Topological properties of cellular structures based on the staggered packing of prisms

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Topological properties of cellular structures based on the staggered packing of prisms

M. A. Fortes

To cite this version:

M. A. Fortes. Topological properties of cellular structures based on the staggered packing of prisms.

Journal de Physique, 1989, 50 (7), pp.725-731. �10.1051/jphys:01989005007072500�. �jpa-00210950�

Topological properties of cellular structures based on the

staggered packing ^of prisms

M. A. Fortes

Departamento ^de Engenharia ^de Materiais, ^Instituto Superior ^Técnico, Av. Rovisco Pais, ¹⁰⁰⁰ Lisboa, Portugal

(Reçu ^{le 9} septembre ^1988, accepté le 8 décembre 1988)

Résumé.

Les structures cellulaires tridimensionelles, formées par des colonnes parallèles ^de prismes empilés base-contre-base,

des bases à différents niveaux dans des colonnes

adjacentes, ^sont analysées dans le but de déterminer certaines propriétés topologiques moyennes, telles que le nombre moyen de faces, F, dans les cellules et la quantité mF, définie

le nombre moyen de faces dans des cellules adjacentes

^cellules

^F faces. Ces structures cellulaires contiennent des sommets tétravalents et trois faces selon chaque ^{arête, étant} topologiquement ^du type ^des structures cellulaires naturelles. On montre que, dans la ^structure cellulaire prismatique, F peut ^{varier entre 8 et l’infini. Quand tous les} prismes ^{sont de même} hauteur,

peut ^obtenir

équation pour mF selon laquelle mF ^est linéaire

1/F, chaque ^fois

que la loi de Aboav ^est observée par le réseau planaire défini par les bases des prismes. ^{La loi de}

Aboav établit

relation linéaire entre mi ^et 1/i, mi étant le nombre moyen de côtés dans les cellules adjacentes

cellules à i côtés dans

réseau planaire aléatoire. Ce résultat est le

premier exemple

d’une relation linéaire entre mF ^et 1/F ^dans

^{structure cellulaire}

tridimensionelle.

Abstract.

The three-dimensional cellular structures, formed by parallel columns of base-to- base packed prisms, ^with staggered ^{bases in} adjacent columns,

analysed for various

topological properties, including the average number of faces in the cells, F, ^{and the} _quantity

mF, defined

the average number of faces in cells adjacent ^to F-faced cells. This class of cellular structures has tetravalent vertices and three faces at each edge, and is therefore of the topological type usually found in natural cellular structures. It is shown that in the prismatic ^{cellular structure} F

vary between 8 and infinite. When all prisms ^{have the}

height,

simple equation

be obtained for mF, which shows that mF is linear in 1/F whenever the random trivalent planar

network that defines the topology of the bases of the prisms follows Aboav’s law, i.e.,

linear relation between _mi and 1/i, where mi is the average number of sides in cells adjacent ^{to an i-}

sided cell in that planar network. This is the first known example ^of

linear relation between mF and 1/F ⁱⁿ

three-dimensional cellular structure.

Classification

Physics ^Abstracts

05.50

Introduction.

There are many systems ^{in nature} which have a cellular structure in the sense that they ^are

formed by polyhedral ^{cells connected at} (entire) faces and filling ^a region ⁱⁿ 3D-space.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01989005007072500

726

Examples range from soap froths ^to polycrystals ^and biological ^tissues [1-3]. The individual cells are, in general, ^convex polyhedra, with three edges meeting ^{at each vertex} (trivalent

convex polyhedra). ^{In the} 3D-aggregate ^{of cells, which can be termed a} poly-polyhedron, ^four edges ^{meet at a vertex and} three faces meet at an edge.

The topology ^{of such} systems ^{is much more difficult to} visualize and analyse ^{than that of} planar networks which can be simply ^drawn. Modelling ^{of 3D cellular} structures is possible (e.g. [4-6]) ^{but not} simple ^{to achieve and to} analyse. This is the main reason why ^the topological properties ^of poly-polyhedra ^are poorly known, particularly ^{of the random ones}

which typify ^{most of the} natural cellular systems. The b.c.c. packing ^of regular ^Kelvin’s polyhedra ^{is often used as a model of 3D cellular} structures but does not exhibit the

complexities ^{of a random} aggregate ^of polyhedra.

The basic topological relations between the numbers of cells (C ), ^faces (F), edges (E) and vertices (V ) ^{in a} poly-polyhedron ^{are the Euler} equation

and the relation

which is a consequence of the tetravalency of the vertices. In the individual trivalent

polyhedra, ^{there are} corresponding relations between the numbers of faces (Fo), edges (Eo ) and vertices (Vo)

In ^a random poly-polyhedron ^{there is a} distribution f (F ) of the number of faces in each

polyhedron, from which the average number of faces per polyhedron, F, ^{can be defined}

The average number of edges ^and of vertices per cell ^are, from (3) ^and (4)

It has been shown [7] ^{that in a convex} tetravalent poly-polyhedron, ^{F can} vary between 8 and infinite. This contrasts with the fixed value, 6, of the average number of edges per cell (façe)

in a planar, trivalent network.

Such planar networks have deserved considerable attention in the literature, particularly

because they ^are obtained in sections of the tetravalent cellular structures found in nature. It has been found that in random planar ^trivalent networks, the average number, mi, of sides in cells adjacent ^{to i-sided} cells, ^varies approximately linearly ^with lli. This is known as

Aboav’s law [2, 8-11]. ^{A linear relation must} have the form [10, 11]

where a is a constant and 1£ 2 () is the second moment of the distribution of cell polygonalities

The form (8) of the linear relation is the ^one compatible ^{with the} identity [10, 11]

Quantities similar ^to the mi ^can be defined for 3D cellular structures, i. e. , the average number, mF, of faces in cells that ^are (face) adjacent ^to F-faced cells. The mF have ^not been measured or calculated for any random 3D cellular structure, and it is uncertain whether a

linear relation between mF and 1 /F provides ^a good approximation ^as Aboav’s linear relation between mi and 1 /i ^{does for} planar ^{trivalent networks.}

The purpose of this paper is ^to analyse ^the topology ^{of a} class of 3D cellular structures formed by ^the staggered packing ^of prisms ⁱⁿ parallel columns. Such structures are of the

topological type ^most frequently ^found in natural systems, i.e., they ^are tetravalent, ^{but are} simple enough ^{to allow the exact} derivation of quantities ^{such as} f (F ), ^F and mF. It will be shown, ⁱⁿ particular, that there is, under certain conditions, ^a linear relation between mF and 1 /F.

This type of cellular structures, ^{which we refer to as} columnar cellular structures, ^{occurs in} cork [12] ^{and other} biological ^tissues [2]. ^{A structure of this} type ^was previously ^{used to} identify ^new types of space filling polyhedra with fourteen faces [13] ^{and to} investigate ^the

range of values of F possible in tetravalent poly-polyhedra [7].

The columnar cellular structure.

The columnar cellular structure (see Fig. 1) ^{is based on the} packing ^of geometrical prisms (not necessarily rectangular). ^The prisms ^are arranged in columns with parallel ^{axes, the} prisms ⁱⁿ

each column being packed ^{base to base} (Fig. la). ^The number, i, ^of lateral faces of the prisms

in a column is therefore the same for all prisms in that column (i-column). ^{All bases are in} parallel planes, ^{but the} prisms ⁱⁿ adjacent columns have their bases staggered, ^{so that a} prism

may be contacted ^at its lateral faces by ^{the bases of} adjacent prisms. ^The heights ^{of the} prisms

Fig. ^1.

The columnar prismatic ^cellular structure : a) ^{cells in}

^column

packed base-to-base ; cells

in adjacent columns have their bases staggered ; b) the cross-section trivalent network ; c) ^section

parallel ^to the column

showing ^the staggered packing ^{of the} prisms.

728

can be chosen at random or subjected ^{to certain rules.} The average height, ^measured parallel

to the column axis, ^{of the} prisms in i-columns will be denoted by hi. ^A section of the structure

by ^a plane parallel ^to the bases is a trivalent network, which will be referred to as the cross-

section network (Fig. 1b). The fraction of i-sided polygons ^{in the} cross-section network is

fi, ^with

and the second moment of the fi distribution is u2. The average number of k-polygons ^{in the}

cross-section network that are adjacent ^to i-polygons ^{is denoted} by nik. We have

and nik 96 nki. Other useful relations ^are

The quantity mi previously defined is given by

When the prisms ^are packed ^to fill space (as ^{shown in} Fig.1c, ^which represents ^{a section of} the structure by ^a plane parallel ^{to the axes of the} prisms), ^their original lateral ^faces may be divided into faces of the poly-polyhedron, ^if they ^{are contacted} by ^{bases of} adjacent prisms.

The cellular structure is tetravalent, and three faces meet at each edge. ^If the cross-section network is _convex, the poly-polyhedron ^{is also convex.}

A complete description of the columnar structure requires giving ^the cross-section network and the heights of the individual prisms. ^{In the} following calculations of _average topological properties, ^{we shall} only require ^the relative values of the average heights, hi. ^{The total}

number of cells in i-columns is then proportional ^to filhi.

Topological properties of the columnar cellular structure.

Consider two adjacent columns i and k. The average number of lateral faces _per i-cell, contributed by ^the adjacencies with cells in k-columns is (1 ⁺ hi/hk). Therefore, the average

number, Fi, ^{of faces} per (separated) ^{i-cell is}

where the term 2 corresponds ^to the cell bases. Since the number of cells in i-columns is

proportional ^to filhi, the average value, F, ^{of the} Fi ^is

Inserting ^the Fi given by (15) ^and using (12) ^and (13) ^{we obtain}

We now consider various special ^cases regarding ^the hi :

Case 1 :

Case 2 :

Case 3 :

If the average heights, hi, ^are independent of i, i. e. , if there is no correlation between the average height and the number of lateral faces of the prisms, the average value of F is 14. If the heights ^{are all} identical, each cell has 14 faces [13]. The average value of F is larger ^{than 14}

if hi ^oc 1 /i and smaller than 14 if hi ^{oc i.} This last conclusion results from the fact that

i - 1> - 1/6 -- 0, ^an inequality ^{that can} easily ^be proved. ^In general, ^if hi increases as i

increases, ^F 14, and vice-versa. In the natural structures, ^{such as} cork, ^{that have a columnar} arrangement ^{of the} cells, it is observed that in the cross-section network the area of cells increases as i increases. This suggests that hi ^{increases with i, which in turn} implies ^{that the}

average number of cells adjacent ^{to a} cell is smaller than 14 (Ref. [13]).

As ^a final special ^case, suppose that the cells in columns of ^some specified polygonality x

have heights, hx, much smaller than the cells in other columns. Then Case 4 :

When x

3, the smaller possible ^{value of F}

8 is obtained. It is of course possible ^{to have}

x -> oo, meaning that F is unbounded in columnar structures. This result was previously

established [7] but without a detailed derivation. The result is in fact general, ^{as shown in}

reference [7] : ^{in convex} tetravalent poly-polyhedra ^{F can} take any value larger ^{than 8.}

When the heights of the cells are all identical, ^the number, Fi, of faces in each cell in a i- column is (Eq. (12) ^and (15)) :

from which we obtain for the second moment of the distribution of the number of faces in

cells :

730

The average number of faces in adjacent ^cells.

In order to calculate the average number ^of faces in cells adjacent ^to F-faced cells we have to make some hypothesis ^on the distribution of cell heights ^{and on the} nik.

We shall admit that all cells have the same height. Then all cells in i-columns have the same

number of faces Fi given by equation (22) ^{and each is} laterally ^contacted by ^{two cells in an} adjacent column. The average number, mF,, of faces in cells adjacent ^to Fi-cells ^{is then}

where the first term corresponds ^to laterally adjacent cells and the second to base adjacent

cells. Combining ^with equation (22) ^and using ^the quantities mi defined by equation (14) yields ^after simple calculations

If Aboav’s linear relation between mi ^and 1/i (Eq. (8)) ^is followed in the cross-section

network, ^we finally ^{obtain, with} equation (23)

where

This means that there is a linear relation between mFi ^and 1 /Fi ^{in the} 3D-network, provided ^a

linear relation between mi and lli is valid for the cross-section network. The linear relation

(26) ^{is of the form}

which is the general ^{form of a} linear relation between MF and 1/F, compatible ^{with the} identity

This identity is the 3D counterpart ^of equation (10) for 2D trivalent networks.

Summary.

The columnar cellular structure, formed by ^the staggered packing ^of (geometrical) prisms, ^is topologically (and sometimes also geometrically) ^{of the} type of the cellular structures found in many natural systems, including polycrystals, soap froths and biological ^{tissues. The columnar}

structures form a topological sub-class of tetravalent poly-polyhedra. ^An arbitrary tetravalent

poly-polyhedron ^{need not be} isomorphic (topologically equivalent) ^{to a} columnar structure ; it is enough ^{to note} that columnar structures contain cells with not less than 5 faces while tetrahedral cells _may occur in an arbitrary poly-polyhedron. The columnar structure has the

advantage ^of being simple enough ^{to make} possible ^{an exact} derivation of various _average

topological characteristics, particularly ^the average number of faces, F, in cells. It was shown,

in agreement ^{with a} general ^result previously ^derived [7], ^{that in the} columnar cellular structure F can vary between 8 and infinite.

When the prism heights ^are uniform, ^a linear relation between mF and 1/F ^is exactly followed, provided ^{a linear relation} between mi ^and 1/i is followed in the cross-section network. It is interesting ^{to note} that the linear relation between mF and 1/F ^{holds even} though ^{the 3D structure is not} completely random in that _case, since the cells in each column

are all of the same topological type, meaning ^{that there is a} correlation between the number of faces of a cell and the number of faces in two of its adjacent cells. This is the first reported example ^{of a random 3D cellular structure} following ^{a law} (Eq. (28)) equivalent ^{to Aboav’s}

law for planar trivalent networks. It can be conjectured ^{that a} linear relation between MF and 1/F (Eq. (28)) applies, ^{at least} approximately, ^{to an} arbitrary random tetravalent

poly-polyhedron, just ^{as Aboav’s} equation (8) ^is approximately ^{followed in random} planar

trivalent networks. A general argument in favour of this conjecture ^{will be} given ^elsewhere.

References

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[4] ^{MARVIN J. W.,} Am. J. Bot. 26 (1939) ^280.

[5] ^{MATZKE E.} B., Am. J. Bot. 26 (1939) ^288.

[6] ^{FERRO A.} C., CONTE J. C. and FORTES M. A., J. Mater. Sci. 21 (1986) ^2264.

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[8] ^{ABOAV D. A.,} Metallography ³ (1970) ^383.

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