Sound Velocities, shear

Notation, k is the wavenumber, Us the sound velocity, F the damping coefficient (11), r the shear viscosity, po the uniform mass density, k the heat conductivity, and Cp the specific heat capacity at constant pressure [4]. 

Silvery-gray metal hexagonal crystal structure malleable, ductile, and soft enough to be cut with a knife density 8.223 g/cm melts at 1,359°C vaporizes at 3,221°C resistivity llhxlCH ohm-cm at 25°C Young s modulus 5.75xl0n dynes/cm2 (from velocity of sound measurements) shear modulus 2.28 dynes/cm2 Poisson s ratio 0.261 thermal neutron absorption cross section, 46 barns insoluble in water soluble in acids. 

Dislocations move when they are exposed to a stress field. At stresses lower than the critical shear stress, the conservative motion is quasi-viscous and is based on thermal activation that overcomes the obstacles which tend to pin the individual dislocations. At very high stresses, > t7crit, the dislocation velocity is limited by the (transverse) sound velocity. Damping processes are collisions with lattice phonons. 

The relation between friction and viscosity goes beyond the Stokes relation. The Navier-Stokes hydrodynamics has been generalized by Zwanzig and Bixon [23] to include the viscoelastic response of the medium. This generalization provides an elegant expression for the frequency-dependent friction which depends among other things on the frequency-dependent bulk and shear viscosities and sound velocity. 

Sound Velocity Through Explosive. Longitudinal and transverse shear sound velocities were... 

The speeds of longitudinal and transverse (shear) sonic waves can be estimated, c.q. predicted via two additive molar functions. From these sound velocities the four most important elastic parameters (the three elastic moduli and the Poisson ratio) can be estimated. 

This approach correctly predicts the observed effective sound velocity, c, in the case of metal inclusions in a soft viscoelastic medium ), but cannot be applied when the medium has a large shear modulus. 

Such behaviour of fracture propagation can clearly be of extreme importance for practical purposes. For example, the fraction r, from the measurement of the sound velocity ratio of compressional and shear waves, can indicate the proximity of the imminent macroscopic failure or fracture of... 

Experimentally, the bulk modulus is the simplest parameter to measure, but the seismological parameters of primary interest, Vp and Vg, both involve the shear modulus as well. It is convenient, therefore, to define a new parameter, the bulk sound velocity (V ), which eliminates all dependence upon the shear modulus (G) through a judicious linear combination of the squares of the two seismic wave velocities =... 

Fig. 5.3.11. Dependence of the sound velocities on the polar angle from Brillouin scattering experiments on (J-methyl butyl p((p-methoxy-benzylidene)amino) cin-namate. (a) Smectic A(T = 60.7 °C), (b) smectic B (F = 48.1 C). The dashed lines are calculated from theory. The presence of a third component in (b) indicates that the shear modulus does not vanish in smectic B at these very hi frequencies. Circles, triangles and squares represent measurements at different scattering angles. (After Liao, Clark and Pershan. )...

Vibration frequencies and phonon dispersion See Figs. 20 - 23. Table 13. Perpendicular vibration frequencies /zcoi and characteristics of the phonon dispersion curves for the noble gas monolayers. The sound velocities c/ and c, were obtained from the initial slope of the dispersion curves for the longitudinal (L) and shear-horizontal (SH) modes, respectively. Where complete or partial dispersion curves are available, oidy the value at the boundary of the surface Brillouin zone is indicated. Abbreviations used F, M, K high syrtunetry points of the 2D adlayer Brillouin zone (BZ) [001], [110] and [112] crystallographic directions of the substrate surface. All data were obtained using inelastic He-atom scattering. (Ad. = adsorbate) ... 

For the volume fractions presented in Figs. (3-6) the shear modulus is on the order of lOdyn/cm and the sound velocity V(= /s/p) = l-5cm/s. The microscopic relaxation time T(-rj/E) 1-10 ms, and the attenuation length A.[= (ImK) =2F/0 t1 1-10cm. For frequencies below IkHz the dissipation is small and the shear waves are propagating. The dimensions of the measuring cell encourage the formation of standing waves. 

Also listed are the bulk moduli values (6 ), measured directly in an anvil cell, and Bf, calculated from the shear and longitudinal sound velocities. The references and values are color-coded the 312s are highlighted yellow, and 413s gray. 

X10. The next three rows present the viscosity rj, the surface tension, and its tenqterature dependence, in the liquid state. The next properties are the coefficient of linear thermal expansion a and the sound velocity, both in the solid and in the liquid state. A number of quantities are tabulated for the presentation of the elastic properties. For isotropic materials, we list the volume compressihility k = —(l/V)(dV/dP), and in some cases also its reciprocal value, the bulk modulus (or compression modulus) the elastic modulus (or Young s modulus) E the shear modulus G and the Poisson number (or Poisson s ratio) fj,. Hooke s law, which expresses the linear relation between the strain s and the stress a in terms of Young s modulus, reads a = Ee. For monocrystalline materials, the components of the elastic compliance tensor s and the components of the elastic stiffness tensor c are given. The elastic compliance tensor s and the elastic stiffness tensor c are both defined by the generalized forms of Hooke s law, a = ce and e = sa. At the end of the list, the tensile strength, the Vickers hardness, and the Mohs hardness are given for some elements. 

---