Displacement tangential

We continue the investigation of the contact problem for a plate under creep conditions. In this section the case of both normal and tangential displacements of the plate is considered. 

The Snoek pendulum, which in the most general sense is a device to measure relaxations, has also been used to measure relaxation caused by tangential displacements at grain boundaries. This application has been the central concern of a distinguished Chinese physicist, Tingsui Ke, for all of the past 55 years. He was stimulated to this study by Clarence Zener, in 1945, and pursued the approach, first in Chicago and then in China. This exceptional fidelity to a powerful quantitative technique was recognised by a medal and an invitation to deliver an overview lecture in America, recently published shortly before his death (Ke 1999). 

Ur, Ur, and u- are the cartesian results Ur and u- are the results in cylindrical co-ordinates. Note u is in the direction of positive P, pressing into the half plane for compressive loading. The tangential displacements of the free surface are towards the origin in agreement with our intuition. 

The calculation of the tangential displacement St requires some special attention and will be addresses in Section III.B.3. 

Suppose that the tangential displacement at tQ is equal to Sto then one would expect that the displacement dt at time t follows by simply integrating the tangential velocity (Hoomans, 2000) ... 

In a 3D system, however, it becomes more complicated. The particle-particle contact now occurs in a plane. The tangential component of the relative velocity is always in this plane and vertical to the normal unit vector according to the definition. Since the normal unit vector is not necessarily situated in the same plane at any time, it is desirable to transfer the old tangential displacement to the new contact plane before we calculate the new tangential displacement. To this end, a 3D rotation of the old tangential displacement should be applied. As... 

Are Eqs. (27) and (28) sufficient to describe the tangential displacement during particle-particle contact In the absence of friction, the answer is yes. When we consider friction during particle particle contact—as pointed out by Brendel... 

When an electric field is applied across the working capillary, the double-layer ions begin to migrate and soon reach the steady-state velocity. In the steady state, electrical and viscous forces balance one another. The forces exerted on the ions by the medium are equal and opposite to the forces exerted on the medium by the ions consequently, the liquid also attains a stationary-state velocity. The tangential displacement of the fluid relative to the wall defines a surface of shear at which the potential equals f. 

Figure 12.7b is a sketch of an apparatus that may be used to measure streaming potential. As was the case with electroosmosis, the capillary can be replaced by a plug of powdered material between perforated electrodes. An applied pressure difference p across the capillary causes the solution to flow through the capillary, thereby tangentially displacing the part of the double layer in the mobile phase from the stationary part. 

A similar relation governs the tangential displacements in the y-direction. For the no-slip region, sx becomes zero so that... 

This distribution leads to a uniform tangential displacement of the contact surface. q0 is the maximum tangential traction at the center of the contact surface. By integrating Eq. (2.94) over the contact surface, q0 is found to be... 

Therefore, a constant tangential displacement induced by the no-slip contact between two spheres is given by... 

Therefore, the tangential displacements in the core region under joint actions portrayed by Eq. (2.100) and Eq. (2.101) can be obtained as... 

Besides the oblique contact, tangential displacements may also be produced in the contact of two elastic spheres under the actions of a compressional twist, as shown in Fig. 2.13. Since the torsional couple does not give rise to a displacement in the z-direction, the pressure distribution is not influenced by the twist and is thus given by the Hertzian contact theory. 

If a torsional couple or twist moment Mz were applied to the two spheres in contact, the contact surface would undergo a rigid rotation relative to distant points in each body if there were no-slip motion at the interface. In this case, the tangential displacements could be expressed by... 

Optical methods can also be utilized to track the displacement of samples surfaces. Various types of interferometers have been employed for measurements of this type, yielding accuracy in the surface displacement between 0.1 and 10-5 A [6,9,15]. Both single and double beam versions have been developed for measuring the d33 coefficient of bulk samples. Cubic samples can avoid some of the difficulties associated with sample flexure on actuation [9], In some cases, one of the probing beams can also be brought to the side of the sample, so that lateral and tangential displacements can be tracked simultaneously [15]. 

The foregoing formulation for crazing focuses on the description along the normal direction. The formulation of Tijssens et al. [20,21] also accounts for a tangential component in the cohesive properties with a tangential traction at and a tangential displacement At being related by... 

For ad the total displacement is the resultant of the radial and tangential displacements according to... 

We will write the linear equations of variation, using, as components of the displacement, the tangential displacement, p, given by ... 

Calculation and analyses of tangential displacements of typical days... 

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