Subject plane

This subject has a long history and important early papers include those by Deijaguin and Landau [29] (see Ref. 30) and Langmuir [31]. As noted by Langmuir in 1938, the total force acting on the planes can be regarded as the sum of a contribution from osmotic pressure, since the ion concentrations differ from those in the bulk, and a force due to the electric field. The total force must be constant across the gap and since the field, d /jdx is zero at the midpoint, the total force is given the net osmotic pressure at this point. If the solution is dilute, then... 

It turns out that many surfaces (and many line patterns such as shown in Fig. XV-7) conform empirically to Eq. VII-20 (or Eq. VII-21) over a significant range of r (or a). Fractal surfaces thus constitute an extreme departure from ideal plane surfaces yet are amenable to mathematical analysis. There is a considerable literature on the subject, but Refs. 104-109 are representative. The fractal approach to adsorption phenomena is discussed in Section XVI-13. 

Therefore, in the RF mode, ions transmitted through the rod guide are subjected to (1) an oscillation in step with the variations of the RF field in the x,y-plane, (2) a drift or guided motion caused by the inhomogeneity of the RF field (x,y-plane), and (3) a forward motion (z-direction) due to any initial velocity of the ions on first entering the rod assembly. The separate motions... 

The deep violet color of pentaphenylbismuth and certain other pentaarylbismuth compounds has been the subject of considerable speculation. It has been shown by x-ray diffraction (173) that the bismuth atom in pentaphenylbismuth is square—pyramidal. WeU-formed crystals are dichromic, appearing violet when viewed in one plane but colorless in another plane. The nature of the chromophore has been suggested to be a charge-transfer transition by excitation of the four long equatorial bonds ... 

The condition for propagation of a mode 1 edge crack, that is, a crack that is subjected to pure opening (tensile) stresses appHed perpendicular to the crack plane, is given by (35) ... 

Since the belt is wrapped snugly around the material, it moves with the belt and is not subject to any form of internal movement except at feed and discharge. In addition, the belt can operate in many planes, with twists and turns to meet almost any layout condition within the fixed hmit of curvature placed on the loaded belt. It can convey and elevate with only a single drive multiple feed and discharge points are relatively easy to arrange. 

If this is subjected to the "Bain strain" it becomes an undistorted b.c.c. cell. This atomic "switching" involves the least shuffling of atoms. As it stands the new lattice is not coherent with the old one. But we can get coherency by rotating the b.c.c. lattice planes as well (Fig. 8.8). 

The resulting mouldings and extrudates are consequently anisotropic and mouldings can be four to five times as strong in one direction as in another direction. This can lead to planes of weakness and easy fracture when subject to shock (impact) stresses. Generally such orientation is undesirable but there are at least two instances of its being of value ... 

The papers which introduced the concept of a dislocation all appeared in 1934 (Polanyi 1934, Taylor 1934, Orowan 1934). Figure 3.20 shows Orowan s original sketch of an edge dislocation and Taylor s schematic picture of a dislocation moving. It was known to all three of the co-inventors that plastic deformation took place by slip on lattice planes subjected to a higher shear stress than any of the other symmetrically equivalent planes (see Chapter 4, Section 4.2.1). Taylor and his collaborator Quinney had also undertaken some quite remarkably precise calorimetric research to determine how much of the work done to deform a piece of metal... 

The previous section has considered the in-plane deformations of a single ply. In practice, real engineering components are likely to be subjected to this type of loading plus (or as an alternative) bending deformations. It is convenient at this stage to consider the flexural loading of a single ply because this will develop the method of solution for multi-ply laminates. 

The in-plane stiffness behaviour of symmetric laminates may be analysed as follows. The plies in a laminate are all securely bonded together so that when the laminate is subjected to a force in the plane of the laminate, all the plies deform by the same amount. Hence, the strain is the same in every ply but because the modulus of each ply is different, the stresses are not the same. This is illustrated in Fig. 3.19. 

If an isotropic material is subjected to multi-axial stresses then the situation is slightly more complex but there are well established procedures for predicting failure. If a,i and Oy are applied it is not simply a question of ensuring that neither of these exceed ar- At values of and Oy below oj there can be a plane within the material where the stress reaches ot and this will initiate failure. 

We showed the possible existence of various forms of helically coiled and toroidal structures based on energetic and thermodynamic stability considerations. Though the formation process of these structures is not the subject of this work, the variety of patterns in the outer and inner surface of the structures indicates that there exist many different forms of stable cage carbon structures[10-19]. The molecules in a onedimensional chain, or a two-dimensional plane, or a three-dimensional supermolecule are possible extended structures of tori with rich applications. 

Deflection of a duct The largest deformation of a duct subjected to an imposed load, given as the measured difference in distance between a plane through the points of support and a plane through the lowest point of the duct under a load. 

Deflection of a joint The largest deformation of a joint subjected to a positive or negative pressure, given by the measured difference in distance from a reference plane outside the joint to the joint with and without pressure. 

Fig. 6.6. One-dimensional, pressure-versus-location predictions at various times are shown for a typical powder compact subjected to baratol plane-wave explosive loading. The pressure is shown to ring-up to a final value between the copper end plates (after Graham [87G03]).

The preceding stress-strain and strain-stress relations are the basis for stiffness and stress analysis of an individual lamina subjected to forces in its own plane. Thus, the relations are indispensable in laminate analysis. 

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