Shear plane system

For an estimate of the ultimate shear strength, r0, of a single domain based on the lattice parameters we use a simple shear plane system proposed by Frenkel [19]. As shown in Fig. 19 it consists of a linear array of periodic force centres resembling the polymer chain. According to this model the relation between the relative displacement x along the shear direction and the shear stress is given by... 

Fig. 19 Shear plane system with periodic force centres spaced at a distance p along the shear direction x and with an interplanar spacing dc according to the model of Frenkel [19]...

Fig. 20 Shear stress r and shear energy as a function of shear displacement x for a simple shear plane system...

A further set of problems which obviously follows from the above discussion concerns the mechanism of shear-plane formation, although we should emphasize that the considerations involved here are quite separate from the thermodynamic ones discussed above. We discuss these mechanistic problems in Section 4 after considering a second structural feature in shear-plane systems, viz. the remarkable long-range ordering that commonly occurs in oxides containing these defects. 

An alternative approach to the problem is provided by the atomistic lattice-energy calculations discussed in Section 2, p. 108. In addition to the obvious advantages of atomistic theories, such calculations also require no assumptions as to the nature of the shear-plane interactions. Thus electrostatic terms, which may play an appreciable role in shear-plane systems are automatically included in such theories. Ordered shear-plane compounds, e.g. the Ti 02 -i, series can be described in terms of large unit cells. If the lattice energies of these structures are calculated as described in E. Iguchi and R. J. D. Tilley, Phil. Trans., 1977, 286, 55. 

The electroosmotic pumping is executed when an electric field is applied across the channel. The moving force comes from the ion moves in the double layer at the wall towards the electrode of opposite polarity, which creates motion of the fluid near the walls and transfer of the bulk fluid in convection motion via viscous forces. The potential at the shear plane between the fixed Stem layer and Gouy-Champmon layer is called zeta potential, which is strongly dependent on the chemistry of the two phase system, i.e. the chemical composition of both solution and wall surface. The electroosmotic mobility, xeo, can be defined as follow,... 

Ti2O3-TiO2 (Ti O2 i) system, the shear operations of (121) [0il] and (132) [0il] are for 3 < n < 10 and 16 < n < 36, respectively. Between these compositions, i.e. between w = 10 and 16 (TiO 89 and TiOj 937), the shear planes seem to pivot around in a continuous manner from (121) to (132), which is unambiguously indicated in the electron diffraction patterns. A similar phenomenon has been observed in VjOg-TiOj, as shown below. 

Fig. 2.112 Reciprocal lattice plane of rutile-typc structure with [111] zone axis. The (s, t) series of shear planes are indicated by dashed lines (see text). White circles are the possible shear planes for this system.

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

Fig. 1 At the level of the approximation we use in this chapter, all experimental shear geometries are equivalent to a simple steady shear. We choose our system of coordinates such that the normal to the plates points along the z-axis and the plates are located at z = j. Between two parallel plates we assume a defect-free well aligned lamellar phase. The upper plate moves with the velocity in positive x direction, the lower plate moves with the same velocity in negative A direction. The y-direction points into the xz-plane. We call the plane of the plates Cry-plane) the shear plane, the x-direction the flow direction, and the y-direction the vortidty direction...

If the inner and outer cylinder rotate together with the fluid as a rigid system, then Q = constant and thus dQ/dr = 0. In that case, the velocity gradient is equal to Q, whereas the shear rate is equal to zero. Hence, in the determination of the shear rate, we have to take into consideration the fact that the velocities of neighbouring shearing planes contain a rigid-body rotation component, Q. We have to subtract this term from the difference in velocities, in order to calculate the shear rate, so that for ... 

The known systems where these shear structures have been found are given in Table III, and a family, or homologous series of compounds, which can be grouped under a common formula is recognizable in each case. The shear planes simply represent discontinuities repeated on the unit cell level at regular intervals. In any one series the same discontinuity is present in all members, where it separates identical blocks of the host structure which vary in size from one member to the next. 

The instrument is operated in a sinusoidal, oscillating mode. The oscillating mode is prefered to minimize shear plane and rate effects associated with continuous 360° rotation. The system computes the torgue response and the resultant phase angle. The phase angle is directly related to the storage (O and loss (G") modulus by the following equations ... 

The measurement of this surface potential (T g or Pq) is impossible due to the hydrodynamic behavior of the system that generates a thin layer of attached liquid around the particles. However, there is a plane where the shear starts (shear plane), and at this plane the surface potential can be measured and the value is known as the zeta potential ( P ). Besides the indifferent counter- and co-ions in solution, there are also so-called potential determining ions (chemists caU them adsorbing ions). For most systems these are and OH ions that can adsorb directly on the particle surface and alter the -potential. There is a pH value for which the potential becomes zero and is called the isoelectric point (lEP), as shown in Figure 11.6. 

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-... 

Shear-plane structures in real systems are invariably more complex. Figure 2 gives a still partially simplified illustration of the shear plane in Ti02-x- However, the essential features of the schematic plane discussed above apply to real systems that is, shear-plane formation eliminates point defects by a change in the mode of linking of MOe octahedra. 

Figure 5 Shear-plane interaction energies vs. inter-plane spacing as obtained by Stoneham and Durham, (a) Systems containing infinite arrays of shear planes (b) Interactions between pairs of shear planes...

Despite their different conclusions, we believe that the approaches of Kittel and of Stoneham and Durham are not incompatible but rather they are complementary. Kittel s approach is probably more appropriate to systems such as the adaptive structures where the lattice parameters of the host and solute may differ considerably. The difference should be much smaller with the non-stoicheiometric systems containing shear planes in which the direct interaction may dominate. The interaction function may clearly differ in these two types of system. 

Shear variants of the Cu3Au structure with two systems of shear planes will be treated later. 

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