Shear mechanical impedance

The amount of attenuation suffered by the pulse echoes depends directly on the shear mechanical impedance, Z, of the polymer film. For... 

The electrical response of a liquid-loaded TSM resonator can be related to the shear mechanical impedance, Z, at the device surface. This mechanical impedance serves as a quantitative measure of the strength of the interaction between the solid and a contacting liquid. 

Figure 3.11 Normalized components of the surface (shear) mechanical impedance Zs (at 5 MHz) vs liquid properties for several surface roughnesses. (Reprinted and adapted with permission. See Ref. [14]. ) 1993 American Chemical Society.)...

The property of a coating which determines how much cyclic shear energy will be transmitted across the coating-substrate interface is the velocity of propagation in the two media. Now G , which depends on the square of velocity, must be replaced by the shear mechanical impedance Z, where... 

The equivalent circuit model of Figure 3.7 can be used to describe the near-resonant electrical characteristics of the quartz resonator coated by a viscoelastic film. The surface film causes an increase in the motional impedance, denoted by the complex element Zg. From Equation 3.19, this element is proportional to the ratio of the surface mechanical impedance Zj contributed by the film to the characteristic shear wave impedance Zq of the quartz. 

The oscillating resonator surface may be considered as a source for shear waves that are radiated into the contacting film. The upper film surface reflects these radiated shear waves downward, so that the mechanical impedance seen at the quartz surface is dependent upon the phase shift and attenuation undergone by the wave in propagating across the film. When the film is rubbery, significant phase shift across the film occurs. Consequently, the coupling of acoustic energy into the film depends upon thin-fllm interference. 

The properties of the impedance spectrum are discussed in detail in Chap. 2 in this volume. Here we present only a relation between the resonant frequency and the mechanical impedance of the mediiun contacting the quartz surface, Zl. The latter is deflned as the ratio of the shear stress acting on the contact medium to the surface velocity [6]. Under the experimental conditions when the surface loading is relatively small, the shift of the resonant frequency with respect to the resonant frequency of the unloaded quartz crystal,/o, can be written as [14,29] ... 

Let us return to the reduction of shear stress at the crack tip due to the emission of dislocations. Figure 14-9 illustrates a possible stress reduction mechanism. It can be seen that the tip of a crack is no longer atomically sharp after a dislocation has been emitted. It is the interaction of the external stress field with that of the newly formed dislocations which creates the local stress responsible for further crack growth. Thus, the plastic deformation normally impedes embrittlement because the dislocations screen the crack from the external stress. Theoretical calculations are difficult because the lattice distortions of both tension and shear near the crack tip are large so that nonlinear behavior is expected. In addition, surface effects have to be included. 

With a test sample on the optically flat top surface of the bar, the pulse echo train is reduced in amplitude. This attenuation is owing to the refraction of part of the ultrasonic wave into the test sample at the frequency used. The ratio of successive peak amplitudes may be measured on the oscilloscope and expressed in decibels loss per echo. From this, the loss per echo with no sample on the bar can be substracted to give a value Adb which is related to the mechanical shear impedance of the sample. Rapid changes can be conveniently monitored by a recorder which follows the peak signal of a selected echo. 

Evidently, there are two distinct frequencies, where either the numerator or the denominator of the complex impedance becomes zero. However, the case of zero impedance is determined exclusively by the serial capacity, whereas the parallel determines the frequency of infinite impedance. These two frequencies thus correspond to the resonance frequencies of the two part circuits mentioned above and are also correctly reproduced in the frequency spectrum of the QCM. Observable side resonances (as shown especially in the insert with lower span) can be traced back to mechanical oscillations that differ from the main one one is the result of antisymmetric thickness shear oscillation, the other of a twist oscillation. The ratio of intensity between the desired thickness shear wave and the side resonances is mainly defined by the ratio between electrode diameter and quartz substrate thickness. This is illustrated in Fig. 4, where the damping spectra for both a 10 MHz and a 5 MHz device are given. In both cases the electrode diameter here is 8 mm (the spectra in Fig. 3 were recorded with 4 mm electrode diameter). Evidently, the 5 MHz QCM shows the desired response pattern, where the shear resonance by far dominates the electrical behaviour. The 10 MHz QCM, however, shows very pronounced side resonances. The rather large electrode diameter (compared to the thickness) very strongly favours the occurrence of torsional motions within the substrate, thus reasonable amplitudes are generated for this mode. 

The secondary flow (in the plane perpendicular to the axial flow, see Figure 9.5) determines, in large measure, the differences in transport rates between the inside and outside walls. The radial velocity is directed toward the wall at the outside, leading to enhancement of transport by a convective mechanism. At the inside of the curvature, the radial velocity is directed away from the wall, and transport is impeded by the convective mechanism. This secondary flow mechanism produces transport rates that are 25 times higher on the outside wall than the inside wafl. This is to be contrasted with wall shear stress values which, for the coronary artery condition, are less than two times higher on the outside wall than on the inside wall [31]. Earlier studies of fufly developed, steady flow and transport in curved tubes [32] are consistent with the above observations. 

Determination of the viscosity coefficients from the mechanical wave propagation and attenuation in the ordered nonatic phase is probably the closest to the first principles methods. The shear impedance technique is based on measuring the reflection and attenuation of ultrasonic shear waves [90-92]. The conqtlex shear impedance of the nematic sample, Zn = Rn + iXn, is determined from the complex... 

When a quartz crystal (or any other solid material) vibrates, there is always a resonance frequency, which we denote fo, at which it oscillates with minimum impedance (that is maximum admittance). The resonance frequency depends on the dimensions and on the properties of the vibrating crystal, mostly the density and the shear modulus. A quartz crystal can be made to oscillate at other frequencies, but as the distance, (on the scale of frequency), from the resonance frequency increases, the admittance decreases, until the vibration can no longer be detected. This is the basis for the analysis of the so-called (mechanical) admittance spectrum of the QCM, which is discussed below. 

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