Plateau’s rules

When Plateau s rule is applied to solids, it may be said that a crack advances whenever the corresponding decrease d (SE)/dx in the strain energy is at least as great as the work necessary to extend the material (in front of the crack) to its maximum strain, that is, to rupture. Two semi-quantitative formulations of this idea are possible. 

Plate 4.5 shows the result of breaking surfaces linked to two edges of the cube. There are numerous minimum surfaces associated with subsets of edges. They can all be obtained by breaking sections of soap film surface, and provide examples which illustrate Plateau s rules. 

In the Steiner problem, Chapter 3, and the minimum surface area problems, discussed earlier in this chapter, the Laplace-Young equation had a simple form. This resulted from the zero excess pressure across any point on the surface of the soap film. In the case of a bubble, or clusters of bubbles, the excess pressure across any surface is not in general zero. However the Laplace-Young equation can be applied under these more general conditions. Plateau s rules concerning the angles at which surfaces and lines of soap films intersect apply also to the surfaces and lines of soap film produced by clusters of bubbles. 

All the bubbles formed inside frameworks satisfy Plateau s rules. Three surfaces meet along a line at angles of 120 ° and four lines intersect at a point... 

Some important mathematical results have recently been proved by Frederick J. Almgren Jr. and Jean E. Taylor. They have been able to show that Plateau s rules for the angles of intersection of soap film surfaces and lines, discussed in Chapter 4, are a general property of the solution of the Laplace-Young equation. These proofs are beyond the scope of this book but references to the work can be found in the popular article by Almgren and Taylor entitled The Geometry of Soap Films and Soap Bubbles. 

The analogy between foams and emulsions and granular media is strongest in the wet limit nevertheless, dry foams are fundamentally significant in their own right, and Plateau s rules for the structure of an ideal dry foam (( ) = 1) can be regarded as a tessellation of space. These rules (for dry 3D foam) [24] are as follows ... 

The quantity X/ called the Euler characteristic, is equal to 1 for a non periodic tessellation. Plateau s rules require Ng = 3/2)Ny in two dimensions (each edge is shared by two vertices) it follows that there are Z = 2Ng/Ny = 6 neighbors per bubble in the limit of large system size. 

Clusters of bubbles illustrate Plateau s three rules. In the two cases studied so far the points, such as O and S in Fig. 4.16 and P, Q, R and O in Fig. 4.18, are formed from the intersection of three surfaces. The angles of intersection of the tangent planes to these surfaces are 120°. 

In Equation 2.2, e"xp is the experimental loss factor value is direct current conductivity d and S are thickness and area of sample, respectively = 2nf(f is frequency) and is the permittivity of vacuum. As a general rule for polymers, is determined from fitting of the real component of the complex conductivity (a j. = where CTq and n are fitting parameters) measured in the low frequency range where a plateau is expected to appear. 

The largest change in acoustic properties of a polymer occurs across Tg, when the material transitions from a hard glassy solid to the mbbery plateau. Above this temperature the moduli and sound speeds drop (a rule of thumb for the latter is that it decreases by a factor of 2) while the absorption increases by an order of magnitude or more, with a maximum some distance above Tg. The temperature derivatives are maximum around Tg and are of order 25 m/s/K [2]. 

The vertices of the polyhedra are called plateau borders after the blind Belgian physicist Joseph Plateau (see picture). At equilibrium the foam obeys the Plateau rules - bubble edges are circular arcs that meet in triplets at 2n/3 angles. According to Laplace s law, their algebraic curvatures (iCjj= -Kjj > 0... 

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