Wavelength flexural

Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product k4, where k, = 27t/A, A, is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets symmetric waves (denoted by So, S],. ..) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (Aq, A, . ..), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (M < 1-6), only two waves exist — the lowest-order symmetric mode (Sq) and the lowest-order antisymmetric mode (Aq). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the Ao mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind.

Figure 3.40 Calculated phase velocity of flexural plate waves vs ratio of plate thickness, d, to wavelength. A, for silicon nitride. Material is assumed to have the elastic properties of Si3N4 and no residual tension. The mode shapes ate illustrated at the right with a greatly enlarged vertical scale for clarity. Ellipses at left show the retrograde elliptical particle motions of the lowest S3rmmetric and antisymmetric modes for d/A = 0.03. (Reprinted with pemtission. See Ref. (621.)...

Figure 3.46 Phase velocity (a) and attenuation (b) of fluid-loaded flexural plate modes plotted vs thickness/wavelength, d/A. The lossless (Scholtc) mode velocity approaches from below the velocity of the fluid, assumed to be water. CPT denotes result of classical plate theory for an unloaded plate. (Reprinted with permission. See Ref. [62].)...

Just as flexural waves can propagate at low speeds in a plate whose thickness is much less than the wavelength, a low-speed flexural wave can propagate in a cylindrical rod whose diameter is much smaller than the wavelength [87]. Because of the low wave speed, operation as a gravimetric sensor in liquids is possible, as with the flexural plate-wave sensor. The gravimetric sensitivity for this sensor is typically S = — l/(2p ), where a is the radius of the rod. 

The wavelength and amplitude of these three structures can be accounted for by the elastic flexure of two cantilevered lithospheric plates if the boundary between East and West Antarctica is taken as a stress-free edge. 

The flexural strength of the specimen before and after oxidation was tested with three-point bending of 3 by 4 by 36 mm bars, using a 30 mm span and a crosshead speed of 0.5 mm min. The specimen was supported on a graphite crucible, and the temperature of the specimen center was measured by a multi-wavelength pyrometer with measurement range of 1000-2500 °C. The isothermal oxidation of the specimens was carried out in static air at constant temperatures of 1000 15, 1200 15 and 1400 15 °C, each for several different times. Our investigation had confirmed that the predominant phases for the as-sintered ceramic were ZrB, SiC and a small quantity of ZrC. 

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