Citation
Applications of topology to sintering

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Title:
Applications of topology to sintering
Creator:
Buteau, Leon Joseph, 1932-
Place of Publication:
Gainesville
Publisher:
University of Florida
Publication Date:
Language:
English
Physical Description:
x, 94 leaves : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Algebra ( jstor )
Mathematics ( jstor )
Matrices ( jstor )
Metallurgy ( jstor )
Sintering ( jstor )
Terminology ( jstor )
Tetrahedrons ( jstor )
Topological properties ( jstor )
Topology ( jstor )
Volume ( jstor )
Dissertations, Academic -- Metallurgical and Materials Engineering -- UF
Metallurgical and Materials Engineering thesis Ph. D
Sintering ( lcsh )
Topology ( lcsh )

Notes

Bibliography:
Bibliography: leaves 92-93.
General Note:
Thesis - University of Florida.
General Note:
Vita.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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000541630 ( ALEPH )
13068720 ( OCLC )
ACW5175 ( NOTIS )

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APPLICATIONS


OF


TOPOLOGY


TO


SINTERING














By


LEON JOSEPH


BUTEAU, JR.


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF


THE UNIVERSITY


OF FLORIDA


IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FTI.ORIA












ACKNODWLEDGMNTS


author wishes


express


his sincere


gratitude


Dr. John Kronsbein


suggest


flang


the subject


matter


and methods


of research.


Without


guidance


encouragement


this


disser-


station would not


have


been


possible.


The author


also wishes


to express


his appreciation


Dr. F.


Rh inca


Head of


the Metallurgy


Department,


II. B.


Reed-Hill,


Mathematics,


Professor


of Metallurgy,


and Dr. W. O.


Smith,


Dr. R. G. Blake,


Professor


Professor


of Mechanical


for serving on


his supervisory


committee.


Finally


, the author


expresses


his appreciation to


Mr. R. A.


Runmel


for developing


the precise


technique


required


to obtain the


serial sections


used


in the


example


calculation.


~ Fneer in














TABLE OF CONTENTS


Page


ACKNOWLEDGM4ENTS . .

LIST OF TABLES .

LIST OF ILLUSTRATIONS . .


NOTATIONS . .

INTRODUCTION . .


CHAPTER


DESCRIPTIVE TOPOLOGY AND SINTERING


Terms


Symbols


The Decomposition of


the Void


Space


NETWORK TOPOLOGY


The Algebra of Networks
The Incidence Matrix
The Loop Matrix


Relations


Between


the Fundamental


Loop


the Incidence Matrix


Cut Matri


ces


SINTERING AND ALGEBRAIC TOPOLOGY


The Conjugate


Algebra


Space Model


--Modulo Two


Knotting Within a Sinter
The Linkage Numbers


Body


The Higher


Dimensional


Betti Numbers


APPLICATIONS


TO SINTER METALLURGY


Some


Relations


Between


the Betti


Numbers of


Sinter


Body


Void Space


filmy


Piramnnl as


r La fsn -- -


III.







TABLE OF CONTENTS


(Continued)


CHAPTER


Page


CALCULATION OF THE


BETTI


NUMBERS FOR AN ACTUAL


SINTER BODY


4 4 4 S 0 0 5 4 0 0 5 0 5 14


SUMMARY OF


THEORETICAL RESULTS


FOR APPLICATION


TO SINTER METALLURGY . .

REFERENCES .

BIOGRAPHY . .














LIST OF TABLES


TABLE

I.


Page


TOPOLOGICAL SYMBOLS AND DEFINITIONS


TOPOLOGICAL AND SINTERING TERMINOLOGY


III.


TOPOLOGICAL CHARACTERISTICS AND


DIMENSIONS


TABULAR DATA OF TOPOLOGICAL ELEMENTS AND


INVARIANTS


TABULAR DATA OF THE


TOPOLOGICAL


INVARIANTS


UNIT VOLUME


SECTION THICKNESSES AND CUMULATIVE


VOLUMES













LIST OF ILLUSTRATIONS


Figure


Page


Rhines


Topological


Model


a Sinter


Body


S S 3


Decomposition


of the Void Space--Illustration


of 6


Decomposition

Decomposition

Decomposition

Decomposition

Decomposi tion

Voids at the


of the Void

of the Void

of the Void

of the Void

of the Void


Vertices


Space--Illustration

Space--Illustration

Space--Illustration

Space--Illustration

Space--Illustration


of 6

of 6


4 of


of 6

of 6


of the Tetrahedron


Branch-node
Section of


Network
a Sinter


Superimposed on a
Body .


Two-dimensional


The Network N


Loops

Trees


of N

of N


* S S S SS S S S S S S S


The Co trees


of N


A Set


of Fundamental


Loops


N *


A Set of Non-fundamental


Loops


of N


The Cuts

The Cuts


of N


of N with an Arbitrary Orientation


Photomicrograph


of Linde Copper


Powder


-170 +200 Mesh,


Sintered


in Hydrogen


for 128


Hours


at 1005


150X


Pho tomi crograph


of Linde Copper


Powder,


-170 +200 Mesh


Sintered


in Hvdronen


for 32 Hours


at 1005


150X


t










LIST OF ILLUSTRATIONS


(Continued)


Figure


Page


Tetrahedron


for Constructing


a Face-


branch


Incidence Matrix


Cubic

Knotte


Tinker


Representation


Tinker


Representation


* S S S S S S S S S

* S S S S S S S S S S U


Knotted


Branch-node


Sinter


Body


Representation


Example of


Examp 1 e


Exainpi


Linkage Number


Zero


of Linkage Number One

of Linkage Number Two


* S S S S S S S S

* S S S S S S S S S S


. Illustration


of Representative


Sinter


Body


Areas


Partial

Pho tomi


Two-dimensional


crograph


Sinter


Linde Copper


Body


Branch-node


Powder


-170 +20


Network

0 Mesh


Sintered


Hypothetical


in Hydrogen


Serial


for 1024 Hours


Sections


at 1005 C


a Sinter


, 150X


Body


. Branch-node Network


for the Hypothetical


Serial


Sections


Section 1

Section 2

Section 3


14 Serial

14 Serial

14 Serial


Sections, 90X

Sections, 90X

Sections, 90X


* S S S S S S S S S

* S S S S S S S S

* 5 S 5 4 5 4 S S S


Section


4 of


14 Serial


Sections,


Section


14 Serial


Sections


U S S S S S 57


. Section


6 of


Serial


Sections,


Section


Section 8


Rae t.{ on


-


14 Serial


Serial


of ia R iial


Sections,

Sections,

Qor.i nna.


90X

90X

QOx


*~~~ ~ ~ U

* 5 5 5









LIST OF


ILLUSTRATIONS


(Continued)


Figure


Page


Section

Section


Section 13


11 of 14 Serial


14 Serial

14 Serial


Sections

Sections


Sections


* a S S S

* a a S a a a

* a S a a a a S


Section


14 of


14 Serial


Sections


Branch-node


Network


for the 14 Serial


Sections


Betti

Betti


Number

Number


Number

Number


of Sections

of Sections


* 5 S S S S S S S

* 5 S S S S S S













NOTATIONS


Area


Matrix element


fundamental


loop matrix


Branch


Branch


connecting nodes


k and


between


sections


and J


Contact


columns


of E2


remaining


after


has been


removed


Incidence matrix

Reduced incidence matrix

Loop matrix


Fundamental


loop matrix


Cut matrix


Fundamental


cut matrix


The columns


of E1


ii a


tree


Rhines

Modulo


genus

two incidence matrix


Unit matrix of


order m


Unimodular matrix

Dimension

Linkage number










Node


Node


on section k


Particles


Betti

Betti


Rank of


number

number


the cut matrix


singular matrix


Tree matrix

Tetrahedron

Volume


Symbol


k k dimensional


a set of nodes


simplex













INTRODUCTION


A sinter


body


is produced


exposing


a large number


of small


metallic


particles


(0.001


to 0.3 inch


diameter)


a temperature


that


near


but below


their melting


temperature.


The particles


arranged


some


type


of geometric


configuration which


allows


interparticle


contacts.


When


these


particles


are maintained


at such


a temperature


for a sufficient


length


of time,


it is found


that


interparticl e


contacts


become


fused


or welded.


As time


increases,


these


welds,


often


referred


to as necks,


grow.


space


between


the particles


constitutes


a void


and in time,


consequence


of heating


is breaking up


of the void


into


separated


parts.


If enough


time


elapses,


all the voids may


disappear


and the metallic mass


will


have


its natural


bulk


density.


It is desirable


to have


a mathematical


apparatus


capable


describing


the characteristics


a sinter


body


at each


stage


of the


sintering process.


This was


first


undertaken


F. N. Rhines


(a)* in


1958


, when


he introduced


the idea


of applying descriptive


topology


to sinter


progress.


Instead


of concerning


himself with


those


geometric


parameters normally


considered


in sintering,


he decided


to represent


*ThP ,undrlrl I nod nimbhors


in nnPrnrhsin ft


.rafr


f-nnf8 ri.P


are










or describe


the sinter


body


terms


one of its topological


invariants.


The first


and simplest model


adopted


Rhines


is shown


in Figure


where


the vertices


of the tetrahedron


represent


sinter particles


the edges


the interparticle


contacts.


advantage of


a topological


description is


invariance with


respect


continuous


deformations.


This means


that


particle


sizes


and shapes


not directly


enter


the problem,


resulting


in considerable


simplification.


The important


of particles


parameters


and the interparticle


in the Rhines model


contacts.


are the number


The invariant


considered


is given


Euler's


formula


for the connectivity


two-


dimensional


manifolds


embedded


in a


three-dimensional


Euclidian


space.


This


G = C


- Interparticle


contacts


- Particles


- Genus


of the surface


bounding


the sinter


body


In topology


term genus may


be used


to mean


the number


handles


on a


sphere


and thus


the number


of self


re-entrant


cuts


that


be made


on the manifold without


dissecting


it into more


than one


connected


surface.


It is restricted


to the surface


and does


not apply


to the three-dimensional


manifold


enclosed


or excluded


In the


case


of sintering,


the manifold


enclosed


the sinter


body


surface


must


be defined as


the sintered material.


can


P~






3


Figure


Rhines


Topological


Model


a Sinter


Body






4


The expression


-P-i


was used


Kicrc hoff


describe


an associated


quantity


that will


be introduced


later


subsequently named


the cyclomatic


number


by Maxwell


who used


a similar


connection.


Veblen


uses


this


term


, but Lefschetz


Alexandrov


and Cairns


refer


to it


as the t Betti


number


and denote


it by


This


terminology will


be used


throughout


this


dissertation.


From a


generalized


point


of view,


the sinter


body


is composed


two three-dimensional


mani folds


one two-dimensional


manifold.


These


are the void


space manifold


and the sinter


body manifold


and they


are related


inasmuch as


they


have


common


same


two-dimensional


manifold,


the topological


entity


first


considered


Rhines.


result


of the


present


research


is a mathematical


model


giving


relationship


of the


two topological


spaces.


Subsequently


DeHoff


in collaboration with


Rhines


introduced


a type


metrical


concept


(i.e.


units


measure


such


as length,


area,


volume,


etc.)


in two-dimensional


topology


of sinter


bodies.


tion


This


, surface


enabled


area


them


uni t


to measure


volume,


such


etc.


parameters


Thus,


as volume


a considerable


frac-


advance-


ment


in the interpretation of


two-dimensional


random measurements


sintering processes was made


and probability


combining


theory.


sinter


body


and associated


void


space


represent


extremely


complicated manifolds


in that


and (C


are very


large


numbers


and may


be of the order


of several


million.


Hence,


visual


P e










present


research


to extend


these methods


so as to make use


of other


topological


invariants


besides


genus


(i8t


Betti


number)


and also


to introduce


algebraic methods


for the study


of wintering.


The first


method


to be considered


is network


topology,


since


this


gives


a model


of an actual


sinter


body.


Investigations


in this


field


lead


to the


introduction of matrix


algebra.


The matrices


of importance


in the


present


problem are


the incidence,


loop-branch,


tree-branch,


cut-branch matrices


, respectively.


The topological


invariants,


such


as the th
as the 0


be defined


later)


and 1st


Betti


number,


are


expressible


terms


of matrix


ranks,


which


can in principle


be obtained


counting procedures


or alternatively


evaluation


of linear


equations.


In addition


some


reference


is made


to linkage


numbers


discussed


Veblen and Alexandrov,


and a new method


classifying


stages


sintering


is introduced,


which


is related


to these


and the


previously mentioned


numbers,















CHAPTER I


DESCRIPTIVE


Terms


TOPOLOGY AND SINTERING


and Symbols


In this


concepts


are


chapter


introduced


some


elementary


and briefly


topological


outlined


to enable


operations


the reader


to correlate


sintering


and topological


terminologies.


It should


observed


that


the topologist


does


not in general


concern himself with


the study


their


of topological


characteristics.


manifolds which are

On the other hand, t


constantly


his


changing


is the principal


concern of


the metallurgist.


Therefore,


it is


necessary


that


topological


study


of a sinter


body


or its associated manifolds


(described


below)


always


be conducted at


a stage where


at least


temporarily none of


the topological


invariants


change.


In Table


we demonstrate


a correlation


between


topological


terminology


symbols


(and


terms


symbolism)


that will


and that


be used


used in


repeatedly


sintering.


throughout


Various


this


work


are presented


therein.


These


terms


are used


in this


disser-


station without


further


definition.


Two other


terms


use in sintering are


defined


below


A one-circuit


is a one-dimensional


complex


such


that


each


i a i nr idnnt"


ui rh onrrt-lv


twn hranchon.


Tn tPrinn


of sintprinu


nnrlP


.




7



TABLE


TOPOLOGICAL


SYMBOLS AND


DEFINITIONS


Node


A point


Branch


A segment
end points


a line excluding its


Two-dimensional


plane


area


bounded


three


non-


simplex


(area)


collinear points


and the branches


joining


them-


-a triangle without


sides


and vertices


Three-dimensional


simplex


(volume)


A three-dimensional


the four


but not


faces of


including


manifold


bounded


a tetrahedron


the faces,


edges,


or vertices


Four


-dimensional


simplex


four-


dimensional


manifold


bounded


five


three-dimensional


simple exes


Zero-dimensional


One node


simplex


Zero-dimensional


More


than one


node


complex


One-dimensional


or more


branches


complex


provided


each


branch


associated


tei th


two nodes


Two-dimensional


or more


areas


including


their


complex


boundaries


Three- dimensional
complex


or more volumes


their


including


boundaries









A two-circuit


a set of two-dimensional


complexes


such


that


each


branch


is incident with


an even number


of two-simplexes.


It is


a set of areas which


form a


closed


surface.


In sintering,


this


a surface


in three-dimensional


space which


bounds


a volume.


Such


a two-circuit


is referred


to as a closed


two- dimensional


manifold.


tion of


use of


the term surface


differential


geometry


the term manifold


for differential


is also used


in topology


geometry.


in sintering,


allows


However,

terms su


reserva-


since


lrface


and manifold may


occasionally


be used


interchangeably.


parameters


of interest


in sintering are


several


of the


topological


invariants.


Invariant,


in this


case,


means


parameter


is unaffected

the manifold


continuous


is undisturbed.


deformation


provided


The invariants


the continuity of


of primary


interest


sintering are


defined


below


--the 0th Betti


number--is


the number


of disconnected


parts


of a manifold.


For the


sinter


body


this will


be assumed


to be 1


hence


* while


for the void


space,


It has been mentioned above


that


network


topology will


be used


initially.


We obtain a


network


for a sinter


body


defining a particle


as a node

of a netw


and a


ork


are related


contact


as a branch.


branches.


of nodes


the following


The sinter


equation which


body


two numbers


applies


is then

introduce


separately


composed

d above


to the


sinter


body


and the void


space,


which explains


the superscripts


have


been


omitted.






9


- n+ P


called


represents


the Ist


the number of


Betti


number


one-circuits


for the sinter


closed


loops)


body


For the


void space manifold


the corresponding P1


depends


upon how


far sintering


progressed,


this will


discussed


later.


Table


II indicates


the parallelism


between


topological


terminology


and sintering


terminology


TABLE II


TOPOLOGICAL AND SINTERING TERMINOLOGY


Topological Topological Sintering Sintering
Symbol Terminology Symbol Terminology


a 0 cell n Node

a 1 cell b Branch

2 2 cell A Area

a 3 cell V Volume
3


To begin with,


the sinter


body will


be regarded


as an assembly


of nodes


sinter


branches.


body


At a later


the void


time


space will


the bounding


also


surface


be considered.


between

A definite


prescription


for the mathematical


construction of


the void


space,


the basis of


the physically


given slnter


body,


will


be given









At the beginning


of the sintering process,


both


the void


space


the sinter


two-dimensional


body


are completely


boundary.


connected and possess a


formation


of interparticle


common


contacts


is equivalent


to the existence of


branches


between particles.


a physical


observation


that


a continuous


path exists


between any


particles.)


As sintering progresses,


particles


fuse


and the places


of fusion are


sometimes


referred


as necks.


In the sintering process


these


necks


or branches


grow,


but from a


topological


branch-node


analysis


this


growth


has no meaning.


As a


result


the topology


of the


problem remains unchanged


until


a branch disappears


in the void


space.


This


so-called channel


closure


in the void


space


causes


one of


things


occur.


Either


the void


space


remains


connected


or segre-


nation of


voids


(isolation of


parts


of the void


space manifold)


initiated.


The fact


that


this


segregation


can occur has


consequences


that will


become


apparent


shortly.


As sintering progresses


the void


space


breaks up


into more and more


separated


parts


so that


in the end


the void


space


consists


only of


isolated


pores.


The Decomposition


of the Void Space


It will


now


be demonstrated how it


is possible


for the


void


space


to decompose


into


isolated parts while


the sinter


body


remains


connected.


Consider


the void


space


breaking up


into distinct


separated


parts


as illustrated


in the following six


figures.


interior of


the cube of Figure


a representation of


a volume


of void


space.


two






11


Figure 3


shows


it under


the influence of


a topological


deformation.


It t.


important


that


one understands


that


Figures


2and 3


are topologi-


call


completely


equivalent.


fact


that


Figure


an additional


two lines


indicated is


on Figure


significant


fact


as it


can


is that


still


be mapped one-to-one


the continuity


of the


complex has


not been changed.


Figure 4


is a different


topological


complex in


that


there


are now


two volumes which are


still


connected.


They


are


connected


because


it is possible


to find a


continuous path


from one


to the other without


leaving


the manifold.


(The


boundary


is here


considered as part


of the manifold in view of


what


has been


stated


before.)


Figure


is different


again in


that


the volumes


remain


connec ted


branch.


an area.


transition


Figure


6 illustrates


to 7


them connected


is significant


a single


in that


the latter


illustrates


two complexes


entirely


disconnected.


view of


the fact


that


two three-dimensional


volumes


are discon-


nected,


it may be


said


that


the void


space is


bounded


in four


dimensions.


A question naturally


arises


as to whether


two voids


in three-


dimensional


this,


space are


we proceed as


in Euclidian n


space,


connected


follows:


is project


in four-dimensional


an n-dimensional s

d linearly into an


space.


implex,


- 1)


answer


embedded

-dimen-


sional


space,


the n-dimensional


simplex fills


entire


(n-1)-


dimensional


space.


is immaterial


what


definition is


ascribed


the infinite points of


this


space;


e.g.,


three -dimensional


projective


space with a


suitable metric


impressed upon it might have


been


chosen.)


irmuateri al


from Figures
























































Figure


Decomposition
Illustration


f the
of 6


Void


Space--


t i
















































Figure


Decomposition
Illustration


of
2 of


Void


Space--














































Figure


Decomposition
Illustration


Void Space--















































Figure


Decomposition
Illustration


Void


Space--


of 6

















































Figure 6.


Decomposition of


Void


Space--


Illustration


of 6




'iNNi~iNNN"flQI''' N: 17""""""


Figure


Decomposition
Illustration


void


space--










Thus,


a four-dimensional


simplex


is projected


into


three-dimensional


space,


four


of the five


three-dimensional


simp 1 exes


bounding it may


be made


to form one


tetrahedron


The remaining


three-dimensional


simplex occupies


the three-dimensional


space


exterior


to this


tetra-


hedron.


To determine whether


the three-dimensional


voids,


referred


previously,


are connected


in four


dimensions,


consider


two isolated


voids


in three


dimensions.


topological


deformation


these voids


can be located


at the vertices of


the tetrahedron


referred


above.


Such a


construction is


illustrated


in Figure


In this way


two vertices


of the tetrahedron are


truncated


as shown.


truncated


tetrahedron


in addition


to the fifth


three-simplex,


which


covers


the remainder


three-dimensional


space,


when projected


into


four


dimensions


forms


the boundary


a four-dimensional


simplex.


The four-dimensional


simplexes,


Assume


the abov


has been so


simplex,


a continuum of

e topological


conducted


that


bounded


points


the five


on its exterior


deformation in


the images


three-dimensional


and interior.


three-dimensional


of the two void


space


three-dimensional


vertices


lie along


the boundary


of the exterior part mentioned above.


Then


there


exists


a path


from


the image


of one of


the voids


to the


other


through


the continuum not


occupied


set of five


three-


dimensional

dimensional


simplexes.


space


Hence


can be connected


any two separated


voids


a continuous


lin


in three-

e in four-


dimensional


space,


which


lies


exterior


to the three-dimensional


manifolds


and meets


them nowhere.






































I
a -


void


- -


void


Figure


Voids


Vertices


Tetrahedron T1














CHAPTER II


NETWORK TOPOLOGY


The Algebra of Networks


The methods of


network


topology


can be


correlated


to the


topology


of a sinter


body


and its associated void space.


Figure


represents


a two-dimensional


section of


a sinter


body.


The material


in white


corresponds


to copper while


the black corresponds


to void


space.


On the


copper


a branch-node network has


been


constructed


this


indicates


clearly


that


the complicated


sinter


body


can be


repre-


sented,


from a


topological


point


of view


, in


tefln8


of branches


and nodes.


It should


be noted


that


a network can also


be constructed


in the black


areas


of the photograph.


Such a network would


be representative of


the void


space manifold.


corresponding


topological


actual


invariants


construction of


can be related


this


so that


is given


later


In this


chapter


the algebra of matrices


as related


to a


sinter


body with


its associated void


space will


be presented.


important matrices


are four in number


and are


named:


Incidence matrix

Loop-branch matrix

Tree-branch matrix
^" __J_ ^.-- --- -- 1. -__ -- t- j- --*,_


on.




D~~DDDDDDKDDD ~ liiaDDD iggjgggg 'N DDDDDDDDDD DmiE:" iiIIIDDDDErhraDINNADD AEffl N E


St.


Figure


Branch-node Network Superimposed on
Two-dimensional Section of a Sinter


a
Body











In addition

be outlined.


to presenting


the matrices


that


their


interrelationships will


introduced in


the previous


chapter.


Statement


- n


This


introduction


equality implies


of matrices


an advancement


in the topology


in that


of sintering,


it allows


which


follows:


All of the following


statements


apply


to both


the void


space


parts


the sinter


body.


this


reason


the superscripts


temporarily


omitted.


Disconnected networks may


be connected


common


node method,


which will


be described


subsequently.


The Incidence Matrix


Consider


the network of


Figure


denoted


byN p


consisting


of four nodes

and to each b


and six branches.


ranch an integer


Assign


to each node


and an orientation.


Note


that


number ri ng


of the branches


and nodes


as well


as the orientation is


completely


arbitrary.


This


does


not imply


that


there may not


most


expedient numbering or


orientation.


incidence matrix,


denoted


is formed as


follows:


Provide


a column


for each


branch and


a row


for each


node.


Atcn ha 4n ai4h a nn^n 4k 4; 4l


The notation used is


now


are


an integer number


P+


3 x kranrrk


ia *~JA



















































Figure


Network









node


and is oriented


away


from


the node,


insert


= +1 in


the matrix.


e= -1 if the orientation


is toward


node


a zero


otherwise.


The incidence matrix for


N is presented


below:


-1 +1


0 +1

-1 -1


0 -1

+1 +1


0 +1


Statement


The rank


of the incidence matrix


is equal


the number


of nodes


in the network minus


the number


of separated


parts.


This


statement


been


proved


by mathematicians


will


not be


repeated


here


The Loop


Matrix


Loop


definition:


A loop,


of a network


is a one-dimensional


complex of N having precisely


two branches


incident


with


each node.


The loop matrix,


denoted


is formed as


follows:


Provide


a column


for each


branch


of the network


anda


row for each


loop.


Arbitrarily


orient


and letter


each


loop.


If branch


is contained


in loop


and is


oriented


in the


same


direction as


the loop,


insert


S+1


in the matrix.


the orientations


are different,


write


= -1,


and if the branch



















































Figure


The


Loops



















0 -1 -1


0 -1


The matrix El


obviously has


property


that


rows


linearly


dependent


because


the summation


of the


rows


zero.


Hence,


one may


delete


row of the matrix and


call


the resultant


matrix,


designated


the reduced


incidence matrix.


The deleted


row is


arbitrary


and therefore


in specific


cases


reference


should


be made


the number


rows


that


have


been


deleted.


Tree


definition:


tree


a connected


network


a one-


dimensional


complex containing all


the nodes


of N


no loops.


From


this


definition and


the proofs


of (4)


and (9),


follows


that:


Every


tree


of N has


a


branches.


number r


trees


in N


equals


the value


of the


determinant


where a prime


denotes


transpose.


An enumeration


all the


trees


of Figure


10 is presented


in Figure


Every


tree


of N divides


the b


branches


into


two sets.


- 1,


branchh.s


in the


treeC


are






















































Figure


The


Trees










The (b


- n + 1


branches


are said


to form


cotree


of the


network.


cotrees


of Figure


are presented


in Figure


Statement


III.


The number


of branches


in a


cotree


of N


equals


the 1st


Betti


number


a network.


This


statement


has been


proved


mathematicians


and will


not be repeated here


(10).


Since


a tree


contains


no loops,


there


a unique


tree


path


between any


two nodes.


This


together with an appropriate


branch,


defines


extr


a loop.

'a branch


The loop is

introduced.


arbitrarily

This set o


oriented


f loops


in the


is called


same


direction


fundamental


they


define


a submatrix of E2


of order


x b,


called


the funda-


mental


loop matrix E


A set of fundamental


loops


corresponding


tree


of Figure


12 is presented


in Figure


14 and the corresponding


0 -1


The term fundamental


has an important


significance.


The number


fundamental


loops


is equal


to the number


of linearly


independent


loops.


It has been


shown


by mathematicians


(e.q


, Veblen,


page


that a


relation exists


be tween


the incidence


and loop


matrix.


relation is





29



















































Figure


A Set
of N


of Fundamental


Loops










Statement


rank of E2


equal a


the ist


Betti


number


of the network N


This


been proved


by mathematicians


A set of three


independent


loops


of the network,


which


fundamental,


are


illustrated


in Figure


The corresponding


matrix,


denoted


0 -1


It should


be noted


that


there


is no unit


submatrix in


This


is implied


the fact


that


it is


not a


fundamental


loop matrix.


Statement


rank


of the loop matrix E2


is equal


the 1st


Betti


number


of the network N.


This


has been


proved


mathematicians


These


two statements


are of


importance


in the study of


a sinter


body


because when


the incidence matrix is


known,


the fundamental loop


matrix can


be determined


from it


and hence


the 0th


Betti


numbers of


the network ascertained.


Thus


a known algebraic


apparatus


is available


to deal


with


the invariants of


the sinter


body


associated void


space.


Relations


Between


the Fundamental


Loop


the Incidence Matrix


For a given


loans


can


tree of


a network N,


be numbered and oriented


the branches and


an that Rn.


and R.


are


fundamental


ai ven


are


~E """" X

















































Figure


A Set
Loops


of Non-fundamental


of N






33


then


(This


is proved elsewhere


Cederbaum


(W.)


where


The fundamental


loop matrix


columns


The columns


of E2

of E1


remaining after


corresponding


has been


to a set of (b


removed


- n + 1)


branches


not


in a


tree


The columns


of E1


corresponding


a tree


The reduced


incidence matrix


unit matrix of


order m


A necessary


and sufficient


condition


for a (


-11+


matrix E2


to be


a fundamental


loop matrix of


a connected network


that


after permuting


its columns,


we can write


( Im,


P F)


The numbering


of the branches


of N must


be chosen


so that E1


can be written in


the form (


T .


This


is always


possible.


the network of


Figure


then


(12)


tU

















Hence


giving


, F)


so that


-I
C


J. UflL


In addition


All of these matrices


can be used


to determine


topological


properties


of the sinter


body


or associated


void


space


any stage


of the


0 0 -1


F









Cut Matrices


Cut Definition:


A cut


of a network


a set of branches whose


removal


increases


the disconnectedness


of the network


one, provided


no proper


sub set has


this


property.*


The cuts


of the network decpicted


in Figure


are given


Figure


Networks


that


are not


connected may


be considered


connected


if they


are joined


the common node method.


This method


consists


of connecting


the Po


separate


parts


at any


arbitrary


reference


node.


Such a process


has the property


leaving


the Ist


Betti


number


unchanged


but obviously not


the 0th


Betti


number


. This


procedure may


be convenient


from


time


to time.


removal


a cut divides


the network N


into


two parts


Each


branch


of the cut was


joined


to a node


in N1


and N2


We define


cut as


an ordering of N1


and N2,


either


NIN2


N2N1


An oriented


as the cut if


node


in N


it i


the cuts


branch of N N2

oriented away


dividing Figure


is said


from its


to have


node


10 into N1


the same orientation


in N1

and N


toward


are


shown in


Figure


Incidence


a matrix E3


between


constructed


the branches


in accordance wi


and cuts may be

th the following


described


convention:


sequel


this


last


phrase will


not be repeated


implied


and N2


N























































Figure


Cuts


of N


































I


I


I


Figure


Cuts


of N with


an Arbitrary


Orientation










Provide


a column


for each


branch and


a row for each cut.


If branch


is contained in


cut j


and oriented in


same


direction as


cut,


write


= +1.


If the branch


is oriented


in the opposite


direction,


write


= -1 and a


= 0 otherwise.


The cut matrix


for Figure


10 is given below:


-1 -1


It is proved


elsewhere


that


= E3B


'=


A cut is a minimal


set of branches


so that


it contains


least


one branch


every


tree


of the network.


has a rank equal


to (n


- 1)


for a connected


network.


Statement


The 1st


Betti


number


a connected network


equals


the number


branches


in the network minus


the rank


(12).


- r


- .


The rank


of El


~b nf






39


If the


cuts


are numbered and


oriented


so that


for a particular


network


-(KIC


is called


the fundamental


cut matrix;


a unit matrix of


- 1)


rows


and columns;


K is


a unimodular matrix.


This matrix has


property


that all


its sub determinants,


including


the elements,


have


the value +


or O.


in addition


, the loops


are numbered and oriented


so that


and E2


are as given


and (9)


then


terms


of the cuts previously illustrated


0 O 1


K


IL is


necessary


to pursue


further


the relations


between


cut, loop,


and incidence matrices.


Given


the incidence matrix,


others


can


be found


solving sets of


linear


equations


conversely


the cut matrix also has


an interpretation in


sintering.


When


sinter body is


broken into


two parts,


a row of


the cut matrix is


determined.


In orinciole


then.


the cut matrix of


a sinter


body


can be


I










ways.


might


a lengthy procedure,


but since most


sinter


bodies


are made with


uniformly


graded


sizes


of metal


particles,


they


essentially


isotropic.


Hence,


a small


specimen,


which must,


however,


still


contain


a relatively


large


number


of particles,


can be considered


to be representative


of the entire


sinter


body


any stage.


This


leads


to the idea


unit volume,


of topologicall


" which has


been used


properties

on occasion


per unit mass


or per


. DeHoff


(13).


are













CHAPTER III


SINTERING AND ALGEBRAIC TOPOLOGY


The Conjugate


Model


In the model


of the sinter


body


proposed


here


the particles


taken


to be nodes


remainder


and the interparticle


of the three-dimensional


contacts


Euclidian


space


to be branches.

e is taken to b


ie void


space.


Thus,


the sinter


body


void


space


have


a common


two-


dimens ional


boundary


(surface) which


is also


a topological


manifold.


These


three manifolds


must


be carefully


distinguished.


From a


topo logical


point


of view,


it is


necessary


to be quite


precise


in the definition of


the void


space.


When considering


sinter


body,


there


is little doubt,


at least


in the early


stages


sintering,


apparatus


as to what


described


a node


previously


and what


is then easily


a branch.


applicable.


algebraic


This


so in the case of


the void


Bpace,


Even cursory


considerations


examination of Figure


show


that


these


e element a


are not obvious


the void space during


later


stages


of sintering.


In addition,


branches


and nodes


in both


the s itter


body


the void space


are not


obvious


in Figure


out as follows:


therefore,


First


construction of


construct


the void space


a regular tetrahedron


carried


in three-


diusetnsonal


space.


The corners of


this


are assumed


to be loaded with


Space

























r


r
'h


I" t...,_,,,,,,,_I..,.. L .,


Figure


Photomicrograph


of Linde Copper


Powder,


-170 +200 Mesh,
128 Hours at 1C


Sintered


)05 C,


in Hydrogen for


150X


"" """"""""""" "ijiEEEE"X("





43


a


SW


-


Figure


Photomicrograph


-170 +200
32 Hours


Mesh,
t 1005


of Linde
Sintered


Copper


Powder,


in Hydrogen


150ox










Now determine


mass


center


of these


particles.


From this


point,


a perpendicular


is dropped


upon each of


the four


faces


of the


tetra-


hedron.


perpendiculars


are extended


beyond


the limits


of the


tetrahedron within which


mass


center


is situated.


extens ion


is made


arbitrarily


far and equal


for all four


perpendiculars,


least


so far


that


the faces


of the regular


tetrahedron


bounded


points


of these


extends ions


nowhere


intercept


the original


tetra-


hedron.


This


new tetrahedron


is the conjugate of


the original.


masses


are actually


taken


to be sinter particles


contact


points


between particles


to be sinter


body


branches


, then


the void space


tetrahedron and as


as nodes


branches


the vert ices


and faces,


of the so constructed new


the corresponding configura-


tions


that

which


belonging


the sinte

assumes


to it.


body


a sinter


cannot


a particle


part icle


be constructed


at each


vertex of


happens


simplicial


the tetrahedro


to be missing so


subdivision,

n. then a


sufficient


number


of fictitious


vertices


are assumed and


the sinter


body


thus


theoretically


is simplicially


subdivided


contain a


part


of the physical


void


space.


This


does


not interfere with


topology


of the problem


because


the fictitious


nodes


are subsequently


removable without


corresponds


distrubing


to Veblen's


the void space


procedure of "regular


node-branch manifold.


subdivision"


This


of a sim-


plicial


subdivision of


a manifold.


The surplus


simplices may


moved whenever


convenient.


The number


of nodes


and branches


in the


void


space may


be considerably


larger


than absolutely


necessary


re-






45


conjugate


to one


another--referred


as void


space


and the associated


s inter


body,


and conversely.


This


is illustrated


in Figure


Both


can


as constructed


be represented


in the previous


terms


of the


where


the branches were


assigned


an orientation.


However,


in the sinter


body


problem,


the use of oriented


branches


not immediately nece


ssary.


Hence,


slightly modified algebra not


requiring


orientation


is used


this


chapter.


Algebra--Modu lo


In the later


stages


of sintering,


the branch-node


identification


method


becomes


more


and more


difficult


and therefore


appears


use-


ful to introduce


the additional


three-dimensional manifolds


conmonn


bounding surface of


the sinter


body


void


space.


For this


purpose


the following algebra


now


introduced.


numbers


0 and


form a Modulo-Two


Algebra when combined


according


to these


rules:


0+0


OxO


= 0 x


= 1xO


Lxi


To understand


this


convention,


consider


the branch-node


complex in Figure


set of nodes


be denoted


the symbol


(x1,


* xn) where n
U


is the total number of


nodes.


Thus,


of nodes


are given


the symbol


- U I.


these manifolds


algebra of matrices,


chapter,


x_ -


*


|1


* *






46


Figure


Conjugate


Space


Model




Df;IIII'lIID r


47


is given


Geometrically


+1(2


represents


set of nodes


that


are in X1


in K2,


not in both X1


Using Modulo-Two


Algebra,


incidence matrices


can


be formed.


Such


matrices,


denoted


H (the notation


is that


used


Veblen),


give


the information required of


a non-oriented network.


matrix gives


information concerning the


nodes


of a network.


Figure


(24)


A second matrix can


formed


denoting the


incidence


branches


and nodes.


This


matrix,


denoted


is termed


the branch-


node


incidence matrix.


convent ion


for forming such


an incidence


matrix


as follows:


Provide


a column


for each


branch


a row


for each


node.


branch


is incident with


node


then aij


=1i,


while


they


are not incident,


= 0.


maliH


matrix


for Figure


10 is presented


below:


(25)


Such an arrangement can


generalized


to higher


dimensions.


and X2.










face


opposite


to it and


designat


so on


face


-branch


incident e


matrix can


constructed.


Such


a matrix


, denoted


for Figure


, is illustrated


below:


(26)


be shown


that


* H2


(27)


as one generalizes


to higher


dimens ions


it is found


that


Hk+l


(28)


There


is a definite


parallelism between


this


representation and


that


of the previous


chapter


From time


to time one


or the other will


used


depending upon expediency


Extending


to higher


d dimensions,


finds


Table


can


be constructed.


Knotting Within a


Sinter


Body


Consider the


two tinker


toy mode ls


shown


in Figures


22 and 23.


In Figure


22 a Simple Cubic Array


is illustrated


= 13)


In Figure


= 13),


a complex is


illustrated


that


iseby


means


siuole.


The significant


difference


is that


the latter


for n2,


can


one


..


i jjj:EEE





49


* S if
5


Figure


Tetrahedron for Constructing
a Face-branch Incidence
Matrix






























































to.


p~igure


Cubic


Tinker


Toy


Representation


~""''"'


r,
:E


EE"
E
":,":
x,





C,


Figure


Knotted


Tinker


Toy


Representation









body


problem and a


branch-node


network


illustrating entanglement,


normally


seen


in early


stage


sintering,


is photographed


in Figure


A number


of the branches


penetrate


faces.


This


property


of the sin-


ter body


not a topological


invariant


and thus


will


not be


considered


further.


The Linkage Numbers


An invariant,


which


COnnon)


to both


the sinter


body


manifold


the void


space manifold,


is termed


the linkage


number.


It follows


from the Alexander


Duality


Theorem


(14)


"The


k dimensional Betti


number


of an arbitrary


polyhedron


lying


in an Rn


is equal


to the


(n-k)-dimensional


Betti


number


of its


complement ary


region R"


In Figures


, (6),


linkage


numbers


of 0


illustrated


respectively.


Statement


VI I.


The linkage


number


, denoted


of the void


space


manifold


summed over


all its


parts,


is equal


to the linkage


number


of the sinter


body


manifold.


Statement


VI II.


Betti


number


the sinter


body


equal


to the


linkage


number


L of


either manifold.


Statement


Betti number


of the void


space,


equal to


the linkage number


only when


the void


space manifold


connected.


Algebraic


manipulation of


these statements


yields


are
















































Figure


Knotted


Branch-node


Sinter


Body


Representation


I'i";;l


























Figure


Example


of Linkage


Number


Zero


Figure


Example


Linkage


Number


Figure


Example


Linkage


Number


Two









P0

j-o a


- n


+ 1)


This


is the basic


topological


relationship


between


three-dimensional


manifolds.


The numbers


and (b.


-nj


+ 1)


imply

these


two essential


- n + 1)
.


invariants


and PV


of the


What


system.


is important


One may

is that


choose

to find


for

any


one invariant


analytically


, two others must


be determined


experi-


mentally.


The Higher


Dimensional


Betti


Numbers


So far all that


has been dealt with


are the 0th


and 1st


order


Betti


numbers.


It is natural


to ask if


any new


information


be obtained


considering


the higher


dimensional


invariants.


example,


one considers


the surface


of the sinter


body


(i.e.,


two-dimensional


manifold


that


croumot


to both


the void


space


the sinter
S


of P


body manifolds)


The Alexander


it is natural


Duality Theorem


tells


to inquire


abou t


us directly:


k
n-k'


= 1,


The two-dimensional


manifold


is embedded


in three-dimensional


space


so that


n = 3


and k


or 2.


Hence


Thus


the branch-node


representation


gives


same


information


terms


of Betti


numbers


that


would have


been


obtained


considering


can


the value


- 1.


S
= P
2










Next


consider


the 3rd


order


Betti


number.


In this


case


n -li ,


k-i.


or 3;


= P


= P
3


Thi s

use o

Betti


type


of procedure


f the higher


numbers


generalizes


dimensions


and their


offers


and thus

no new


connection with


it is concluded

information as f


sinter


bodies


that


ar as the


and their


associated


void


spaces


are concerned.


; ""'E':Eiii;iii"E"E'"" x














CHAPTER


APPLICATIONS


TO SINTER METALLURGY


Some
Body


Relations
and Void


Between


the Betti.


Numbers


of the Sinter


Space


A method of


characterizing


a sinter


body


is the determination


of its incidence


matrix.


This


problem has


several


aspects:


In the early


stages


of sintering,


if individual


sinter


body


layers


were


removed,


say of particle


size


thickness


representative


could


incidence


matrix


A typical


for the sinter


section


a sinter


body

body


manifold

is presented


in Figure


The incidence matrix would


not be


exact


as every


branch


and node


possibly


might


not be found.


However


if homo-


geneity


and isotropy


are assumed,


of the sinter


body


could


be evaluated


on a


unit


mass


or volume


bas is.


This


is done


in practice


metallurgists.


In Figure


28 the 2


equal


areas


of 2.09


square


inches


each


enclose


approximately


10 particles.


The entire


photograph


con-


tains


particles


per square


inch


and the sub areas


have


same


number


of particles


per square


inch.


Therefore,


the 2


areas,


although


small


are representative


of the entire


sinter


body.


criterion


of whether


a part


of the sinter


body


is representative




59


Figure


Illustration


Representative


Sinter


Body


Areas










par tial


areas


1 and


in Figure 28


have


approximately


10 particles


and whether


or not this


is sufficient


depends


upon the


purpose


which


the sinter


body


is to be used or upon other


considerations.


A partial


incidence matrix


for the network constructed


Figure


29 is presented


approximately


a half


be low.


layer


taking


lower


the next


section


, the three-dimensional 1


network


can be constructed.


000


0 0

1 0


000


0000000000

0000000000


0


0


000


0000


0


0000000


00000000000


0000000


0000


00000


00000


0000


00000000


00000


000000000


00000000


0000000


00000000


000000000


0000000000


00000000000


00000000000


0000000000


000


000


000


000000

100000

111010


0000


0000000000


IInl~l I. S


n nn


nnnnn


000


0 0


(33)


I II


I


I|




































"12



nil


'3io


"1,


~~''"'"


b12











000


000


001


000


000


In late


cult


stages


to distinguish


of sintering


individual


(Figure


particles


it becomes


and interparticle


diffi-


con-


tacts


due to the coalescence


of the material.


Thus


, in studying


the topology


sintering during


late


stages


may be simpler


to consider


the topology


of the void


space.


This


is known


become


disconnected


as sintering


progr


esses


which means


there


will


a large


number


parts.


A practice


al method of


estab-


lishing a


branch-node


incidence


matrix of


the void


space


is out-


lined


in the following:


Consider


the six sketches


shown


in Figure


31 and


assume


the sections


to be taken successively


so that


isolated


void s


lie between


them.


validity


of this


assumption


depends


upon particle


size


and section


thickness.


First


number


all the voids


taking


individual


voids


to be


nodes.


Subscripts


on nodes


refer


to particular voids


and superscripts


to the sections


on which


they


appear


i.e.,


is void


one on


section


two.


Then on


the bas is


of the


geometry


a void


and its


location on


a section


decide


whether


is connected


k+1
n.
3


Figure


31 shows


the following:


is connected


is connected








is connected


3
to n1


3
and n2


is connected


is connected


3
to n3

3
to n4


are connected


4
to n1


and ni


respectively


is connected


is connected


is connected


is connected


n an


to n1


to n2


to n5
3


and n4


and n5


are connected


are connected


is connected


ton6
2


to n5
4

to n6


and n6


is connected


to n
4


Branches


scripts.


will


Superscripts


also


be numbered


i and


indicate


with subscripts


sections


super-


and subscripts


and 1


indicate


the nodes


at the end of


the branch.


Reference


to the


branch-node


representation of Figure


32 will


clarify


this


terminology.


incidence matrix


(35)


for Figure


32 follows


subsequently.


It should


be emphasized


once


again


that


both


the loop-branch






64


Figure


Photomicrograph of Linde Copper Powder,
-170 +200 Mesh, Sintered in Hydrogen for
1024 Hours at 1005 C, 150X






65


Cl ~ ~ ~ ~ ~ ~ ~ a ra la-4.4a


nr


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45
bl
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3 35


34 4 4;- 5 56 6j
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44 4 44 4 4


Figure


Branch-node


Serial


Network


Hypothetical


Sections


12
bl
11















la-

teal
anc


'On3
in e


0 0


0 0


0 0 0 0 0


0 0 0 0 0 0 0 0


f lNe
a


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Other Examples


The incidence matrices


for Figures


22 and 24


respectively


are constructed


subsequently.


The numbering


individual


branches


the related


and nodes


has been omitted


configuration


is desired.


because


only


It is


the character


necessary


to assign


specific


numbers


because


they


are arbitrary


and the columns


rows


are subject


to arbitrary


permutations.


001


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000


000


000


0000


000


000


000


0000


000


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000


000


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000


000


000


000


000


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0001


000


000


000


000


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00000


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:N~i~ihiEE~ii

69


000


000


00000000000


000


001


00000


000000


000


0000


000

000

110


000


00000


0000

0000

0000


0000


000


000


0 000000000


0000


000000


000


001


000


000


00000

00000


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00000000


000


000


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000


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0000


000


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000


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001


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0000


000


000


000


000


000


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000


0000


In Chapter


a sample


calculation


is presented


an actual


sinter


body


to be PV=0.40


The number

X 105/cubic


of separated


inch.


parts


of void


Evaluation of


space


is computed


-n+1)


yields


cubic


P=105/


inch.


Hence a method


of determining


the Betti numbers


a sinter


body


has been


presented and


a numerical


example


com-


puted.


(37)


T(b





- -- --











-n1


V
-a1


This


true


because of


the method of


constructing


the conjugate


void


space


and the fact


that


they


have


a common


two-dimens ional


manifold.


As sintering progresses


and the void


space


decomposes


into


two parts


S


S
-n2


- n


+ 1)


I
-a"2


+ 1)


where


now the


subscripts


on the right


hand


side


refer


to the


separated


parts.


The above


equation


follows


from


the fact


that


each


component


of the void


space


has its own


Betti


number;


at the


same


time


the ist


numbers


body


has its


own


space


Betti


equal s


number;

the Ist


sum of


Be tti


number


of the sinter


body


because of


the equality


of the ranks


appropriate matrices.


An alternate


interpretation


of (39)


obtained


from Figure


in the sinter


body


20 if


there


one notes


exists


that


a closed


loop


every


closed


in the void


loop


space


Thus


* n


+ 1)


over


the void


space parts must


equal


Betti


number of


the sinter


body.


With further


wintering


- n
en3


- n


+ 1)


-"2


+ 1)


- n3


+ 1)


(40)


S
-Ut


(i-1)
- n


+ 1)


the sinter


Betti


of the void


the ist


Y
b
1


+ (b2


+ (bg


f(b2


~ (bi


=~(bj




U~


superscript


v has


been omitted


in order


to avoid


confusion


but the


right


hand


side


refers


to the void


space


components.


left


hand


side constitutes


a monotonically


decreasing sequence


of integers.


same


is true of


each


- nji-1)


+ 1)


on the right


hand side


however


the number


is monotonically


increasing


thus


the rate of


decrease


the difference


-II.)


must


greater.


It is of interest


to note


the change


in structure of


the sinter


body


and void


space matrices.


As sintering progress


the elimination


of sinter


body


nodes


and branches


trans forms


the sinter


body


incidence


matrix as


shown


be low.


where


the solid


lines within


the matrix represent


the elimination


of rows


and columns.


In the


case of


the void


space matrix two


possibilities


must


be considered:


The void


space decomposes


into several


parts.


The void


space


remains


connected.








A partitioned matrix can always


be obtained,


one for each


component


the void


space.


Experiment 1


Proposals


During


course


of this


research


several


experimental


pro-


posals


were


forward


and are enumerated


below:


Determination of


the 0th


Betti


number


of the void


space.


procedure


is to consist


of innmmersing a


copper


sinter


body


molten


void


lead


so that


space


parts


the void


that


space manifold would


are located


become


completely within


flooded.


the sinter


body


not accessible would


course


not be flooded


but this


be avoided


choosing


a sufficiently


emil1,


but still


repre-


sentative,


sinter


body


volume.


At liquid


helium temperatures


lead


would


become


superconducting while


the copper would not


for practical


purposes


would


an insulator.


applying


a poten-


tial


difference


to probes


and then


checking


current


the number


of separated


parts


of the void


space


is determined.


Determination of


the Oth


Betti


number


the fountain effect.


If one face

helium then

exhibit the


of the sinter


the void


fountain


body


space which

effect. T


is brought int

h is connected


he number


o contact with


to that


of fountain


liquid


face will


streams


dicates


the number


of connected


voids.


All void


space


aperatures


except


one)


be blocked


so connectedness


to the


aperatures


the other


faces


will


be exhibited.


The rate


of growth


of the necks


or branches


as they


have


been


can







(15).


The maximum-flow minimum-cut


theorem is


fundamental


in this


connection.


is proposed


to investigate


this


in a subsequent


communicate ion.


Experimental methods


suitable


for these


theories


which


might


be considered are


the following:


Heat


conduction


enables


the measurement


of maximum heat


flow


from particle


body with


to particle


the exception of


insulating


two separated


the entire


particles


sinter


to which


probes


are attached with an arbitrary


temperature difference.


Repeated


application of


this


method


enables


one to compute


the maximum-


flow minimum-cut


property.


Electrical


known


conduction


to provide maximum flow.


from one or more


Here


particles


an additional


is also


natural


be brought


to bear


in the sinter


body


problem


because


in any


network

Thus, i


the distribution of


t is possible


current


by measuring


causes


heat


minimum heat


generated within


generation.

a given


sinter


body


(suitably


insulated)


to obtain


information about


current


distribution within


the sinter


body.


I: :: ::: 39;















CHAPTER


CALCULATION OF


THE BETTI


NUMBERS


FOR AN ACTUAL


SINTER BODY


The sinter


body


used


for the following


calculation was made


with


Linde


copper powder


-170 +200 mesh).


was sintered


1024


hours


at 1005


in hydrogen and its


final


density was


calculated


to be 8


12 gms./cc.


In Figures


33 through


46 microstructures


the above mentioned


are presented.


These


are serial


sections


and on the


average


are approx-


imately 0.0005


inches


apart


A close


examination of


the sections


illustrates


that


specific


voids


can be followed


from one


section


next


considering


both


the void


geome t ry


and the surrounding


structure.


It is


to be noted


that


situations


do occasionally


arise


where


one cannot


sure


what


happens


a void


between


particular


sections.


In such


cases


where


the frequency


occurrence


is high,


thinner


sections


are required.


The boundary


illustrated


on the figures


is arbitrary


and in


this


case


has been


taken


to be


inch


square.


Since


the twins


occurring within


the grain


are parallel to


each


other


on the various


sections


, it is reasonable


assume


that


section


not rotated


relative


to section


(i


The total


vo lume


material


cons idered


in the example





tp
'/)

t


Figure


Section


Serial


Sections,


75












>7


Figure


Section


Serial


Sections,


4




~1:


iS*





















S-


*i- *.-rw


-at~


Figure


Section


of 14


Serial


Sections,




























K.P


Figure


Section


Serial


Sections,


90X


I I
Iz


': iiEilliiiiii"::::


F~


7;5C"





xxxx x7 9


Figure






2'


Figure


Section


Section


of 14


10 of


Serial


Serial


Sections,


Sections,


90X


90x





























Figure


Section


11 of


Serial


Sections,


90X


cc1F


f

1"
~P"~

r, ,r


"" "" :::iii~ : "4





















,- ----


Figure


Section


of 14


Serial


Sections,


90o


IEE'"""""":C:ii""iE"""""


5,









whether


other


randomly


chosen volumes


of the sinter


body would possess


same


values


and P


deferred.


The calculation


of the Betti


numbers


proceeds


as follows:


Consider


a set of serial


sections


and rule


an arbitrary


area


on each.


It i


advisable


to pick an area where


sufficient


struc-


ture


present


so that


following


the individual


voids


facilitated.


Arbitrarily


number


each and


every void


that


occurs entirely


within


the boundary


a particular


section and


continue


process


for all sections.


a network


as shown


Figure


47 and


connect


branches


between node


node


1+1


(where


subscripts


and superscripts


are as defined


previously)


a continuous


path


exists


be tween


them.


If a node


occurs


on section i


that


connected


to the


external


void


space


between


sections


and (


then


connect


a branch


and let this


branch


terminate


between


sections


+ 1)


without


a node


at its other


end.


Thi s


same


as con-


i
nesting nj


to the common node


of the external


void


space.


When


sections


and i


are in


contact


with


the external


void


space


the total


number of


branches


consists


of those


between


sections


of sections


and i


aud i


addition to

to the void


the branches


space


connecting


common node.


the nodes


In addition


it must


be remembered


that


the total


number


nodes must


include


- t


" ::~ BE" I


r


_ *










It is incorrect


over


to count


all the separated


the total


parts


number


and take


of branches


total


- n


and nodes


total


In view of


the above


the network* is presented


in Figure 47


while


in Table


the numbers


of individual


branches


and nodes


listed with


the appropriate


Table


V lists


and P1
1


on a


uni t


volume


basis,


while


Table


VI lists


the indi-


vidual


section


thicknesses with


the cumulative


volumes


over


given


sections.


In Figures


48 and


49 the Betti


numbers


per unit


volume are


plotted


vs.


the number


of sections.


It is


to be noted


that


curves


tend


zero


slope


as more


sections


are considered.


Thus


ques-


tion


of how many


sections


are required


can be answered


graphing


appropriate


invariant


vs.


the number


sections


determining where


the slope


tends


zero.


*Since


calculation made


the complete


from


this


network
network


presented


and the resultant


the appropriate matrices


will


be constructed.


are


1""""EEEEEBEEiii ""

QXI


n 'r












































Figure 47.


Branch-node


Network






85





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TABLE VI


SECTION THICKNESSES AND CUMULATIVE VOLUMES


Number of


Sections


Volume


Thickness
(Inches)


(Inched3 x 105)


0.0007

0.0015


0.0021

0.0024

0.0029

0.0035

0.0038

0.0042


11.4

12.8

15.4

18.6

20.2

22.4


0.0048


0.0053


28.2


0.0057

0.0067

0.0071


30.4

35.6

37.8





88






89


e-














CHAPTER


SUMMARY


OF THEORETICAL RESULTS


FOR APPLICATION TO SINTER METALLURGY


space


t il


has been

s space


introduced

conjugate t


as a topological


o the sinter


manifold.


body which


bears


a definite


quantitative


relation


to the sinter


body


has been


effected.


Network and


the associated


algebraic
r


topology


have


been


introduced


and a physical


interpretation,


terms


of the sinter


body


and its associated- void


space


or spaces


, of the following


matrices


has been given:


Incidence


of the branch-node


configuration.


Loop-branch.

Cut-branch.

Tree-branch.


It has been shown


that


the introduction of


the higher


dimensional


Betti numbers


gives


no additional


information.


A topological


relationship


between


two three-dimens ional


manifolds


was found


to be


At the beginning of


PS
1


s inter ing


and P"
0


While


in general


a' S a -


The void


The construction of


p0


C- ** r 'L




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