Curvature criterion

Except for some in special cases, the canonical curvatures are finite properties that can be computed everywhere on the surface. Suppose we now group all the points on G a,K) that satisfy a curvature criterion, say those for which the two curvatures /7j(r) and h2 i) are negative. Such a domain on the surface can be indicated as Dc(a, K), where C indicates the criterion followed for classification. With the criterion of two negative curvatures, Dda, K) [e.g., Diia, K)] corresponds to the regions on the surface that are locally convex. [Note that Dcia, K) can be empty or composed by several disjoint pieces.] Finally, if we now remove (i.e., cut away) this region, we derive a truncated surface from the original one ... 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

Here b is an adjustable constant. This criterion can be written in a more general form, which will also include effects of regions with high curvature ... 

Figure 2.2 Bubble activation (a) change of radius of curvature of a bubble as it grows out from a cavity (b) criterion for activation of ebullition site. (From Hsu and Graham, 1976. Copyright 1976 by Hemisphere Publishing Corp., New York. Reprinted with permission.)...

The extrapolation of this ratio to m2 = 0 is shown in Figure 17.2. The curve in Figure 17.2 should have a horizontal limiting slope. If an experimental curve does not have a horizontal limiting slope, the data have not been obtained in sufficiently dilute solution to show Henry s-law behavior. Very few data are available that satisfy this rigorous criterion [1]. We frequently settle for a condition in which the plot of 2/ 2 has a clear linear extrapolation to m2 = 0, as opposed to a continuing curvature that makes an extrapolation impossible. 

For the Eikonal theory to be valid, the distortions must be suffieiently small that the eoneept of a ray is retained. The criterion is that the radius of curvature of the reflecting planes does not exceed a critical value R. approximately equivalent to an angular rotation of the Bragg planes by half the reflecting curve width in an extinction distance. The critical radius of curvature is thus... 

As shown in the figure, the curvature of the entropy function always causes it to fall below its tangent planes. A mathematical object having such distinctive global curvature (such as an eggshell or an upside-down bowl) is called convex. Accordingly, we may restate the Gibbs criterion in terms of this intrinsic convexity property of the entropy function S = S(U,V,N) ... 

As the second form of the criterion shows, an equivalent statement is that the curvature of the free energy along the direction 5p vanishes. Equation (50) can also be written as... 

The performance and large-scale feasibility of a TN algorithm depend on the precise formulation of a truncation criterion and implementation of the inner PCG loop. The PCG process can be terminated when either one of the following conditions is satisfied (1) the residual rk is sufficiently small, (2) the quadratic model qk p ) as defined in Eq. [35] is sufficiently reduced, or (3) a direction of negative curvature d is encountered (i.e., < 0). A negative... 

This sharp curvature is particularly pronounced when is large, which is to say when the product of activation energy and heat of reaction is large. Barkelew proposed as a condition for sensitivity that the combination of N and be such that the value of N/S is above the point of contact with the envelope. Given this criterion, other families of curves were constructed, showing the critical relationship between N/S and for various combinations of the parameters a, /3, and t0, the initial dimensionless temperature. Several such families of curves are shown in the original paper. 

To illustrate the application of the packing equation we now show that spheroids cannot accommodate amphiphiles in their regions of greatest curvature once these amphiphiles can no longer pack into spheres, /.e., once y/uo/c exceeds Consider the oblate spheroid of fig. A b). Since by assumption spheres are prohibited, we have /-j > /c, and by criterion (2), r2 < 4- In the region of greatest curvature P,... 

As a general result, we conclude that micelle and vesicle formation cannot be explained by cone or cylinder shapes of the monomeric amphiphiles. The key criterion for the curvature of molecular assemblies lies in the saturation solubility or cmc of the amphiphile. A cmc above 10 M usually means appreciable dissociation leading to small aggregation numbers of micelles. A cmc below 10 M means large planar bilayers or, upon their disruption, vesicles. 

The dependence of the curvature of the freezing point curve at the origin upon the entropy of fusion of the solvent was first pointed out by van Laar. The most common case corresponds to (22.17) and fig. 22.1 (c/. chap. XIV, 5, table 14.6). On the other hand, for spherical solvent molecules (c/. table 14.5) for which the entropy of fusion is abnormally small, the case (22.18) is realized. The form of the freezing point curve at the origin is thus a useful criterion, in the absence of calorimetric data, for the identification of those compounds which have a low entropy of fusion, f... 

The practical aim of this work is to segment a 3-D numerical image into physically relevant grains by using the Gaussian curvature G as a criterion. 

Some visual stress grading rules limit the growth rate and exclude all pith. There are two separate issues. The exclusion of with-pith material is understandable as corewood in its vicinity has a number of undesirable characteristics, including low density, low stiffness, and dimensional instability. However, a restriction on growth rate is less satisfactory unless it is coupled with a criterion that addresses ring curvature to ensure that such sorting only excludes material in proximity to the pith. 

The daughter bubble size is thus limited by two constraints. The capillary pressure constraint states that if the dynamic pressure of the turbulent eddy Pc v x exceeds the capillary pressure aijd", the fluid particle is deformed and finally breaks up resulting in a minimum breakage fraction /vm, (or bubble size dj min) [69]. d denotes the diameter of the smaller daughter size (or two times the minimum radius of curvature). When breakage occurs, the d3mamic pressure induced by the eddy turbulence kinetic energy satisfy the criterion ... 

Screen-Printable Polyimide. Bending beam results for a screen-printable polyimide used for chip protection are presented in Figure 8. The EPO-TEK 600 polyimide paste was applied to a quartz beam, then the beam was spun at 5000 rpm to achieve high uniformity. Results are presented as inverse radius of curvature, 1/R, because the 50 m coating thickness on the 84 im substrate violates the thin film criterion of Equation 2. We have no knowledge of the polyimide s viscoelastic modulus, which is needed in order to convert 1/R into interfacial stress using Equation 1. 

If the maximum principal stress criterion is to hold, then the ratio of the apphed stress to cause fracture to the fracture stress should approach zero as the radius of curvature is reduced to zero (i.e., a/a/ 0 as p 0) in accordance with the following relationship ... 

Mezin, A. (2006) Coating internal stress measurement through the curvature method a geometry-based criterion delimiting the relevance of Stoney s formula. Surf. Coat. Technol., 200 (18/19), 5259-5267. 

Low conversion in the sense used here is a matter of judgment, and depends on the level of distortion from initial behaviour introduced by a given non-zero level of conversion. Often conversions below 5% are quite acceptable but it is not impossible that conversions as low as 1% will show the effects of severe inhibition by products, or some other exaggerated effect, that induces significant curvature to the plot of conversion vs. time. This means that the criterion of low conversion can vary greatly, depending on what we wish to call the initial reaction rate. The difficulties described above will vary with the acceptable level of initial conversion. 

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