilute solutions of linear chains (see Problem 8.16) ... 

Calculate the storage and loss moduli corresponding to the Doi fluctuation model with stress relaxation modulus... 

Figure 13. Excess microbending losses measured in the experiment of Figure 12 for various single mode and multimode fibers as a function of the 30-min, 23 °C tensile relaxation modulus of the coating material. The relaxation modulus-time relationships of the coating materials are depicted in the inset. (Reproduced with permission from Ref. 6. Copyright 1980 IEEE.)...

Because the shear relaxation modulus G t) with both its storage G (d) and loss parts G"(ffl) of the complex modulus are connected by Fourier transform it follows... 

Relaxation curves of loaded vulcanizates with similar crosslink densities, when cured with stable systems such as sulfurless TMTDS or with dicumyl peroxide, indicate losses of modulus of the order of 5% over periods of 107 sec at room temperature16). Comparison with Fig. 18 indicates that... 

Figure 4.2 shows the DMA results of a PP/talc (85/15) system represented in the standard way i.e. the log G and log G plotted as a function of the temperature. The loss shear modulus curve shows relaxation maxima at about 60°C (the crystalline phase [a] relaxation) and at about 0°C the amorphous phase glass-rubber [S] relaxation). Blending such a PP sample with a C2C3 rubber results in an extra (rubber) relaxation maximum at about -50°C. 

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