Shear wave quasi

Figure 9 Slowness diagram At the interface between isotropic steel and a V-bntt weld with 10° inclination and perpendicular grain orientation the incident 45° (with regard to the sample top surface) shea.r wave will split into two quasi shear waves qSV and qSV( 2.)...

Splitting into two quasi shear waves If the transducer is coupling to the isotropic steel the incident shear wave may split into two independent quasi shear vertical wave-... 

Using now the phase matching condition, it can be seen that besides the quasi shear wave (qSV) which is obtained as usual, a second quasi shear wave (qSV(2)) results from the upper quasi shear wave part. Since the direction of the group velocity vector points downwards this wave is able to propagate and can be seen in the snapshot (see Fig. 10) if a is properly adjusted, i.e. is pointing upwards as in Fig. 2. 

A piezo-composite consists of a piezoelectric active phase and a passive plastic phase [2]. In the 1-3-configuration adopted in our case, piezoelectric rods parallely aligned in thickness direction are imbedded in a three-dimensional plastic matrix (Fig. 1). The distance between the rods has to be chosen inferior to the half wave length of the shear wave in the matrix material ensuring that the whole compound is vibrating as a quasi-homogeneous material. 

For each section, the quantity p(w/k)2 is given for the pure shear wave, which is polarized perpendicular to the plane of the section, and for the quasi-shear and quasi-longitudinal waves, which each have particle motion in the plane of the section. The angle between the wavevector k and the lowest symmetry direction in the plane is denoted by 0. 

The isotropic part has not changed. The quasi pressure (qP) curve splits up into a real and an imaginary branch . During this real part the transversal share of the polarization increases until the wave becomes a quasi shear vertical wave. Furthermore, the wave is not anymore a propagating but an evanescent wave in this part. The branch is again only real, it is part of the quasi shear vertical (qSV) curve of the homogeneous case (dotted line), its polarization is dominated by the transversal share and the wave is a propagating one. For the branches (real) and... 

Twinning is itself a kind of solid solid phase transformation with the first and seeond phases having the same erystal strueture (henee the same pressure-volume response) but with prineipal erystallographie orientation at some angle to the original orientation. This results in a eontribution to plastie shear deformation under both quasi-statie and shoek-wave eonditions [61]. 

Like the isotropic wave equation the Christoffel equation has three solutions, although in general there is no degeneracy except along symmetry directions. The motions of the particles are orthogonal for the three solutions, but not necessarily exactly parallel or perpendicular to the propagation direction, and so the waves are described as quasi-longitudinal or quasi-shear. 

In the surface of an anisotropic solid the situation is more complicated. Pure Rayleigh waves can exist only along certain symmetry directions in which pure SV waves exist. Away from these directions, however, the two quasi-shear polarizations are not pure SV and SH therefore, although the particle motions are orthogonal, at the surface they can be weakly coupled. If the SH mode has a higher velocity than the SV, then there can be no real solution to Snell s law... 

In the first part of this chapter we studied the radial vibrations of a solid or hollow sphere. This problem was considered an extension to the dynamic situation of the quasi-static problem of the response of a viscoelastic sphere under a step input in pressure. Let us consider now the simple case of a transverse harmonic excitation in which separation of variables can be used to solve the motion equation. Let us assume a slab of a viscoelastic material between two parallel rigid plates separated by a distance h, in which a sinusoidal motion is imposed on the lower plate. In this case we deal with a transverse wave, and the viscoelastic modulus to be used is, of course, the shear modulus. As shown in Figure 16.7, let us consider a Cartesian coordinate system associated with the material, with its X2 axis perpendicular to the shearing plane, its xx axis parallel to the direction of the shearing displacement, and its origin in the center of the lower plate. Under steady-state conditions, each part of the viscoelastic slab will undergo an oscillatory motion with a displacement i(x2, t) in the direction of the Xx axis whose amplitude depends on the distance from the origin X2-... 

Because vertical rock stress is reducing with depth compared to the horizontal ones beneath the first 4 km, Bradshow Zoback (1988), the faults plateau and LVZ are quasi-horizontal and compose wave-guides between the Forsch boundary (F) and the Conrad (C) boundary. They can be understood also as large scale localization of shear bands. 

Target size. In order to ensure that the flexure stress wave and shear stress wave be transmitted and reflected many times by the boundary, and to homogenize the stresses sufficiently, a smaller sized target tends to result in a quasi-static response. 

The general structure of stability Equation 18 remains unchanged when different quasi-steady models are applied for the various shear stresses terms. Moreover, even when the viscous effects are completely ignored, resorting to an inviscid K-H stability type of analysis, the structure of the resulting stability condition. Equation 18, is still maintained while Equation 19 for attains different expression. For instance, the long wave K-H stability analysis on two inviscid layers (rectangular channel) yields ... 

The stability conditions in Equations 18-19 correspond to a quasi-steady modelling of the various shear stresses hence, the effect of axial convection of the wave-... 

Recently, Hanratty presented a comprehensive review of the attempts to account for the interfacial waviness in modelling the interfacial shear stress for the stability analysis of gas-liquid two-phase flows [53]. Basically, the approach taken was to implement the models obtained for the surface stresses in air flow over a solid wavy boundary as a boundary condition for the momentum equation of the liquid layer over its it mobile wavy interface. Craik [98] adopted the interfacial stresses components which evolve from the quasi-laminar model by Benjamin [84]. Jurman and McCready [99], Jurman et al. [100], and Asali and Hanratty [101] used correlated experimental values of shear stress components (phase and amplitude) based on turbulent models which consider relaxation effects in the Van Driest mixing length. Since the characteristics of the predicted surface stresses are dependent on the wave number, Asali and Hanratty picked the phase and amplitude values which correspond to the wave lengths of the capillary ripples observed in their experiments of thin liquid layers sheared by high gas velocities [101]. It was shown that the growth of these ripples is controlled by the interfacial shear stress component in phase with the wave slope. 

Equation 23 implies a shear stress augmentation at the windward side of the wave and a relaxation at its leeward side. With reference to Figure 4a, for the case of u > u, X > 0. In this case, dh/3x > 0 at the wind-side, and thus. Equation 23 yields augmentation of the interfacial shear stress, Xj > x9 as expected at the wind-side. Clearly, shear stress relaxation occurs at the lee-side where 3h/9x < 0. On the other hand, in the case of u < u. Figure 4b, corresponds to a negative quasi-steady interfacial shear, x < 0 and thus the (negative) shear stress is augmented at 9h/3x < 0, which is now the wind-side, and is relaxed at the lee-side, where dh/dx > 0. Hence, a positive dynamic coefficient in Equation 23, > 0, consistently... 

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