General dynamic modulus

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. 

Moreover, if for pure polymer melts the correlation of the behavior of the functions ri (co) andrify) under the condition of comparing as y takes place, as a general rule, but for filled polymers such correlation vanishes. Therefore the results of measuring frequency dependences of a dynamic modulus or dynamic viscosity should not be compared with the behavior of the material during steady flow. 

Dynamic Mechanical and Thermomechanical Analysis. A DuPont Model 981 DMA was used to determine the dynamic modulus and damping characteristics of baseline and irradiated specimens. Transverse composite samples 1.27 cm x 2.5 cm were used so that the modulus and damping data were primarily sensitive to matrix effects. Data were generally determined from -120°C through the glass transition temperature (Tg) of each material using a heating rate of 5°C/min. 

Figure 14.11 Elastic (<30 and loss (G 0 components of the dynamic modulus of the first seven generations of bulk PAMAM dendrimers as a function of shifted shear frequency [aT co] at 40°C. Solid lines indicate a fit for each curve to the generalized Maxwell model...

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

Studies of rheokinetics over the whole range of polyester curing is based (as for other materials) on a dynamic method, i.e., on measurements of the time dependence of the dynamic modulus at a fixed frequency, from which the time dependence of the degree of conversion (3(t). The observed dependence P(t) for polyester resins can be analyzed by an equation of the type used for other materials. Thus the following general equation was proposed for the kinetics of curing polyester and epoxy resins 69 72... 

The results of measurements of the dependencies G (w,t) for three circular frequencies w0 = 27tf0, wi= 4rcwo, and w2 = 16jtwo are shown in Fig. 3.1. The lack of coincidence in the shapes of die time dependencies of the dynamic modulus components for different frequencies is obvious. This phenomenon is especially true for G", because the position of the maximum differs substantially along the time axis. In the most general sense, this reflects the contributions of the main relaxation mechanisms of the material to its measured viscoelastic properties. 

To extend the theory for higher frequencies, we have to consider the general case, when the micro-viscoelasticity is given by (6.47). Using equations (4.28) and (4.29), after some rearrangement, one can find the dynamic modulus... 

The strain-optical coefficient O(w), defined by equation (10.22), can be corresponded to dynamic modulus calculated in Section 6.4.2. Taking all the previous speculations into account, an expression for the strain-optical coefficient can be written in general way as... 

All concretes generally exhibit drastic loss in compressive strength, bond strength, and dynamic modulus of elasticity after 1000 freeze-thaw cycles. 

Effects of vulcanization (Bateman et al., 1963 Hofnoann, 1967 Coran, 1978) on end-use properties are illustrated in Figure 7.2. It should be noted that static modulus increases with vulcanization to a greater extent than does the dynamic modulus. (Here, static modulus is more correctly the equilibrium modulus, approximated by a low-strain, slow-strain-rate modulus. Dynamic modulus is generally measured with the imposition of a sinusoidal, small strain at a frequency of 1-100 Hz.) The dynamic modulus is a composite of viscous... 

Equipment for measuring mechanical properties has been reviewed extensively. General descriptions are given in Nielsen, Ferry and Ward, " and dynamic modulus techniques are discussed in the above, and also more particularly by Hillier and McCrum et al Apparatus is described in most of the experimental investigations cited in this chapter, so that only a brief review need be given here. 

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. 

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... 

For the generalized viscoelastic model given in Problem 4.12, show that real and imaginary parts of the complex dynamic modulus G oi) are given by... 

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