Sound speed modulus

The measurement of the sound speed in filaments and yams requires special techniques, viz. by means of a magnetostrictive oscillator (see Ballou et al. 1944/1949). The dynamic modulus, determined in this way is considerably larger than the Young modulus from stress-strain experiments Edyn 1-5 Estat-... 

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... 

In each case the attenuation of sound can be formally represented by defining a complex wavenumber, where the sound attenuation coefficient is the imaginary part of the wavenumber. The complex wavenumber also leads to the definition of a complex sound speed and a complex dynamic elastic modulus. 

The elastic moduli of a lossy material can also be represented as complex quantities [3,4]. The complex dynamic modulus M is related to the corresponding complex sound speed c as follows [3],... 

Eg.8 has the same form as the relation between the (real) elastic modulus and sound speed in a lossless medium. From Eqs.6-8 it follows that the sound attenuation coefficient is related to the complex elastic modulus,... 

While the design of such coatings is straightforward, selection of appropriate materials is not. Usually materials with the properties required for a particular application are not readily available, and some custom laboratory fabrication is necessary. This usually involves selecting a polymer composite which somewhat approximates the required physical properties. Then minor alterations to the chemical constituents or fillers are used on a trial basis and the acoustic properties (some combination of Young s Modulus and damping factor, sound speed, attenuation, density, and front-face reflectivity) of these sample formulations are measured. This continues until a suitable formulation is achieved. 

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. 

Techniques for measuring the complex sound speeds and moduli of polymers are described in the section on test methods. The data shows that the real and imaginary components of the elastic moduli are frequency dependent. The frequency dependence is strongest for materials with high values of the loss factor r. Materials with frequency-dependent elastic moduli are called dispersive, and measurements and theory show that sound absorption mechanisms lead to dispersion. The real and imaginary part of an elastic modulus are related by the Kramers-Kronig relations, which are presented in the next section. 

By making reasonable assumptions about the form of the intermolecular potential, it was possible to calculate bulk modulus as an analytic function of volume (18). The calculation agrees fairly well with experimental data, and with the assumption that volume is the primary factor determining bulk modulus and, by extension, sound speed. 

In applying Rao s rule to solid polymers, it should be kept in mind that the longitudinal sound speed (eq. 1) includes not only a bulk modulus term, but also a shear modulus term that must be taken into account (5). This has been done for both linear and cross-linked polymers (44-47). Van Krevelen has named the additive variable for bulk modulus the Rao function, Ur, and for shear modulus the... 

The sound speed in polymer blends (qv) varies with composition in a manner similar to that in copolymers. For blends of polystyrene and poly(vinyl methyl ether), the sound speed increases as the weight percent of the higher-soimd-speed polystyrene increases, but the relation may not be linear because of phase inversion caused by polymer incompatibility (90) (see Compatiblity). The effect of carbon black (qv) on sound speed in rubber is more complicated than in blends, but the qualitative effect is to increase the sound speed (91). In contrast, the addition of iron oxide to rubber decreases the sound speed (92). In this case, the higher density of the filler dominates, rather than the higher modulus. A decrease in sound speed is also observed when voids are present (79). In this case, the lower modulus of the voids dominates, rather than the lower density. 

Additive Properties. It is reasonable to assume that different chemical groups should contribute differently to the macroscopic properties of polymers. Rigid aromatic groups would be expected to raise the elastic modulus (and sound speed), while flexible aliphatic groups would be expected to lower the modulus. A quantitative expression of this idea is known as the method of additive properties and has been extensively developed and reviewed by Van Krevelen (42). In this method it is assumed that various polymer properties are determined solely from the additive contributions of their constituent groups. The groups are... 

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. 

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