Shear motion

Figure 3.12. Experimental configuration and velocity profiles demonstrating the use of VISAR interferometric techniques in pressure-shear instrumentation to determine in-plane shear motion as well as longitudinal (P-wave) motion (Chhabildas and Swegle, 1980).

If one side of the quartz is coated with material, the spectrum of the resonances is shifted to lower frequencies. It has been observed that the three above mentioned modes have a somewhat differing mass sensitivity and thus experience somewhat different frequency shifts. This difference is utilized to determine the Z value of the material. By using the equations for the individual modes and observing the frequencies for the (100) and the (102) mode, one can calculate the ratio of the two elastic constants Cgg and C55. These two elastic constants are based on the shear motion. The key element in Wajid s theory is the following equation ... 

The use of a rotating vane has become very popular as a simple to use technique that allows slip to be overcome (33,34). Alderman et al (35) used the vane method to determine the yield stress, yield strain and shear modulus of bentonite gels. In the latter work it is interesting to note that a typical toique/time plot exhibits a maximum torque (related to yield stress of the sample) after which the torque is observed to decrease with time. The fall in torque beyond the maximum point was described loosely as being a transition from a gel-like to a fluid-like behavior. However, it may also be caused by the development of a slip surface within the bulk material. Indeed, by the use of the marker line technique, Plucinski et al (15) found that in parallel plate fixtures and in slow steady shear motion, the onset of slip in mayonnaises coincided with the onset of decrease in torque (Fig. 8). These authors found slip to be present for... 

When the particle concentration is high, the shear motion of particles leads to interparticle collisions. The transfer of momentum between particles can be described in terms of a pseudoshear stress and the viscosity of particle-particle interactions. Let us first examine the transfer of momentum in an elastic collision between two particles, as shown in Fig. 5.8(a). Particle 1 is fixed in space while particle 2 collides with particle 1 with an initial momentum in the x-direction. Assume that the contact surface is frictionless so that the rebound of the particle is in a form of specular reflection in the r-x-plane. The rate of change of the x-component of the momentum between the two particles is given by... 

Figure 5.8. Impaction due to shear motion (a) Elastic collision between two particles (b) Elastic collision of a single sphere with a cloud of particles.

The dilational rheology behavior of polymer monolayers is a very interesting aspect. If a polymer film is viewed as a macroscopy continuum medium, several types of motion are possible [96], As it has been explained by Monroy et al. [59], it is possible to distinguish two main types capillary (or out of plane) and dilational (or in plane) [59,60,97], The first one is a shear deformation, while for the second one there are both a compression - dilatation motion and a shear motion. Since dissipative effects do exist within the film, each of the motions consists of elastic and viscous components. The elastic constant for the capillary motion is the surface tension y, while for the second it is the dilatation elasticity e. The latter modulus depends upon the stress applied to the monolayer. For a uniaxial stress (as it is the case for capillary waves or for compression in a single barrier Langmuir trough) the dilatational modulus is the sum of the compression and shear moduli [98]... 

Let us imagine, following Pokrovskii and Pyshnograi (1988), the shear motion of the system as a motion of overlapping macromolecular coils, each of which is characterised by the function (1.24) of the mean number density of Brownian particles... 

In this section we shall continue to investigate shear motion, while, in contrast to the previous section, we shall assume that the velocity gradient depends on the time but, as before, does not depend on the space coordinate. We shall consider a simple case of ideally flexible chains, for which the stress tensor and relaxation equations are defined by equations (9.3) and (9.4). 

These expressions describe the establishment of stresses for given uniform shear motion. 

The ratio of the coefficients is a function of the invariant D which, for shear motion, has the form... 

Further on we shall consider the simple case when the relaxation equation is given by equation (9.58) and we shall assume the shear motion to be a steady-state one. So, we have the expressions... 

See, however, the postulate of Maxwell [2] about the anisotropic distribution of velocities in gases having a stationary shearing motion, which he states without any attempt at proof ... 

Q. Jiang, Fractal Structure of Aggregates Induced by Shear Motion, Ph.D. Dissertation, Department of Civil Engineering and Engineering Mechanics, University of Arizona, 1993. 

Equivalent considerations for nonstatic, sheared systems demonstrate the kinematical possibility of such shearing motions. This requires, inter alia, that the distance between any two sphere centers remains larger than 2a. The static viewpoint can be generalized to such circumstances as follows Rather than considering the lattice deformation, it suffices to examine the deformed collision sphere. The latter body 3 is defined as the set of points... 

When the lattice points initially lie entirely outside <3 (except 0, which lies at the center of 6), it can be shown (Adler and Brenner, 1985a) that macroscopic shearing motion is possible for all times. 

This form significantly simplifies the algebraic structure of the kinetic problem. In particular, no direct coupling now exists between the translational motion and the angular and shearing motions, suggesting the possibility of treating the translational motion independently of the latter two. 

Admittance-vs-frequency measurements made at several temperatures on a polyisobutylene-coated TSM resonator were fit to the equivalent-circuit model of Sections 3.1.3 and 3.1.9 to determine values of G and G for the film [66]. These extracted values are shown in Figure 4.4, along with 5-MHz values obtained from the literature for polyisobutylene having an average molecular weight of 1.56 X 10 [44]. We note excellent agreement between the extracted and literature values of G from —20°C to 60°C, and in G" from —20°C to 10°C. Above 10°C, the extracted G" values are approximately 30% higher than the literature values. These results illustrate how AW devices can be used to quantitatively evaluate the viscoelastic properties of polymer films. Similar models for other AW devices, such as the model for SAW devices coated with viscoelastic layers (Section 3.2.7 and [61]), can enable these other devices also to be used to determine modulus values. However, the pure shear motion of the TSM does simplify the model, and the evaluation of the modulus values as compared with the more complex displacements of other AW devices such as the SAW device (a comparison of the models of Section 3.1.9 for the TSM and Section 3.2.7 for the SAW demonstrates this point). 

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