High-frequency elastic modulus

To see how these new rheological features appear, we calculate the number figff of elastically effective junctions in a unit volume from (8.31), and the number of elastically effective chains Veff from (8.32). 

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The high frequency elastic modulus does not appear in the retardation spectrum but is an intrinsic part of the relaxation spectrum. These features are reinforced when the interrelationship between the spectra are considered. 

Figure 20 The VE theory of dephasing fit to Raman-FID data on toluene (see Fig. 15). The entire range of data is fit with only three temperature-independent parameters the solvent s high-frequency elastic modulus, a solvent-solute coupling constant, and a homogeneous dephasing time. A temperature-independent homogeneous dephasing process is assumed in addition to the VE mechanism.

Ultrasonic interferometry, in which the travel time of high-frequency elastic waves through a sample is measured, also yields elastic moduli. Because it is a physical property measurement, rather than an optical spectroscopy, it can be used equally well on poly-crystalline samples as single-crystals, although polycrystalline measurements only yield the bulk elastic properties, bulk modulus and shear modulus, G. High-pressure ultrasonic interferometry techniques were initially developed in the piston cylinder... 

Fig. 2). If the excitation frequency (oj) is much faster than tu , then the molecules do not have time to respond and there is no flow if to < o , then the molecules slide easily past one another (i.e., the viscosity is low) and little energy is dissipated. Similarly, if the frequency is fixed, but the degree of reaction (extent of cross-linking) is low, the sol is fluid and W (or G") is small as the reaction proceeds p increases), the viscosity increases and G" passes through a maximum before the gel becomes purely elastic (when the network is too stiff to flow) and dissipation is arrested. The storage modulus, G, is the familiar elastic (static) shear modulus when (o> u>o, but G (

The peculiarities of dynamic properties of filled polymers were described above in connection with the discussion of the method of determining a yield stress according to frequency dependence of elastic modulus (Fig. 5). Measurements of dynamic properties of highly filled polymer melts hardly have a great independent importance at present, first of all due to a strong amplitude dependence of the modulus, which was observed by everybody who carried out such measurements [3, 5]. 

Fig. 14. Amplitude dependences (y0 is the amplitude of cyclic deformations) of the elastic modulus for frequency a) = 63 s 1 13% dispersion of acetylene carbon black in low- (/) and high-molecular (2) poly(isobutylene)s...

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

Some specimens have withstood over 500 cycles. Untreated specimens of these concretes commonly failed the test at approximately 80 cycles. The elastic modulus criterion (less than 60% of original by resonant frequency analysis) was used to determine failure. Specimens often failed the test with no visible cracking or spalling. Strength retention in these specimens was usually high (50% or more). 

Rheologically, the BCC phase behaves as a solid, as expected, i.e. the dynamic elastic modulus is independent of frequency at high frequency in the linear... 

In materials which are highly attenuating, the particle velocity and particle displacement are out of phase, so the elastic modulus and density of the material are complex and dynamic ( .e. frequency dependent). For many materials, the attenuation coefficient is fairly small ( .e. a co/c), so the particle velocity and displacement are in phase and Eq. 9.4 can be replaced with ... 

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