Geometry of the neck

Taking into account relation [12.3], if we bring back this area to a grain by considering its coordination number, the total area of the necks on grain will be  

The desired volnme is F = f] - Fj - Fj ---. Thus, taking into account equation 

The area of the desired surface is S = r O. where, Q. = 2ji(l -cos6 ), the solid angle under which we see the sector, and thus, considering equation [12.3], we calculate the desired area as  

Relation between thefractional extent and the radius x of the bridge 

As the radius x of the bridge varies with time, there exists a relation between the fractional extent of the transformation and this radius. Using the definition of a from relation [12.1] and the transferred volume from relation [12.5], and remembering 

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F. 5.20 Geometry of the neck formed between two cylindrical particles during sintering. Reproduced with permission from [40]. Copyright 1979, Elsevier... 

Depending on the geometry of the dewar neck and of the other gas-filled tubes connecting cold with warm parts, the excitement of thermoacoustic oscillation is possible above the helium bath. 

The variation in the measured electron mobilities from sample to sample in sintered materials (also observed by Hahn, ref. 24), may be due to any of several effects. The most probable reason for this variation in the well-sintered samples studied is a difference in history the individual samples are obtained with different numbers of conduction electrons per cm. frozen in in the necks. That is, the different history has allowed different amounts of oxygen to be adsorbed on the surface. Thus the concentration of electrons in the grain, as measured by the Hall coefficient, will have little relation to the concentration of electrons in the neck, as measured by the conductivity, and the mobility, obtained from the product of the Hall coefficient and the conductivity, will be neither the true mobility nor constant from sample to sample. The different samples may also end up with varying geometry of their necks, according to their previous treatment. 

The reason fi>r the limitation ofx/a 0.3 on these definitions of A, V, and K is that a simplified spherical geometry has been used for these calculations. This limits the approach to small simtering times. Substituting these geometric definitions and the condensation rate, m, into the earlier equation, we obtain a relationship for the growth rate of the neck diameter, x, with time ... 

A large number of neck and spinal models also have been developed over the past four decades. A paper by Kleinberger [1993] provides a brief and incomplete review of these models. However, the method of choice for modeling the response of the neck is the finite-element method, principally because of the complex geometry of the vertebral components and the interaction of several different materials. A partially validated model for impact response was developed by Yang et al. [1998] to simulate both crown impact as well as the whiplash phenomenon due to a rear-end impact. 

Kraemer " and McBain explained hysteresis on the basis of pore geometry of the adsorbent These workers considered the pores of the adsorbent as ink-bottle shaped, so that the vapor pressure during adsorption is determined by condensation in larger diameters of the bottle, while the pressure at which desorption occurs corresponds to the narrow neck. Rao and Katz found this hypothesis to furnish a qualitative explanation for their observations on adsorption-desorption hysteresis with several gels and sorbates. However, this concept could not explain the complete elimination of the hystereus effect without a marked fall in the adsorption capacity. 

Now it cannot escape the casual reader that the various theories of necking in cold-drawing all describe the comportment of fibers and thin strips. This is not accidental indeed, to the untutored eye the specimens used by Zapas and Crissman were about 15 cm. long, only several centimeters wide, and of negligable thickness. How Is this related to the necking Can a full three dimensional theory based on appropriate principles also predict this instability or Is the geometry of the specimen also very important The mechanical phenomenon is not restricted in this way it occurs in tubes for example. Some work of Spector [23] may be of use here. From a different perspective, it is possible to ask if there are families of time dependent St. Venant-type solutions for this sort of material which display the appropriate behavior. 

The junction region of the bulb of a CRT with the neck section is critical to the geometry of the device. Tubes made with these separate sections are intended for electromagnetic deflection, and this region is where the deflection yoke is located. 

The most dramatic consequence of yield is seen in a tensile test when a neck or deformation band occurs, as in Figure 12.1, with the plastic deformation concentrated either entirely or primarily in a small region of the specimen. The precise nature of the plastic deformation depends both on the geometry of the specimen and on the form of the applied stresses, and will be discussed more fiilly later. 

Exner [17]), the centers of the spheres approach one another as the neck between them widens the change in distance between the centers of the two spheres is assumed to equal the linear contraction of a compact of such particles. This geometry is not as simple as it may seem, because the shape of the neck between the particles changes considerably by the time the centers move significantly. The mathematical description of the shape of the neck is therefore difficult, and is handled with severe simplifying assumptions. In addition, the flow pattern in the spheres is expected [18] to resemble that... 

To account for this effect Buist et ah ( ) pointed out that in an assemblage of spherical grains with finite contact areas the surface curvature would tend to decrease as 0 increased, and it was assumed that the diameters of the necks (contact areas) between the grains remained constant they showed that if a suitable geometry was assumed the observed effects on grain growth could be accounted for. 

These results agree with the concept that initially the geometry of the powder compact is such that there are many particle contacts which have small radius of curvature, and therefore high stress exists to move material into the neck area. When the temperature is raised dislocations and dislocation sources which were inactive at the lower temperature become operational, with the result that there is a burst of densification. As the geometry of the... 

The radius of the neck varies with time so that the reactivities defined classically by relations [7.3], [7.7] or [7.9] become, even in pseudo-steady state mode, functions of the geometry at the time t and thus functions of the time. Therefore, it is necessary to re-examine the definitions in order to integrate otdy the geometry in the space function. For that, we will take again the two cases of reactions or diffusions as rate-determining steps. 

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