Elastic wave

Acoustic emission is a naturally occurring phenomenon within materials, and the term Acoustic Emission is used to define the spontaneous elastic energy released within material or by a process, in the form of transient elastic waves. (2)... 

Numerical Modeling of Elastic Wave Propagation in Inhomogeneous Anisotropic Media. 

Recently, EFIT has been extended to simulate elastic waves in homogeneous anisotropic media [3, 4] and inhomogeneous anisotropic media [5, 6]. Since Waite et al. [7,... 

Restoring of SD of parameters of stress field is based on the effect of acoustoelasticity. Its fundamental problem is determination of relationship between US wave parameters and components of stresses. To use in practice acoustoelasticity for SDS diagnosing, it is designed matrix theory [Bobrenco, 1991]. For the description of the elastic waves spreading in the medium it uses matrices of velocity v of US waves spreading, absolute A and relative... 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

The energy of an elastic wave in a solid is quantized just as the energy of an electromagnetic wave in a cavity. 

The quanta of the elastic wave energy are called phonons The themral average number of phonons in an elastic wave of frequency or is given, just as in the case of photons, by... 

Phonons are nomial modes of vibration of a low-temperatnre solid, where the atomic motions around the equilibrium lattice can be approximated by hannonic vibrations. The coupled atomic vibrations can be diagonalized into uncoupled nonnal modes (phonons) if a hannonic approximation is made. In the simplest analysis of the contribution of phonons to the average internal energy and heat capacity one makes two assumptions (i) the frequency of an elastic wave is independent of the strain amplitude and (ii) the velocities of all elastic waves are equal and independent of the frequency, direction of propagation and the direction of polarization. These two assumptions are used below for all the modes and leads to the famous Debye model. 

There are differences between photons and phonons while the total number of photons in a cavity is infinite, the number of elastic modes m a finite solid is finite and equals 3N if there are N atoms in a three-dimensional solid. Furthennore, an elastic wave has tliree possible polarizations, two transverse and one longimdinal, in contrast to only... 

The Debye model is more appropriate for the acoustic branches of tire elastic modes of a hanuonic solid. For molecular solids one has in addition optical branches in the elastic wave dispersion, and the Einstein model is more appropriate to describe the contribution to U and Cj from the optical branch. The above discussion for phonons is suitable for non-metallic solids. In metals, one has, in addition, the contribution from the electronic motion to Uand Cy. This is discussed later, in section (A2.2.5.6T... 

Phonon transport is the main conduction mechanism below 300°C. Compositional effects are significant because the mean free phonon path is limited by the random glass stmcture. Estimates of the mean free phonon path in vitreous siUca, made using elastic wave velocity, heat capacity, and thermal conductivity data, generate a value of 520 pm, which is on the order of the dimensions of the SiO tetrahedron (151). Radiative conduction mechanisms can be significant at higher temperatures. 

Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflec ted, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through sohds in a complex array of types of elastic waves. Sound is charac terized by its amplitude, frequency, phase, and direction of propagation. 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

Figure 4.23. Elastic wave velocities as a function of pressure along the Hugoniot of iron. The solid curve is the calculated bulk sound velocity. (From Brown and McQueen (1982).)...

A typical shock-compression wave-profile measurement consists of particle velocity as a function of time at some material point within or on the surface of the sample. These measurements are commonly made by means of laser interferometry as discussed in Chapter 3 of this book. A typical wave profile as a function of position in the sample is shown in Fig. 7.2. Each portion of the wave profile contains information about the microstructure in the form of the product of and v. The decaying elastic wave has been an important source of indirect information on micromechanics of shock-induced plastic deformation. Taylor [9] used measurements of the decaying elastic precursor to determine parameters for polycrystalline Armco iron. He showed that the rate of decay of the elastic precursor in Fig. 7.2 is given by (Appendix)... 

When the elastic shock-front speed U departs significantly from longitudinal elastic sound speed c, immediately behind the elastic shock front, the decaying elastic wave amplitude is governed by (Appendix)... 

Y.M. Gupta, Stress Dependence of Elastic-Wave Attenuation in LiF, J. Appl. Phys. 46, 3395-3401 (1975). 

If we accept the assumption that the elastic wave can be treated to good aproximation as a mathematical discontinuity, then the stress decay at the elastic wave front is given by (A. 15) and (A. 16) in terms of the material-dependent and amplitude-dependent wave speeds c, (the isentropic longitudinal elastic sound speed), U (the finite-amplitude elastic shock velocity), and Cfi [(A.9)]. In general, all three wave velocities are different. We know, for example, that... 

Figure 8.7. Propagation of wave profile in an elastic-plastic material from the spall plane to the monitoring interface. The wave front propagates at a plastic wave speed whereas the wave release propagates at an elastic wave speed and complicates the analysis of the material spall strength.

The data shown in Fig. 8.11 are for an 80 ml/kg grade oil shale obtained from a mine near central Colorado. Oil shale grades from this region vary from 40-320 ml/kg. Properties such as fracture toughness and elastic constants are found to depend on oil shale grade. For the oil shale studied in Fig. 8.11, a fracture toughness of x 0.9 MN/m, a density of p = 2000 kg/m and an elastic wave speed of c = 3000 m/s are representative. 

The rise times of the elastic wave may be quite narrow in elastic single crystals, but in polycrystalline solids the times can be significant due to heterogeneities in physical and chemical composition and residual stresses. In materials such as fused quartz, negative curvature of the stress-volume relation can lead to dispersive waves with slowly rising profiles. 

Wave profiles in the elastic-plastic region are often idealized as two distinct shock fronts separated by a region of constant elastic strain. Such an idealized behavior is seldom, if ever, observed. Near the leading elastic wave, relaxations are typical and the profile in front of the inelastic wave typically shows significant changes in stress with time. 

Fig. 2.7. Elastic precursor decay in which elastic waves are observed to decrease in amplitude with propagation distance is a typical behavior. The data of this figure describe the behavior of crystalline LiF samples of different yield strengths (after Asay et al. [72A02]).

Fig. 2.20. The release wave portion of time-resolved velocity profiles in porous aluminum is shown as measured with VISAR instrumentation. At pressures near that required to cause melt, the release changes from that of an elastic wave to that of a bulk plastic wave, indicating the change to a melt condition (after Asay and Hayes [75A01]).

Fig. 5.1. The electrostatic configurations of the Neilson-Benedick three-zone model describe a piezoelectric solid subject to elastic-inelastic shock deformation which divides the crystal into three distinct zones. Zone 1, ahead of the elastic wave, is unstressed. Zone 2 is elastically stressed at the Hugoniot elastic limit. Zone 3 is isotropically pressurized to the input pressure value (after Graham [74G01]).

P. Constance Yang, Charles H. Norris, and Yehuda Stavsky, Elastic Wave Propagation In Heterogeneous Plates, International Journal of Solids and Structures, October 1966, pp. 665-684. 

Under development are intelligent vehicles for crack detection. An elastic-wave version (developed by British Gas and the Harwell Laboratory) is currently being evaluated in a test-loop. This vehicle has successfully detected stress-corrosion cracks in the test-loop. The Gas Research Institute (USA) is sponsoring development work with intelligent vehicles at the Battelle Columbus Division (Ohio). Facilities for testing vehicles were commissioned in 1991... 

Fedorov A. F. I., Theory of elastic waves in cristals, Plenum... 

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