P Radius of curvature

E. Winkler, F. Grashof, H. Hertz,8 etc., have studied the stresses which are set up when two elastic isotropic bodies are in contact over a portion of their surface, when the surfaces of contact are perfectly smooth, and when the press, exerted between the surfaces is normal to the plane of contact. H. Hertz showed that there is a definite point in such a surface representing the hardness defined as the strength of a body relative to the kind of deformation which corresponds to contact with a circular surface of press. and that the hardness of a body may be measured by the normal press, per unit area which must act at the centre of a circular surface of press, in order that in some point of the body the stress may first reach the limit consistent with perfect elasticity. If H be the hardness of a body in contact with another body of a greater hardness than H, then for a circular surface of pressure of diameter d press. p radius of curvature of the line p and the modulus of penetration E,... 

Here p/p° is the relative pressure of vapour in equilibrium with a meniscus having a radius of curvature r , and y and Vi are the surface tension and molar volume respectively, of the liquid adsorptive. R and T have their usual meanings. 

When the surfaces are in contact due to the action of the attractive interfacial forces, a finite tensile load is required to separate the bodies from adhesive contact. This tensile load is called the pull-off force (P ). According to the JKR theory, the pull-off force is related to the thermodynamic work of adhesion (W) and the radius of curvature (/ ). 

Israelachvili and coworkers [64,69], Tirrell and coworkers [61-63,70], and other researchers employed the SFA to measure molecular level adhesion and deformation of self-assembled monolayers and polymers. The pull-off force (FJ, and the contact radius (a versus P) are measured. The contact radius, the local radius of curvature, and the distance between the surfaces are measured using the optical interferometer in the SFA. The primary advantage of using the SFA is its ability to study the interfacial adhesion between thin films of relatively high... 

The force p needed to compress a single asperity and the displacement 8 of its tip relative to the undeformed region of the substrate was calculated using JKR theory and determined to be related to the radius of curvature of the asperity and the contact radius a by... 

The radius of curvature R of a plane curve at any point P is the distance along the normal (the perpendicular to the tangent to the curve at point P) on the concave side of the curve to the center of curvature (Figure 1-33). If the equation of the curve is y = f(x)... 

Something that is not obvious from the simple sag equation is how the local radii of curvature change with radial distance, p. Using analytical geometry one can show that the local radius of curvature in the radial direction goes as... 

In many cases the potential P(z) is small compared with the surface energy of the liquid and the droplet shape is very close to a spherical cap. If the height e and the radius of curvature R at the top of the droplets can be measured, an effective contact angle can be defined through the expression ... 

Stresses can can be concentrated by various mechanisms. Perhaps the most simple of these is the one used by Zener (1946) to explain the grain size dependence of the yield stresses of polycrystals. This is the case of the shear crack which was studied by Inglis (1913). Consider a penny-shaped plane region in an elastic material of diameter, D, on which slip occurs freely and which has a radius of curvature, p at its edge. Then the shear stress concentration factor at its edge will be = (D/p)1/2.The shear stress needed to cause plastic shear is given by a proportionality constant, a times the elastic shear modulus,... 

The formation of two-layer PS on p-Si involves two different physical layers in which the potential-current relations are sensitive to the radius of curvature. The space charge layer of p-Si under an anodic potential is thin, which is responsible for the formation of the micro PS. The non-linear resistive effect of the highly resistive substrate is responsible for that of macro PS. The effect of high substrate resistivity should also occur for lowly doped n-Si. However, under normal conditions, the thickness of the space charge layer under an anodic potential, at which macro PS is formed, is on the same order of magnitude as the dimension... 

The smaller the radius of curvature, the lower the vapor pressure. Here is the radius of curvature, which is equal to the pore radius minus the thickness of the adsorbed N2 layer, 7 is the surface tension and Vi is the molar volume. This capillary condensation as a function of pres sure helps establish the pore size distribution when the volume of adsorbed N2 is plotted against P/P . A sharp increase in the N2 uptake is then observed at the pressure corresponding to the filling of mesopores. This type of isotherm is known as type IV, as illustrated in Figure 13.1. 

Another procedure is to rewrite the relation in Equation 2 in terms of quantities which can be accurately measured from a photographic image (1,2,20). At point P we can let the radius of curvature be Rx p of curve Vj, Figure 1. The curve V2, which is perpendicular to Vj, and passes through P, will be such that OP is normal to both curves at P. Further, since OP is on the axis of revolution, P remains on curve Vj when OP rotates about the axis BO. This gives the relation OP = X/sin( ), which is the other radius of curvature of the interface at point P = R p. We can now rewrite Equation 2 as ... 

In the abstract of the paper (Ref 36a, p 1920) it is stated The limiting slope of he detonation velociry-wave front curvature locus for small- velocity deficits is obtained under an assumption concerning the "reaction zone length as related to the charge diameter and the radius of curvature of the wave front. The model is an extension to two dimensions of von Neumann s classical theory of the plane wave detonation... 

Equating the molar free-energy terms in (12.24) and (12.25) affords an expression which relates the hydraulic pressure P required to force mercury into pores to the relative pressure, PJPq, exerted by the liquid with radius of curvature, r. That is. 

Experimentally, work-function measurements which rely on the cold emission of electrons are carried out in the field-emission microscope (F.E.M.) (21). The apparatus, as shown in Fig. 11, consists of a W tip P sharpened by electrolytic polishing so that the radius of curvature is 10 cm., and an anode in the form of a film of Aquadag. A variable potential of 3 to 15 kv. is applied to the anode, and the electrons, pulled out from the point, travel in approximately straight lines to the fluorescent screen. The linear magnification obtained is of the order of 10 to 10. The secondary electrons from the screen are collected by the anode, and the field-emission current is measured by a sensitive microammeter. 

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