Of dynamic modulus

Fig. 5. Typical forms of frequency dependences of dynamic modulus. The content of the filler increases upon the transition from durve 1 to 2 and to 3. The discussion of regions I-VI, displayed on the curves, see text...

A typical behavior of amplitude dependence of the components of dynamic modulus is shown in Fig. 14. Obviously, even for very small amplitudes A it is difficult to speak firmly about a limiting (for A -> 0) value of G, the more so that the behavior of the G (A) dependence and, respectively, extrapolation method to A = 0 are unknown. Moreover, in a nonlinear region (i.e. when a dynamic modulus depends on deformation amplitude) the concept itself on a dynamic modulus becomes in general not very clear and definite. 

FIGURE 1.7 Construction of a master curve of dynamic modulus /a versus log (frequency) by lateral shifting of experimental results made over a small frequency range but at several different temperatures. 

Figure 1. Temperature dependence of dynamic modulus and loss tangent (11 Hz) of monomer I cured for 7 days at 280°C.

Fig. 1.2 Richness of dynamic modulus in a bulk polymer and its molecular origin. The associated length scales vary from the typical bond length ( A) at low temperatures to interchain distances ( 10 A) around the glass transition. In the plateau regime of the modulus typical scales involve distances between entanglements of the order of 50-100 A. In the flow regime the relevant length scale is determined by the proper chain dimensions...

Age (days) Ratio of dynamic modulus Cement Average... 

The real and the imaginary components of dynamical modulus of a dilute suspension of macromolecules in a viscoelastic liquid are calculated at values of B shown at the curves and at % = 1. Adapted from the paper of Pokrovskii and Volkov (1978a). 

There are plenty of measurements of dynamic modulus of nearly monodisperse polymers starting with pioneering works of Onogi et al. (1970) and Vinogradov et al. (1972a). The more recent examples of the similar dependencies can be found in papers by Baumgaertel et al. (1990, 1992) for polybutadiene and for polystyrene and in paper by Pakula et al. (1996) for polyisoprene. 

Though the reptation relaxation times are defined by equation (4.37), the weights pa of the contributions of separate relaxation processes remain unknown, and in fact, the replacement is forbidden, so that we prefer, as an initial approximation, to consider evaluation of dynamic modulus without any modification. 

The estimation of the main terms of expansion of dynamic modulus (6.49) determine the expressions for the terminal quantities... 

The considered system contains no macromolecules of the matrix and n macromolecules of the additive per unit volume and can be characterised by dynamic modulus G(u). The medium, in which the macromolecules of the additive move, is a system consisting of a linear polymer of molecular weight Mo, which is characterised by the modulus Gq(u) = — iurjo(u). The change of dynamic modulus, taking into account the fact that some of the macromolecules of the matrix have been replaced by impurity macromolecules, can be written as... 

Then, one can make use of the expression (9.19) for the stress tensor to obtain the coefficient of dynamic modulus... 

This equation, also as equation (6.49) gives description of the frequency dependency of dynamic modulus at low frequencies (the terminal zone). Both in equation (6.49) and (9.35), the second terms present the contribution from the orientational relaxation branch, while the first ones present the contribution from the conformational relaxation due to the different mechanisms diffusive and reptational. 

Relations (10.22) are quite similar to the definitions of dynamic viscosity r)(u>) and dynamic modulus G(u>), so that relations (10.23) are similar to the relations between the components of dynamic modulus and dynamic viscosity (equations (6.10)). 

Very useful properties of GPO include outstanding room temperature hysteresis and good dynamic properties over a wide temperature range. For example in measurements of dynamic modulus the flatness of the curve is observed between —40° C and 140° C and this property is maintained even after aging the polymer 7 days at 150° C, which is not the case with natural rubber. 

Plexiglas GP Poly(methyl methacrylate) 111 107.8 725 138 134.6 132 Measurements of Dynamic Modulus by DMA Multiplex 2 C step heating... 

Fig. 8. Time dependence of dynamic modulus (E ) and Poisson s ratio (v ) for POM.

Okonishnikov GB, Blednykh El, Skripov VP. The study of dynamic modulus of PMMA saturated with carbon dioxide. Mehk Polim 1973 2 370-372. 

Physical test properties on some cured rubber stocks prepared from lithium-catalyzed butadiene polymers are listed in Tables V and VI with appropriate controls. The results are only roughly indicative of the potential properties of rubbers made from lithium-catalyzed butadiene polymers because of the limited quantity of polymer available. The tensile data in Table VI indicate that compounded stocks from the lithium polymers are about equal or slightly inferior to the emulsion and sodium polymer controls in regard to these properties however, a hot tensile (lOO C.) on a cured compound from lithium polybutadiene was 325 pounds per square inch compared to 200 to 250 for an emulsion polybutadiene control. The internal friction of cured stocks from the lithium-catalyzed butadiene polymers is similar in magnitude to the emulsion or sodium polymer controls at 50 C. but higher at 100 °C. All lithium polymers, even those with low Mooney viscosities, gave cured compounds with high values of dynamic modulus. 

In attempting to predict the direction that future research in carbon black technology will follow, a review of the literature suggests that carbon black-elastomer interactions will provide the most potential to enhance compound performance. Le Bras demonstrated that carboxyl, phenolic, quinone, and other functional groups on the carbon black surface react with the polymer and provided evidence that chemical crosslinks exist between these materials in vul-canizates (LeBras and Papirer, 1979). Ayala et al. (1990, 1990) determined a rubber-filler interaction parameter directly from vulcanizatemeasurements. The authors identified the ratio a jn, where a = slope of the stress-strain curve that relates to the black-polymer interaction, and n = the ratio of dynamic modulus E at 1 and 25% strain amplitude and is a measure of filler-filler interaction. This interaction parameter emphasizes the contribution of carbon black-polymer interactions and reduces the influence of physical phenomena associated with networking. Use of this defined parameter enabled a number of conclusions to be made ... 

Table 2 and Table 3 respectively. The equation (Eqn. 1) provided by Barton (1991) was used for calculation of dynamic modulus of elasticity. 

Figure 3.12. Temperature dependence of dynamic modulus E and dynamic loss modulus E" for PVC/ PBD blends (an incompatible system) (—) 100/0 (---) 100/5 (-----) 100/15. Note the two dis-...

A more simplified predictive model has been developed by Al-Khateeb et al. (2006), which, for the determination of dynamic modulus, uses only two parameters the voids in the mineral aggregate and the dynamic shear modulus of the binder. As concluded, the model is capable of predicting the dynamic modulus of an asphalt concrete at a broader range of temperatures and loading frequencies than the Hirsch model. It also has the advantage of estimating the dynamic modulus of an asphalt concrete with modified bitumen (Al-Khateeb et al. 2006). The mathematical formulation of the model developed is as follows ... 

Evaluation studies on all predictive models carried out by various researchers have indicated that, in general, they provide reasonable predictions of dynamic modulus l l. Their predictive power varies with one or more parameters such as nominal size aggregate, temperature, frequency or air voids. 

Robbins M.M. and D. Timm. 2011. Evaluation of dynamic modulus predictive equations for NCAT test track asphalt mixtures. Proceedings of the Transportation Research Board 90th Annual Conference, January 23-27. 

Figures 9-8 and 9-9, respectively, demonstrate the effect of changing only the chain extender on PPDI TPUs. The use of BDO alone (Fig. 9.8) gives a long region of no change of dynamic modulus from 20-170°C Dianol 22 alone (Fig. 9.9) does not produce a region of constant modulus as the temperature rises.

In the solicitation stage, in the force range from 30 to 70 cN, the both parameters tends to constant values after that a rapid increase of their valued is recorded with the increase of solicitation. This evolution demonstrates that in the first moments of solicitation (10-30 cN/thread) a molecular orientation in amorphous phase takes place, concretised in the increase both of dynamic modulus and ultrasonation rate. It follows a structural stabilisation, in the force range from 30 to 70 cN/thread, after which the orientation occurs predominantly in higher organised structural formations, i.e. crystalline regions and microfibrils. 

Carbon black or silica filled NR generally demonstrates viscoelastic behaviour that is usually evaluated by dynamic viscoelastic measurements. One of the most widely discussed in the literature is the strain dependency of dynamic modulus known as the Payne effect. In such cases a high dynamic modulus at low strains (< 1%) is measured, which decreases at higher strains (> 10%), as shown in Figure 24.1. The reason for this phenomenon is the formation of a network created by filler-filler interactions. For carbon black the interactions are Van der Waals forces and for silica, much stronger hydrogen... 

---