Curvature shapes

Fig. 31. Seven axes of motion on fiber-placement system can steer prepreg tow or slit tape around compound-curvature shapes to make highly contoured parts. Courtesy of Cincinnati Machine.

The interest in vesicles as models for cell biomembranes has led to much work on the interactions within and between lipid layers. The primary contributions to vesicle stability and curvature include those familiar to us already, the electrostatic interactions between charged head groups (Chapter V) and the van der Waals interaction between layers (Chapter VI). An additional force due to thermal fluctuations in membranes produces a steric repulsion between membranes known as the Helfrich or undulation interaction. This force has been quantified by Sackmann and co-workers using reflection interference contrast microscopy to monitor vesicles weakly adhering to a solid substrate [78]. Membrane fluctuation forces may influence the interactions between proteins embedded in them [79]. Finally, in balance with these forces, bending elasticity helps determine shape transitions [80], interactions between inclusions [81], aggregation of membrane junctions [82], and unbinding of pinched membranes [83]. Specific interactions between membrane embedded receptors add an additional complication to biomembrane behavior. These have been stud-... 

K (bottom) and 480 K (top). The curvature of the isothenns is interpreted as a temperature-dependent barrier shape [89],... 

Figure C2.17.2. Transmission electron micrograph of a gold nanoneedle. Inverse micelle environments allow for a great deal of control not only over particle size, but also particle shape. In this example, gold nanocrystals were prepared using a photolytic method in surfactant-rich solutions the surfactant interacts strongly with areas of low curvature, thus continued growth can occur only at the sharjD tips of nanocrystals, leading to the fonnation of high-aspect-ratio nanostmctures [52].

Finally, also size and shape of the nonpolar solute seem to influence the formation of hydrophobic hydration shells. Particularly the curvature of the nonpolar surface has been suggested to be... 

Unless extremely high potentials are to be used, the intense electric fields must be formed by making the radius of curvature of the needle tip as small as possible. Field strength (F) is given by Equation 5.1 in which r is the radius of curvature and k is a geometrical factor for a sphere, k = 1, but for other shapes, k < 1. Thus, if V = 5000 V and r = 10 m, then, for a sphere, F = 5 x 10 V/m with a larger curvature of, say, Iff m (0.1 mm), a potential of 500,000 V would have to be applied to generate the same field. In practice, it is easier to produce and apply 5000 V rather than 500,000 V. 

Curvature and rotation lenses correct for any imperfections (aberrations) in the cross-sectional shape of the beam before it reaches the collector slit. The curvature lens provides a means of changing any banana-shaped beam cross-section into a rectangular shape (Figure 24.8). The rotation lens rotates the beam such that the sides of the beam become parallel with the long axis of the collector slit (Figure 24.8). 

Textile fibers must be flexible to be useful. The flexural rigidity or stiffness of a fiber is defined as the couple required to bend the fiber to unit curvature (3). The stiffness of an ideal cylindrical rod is proportional to the square of the linear density. Because the linear density is proportional to the square of the diameter, stiffness increases in proportion to the fourth power of the filament diameter. In addition, the shape of the filament cross-section must be considered also. For textile purposes and when flexibiUty is requisite, shear and torsional stresses are relatively minor factors compared to tensile stresses. Techniques for measuring flexural rigidity of fibers have been given in the Hterature (67—73). 

As the size or the pressure goes up, curvature on all surfaces becomes necessary. Tariks in this category, up to and including a pressure of 103.4 kPa (15 Ibf/in"), can be built according to API Standard 620. Shapes used are spheres, ellipsoids, toroidal structures, and circular cylinders with torispherical, elhpsoidal, or hemispherical heads. The ASME Pressure Vessel Code (Sec. TII of the ASME Boiler and Pressure Vessel Code), although not required below 103.4 kPa (15 Ibf/in"), is also useful for designing such tanks. 

One further effect of the formation of bands of electron energy in solids is that the effective mass of elecuons is dependent on the shape of the E-k curve. If dris is the parabolic shape of the classical free electron tlreoty, the effective mass is the same as tire mass of the free electron in space, but as tlris departs from the parabolic shape the effective mass varies, depending on the curvature of tire E-k curve. From the dehnition of E in terms of k, it follows that the mass is related to the second derivative of E widr respect to k tlrus... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

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