Shear sound speed

TABLE 14.2 Frequency derivatives of longitudinal and shear sound speeds and sound absorptions... 

Elastic Constants. For the polymers listed in Table 2 for which both longitudinal and shear sound speeds are given, the elastic constants have been calculated at room temperature, ambient pressure, and a frequency of 2 MHz these are listed in Table 7. The moduli values are approximately 1 order of magnitude lower than those for metals. The range of Poisson s ratio values is somewhat higher than that for metals. A review of elastic properties of polymers is given by Hartmann (130). 

Immersion Technique. A test method known as the immersion technique will be described in some detail. Other test methods will then be discussed in contrast to this method. The immersion technique is relatively simple to use and provides both longitudinal and shear sound speeds and absorption over a range of temperatures, generally in the MHz frequency range, to 1% accuracy in sound speed. 

Elastic Constants. One of the purposes of measuring sound speeds is to determine the elastic constants of the polymer. Longitudinal sound speed vi, and shear sound speed Vg are related to elastic constants by... 

Results have also been obtained for the density and bulk modulus of cross-linked epoxies (44). Agreement is about the same as for linear polymers. Finally, shear modulus has been related to an additive property (47). These results, along with the additive results for density and bulk modulus, show how both longitudinal and shear sound speeds are related to molecular components. 

The Lagrangian sound speed is obtained in the following heuristic way. We consider small departures from the shock-compressed state, where the bulk and shear moduli are K and G. The Eulerian sound speed c is then given by... 

The motion of disloeations under eonditions of shoek-wave eompression takes plaee at sueh high veloeities (approaehing the elastie sound speed) that many vaeaneies and interstitials are left behind. However, these point defeets ean anneal out at room temperature and are thus diflieult to study by shoek-reeovery teehniques. The presenee of point defeets has little effeet on the material eompressibility and other properties related to equation of state. While they also have little direet influenee on the relief of shear stresses, point defeets do influenee the mobility and multiplieation of disloeations. This, in turn, affeets most of what happens under shoek-wave loading eonditions. 

Some experimental techniques yield a modulus rather than a sound speed, but the underlying physical properties are the same. In a torsional pendulum, for example, the shear modulus, G, is measured while in a Rheovibron, Young s modulus, E, is measured. When there... 

The quantities a, c, f, F, r, and p are the thermal diffusivity, sound speed, heat capacity ratio, bulk viscosity coefficient, shear viscosity coefficient, and density of the sample, respectively and Eo, a, P and Cp are the energy fluence of the laser beam, the optical absorption coefficient, the volume expansion coefficient, and the isobaric heat capacity, respectively, of the fluid. Tlie first and second terms in Eq. 2 describe the time dependences of the thermal and acoustic modes of wave motion, respectively. Since the decays of the acoustic and thermal mode densities back to their ambient values take place on such different time scales (microsecond time scale for acoustic mode and millisecond time scale for thermal mode), they were recorded on the oscilloscope using different time bases. 

Figure 5a. Sound speed in PDMS networks plotted as a function of shear modulus, G (O), 52° scattering angle (X), 123° scattering angle. All data is at room temperature (293 K).

FIGURE 60.5. (a) Sound speed and absorption vs. temperature for poly (carborane siloxane) and (b) log plot of shear modulus and loss factor vs. frequency for polyurethane [1]. Reprinted from [1] Copyright 1996, with permission of Springer Science + Business Media. 

Associated with each of the two modes of propagation, there is a sound speed and an absorption. Thus, four parameters are required to specify the acoustic properties of a solid isotropic polymer longitudinal speed, shear speed, longitudinal absorption, and shear absorption. 

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