VSEBINA – CONTENTS
[estdeset let prof. dr. Vasilija Pre{erna
Laudation in honour of Professor Dr. Vasilij Pre{ern on the occasion of his 60th birthday
M. Jenko . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
PREGLEDNI ZNANSTVENI ^LANEK – REVIEWED SCIENTIFIC ARTICLE
The oxidation and reduction of chromium during the elaboration of stainless steels in an electric arc furnace
Oksidacija in redukcija kroma iz `lindre med izdelavo nerjavnih jekel v elektrooblo~ni pe~i
B. Arh, F. Tehovnik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
IZVIRNI ZNANSTVENI ^LANKI – ORIGINAL SCIENTIFIC ARTICLES
A new topology for the trajectories of the meniscus during continuous steel casting
Nova topologija trajektorij meniskusa pri neprekinjenem litju jekla
I. B. Risteski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Multiscale modelling of short cracks in random polycrystalline aggregates
Ve~nivojsko modeliranje kratkih razpok v naklju~nih ve~kristalnih skupkih
L. Cizelj, I. Simonovski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Changes to the fracture behaviour of medium-alloyed ledeburitic tool steel after plasma nitriding
Spremembe v na~inu preloma srednje legiranega ledeburitnega jekla zaradi plazemskega nitriranja
J. Peter, F. Hnilica, J. Cejp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
The fracture and fatigue of surface-treated tetragonal zirconia (Y-TZP) dental ceramics
Prelom in utrujenost povr{insko obdelane tetragonalne (Y-TZP) dentalne keramike
T. Kosma~, ^. Oblak, P. Jevnikar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Povr{ina zlitine Cu-Sn-Zn-Pb po obsevanju z ultravijoli~nim du{ikovim laserjem
Surface of Cu-Sn-Zn-Pb alloy irradiated with ultraviolet nitrogen laser
F. Zupani~, T. Bon~ina, D. Pipi}, V. Hen~ - Bartoli} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
A preliminary S-N curve for the typical stiffened-plate panels of shipbuilding structures
Preliminarna krivulja S-N za toge plo{~ate panele za ladjedelni{ke strukture
L. Gusha, S. Lufi, M. Gjonaj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
ISSN 1580-2949
UDK 669+666+678+53 MTAEC9, 41(5)197–253(2007)
MATER.
TEHNOL.
LETNIK
VOLUME 41
[TEV.
NO. 5
STR.
P. 197–253
LJUBLJANA
SLOVENIJA
SEP.-OKT.
2007

[ESTDESET LET PROF. DR. VASILIJA PRE[ERNA
LAUDATION IN HONOUR OF PROFESSOR DR. VASILIJ PRE[ERN
ON THE OCCASION OF HIS 60th BIRTHDAY
Rodil se je 4. julija 1947 v Mariboru. Osnovno {olo
in prvi letnik gimnazije je obiskoval na Jesenicah. Po
preselitvi v Ljubljano je leta 1966 z odliko maturiral na
Gimnaziji Vi~ v Ljubljani. Vpisal je {tudij metalurgije na
Univerzi v Ljubljani, kjer je leta 1971 kot prvi iz letnika
tudi diplomiral. Naziv magistra znanosti je pridobil leta
1974, naziv doktorja metalur{kih znanosti pa `e leta
1978. Leta 1992 je bil habilitiran za izrednega profe-
sorja.
Po diplomi se je zaposlil na Metalur{kem in{titutu v
Ljubljani (danes IMT – In{titut za kovinske materiale in
tehnologije), kjer je ostal do leta 1991.
Na in{titutu se je ukvarjal z raziskavami termo-
dinamike procesov rafinacije z dodatkom kalcija v z
aluminijem pomirjenem teko~em jeklu in s formiranjem
nekovinskih vklju~kov, z raziskavo reakcije kalcija v
sistemu talina-vklju~ki-`lindra. Podro~je raziskav je `e
takrat zahtevalo tesno sodelovanje s slovensko oziroma
jugoslovansko kot tudi z avstrijsko jeklarsko industrijo.
[tudij termodinami~nih reakcij v talini pri dodajanju Ca
je bilo izredno pomembno tako z raziskovalnega vidika
kot tudi s proizvodnega, zato so ga povabili k sodelo-
vanju strokovnjaki z National Standardization and
Technology – NIST, Gaithersburg, ZDA. V ve~letnem
bilateralnem projektu YU-USA je bil pojasnjen vpliv
CaO in Al2O3 v talini, dobljeni rezultati pa so bistveno
Materiali in tehnologije / Materials and technology 41 (2007) 5, 199–201 199
Professor Dr. Vasilij Pre{ern, scientific councillor
and former general director of the Acroni steelworks, is
celebrating his 60th birthday. This provides us with an
excellent opportunity to look at the background and the
development of this well-known scientist – and even
more successful economist – and his influence on
research and economics in the field of steelmaking in
Slovenia and abroad.
Vasilij Pre{ern was born in Maribor, on the 4th of
July 1947. After completing his secondary-school
education in Ljubljana with a distinction, he studied
metallurgy at the University of Ljubljana and finished in
1971, as the first in his class. He joined the Metallurgical
Institute, now the Institute of Metals and Technology
(IMT), Ljubljana, where he worked until 1991. In 1974
he finished his master's degree. He then went on to his
doctoral thesis and graduated in 1978 at the University
of Ljubljana. In 1992 he was habilitated at the university
as a professor.
In 1981 he received a six-month scholarship from the
Confederation of British Industry (CIB) to develop
subsidiary steelmaking agents for molten steel treatment
at Foseco in Birmingham, UK.
Vasilij Pre{ern is an enthusiastic developer, with
creative ideas and tremendous energy. During his time at
IMT he worked on the thermodynamic conditions for the
modification of inclusions in calcium-treated alumi-
nium-killed molten steel. The topic of his research was
very innovative and he was invited to take part in a
pripomogli k proizvodnji jekla z manj{o vsebnostjo
`vepla in nekovinskih vklju~kov.
V `elji, da bi svoje bogato teoreti~no znanje in
poznanje svetovne metalurgije ~im bolj povezal s prakso,
se je leta 1991 zaposlil v takratnem Holdingu Slovenske
`elezarne (danes SIJ – Slovenska industrija jekla).
Leta 1998 je sprejel zelo zahtevno nalogo sanacije
najve~je slovenske jeklarske dru`be Acroni. Kot glavni
direktor Acronija je bil postavljen pred zahteven izziv
sestaviti skupino klju~nih sodelavcev, ki bo sposobna
prenoviti in usposobiti podjetje za konkuren~ni boj na
zahtevnih svetovnih trgih.
Z dobrim poznanjem svetovnega jeklarskega trga in
konkurence ter s svojo {irino in pozitivnim na~inom je
Pre{ernu uspelo prepri~ati lastnika, da je jeklarstvo
panoga, ki ima dolgoro~no prihodnost tudi v Sloveniji in
ji je vredno zagotoviti vso podporo za uspe{no sanacijo
in tudi nadaljnji razvoj Acronija. Tako so z ve~letnim
investiranjem v spremembo in optimizacijo proizvodnih
poti, v odpravljanje ozkih grl in v naprave, ki omogo~ajo
ve~jo proizvodnjo izdelkov z visoko dodano vrednostjo,
uspeli sanirati Acroni, in leto 2005 je bilo v vseh
pogledih najuspe{nej{e, saj je postal najve~ji proizva-
jalec specialnega jekla v Sloveniji z dodano vrednostjo
ve~ kot 38 000 EUR na zaposlenega.
Za uspe{no saniranje Acronija je bil prof. dr. Pre{ern
prejemnik nagrade GZS za leto 2006. V utemeljitvi so
navedli:
"ACRONI je dru`ba z eno najve~jih hitrosti v rasti
dodane vrednosti na zaposlenega in hitre rasti produk-
tivnosti! Ima mo~no izra`eno razvojno komponento,
dolgoro~no in moderno, atraktivno strategijo, rekordno
izpolnjevanje tr`nih, razvojnih, R&D in splo{nih meril za
nagrado GZS ter tudi meril internacionalizacije. Velja
podobno kot za Krko: model nagrad GZS najbolj ustreza
prav tak{nim dru`bam.
Zelo dobri poslovno-finan~ni rezultati, tudi ~e se jih
zaradi razmeroma velikega tr`nega dele`a in majhnega
{tevila primerljivih konkurentov primerja na osnovi
celotne panoge 27 – Proizvodnja kovin.
Predsednik uprave prof. dr.Vasilj Pre{ern je
opravljal funkcijo od leta 1998 do 2007 in je med-
narodno priznan strokovnjak na svojem podro~ju ter
optimisti~en, nadvse uspe{en gospodarstvenik.
Ves ~as je prof. dr. Vasilij Pre{ern skrbel tudi za
prenos znanja, saj je sodelovanje med In{titutom za
kovinske materiale in tehnologije vzor za sodelovanje
med akademsko sfero in industrijo. Mladi raziskovalci so
se usposabljali od pol do 1 leta v Acroniju in delali
skupaj s strokovnjaki iz prakse ter si tako pridobili
izredne izku{nje, in stkale so se prijateljske vezi, ki
zagotavljajo dolgotrajno sodelovanje.
Po prodaji ve~inskega dele`a SIJ je prof. Vasilij
Pre{ern prepustil mesto glavnega direktorja mlaj{im
200 Materiali in tehnologije / Materials and technology 41 (2007) 5, 199–201
Yugoslav-USA bilateral project, where he worked
together with world-recognized experts from the
National Institute of Standards and Technology in
Gaithersburg, USA. He investigated the thermodynamic
relations between calcium-treated aluminium-killed
molten steel and non-metallic inclusions and demon-
strated the importance of having the right Ca content.
The results of this research were also very important for
the steelmaking industry. The influence of CaO and
Al2O3 in molten steel was explained during the course of
the bilateral project, and the results were very helpful for
the production of cleaner steels containing smaller
amounts of sulphur and non-metallic inclusions.
During his time at IMT he published more than 50
papers in leading scientific journals and had more than
50 invited lectures at the most important international
conferences. He has also been involved in more than 250
projects with industry.
Because of his wish to combine his deep theoretical
expertise and his knowledge of world metallurgy with
industrial work, in 1991 he joined the then Holding of
Slovenian Steelworks (presently, the SIJ, the Slovenian
Steelwork Industry)
In 1998 he accepted the very difficult task of
reorganizing the biggest Slovenian steelworks company,
Acroni. As the general manager he faced a great
challenge: to form a highly qualified group of leading
co-workers, qualified to modernize and reposition the
company to be competitive on the global market.
His expertise in the global steelmaking market and
competition and his positive approach enabled him to
persuade the owners that the steelmaking branch had
great promise – also in Slovenia – and was worth
supporting during the successful reorganization and
further development of Acroni.
The investments over several years led to the
modification and optimisation of the production lines
and equipment; this enabled increased production levels
with a higher added value and made possible the
successful reorganization of Acroni. The year 2005 was,
in all respects, the most successful year for Acroni,
which produced special steels in Slovenia with an added
value of  38,000/employee.
In 2006 Vasilij Pre{ern was the winner of an award
from the Chamber of the Economy of the Republic of
Slovenia (GZS-gospodarska zbornica RS) for his
successful reorganization of Acroni. At the awards
ceremony Acroni was recognised as a company with a
rapid growth in added value per employee and increasing
production levels. The company was described as having
a strong development component, long-range and
advanced strategies, a record of satisfying the market,
and good R&D standards.
Vasilij Pre{ern took great care of the knowledge
transfer and cooperation involving on of Slovenia's
leading research institutes, IMT, representing an ideal
model of cooperation between the academic sphere and
industry in Slovenia. IMT's young researchers were able
to work together with Acroni's experts for six months to
one year at the Acroni plant, where they gained
sodelavcem, ki jih je vzgojil in ki jim zaupa, da bodo
tako zagnani in uspe{ni, kot je bil sam.
Prof. dr. Vasilij se privaja na novo odgovornost v
okviru SIJ kot direktor za investicije in razvoj celotnega
SIJ-a. Tako mu ostaja ~as, ki ga je vsak dan porabil za
vo`njo na Jesenice, da ga pre`ivlja s svojo vnukinjo
Julijo.
Vendar pa {e vedno najde ~as za stroko, saj v okviru
Mednarodne podiplomske {ole Jo`efa Stefana prena{a
svoje bogate izku{nje na mlade podiplomce.
Dragi kolega in prijatelj Vasilij! Ob 60-letnici ti
iskreno ~estitamo za vse dose`ke in ti `elimo tudi v
prihodnje veliko uspehov, sre~e, predvsem pa osebnega
zadovoljstva v krogu svoje dru`ine in prijateljev.
Materiali in tehnologije / Materials and technology 41 (2007) 5, 199–201 201
invaluable experience. Friendly links were formed and
these links have ensured the long-lasting cooperation
between these two partners
After a takeover of the majority share of the
Slovenian Steel Industry, Vasilij Pre{ern left his position
as Acroni's general manager to younger co-workers, who
he brought up in previous years and who he has trusted
to be even more successful than he was.
Professor Vasilij Pre{ern is getting accustomed to his
new position in the SIJ – as the director responsible for
investments and the development of the whole SIJ. This
means he now has the time, which he previously used for
a daily commute to Jesenice, to spend with his grand-
daughter Julija.
Dear colleague and friend, Vasilij, on the occasion of
this 60th anniversary we congratulate you on your
excellent results and we wish you and your family many
years of successes and happiness in good health.
Monika Jenko

B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
THE OXIDATION AND REDUCTION OF CHROMIUM
DURING THE ELABORATION OF STAINLESS STEELS
IN AN ELECTRIC ARC FURNACE
OKSIDACIJA IN REDUKCIJA KROMA IZ @LINDRE MED
IZDELAVO NERJAVNIH JEKEL V ELEKTROOBLO^NI PE^I
Bo{tjan Arh, Franc Tehovnik
Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana, Slovenia
bostjan.arhimt.si
Prejem rokopisa – received: 2007-07-13; sprejem za objavo – accepted for publication: 2007-08-30
The oxidation of chromium during the elaboration of stainless steels occurs with oxygen in solution blown in the melt and with
oxides in the slag. The loss of chromium during the steel bath processing increases the production costs and generates problems
because of the high content of chromium oxide in the slag. A higher content of silicon in the furnace charge decreases the extent
of the oxidation of chromium; however, the efficient reduction of chromium from the slag is very important for a minimal loss
of chromium. In this survey, the theory of the oxidation of chromium, its reduction from the slag and the conditions for the
formation of foaming slag are discussed.
Key words: electric arc furnace, chromium oxidation, foaming slag, slag reduction, stainless steel
Oksidacija kroma med izdelavo nerjavnih jekel poteka zaradi topnega kisika v talini, vpihanega kisika in zaradi oksidov v
`lindri. Izguba kroma med izdelavo nerjavnega jekla v EOP nima za posledico samo vi{ji materialni stro{ek, ampak tudi
dolo~ene operativne posege zaradi prekomerne koli~ine kromovega oksida v `lindri. Ve~ja vsebnost silicija v zalo`enem
legiranem vlo`ku zadr`i oksidacijo kroma med njegovim taljenjem, vendar je le u~inkovita redukcija `lindre po oksidaciji taline
klju~nega pomena za minimalno izgubo kroma. V prispevku bomo predstavili predvsem teoreti~ne osnove oksidacije kroma,
redukcijo kroma iz `lindre in pogoje za tvorbo pene~e se `lindre pri izdelavi nerjavnih jekel v elektrooblo~ni pe~i.
Klju~ne besede: elektrooblo~na pe~, oksidacija kroma, pene~a se `lindra, redukcija `lindre, nerjavna jekla
1 INTRODUCTION
The content of chromium in austenite stainless steels
is generally in the range from 16 % to 24 %. More than
97 % of the chromium is lost during the melting of steels
from scrap in an electric arc furnace (EAF). The oxi-
dation of chromium occurs during the melting, and to an
even greater extent it occurs during the blowing in of
oxygen, aimed at decreasing the content of carbon in the
bath. A smaller part of chromium is also oxidised while
discharging the melt from the EAF. A high content of
chromium increases the crusting of the slag, decreases its
reactivity and impairs the formation of the foaming slag
and the slag reduction during the process of steel
elaboration. Stainless slags with a high content of
chromium oxide cannot be recycled or used, and are for
this reason also an ecological problem.
2 OXIDATION OF CHROMIUM IN THE MELT
The oxidation of chromium with oxygen in solution
in the melt occurs in parallel with the oxidation of other
elements, e.g., carbon, aluminium, silicon and manga-
nese, and it depends on the temperature and the activity
of these elements and the oxygen. The standard free
energy of oxidation (∆G°) for these elements is given by
the following relations1:
3
Cr + 4
O = (Cr3O4) ∆G° = –244.800 + 109.6 T (1)

C + 
O = CO(g) ∆G° = –5.600 – 9.37 T (2)

Si + 2
O = (SiO2) ∆G° = –139.070 + 53.09 T (3)
2
Al + 3
O = (Al2O3) ∆G° = –286.900 + 89.05 T (4)

Mn + 
O = (MnO) ∆G° = –57.530 + 24.73 T (5)
The equilibrium constant is deduced for every one of
these reactions from the change in the standard free
energy, and it is written for the oxidation of chromium
as:
lg K1 =
53521
23 96
.
.
T
− (6)
and
K1 =
a
a a
a
f f w w
Cr3O4
Cr
3
O
4
Cr3O4
Cr
3
O
4
Cr
3
O
4⋅
=
⋅ ⋅ ⋅
(7)
with aCr2O3, aCr, aO being the activity of (Cr2O3), 
Cr,

O; fCr, fO being the activity coefficient for Cr and O,
and wCr/%, wo/% being the mass fractions of Cr and O
in the steel melt.
The equation giving the content of oxygen in solution
in the steel bath of known composition (T304L) as a
function of the temperature and the activity of chromium
oxide is:
wo/% =
a
K f f w
Cr3O4
Cr
3
O
4
Cr
3
1
1 4
⋅ ⋅ ⋅
⎡
⎣
⎢
⎤
⎦
⎥
/
(8)
Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211 203
UDK 669.18:669.14.018.8 ISSN 1580-2949
Reviewed scientific article/Pregledni znanstveni ~lanek MTAEC9, 41(5)203(2007)
For a high content of chromium in the melt the
activity of the oxygen and of the carbon is lower, as both
interaction coefficients are negative. The effect of
chromium on the content and the activity of the oxygen
in the melt at 1600 °C is shown in Figure 12.
The dependences were calculated using an equation
valid for a content of 9 % chromium and more in the
steel bath (eq. 9)2,3. It is evident that for a high content of
chromium the solubility of oxygen in the melt is greater
(Figure 1). For this reason, and with an equal content of
carbon, the content of oxygen is higher in the bath with
chromium than in the bath without this element.
lg
wCr
3/4 · (fO · wO) = –13380/T + 5,99 (9)
Chromium is a relatively strong deoxidiser. With a
higher content of chromium, the oxidation products
consist of iron-chromium spinels of the type FeO·Cr2O3.
The transition from spinel to the saturation with Cr2O3 in
the system Fe–O–Cr occurs when the critical content of
chromium is exceeded4.
Fe(s) + ½ O2 (g) + Cr2O3(s) =
= FeO·Cr2O3(s) ∆G° = –307.600 + 66.82 T (10)
In Figure 2 the equilibrium between FeO.Cr2O3 and
oxygen is shown for both a low and a high content of
chromium in the bath. The critical content of chromium
is shown for different temperatures when the equilibrium
oxide phase changes from FeO·Cr2O3 to Cr2O3. The
phase FeO·Cr2O3 in equilibrium with the Fe–Cr melt is
changed to Cr2O3 when the actual content of chromium
in the bath increases above the critical content. Pure
Cr2O3 is the phase in equilibrium with the Fe–Cr bath for
more than 7 % Cr in the bath.
The activity of Cr2O3 and of FeO·Cr2O3 in the
FeO·Cr2O3 solid solution in equilibrium with the Fe–Cr
melt decreases when decreasing the content of chromium
below the wCr critical.
In addition to the thermodynamic activity of different
elements in the melt, the connection temperature-activity
of chromium oxide in the slag is of great importance,
since this effect is significantly greater at lower tempe-
ratures. The activity of Cr2O3 depends on the alkalinity
of the slag and on the content of CaO and SiO2. The
dependence of the free enthalpy of oxidation of
chromium, silicon and carbon on the temperature shows
that, according to the affinity for oxygen in solution,
silicon lies between carbon and chromium and, for this
reason, it is oxidised first and its oxidation delays the
oxidation of chromium.
The deoxidation of the Fe-Cr melt with silicon is
shown in Figure 35 as the equilibrium of the contents of
chromium and oxygen in the bath at 1600 °C in depen-
dence of the silicon content. The initial content of
oxygen in the Fe-Cr melt before the addition of silicon is
shown by the upper dashed line, which represents the
equilibrium Fe-Cr/Cr2O3 or Fe-Cr/FeO·Cr2O3 for the
content of wSi/% = 0. The addition of silicon lowers the
content of oxygen by a constant amount of chromium.
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
204 Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211
Figure 2: Equilibrium between chromium and oxygen in the steel
melt saturated with pure solid Cr2O3 and FeO·Cr2O3 4
Slika 2: Ravnote`je med kromom in kisikom v teko~em `elezu, nasi-
~enem s ~istim trdnim Cr2O3 in FeO·Cr2O3 4
Figure 1: Effect of chromium on the activity and the content of
oxygen in the system Fe-Cr at 1873 K 2
Slika 1: Vpliv kroma na aktivnost in vsebnost kisika v sistemu Fe-Cr
pri 1873 K 2
Figure 3: Equilibrium between the contents of chromium and oxygen
in the steel bath at 1873 K for different contents of silicon5
Slika 3: Ravnote`je med vsebnostjo kroma in kisika v teko~em `elezu
v odvisnosti od vsebnosti silicija pri 1873 K 5
The deoxidation power of silicon is lower with a higher
content of chromium, e.g., by the addition of 1 % of
silicon, the content of oxygen in the Fe + 5 % Cr bath is
lowered from 350 µg/g to approximately 73 µg/g and for
Fe + 18 % Cr it is lowered from 450 µg/g to 110 µg/g.
For this reason, a low content of oxygen cannot be
achieved with the addition of silicon alone. The
oxidation-reduction equation for silicon and chromium
is:
2Cr2O3 + 3
Si = 4
Cr + 3SiO2 (11)
The thermodynamics indicates that the content of
silicon in the melt is proportional to the activity of SiO2
in the slag; thus, the content of silicon in the melt is
higher with a higher content of SiO2 in the slag. The
thermodynamics of the relations content of Cr2O3 in the
slag, the content of chromium and silicon in the melt, the
melt temperature and the activity of SiO2 in the slag
were investigated by McCoy and Langerberg6. The
results of these investigations show that the loss of
chromium is higher with a low content of silicon in the
melt and with a higher bath temperature6.
Furthermore, carbon, with an over critical content,
delays the oxidation of chromium2. The equilibrium
between chromium and carbon was, for the pressure of
pco = 1 bar and the different temperatures in Figure 4,
determined by applying the following relations and
considering the coefficient of interaction of the first
order:
Cr2O3 + 4C = 3Cr + 4 CO (12)
lg
(Cr)3/4 · pCO/aC = –11520/T + 7.64 (13)
2.1 Stainless slags
The behaviour of chromium oxides (CrOx) in
metallurgical slags is complex because of the existence
of ions with several valences and of the high melting
temperature of the slags containing chromium7. In slags
the non-stoichiometric quantity of CrOx depends on the
temperature, on the slag’s alkalinity and on the content
of chromium oxides in equilibrium with the metallic
chromium. For the mixture of Cr2O3 with the other
components of the slag in contact with metallic
chromium at high temperature, the equilibrium of the
oxides is:
2CrO1,5 + Cr = 3CrO (14)
Chromium is found in the slag as two- and
three-valence ions. The activity of CrO and CrO1.5
increases with the increase of the total content of CrOx
and the basicity. In Figure 5 the activity of CrO and
CrO1.5 in the system CaO–SiO2–CrOx at 1873 K is
shown. At higher temperature, the activity is lower and
the content of the bivalent chromium in the slag is
increased. In contrast, with increased basicity the activity
of chromium oxides is increased and the content of
bivalent chromium in the slag is diminished (Figure 6).
A greater activity of chromium oxides in the slag is
found with greater alkalinity, a higher ratio of CaO/MgO
and a lower temperature. The lower content of MgO in
the slag increases the activity of the chromium oxides in
the slag up to a limit; however, it does not affect the
oxidation state and, as a result, the content of Cr2+ and
Cr3+. Small additions of Al2O3 affect the activity of CrOx,
while larger amounts have little effect. Slag with greater
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211 205
Figure 5: Iso-activity diagrams for CrO1.5 and CrO in the CaO-SiO2-
CrOx quasi-ternary system at 1873 K 7
Slika 5: Izoaktivnostni diagram molskih dele`ev (x/%) CrO1,5 in CrO
v CaO-SiO2-CrOx v kvasiternarnem sistemu pri temperaturi 1873 K 7
Figure 4: Thermodynamic equilibrium between carbon and chromium
for different temperatures and a pressure of pco = 1 bar 2
Slika 4: Termodinami~no ravnote`je med ogljikom in kromom v
odvisnosti od temperature pri tlaku pco = 1 bar 2
activity improves the reduction of chromium from the
slag and increases the yield of chromium during the
elaboration of stainless steels.
The stainless slag in the EAF consists mostly of CaO,
MgO, Al2O3 and SiO28. The optimal composition in the
system, CaO–MgO–SiO2, shown in Figure 7, is the
saturation with CaO and MgO9, which is the necessary
condition for a minimal content of Cr2O3 in the liquid
slag and it is acceptable for the resistance of the
refractory lining, also. The solubility of CrOx in these
slags is small. During the blowing of oxygen in the bath,
the elements are oxidised according to their affinity to
oxygen and their activity in the melt. Al and Si have a
greater affinity for oxygen; however, chromium is also
oxidised due to its greater activity. If CrOx is absorbed in
the molten slag, the equilibrium conditions change after
the blowing of oxygen and the chromium is returned in
the bath on the condition that there is a sufficient content
of silicon. In the presence of chromium in the solid
phase (MgCrO4, CaCrO4), the slag is rigid and the return
of chromium in the bath is lower, also in the case of a
higher content of silicon in the bath.
3 REDUCTION OF CHROMIUM FROM THE
SLAG
Several procedures were developed for the reduction
of chromium from the slag during the elaboration of the
steel in the furnace and during the discharge in the ladle
using silicon and aluminium. It is, however, very
important that the loss of chromium is minimised during
the oxidation melting and blowing of different
reductants. In Figure 8 and 9 the equilibrium curves are
shown for the oxidation of chromium with respect to the
reactivity of silicon, carbon, aluminium and manganese
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
206 Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211
Figure 6: Effect of temperature on the activity of CrO1.5 in the system
CaO-SiO2-CrOx and the effect of slag basicity on the activity of
CrO1.5 in the system CaO-SiO2-Al2O3-CrOx at 1873 K 7
Slika 6: Vpliv temperature na aktivnost CrO1,5 v sistemu CaO-SiO2-
CrOx in vpliv bazi~nosti `lindre na aktivnost CrO1,5 v sistemu
CaO-SiO2-Al2O3-CrOx pri temperaturi 1873 K 7
Figure 7: Isothermal section of the system CaO-MgO-SiO2 at 1600
°C 9
Slika 7: Izotermi~ni prerez sistema CaO-MgO-SiO2 pri 1600 °C 9
Figure 8: Effect of temperature on the ratio wCr/wSi and wCr/wC in the
steel bath with 18 % Cr 10
Slika 8: Vpliv temperature na razmerji masnih dele`ev wCr/wSi in
wCr/wC v talinah jekel z 18 % Cr 10
at the temperature of the elaboration of the steel
T304L10.
The thermodynamics of the relation wCr/wSi in the
melt with 18 % Cr shows that with a temperature of
approximately 1500 °C the oxidation of silicon starts at
0.2 % Si and at 1700 °C at 0.4 % Si. Thus, at 1500 °C a
content of 0.2 % Si in the melt already arrests the
oxidation of chromium, while at higher temperatures a
greater content of silicon is required. The ratio of wCr/wC
for 18 % Cr in the melt shows that the reactivity of the
chromium increases with the temperature. For this
reason, at higher temperatures a lower equilibrium
content of carbon impairs the oxidation of chromium.
Figure 9 shows that the affinity of aluminium for
oxygen is very high and that its activity decreases with
increasing temperature. The activity of manganese is not
affected by the temperature and in stainless steels with 2
% Mn the behaviour of manganese is different from that
of chromium, carbon, aluminium and silicon.
Silicon impairs the oxidation of chromium at low
temperatures, thus, during the melting of the charge,
while carbon is efficient in this role and with sufficient
concentration, at higher temperatures. The blowing in of
the oxygen at high temperature (>1550 °C) increases the
oxidation of carbon and strongly decreases the oxidation
of chromium.
The slag basicity ((CaO+MgO)/(SiO2+Al2O3)) has a
strong effect on the reduction of chromium oxides, and
with a higher basicity the content of chromium oxide in
the slag is lower. The activity of SiO2 and Al2O3 is also
lower, while the activity of CrOx is higher. All these
levels of activity increase the rate of reduction of
chromium oxide from the slag (Figure 10)11. According
to this figure, the basicity should be approximately at the
level of B = 1.8.
The efficient reduction of the furnace slag depends
on the selection of a suitable reductant, dependent on the
used procedure and the control of the furnace slag, which
should ensure a high level of reduction of chromium
oxide from the slag and should not contaminate the melt.
The selection of the reductant depends on the furnace
and the raffination practice.
However, of special importance is the decrease of the
chromium losses during the oxidative melting with the
blowing in of the different reductants. The first measure
for the control of the content of Cr2O3 is the use of a
charge and alloys (FeSi) with a high content of silicon,
ensuring a content of 0.3 % Si after the melting of the
charge. During the discharge, the reduction of chromium
oxides is continued with silicon in solution.
For a decrease of the chromium oxidation, the
injection of carbon is widely used because it is
economically more suited than the reduction with Si and
Al. These two elements are bound to form oxides,
thereby decreasing the slag basicity, which requires the
further addition of lime, resulting in the increase of the
slag volume. With slag reduction as a result of the
blowing in of carbon, carbon monoxide is formed, which
is necessary for the formation of the foaming slag.
However, for an efficient reduction of Cr2O3 with the
blowing in of carbon, a high slag temperature is also
necessary. For this reason, the blowing in of carbon is
carried out with the parallel blowing in of oxygen. The
reduction of chromium oxide with Si and C occurs by
the reactions:
(Cr2O3) + 1,5 
Si = 2 
Cr + 1,5 (SiO2) (15)
(Cr2O3) + 3C(S) = 2
Cr + 3CO(g) (16)
The reduction of chromium oxide is more efficient
with conditions of:
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211 207
Figure 10: Effect of slag basicity on the content of Cr2O3 in the slag11
Slika 10: U~inek bazi~nosti `lindre na vsebnost Cr2O3 v `lindri11
Figure 9: Effect of temperature on the ratio wCr/wAl and wCr/wMn in
the steel melt with 18 % Cr 10
Slika 9: Vpliv temperature na razmerji masnih dele`ev wCr/wAl in
wCr/wMn v talinah jekel z 18 % Cr 10
• a high activity of Cr2O3 with increased alkalinity,
• a high activity of carbon, ac = 1 during the blowing in
of carbon,
• a low partial pressure,
• a low activity of chromium in the metallic phase,
• a high temperature.
Besides the used technology for the reduction with
silicon and aluminium and the blowing in of carbon, the
use of the blowing in of calcium carbide was reported
also12. When this carbide reacts with oxides in the slag,
the products of the reaction are chromium, lime and car-
bon monoxide. CaO has the function of a non-metallic
addition for the formation of the slag, while carbon
monoxide improves the slag foaming in comparison to
the blowing in of carbon powder. The reduction of
chromium oxide with calcium carbide (CaC2) proceeds
according to the reaction:
Cr2O3 + CaC2 = 2Cr + 2CO(g) + CaO (17)
The change of Gibbs free energy with temperature
for the reactions of chromium oxide with silicon, carbon
and calcium carbide are shown in Figure 11. It is clear
that the reduction of chromium with carbon is more
efficient at high temperature; therefore, in practice it is
performed with the parallel blowing in of oxygen. The
reaction between silicon and chromium oxide is not
strongly temperature dependent and occurs, for this
reason, also at lower temperatures. The suitability of
calcium carbide as a reductant is shown by the reaction
between this carbide and chromium oxide, which has a
lower Gibbs energy in the temperature range of 1550 °C
to 1700 °C.
It is characteristic for slags containing iron and
chromium oxides, that on the boundary between the slag
and the carbon the reduction starts immediately and
without any incubation period, while the reduction of
Cr2O3 has an incubation period and is slower than the
reduction of FeO13. The reduction of Cr2O3 begins only
after the reduction of a considerable amount of FeO
(Figure 12).
In Figure 13 the ratio ln(wCr/wCro) with respect to
time is shown for a slag with a different content of FeO
with (%Cr) as the total content of chromium in the slag
and with (Cro) as the reduced chromium in the slag at the
time t13. The slope of the curves for the content of 5 % or
10 % of FeO shows that the reduced part of Cr2O3 is
greater with a higher content of FeO in the slag. For a
content of FeO = 0.5 % the incubation for the reduction
becomes longer and the Cr2O3 reduction time is
lengthened.
3.1 Foaming slag
The formation of foaming slag is a process dependent
on a sufficient evolution of gases in a slag with a proper
viscosity, which should not be too small, as a determined
time of presence of gas bubbles in the slag is necessary
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
208 Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211
Figure 13: The kinetics of the reduction of chromium oxides for a
different content of FeO in the slag and the slag alkalinity of
CaO/SiO2 = 1.15 13
Slika 13: Redukcija kromovega oksida v odvisnosti od vsebnosti FeO
pri bazi~nosti `lindre CaO/SiO2 = 1,15 13
Figure 11: Temperature dependence of Gibbs free energy for the
reduction of chromium with calcium carbide, silicon and carbon12
Slika 11: Temperaturna odvisnost Gibbsove proste energije za
redukcijo kroma s kalcijevim karbidom, z ogljikom in s silicijem12
Figure 12: Content of FeO and Cr2O3 with respect to time at 1600 °C
and 1650 °C 13
Slika 12: Vsebnosti FeO in Cr2O3 v `lindri v odvisnosti od ~asa pri
temperaturi 1600 °C in 1650 °C 13
to maintain sufficient foaming. If the viscosity is too
great, the foaming cannot occur, or occurs with
insufficient intensity.
The foaming characteristics are improved with the
lowering of the surface tension (σ) and density (ρ) and
with the increase in the slag viscosity (η)14. The foaming
stability and the foaming index (Σ) are determined from
the average size of the bubbles (Db).
Σ = 115 η /( σ · ρ)0,5 (18)
Σ = η1,2/( σ0,2 · ρ · Db
0,9) (19)
Db is the increase of volume due to the arising CO
bubbles in the slag. Small bubbles are of special
importance for the slag stability. The presence in the slag
of a suspension of different solid phases has a stronger
effect on the foaming than the slag surface tension or the
viscosity. An optimised slag is not entirely liquid;
however, it is saturated with CaO (Ca2SiO4) and MgO.
The particles in suspension act as nuclei for gas bubbles
and enable the formation of a large number of small
bubbles.
For the effect of particles in suspension on the slag
viscosity, the following relation was developed:
η e = η(1–1,35θ)
–5/2 (20)
With: η e – the effective slag viscosity
η – the viscosity of the liquid gas without particles in
suspension
θ – the volume share of the solid phase in the slag
In Figure 14 the dependence between the foaming
index and the relative effective slag viscosity is shown15.
If the relative effective slag viscosity is increased, the
time of the presence of bubbles in the slag is lengthened,
the foaming stability is increased and the foaming time is
lengthened, also. As shown in Figure 15, the optimal
quantity of solid particles is at the point G. Away from
this point the slag becomes too crusty and the index of
foaming is decreased.
The foaming slag has a positive effect in the EAF,
since it decreases the electrodes radiation loss, decreases
the spraying of the slag on the lining and on the water
panels, decreases the loss of electrodes and lowers the
effect of variation of electrical tension with a longer
electric arc16. It was found, also, that the foaming slag
increases the return of chromium from the slag in the
metallic bath (Figure 15).
By the elaboration of the carbon steels with FeO as
major oxidation product (> 20 % of FeO in the slag), the
propensity of the slag to foam is achieved simply with
the blowing in of carbon in the slag. The base reaction
for the formation of gas bubbles in the slag is:
FeO(s, l) + C(s) = Fe(l) + CO(g) (21)
The foaming of stainless slag is different from that
during the elaboration of carbon steels. The foaming
capacity of stainless slags is decreased by a low content
of iron oxide and a high content of chromium oxide in
the slag. Solid particles of chromium oxide in the slag
with a high melting point increase the slag viscosity and
impair the foaming. With a low FeO content and with
slow kinetics of reduction of chromium oxide, the
addition of carbon, which increases the quantity of CO,
does not fulfil the condition for an efficient foaming17.
The controlled formation of gas bubbles is very
important for the formation of the foaming slag. During
the elaboration of stainless steels, CO is produced,
mainly with the carbon reduction of CrOx and FeOx. In
stainless slags the Cr-O buffer appears at a much lower
oxygen potential than the Fe-O buffer and, for this
reason, chromium is oxidised before iron18. During the
blowing in of oxygen in the melt CrOx is formed as an
oxidation product in place of FeO and for this reaction it
is characteristic that:
• the solubility of CrOx in the slag is low and it
depends upon the ratio CaO/SiO2 and the tempe-
rature,
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211 209
0
5
10
15
20
25
0 2 4 6 8 10
Cromium Oxide, WCr2O3 /%
F
o
a
m
in
g
S
la
g
s
In
d
e
x
,
Σ
Figure 15: Content of chromium oxide in the slag after discharge of
the furnace in dependence of the slag foaming index16.
Slika 15: Vsebnost kromovega oksida v `lindri po prebodu na pe~i v
odvisnosti od indeksa penjenja `lindre16
Figure 14: Foaming index versus the effective slag viscosity15
Slika 14: Indeks penjenja v odvisnosti od efektivne viskoznosti `lin-
der15
• the share of reduction of CrOx with carbon is small in
comparison to the reduction of FeO. Consequently,
the rate of formation of CO is small, since it forms
mostly with the reduction of FeO.
In presence of FeOx a greater quantity of CO is
formed, and it can also be achieved with a greater degree
of oxidation of the melt or the addition of FeO to the
slag.
The second condition for the formation of the
foaming slag is the proper slag fluidity. During the
elaboration of stainless steels in the EAF the SiO2 and
Cr2O3 are the main oxidation products. SiO2 is a flux
component, while Cr2O3 is a component of the lining and
it increases the rigidity of the slag. For this reason, the
control of the Cr/Cr2O3 equilibrium in the melt is very
important for achieving a foaming slag. The solubility of
CrOx in these slags is small (< 5 % CrOx), and once the
solubility is reached, the oxide CrOx is precipitated.
If a moderate amount of secondary particles is
available in the slag, the viscosity and thus the foaming
index increases. However, at higher chromium contents,
large solid particles destroy the bubble network and
become detrimental to the slag’s foamability. The data
from the investigation shows that the Cr2O3 content
should be kept lower than 16 % for the slag composition,
temperature and oxygen potential used in the research15.
The next parameter affecting the slag foaming is the
temporal addition of additives during the elaboration of
the steel in the furnace, since the composition of the slag
is very important for the control of chromium and the
achieving of the optimal degree of foaming. The
formation of a fluid slag during the melting depends on
the oxidation-reduction kinetics in the steel bath. An
early fluid slag is achieved with the addition of
wollastonite (CaSiO3). The addition of MgO in the
system CaO-SiO2 increases the fluidity and decreases the
melting point of the slags up to a content of 15 % MgO,
above this level periclase or spinel are precipitated.
Considering the above analysis, it is understandable
that achieving foaming in EAF slags rich in Cr2O3 is a
difficult task, especially with regard to the formation of a
sufficient quantity of gas bubbles. The investigation of
the foaming and the reduction confirm that the area of
stable foaming for EAF stainless slags is small19. This
area and the area of the formation of gases are schema-
tically shown in Figure 16.
Area I: A low basicity slag with a high viscosity
gives a good foamability, according to Equation (18).
Unfortunately, the gas generation is very small. The
overall result of this is a poor foaming without any
industrial interest.
Area II: This area shows the composition of the slag
after a poorly controlled oxidation. The slag has a high
viscosity and it has a large content of Cr2O3 solid
particles. The great viscosity and the share of the solid
phase in the slag decrease the kinetics of the formation
of gases.
Area III: This part of the slag system is optimal. The
basicity is high and the conditions for the formation of
the CO gas are optimal. With particles of solid CaO and
Cr2O3 the viscosity is increased and the index of foaming
is increased, also.
The reduction of chromium oxide in the foaming
occurs according to the following reactions18:
(CrO1,5) + ¾ 
Si = 
Cr + ¾ (SiO2) (22)
(CrO) + ½ 
Si = 
Cr + 1/2 (SiO2) (23)
(CrO) + xC = xCO + 
Cr (24)
4 CONCLUDING REMARKS
During the elaboration of stainless steels in an EAF
the loss of chromium is reduced with a decrease of the
oxidation during the melting of the charge, the control of
the blowing in of the oxygen by partial oxidation of the
melt, with the slag composition and with efficient slag
reduction.
The share of the oxidation of chromium depends on
the composition of the charge, which may contain
chromium oxides. During the melting, chromium is
oxidised with other oxides present and with oxygen in
the air. The dependence of the oxidation enthalpy on
temperature shows that silicon and aluminium prevent
the oxidation of chromium and that the sequence of
oxidation depends on the activity of the element.
The 0.3 % content of silicon and the addition of
silicon decrease significantly the oxidation of chromium
during the melting because silicon is oxidised first and,
in this way, the content of chromium in the slag is
reduced. The oxidation of silicon is efficient during the
melting and the heating of the charge and during the
discharge of steel and slag from the furnace. The
decrease of the oxidation of chromium with the addition
B. ARH, F. TEHOVNIK: THE OXIDATION AND REDUCTION OF CHROMIUM ...
210 Materiali in tehnologije / Materials and technology 41 (2007) 5, 203–211
Figure 16: Areas of the tri-phase diagram with different activities of
foaming/reduction17
Slika 16: Podro~ja trifaznega diagrama s prikazom razli~nih lastnosti
penjenja/redukcije17
of FeSi requires the addition of lime to maintain the right
slag basicity (CaO/SiO2). These additions increase the
volume of the slag.
During the refining period the reducing conditions
must be secured by controlling the C/Otot injection ratio
and the temperature of the blowing of oxygen. For this
reason it is recommended that the blowing in of oxygen
is carried out at a high temperature > 1600 °C, when the
oxidation of carbon is strong and the oxidation of
chromium is minimised.
After the charge was melted and the blowing in of
oxygen stopped, a significant content of chromium
oxides in the slag is obtained. The chromium is returned
in the bath with the reduction of slag during the heating
up and the raffination of the bath and during the
discharge from the EAF. The selection of the reductant
and the procedure of reduction are significant for
achieving a maximal return of chromium and a minimal
content of reductant in the melt.
Suitable reductants are ferrosilicon, aluminium,
calcium carbide and carbon. However, silicon and
carbon are used the most. A second possibility is the
addition of ferrosilicon with parallel blowing in of
carbon powder in the slag when CO and gas bubbles are
formed. Both are necessary to obtain a foaming slag.
However, the blowing in of carbon is efficient at a high
bath temperature. The alternative is the injection of
calcium carbide into the slag, when CO and CaO are
obtained as products. The advantage of calcium carbide
is that it is an efficient reductant for chromium oxide
even at lower bath temperatures.
Good mixing between the steel and the slag during
tapping provides excellent conditions for dissolving the
silicon to reduce the Cr2O3 in the slag. The chemical
composition and the basicity of the slags affect strongly
the reduction of chromium oxides. For furnace slag, the
lowering of chromium oxide depends also on the
lowering of the activity of SiO2 and Al2O3. For a higher
slag basicity the activity of SiO2 and Al2O3 is lower and
the reduction of chromium oxide greater. An basicity of
1.4 to 1.8 is the optimum range.
For the formation of the foaming slag an early
presence of a sufficient volume of fluid slag is necessary.
The final slag should be saturated with CaO and MgO
(5–10 %) and have a viscosity suitable for the formation
of the foaming process. The appropriate content of Si in
the steel bath is necessary for the control of the ratio
Cr/Cr2O3, since the solid chromium oxide increases the
viscosity of the slag and impairs the foaming.
Furthermore, the reactions should ensure the formation
of small gas bubbles. The injection of carbon into the
slag with a high potential of oxygen in the bath and a
high content of FeO generates the formation of CO
bubbles. For the stable foaming of stainless slags with a
low content of FeO it is necessary to maintain the
generation of gas bubbles with the injection of carbon
and iron oxide in the slag.
5 REFERENCES
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halten von Schlaken aus der Produktion chromhaltiger Stähle im
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stainless steelmaking, Electric furnace conference proceedings, 1997
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Prejem rokopisa – received: 2007-04-13; sprejem za objavo – accepted for publication: 2007-07-10
The theoretical basis for a new computational method is given for the topology of the meniscus trajectories during continuous
steel casting. The method is based on the solution of the meniscus equation in ℜ3. Here the topology is treated only in the sense
of the categorization of trajectories in an orientation space. This suggests a new type of efficient self-adaptive scheme suitable
for the solution of the shape of the meniscus. In the present work a new approach is used to overcome previously unknown
pathological, non-physical predictions in various constitutive models derived using closure approximations. The generalized
meniscus equation as well as its stability is solved. Here it is shown, for the first time, that the cyclic change of the shape of the
meniscus depends on the coordinates, while up to now the cyclic change of the meniscus was presented only as a function of the
time over the expression of the mould velocity.
Key words and phrases: topology of trajectories, shape of the meniscus, meniscus equation, generalized meniscus equation,
meniscus stability.
Dana je teoreti~na podlaga za nov izra~un topologije trajektorij meniskusa pri neprekinjenem litju jekla. Podlaga metode je
re{itev ena~be meniskusa ℜ3. Topologija je obravnavana le v smislu kategorizacije trajektorij v nekem orientacijskem prostoru.
To navaja na novo shemo, ki se sama primerno prilagaja za re{itev oblike meniskusa. V tem delu je uporabljen nov na~in, da bi
se obvladalo preje neznane patolo{ke, nefizikalne napovedi v razli~nih konstitutivnih modelih, ki so bili razviti z uporabo
kon~nih aproksimacij. Re{eni sta splo{na ena~ba meniskusa in njena stabilnost. Prvi~ je prikazano, da je cikli~na sprememba
oblike meniskusa odvisna od koordinat, medtem ko se je dosedaj cikli~ne spremembe meniskusa prikazalo samo kot funkcijo
razmerja ~as proti hitrosti kokile.
Klju~ne besede in stavki: topologija trajektorij, oblika meniskusa, ena~ba meniskusa, splo{na ena~ba meniskusa, stabilnost
meniskusa
1 INTRODUCTION
The study of the changes of the meniscus during
continuous steel casting is neither an easy nor a simple
task. It is, in fact, very complicated and requires a
serious approach and hard work, because the meniscus’s
appearance depends on many factors of the continuous
steel casting process. Generally speaking, this type of
multidisciplinary research looks for a sufficient
knowledge of steel metallurgy as well as the highest
level of mathematics.
Many efforts have been spent to describe the shape of
the meniscus during continuous steel casting. For
instance, in 1 and 26 the authors considered the shape of
the meniscus as a linear function and developed a model
based on the Navier-Stokes equation for a hydrodynamic
fluid. In 12, by virtue of complex functions, the
movement of molten powder between the strand and
mould wall is presented as a Newtonian fluid flowing
between two parallel plates by neglecting the thermal
contact resistance between the solidifying metal and the
mould 11. Particular cases of this complex model appear
in the results given in 13,18. In later research works 14,16 the
meniscus is shaped with an exponential function by
virtue of the meniscus’s dimensions 17.
In 24,25 the authors developed a dimensionless model
for the meniscus, introducing Reynolds’ lubrication
theory. A model closely related to the free coating
problem 27, is solved numerically and is compared with
the published data.
On other hand, in the simulation model 2 a fixed
shape of the meniscus is used to calculate the fluid flow
and heat transfer. The authors in 5 account for the
interdependence of the shape of the flux gape and the
fluid flow therein, but still require some parameters to be
selected rather arbitrarily, if impossible, to determine the
experimental measurements.
Research in 6,7 showed that the movement of molten
powder in the flux space may be determined by a
pseudo-transient analytical solution of the Navier-
Stokes equation. The validity of this solution is verified
using an explicit finite-difference discretization method
and the MATLAB software package. The simulation and
behavior of interfacial mould slag layers in the
continuous casting of steel are investigated in 8.
In 28,29,30 the authors modified the model for lubri-
cation on the meniscus, given in 14,16, with the difference
between steel and flux density and extended it with the
heat-transfer phenomena. They do not use the natural
logarithm with base e for the description of the expo-
Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 213
UDK 621.74.047:519.68 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)213(2007)
nential shape of the meniscus, and use instead the decade
logarithm with base 10, without considering the correla-
tion ln x ≈ 2.303 lg x.
The newest research 4,10 is directed to cold model
experiment on the infiltration of mould flux during the
continuous casting of steel, neglecting the mould
oscillations and the infiltration phenomena of molten
powder derived from an analysis using the Reynolds
equation.
Generally speaking, up to now in the literature the
meniscus changes during continuous steel casting were
approximately treated with one-dimensional mathema-
tical models by virtue of some real function as a fixed
shape. The treatment was adopted because the meniscus
equation as a function of several variables was not
known. In the present work, the introduced meniscus
equation as a quasicyclic real function equation, i.e., its
solution, can be used for all the possible cases of
meniscus changes occurring cyclically during the mould
cycle. In this way the mathematical description of the
shape of the meniscus is much better, because the shape
of the meniscus is closer to its own real shape.
Up to now in relevant references the form of the
meniscus was presented as a cyclical change inde-
pendent of time over the mould speed only, while, in this
article, it is shown that the change of the form of the
meniscus depends on other coordinates too.
The present work gives a new approach to meniscus
vicinity in a sufficiently sophisticated way, which is
more complete than previous treatments. With the goal
to shed new light on this topic, with this article a new
shape of the meniscus is introduced by virtue of the
solution of the meniscus equation. With the intention to
better understand this approach, emphasis is given to the
engineering experience and the theoretical knowledge of
mathematical modeling of the continuous steel casting
process.
2 PRELIMINARIES
Let A = aij
n×n be a real matrix. Suppose that by ele-
mentary transformations the matrix A is transformed into
A = P1DP2, where P1 and P2 are regular matrices and D is
a diagonal matrix with diagonal entries 0 and 1, such that
the number of units is equal to the rank of the matrix A.
The matrix B = P2–1DP1–1 satisfies the equality ABA = A.
This means that the matrix equation AXA = A has at least
one solution for X.
If A satisfies the identity
Ar + k1A
r–1 + · · · + kr–1A = O
where kr–1 ≠ 0 and O is the zero n×n matrix, then the
matrix
X = – (Ar–2 + k1A
r–3 + · · · + kr–2I)/kr–1
where I is the unit n×n matrix, is also a solution of the
equation AXA = A.
Now we recall the following theorem proven in 20.
Theorem 2.1. If B satisfies the condition ABA = A,
then
1° AX = O ⇔ X = (I – BA)Q
(X and Q are n × m matrices),
2° XA = O ⇔ X = Q(I – AB)
(X and Q are m × n matrices),
3° AXA = A ⇔ X = B + Q – BAQAB
(X and Q are n × n matrices),
4° AX = A ⇔ X = I + (I – BA)Q,
5° XA = A ⇔ X = I + Q(I – AB).
Throughout this paper, ℜ is a finite-dimensional real
vector space. Vectors from ℜ will be denoted by Xi =
(x1i,x2i,…,xni)T, and also we denote with O = (0,0,…,0)T
the zero vector in ℜ. Let ⊗ denote the exterior product in
ℜ and let k (1 ≤ k ≤ n) be an integer. With respect to the
canonical basis in the k-th exterior product space ⊗k ℜ,
the k-th additive compound matrix Ak
 of A is a linear
operator on ⊗k ℜ whose definition on a decomposable
element x1 ⊗ · · · ⊗ xk is
Ak
(x1⊗ · · · ⊗xk) = x
i
k
1
1=
∑ ⊗ · · · ⊗Axi⊗ · · · ⊗xk. (2.1)
For any integer i = 1, 2,…, n!/k!(n – k)!, let (i) =
(i1,…,ik) be the i-th member in the lexicographic ordering
of integer k-tuples such that 1 ≤ i1 < · · · < ik ≤ n. Then
the (i, j)-th entry of the matrix Ak
 = qij
 is
qij = ai1,i1 + · · · + aik,ik if (i) = (j)
qij = (–1)
m+sajm,is (2.2)
if exactly one entry is of (i) does not occur in (j) and jm
does not occur in (i),
qij = 0
if (i) differs from (j) in two or more entries.
As special cases, we have A1
 = A and An
 = trA 9.
Let σ(A) = λi, 1 ≤ i ≤ n
 be the spectrum of A. Then
the spectrum of Ak
 is σ(Ak
) = λi1 + · · · + λik , 1 ≤ i1 < ·
· · < ik ≤ n
.
Let · denote a vector norm in ℜn. The LozinskiÇ
measure µ on ℜn with respect to  is defined by
µ(A) = lim
ρ→ +0
(I + ρA – 1)/ρ (2.3)
The LozinskiÇ measures of A = aij
n×n with respect to
the three common norms
x

= supi xi
x1 = Σi xi
x2 = (Σi xi
2)1/2
are
µ

(A) = supi (aii + Σk,k≠i aik)
µ1(A) = supk (akk + Σi,i≠k aik) (2.4)
µ2(A) = stab(A + A
T)/2

where
stab(A) = maxλ, λ∈σ(A)

ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
214 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
is the stability modulus of the matrix A, and AT denotes
the transpose of A 3, p.41.
Definition 2. 2. A stable system is that system in
which after the transitive action appearance a constant
position is achieved 15, p.38.
3 TOPOLOGY OF THE SOLUTIONS OF THE
MENISCUS EQUATION
In this section we will give a complete analysis of the
meniscus equation represented by a quasicyclic real
functional equation for all possible cases. For that
purpose we will use techniques for the solution given in
19,21,23.
Let us consider now the equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) + a3f(x3,x1,x2) =
= α1f(x1,x1,x2) + α2f(x2,x2,x3) + α3f(x3,x3,x1) (3.1)
f: ℜ3 → ℜ, where ai, αi (1 ≤ i ≤ 3) are real constants.
For equation (3.1) we suppose that a1 + a2 + a3 > 0.
If we permute cyclically the variables in the equations
(3.1), we obtain
a1f(x2,x3,x1) + a2f(x3,x1,x2) + a3f(x1,x2,x3) =
= α1f(x2,x2,x3) + α2f(x3,x3,x1) + α3f(x1,x1,x2) (3.2)
a1f(x3,x1,x2) + a2f(x1,x2,x3) + a3f(x2,x3,x1) =
= α1f(x3,x3,x1) + α2f(x1,x1,x2) + α3f(x2,x2,x3) (3.3)
The determinant for the system of the equations
(3.1), (3.2) and (3.3) is
∆1
1 2 3
2
1
=
a
a a
a a a
3 1
2 3
a a
a
Let us note the identity
∆ = (a1 + a2 + a3)
(a1 – a2)
2 + (a2 – a3)
2 +
+ (a3 – a1)
2
/2 (3.4)
First we consider the case
1° Let α1 = α2 = α3 = 0. Now the system (3.1), (3.2)
and (3.3) takes the form
a1f(x1,x2,x3) + a2f(x2,x3,x1) + a3f(x3,x1,x2) = 0
a3f(x1,x2,x3) + a1f(x2,x3,x1) + a2f(x3,x1,x2) = 0 (3.5)
a2f(x1,x2,x3) + a3f(x2,x3,x1) + a1f(x3,x1,x2) = 0
If ∆ ≠ 0, then the system (3.5) implies f(x1,x2,x3) = 0.
Now let ∆ = 0. According to (3.4), this is possible if
the real constants a1, a2, a3 satisfy either a1 = a2 = a3 or a1
+ a2 + a3 = 0.
First, let us suppose that a1 = a2 = a3 (≠ 0). Then
system (3.5) is equivalent to the equation
f(x1,x2,x3) + f(x2,x3,x1) + f(x3,x1,x2) = 0 (3.6)
whose general solution according to 22 is
f(x1,x2,x3) = F(x1,x2,x3) – F(x2,x3,x1) (3.7)
where F: ℜ3 → ℜ, is an arbitrary function.
Now let us suppose that ∆ = 0, although the real
constants are not all equal. Then necessarily a1 + a2 + a3
= 0 and we can suppose, without any loss of generality,
that a1 ≠ a2. In this case we set a3 = – a1 – a2 and system
(3.5) can be written in the form
a1
f(x1,x2,x3) – f(x2,x3,x1) = a2
f(x3,x1,x2) – f(x1,x2,x3)
a1
f(x2,x3,x1) – f(x3,x1,x2) = a2
f(x1,x2,x3) – f(x2,x3,x1)
a1
f(x3,x1,x2) – f(x1,x2,x3) = a2
f(x2,x3,x1) – f(x3,x1,x2)
From this we derive easily
(a1
3 – a2
3)
f(x1,x2,x3) – f(x2,x3,x1) = 0
With the assumption that a1 ≠ a2 and they have real
values, equation (3.4) reduces to
f(x1,x2,x3) – f(x2,x3,x1) = 0 (3.8)
According to 20, the general solution of the above
functional equation is
f(x1,x2,x3) = F(x1,x2,x3) + F(x2,x3,x1) + F(x3,x1,x2) (3.9)
where F: ℜ3 → ℜ, is an arbitrary function.
The above results concerning the cyclic functional
equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) + a3f(x3,x1,x2) = 0
can be derived from those in 20, where a1, a2, a3 are real
constants.
From now we suppose that α1 + α2 + α3 > 0. We
can distinguish the following two cases:
2° Let ∆ ≠ 0, from (3.1), (3.2) and (3.3) we obtain
⏐α1F(x1,x2) + α2F(x2,x3) + α3F(x3,x1) a2 a3⏐
f(x1,x2,x3) =⏐α1F(x2,x3) + α2F(x3,x1) + α3F(x1,x2) a1 a2⏐
⏐α1F(x3,x1) + α2F(x1,x2) + α3F(x2,x3) a3 a1⏐
where F: ℜ2 → ℜ is defined by F(u,v) = f(u,u,v)/∆.
If we introduce the notations
∆1
1 2 3
2
1
=
α α α
a a
a a a
3 1
2 3
a ∆ 2
1 2 3
1
3
=
α α α
a a
a a a
2 3
1 2
a ∆ 3
1 2 3
3
2
=
α α α
a a
a a a
1 2
3 1
a
then we can write
f(x1,x2,x3) = ∆1F(x1,x2) + ∆2F(x2,x3) + ∆3F(x3,x1) (3.10)
For (3.10) to be a solution of the functional equation
(3.1), the following condition must be satisfied:
α1
(∆ – ∆2)F(x1,x2) – ∆3F(x2,x1) – ∆1F(x1,x1) +
+ α2
(∆ – ∆2)F(x2,x3) – ∆3F(x3,x2) – ∆1F(x2,x2) +
+ α3
(∆ – ∆2)F(x3,x1) – ∆3F(x1,x3) – ∆1F(x3,x3) = 0
(3.11)
By a cyclic permutation of the variables x1, x2, x3 in
(3.11) we obtain two new equations. The system of these
three equations has a nontrivial solution with respect to
(∆ – ∆2)F(xi,xi+1) – ∆3F(xi+1,xi) – ∆1F(xi,xi), i = 1, 2, 3
(with the convention x4 ≡ x1) if the following condition
is satisfied
α α α
α α α
α α α
1 2 3
1
2
02 3
3 1
= (3.12)
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 215
By virtue of an equality of the type of (3.4) this is
true if
α1 + α2 + α3 = 0 or α1 = α2 = α3
First, we will consider the case
α1 + α2 + α3 = 0 (3.13)
Since a1 + a2 + a3 ≠ 0 (because of the assumption ∆
≠ 0 and (3.4)), by putting into equation (3.1) x1 = x2 = x3
we derive F(x1,x2) = 0. By using the last equality, the
equation (3.11) for x3 = x1 becomes
α1(∆ – ∆2) – α2∆3
F(x1,x2) –
– α1∆3 – α2(∆– ∆2)
F(x2,x1) = 0 (3.14)
If we change the places of x1 and x2, the equation
(3.14) is transformed into
– α1∆3 – α2(∆ – ∆2)
F(x1,x2) +
+ α1(∆ – ∆2) – α2∆3
F(x2,x1) = 0 (3.15)
Let
(α1
2 – α2
2)(∆ – ∆2)
2 – ∆3
2

 ≠ 0
then from (3.14) and (3.15) it follows that F(x1,x2) = 0
and then from (3.10) f(x1,x2,x3) = 0.
The condition
(α1
2 – α2
2)(∆ – ∆2)
2 – ∆3
2

 = 0 (3.16)
implies (∆ – ∆2)2 – ∆32 = 0. Let us suppose that the last
equality is not true. Now, if we set x3 = x2 into (3.11),
we obtain
(α1
2 – α3
2)(∆ – ∆2)
2 – ∆3
2

 = 0
The last equality, with (3.16), gives α12 = α22 = α32
which, by virtue of the assumption (3.13), yields α1 = α2
= α3 = 0, and this contradicts the hypothesis α1 + α2 +
α3 > 0.
Thus, we have ∆ – ∆2 = ± ∆3. For the case ∆ – ∆2 =
∆3 (≠ 0), equation (3.14) yields
(α1 – α2)∆3F(x1,x2) – ∆3F(x2,x1)
 = 0
so that, for α1 ≠ α2, we have
F(x1,x2) = G(x1,x2) + G(x2,x1) (3.17)
where G is an arbitrary function ℜ2 → ℜ such that
G(x1,x1) ≡ 0
If α1 = α2, then necessarily we must have α1 ≠ α3,
(because otherwise we would have α1 = α2 = α3 = 0) and
by a procedure analogous to the one above we obtain
(3.17).
Let ∆ – ∆2 = – ∆3 (≠ 0), from (3.14) we obtain
(α1 + α2)∆3F(x1,x2) + F(x2,x1)
 = 0 (3.18)
For α1 + α2 ≠ 0, the general solution of equation
(3.18) is given by
F(x1,x2) = G(x2,x1) – G(x2,x1), G: ℜ2 → ℜ (3.19)
If α1 + α2 = 0, then from (3.13) we deduce α3 = 0,
and then α1 + α3 = α1 ≠ 0 and we obtain (3.19) by an
analogous procedure.
The condition (3.11), for the case ∆3 = ∆ – ∆2 = 0, is
satisfied for every function F(x1,x2) with the property
F(x1,x1) ≡ 0.
If (∆ – ∆2)2 ≠ ∆32, then, as was mentioned above,
equations (3.14) and (3.15) have a trivial solution as a
general solution. According to (3.10) we obtain
f(x1,x2,x3) ≡ 0
Now we suppose that (3.12) is satisfied but (3.13) is
not. This means that
α1 = α2 = α3 ≠ 0 (3.20)
It immediately follows that
∆1 = ∆2 = ∆3 (≠ 0)
The quasicyclic equation (3.11) implies
(∆ – ∆1)F(x1,x2) – ∆1F(x2,x1) – ∆1F(x1,x1) =
= ∆1P(x1) – ∆1P(x2) (3.21)
where P is an arbitrary function ℜ → ℜ
For α1 = α2 = α3 and x1 = x2 = x3 equation (3.1)
becomes
(a1 + a2 + a3 – 3α1)F(x1,x1) = 0
Let a1 + a2 + a3 = 3α1, then 3∆1 = ∆ and the equality
(3.21) takes the form
2F(x1,x2) – F(x2,x1) = P(x1) – P(x2) + R(x1)
where R(x1) = F(x1,x1).
By a permutation of the variables x1 and x2 it follows
that
– F(x1,x2) + F(x2,x1) = P(x2) – P(x1) + R(x2)
From the last two equalities we obtain
F(x1,x2) = P(x1) – P(x2) + 2R(x1) + R(x2)
/3
By using the last equality, from (3.10) it follows that
f(x1,x2,x3) = Q(x1) + Q(x2) + Q(x3) (3.22)
where Q(x1) = ∆1R(x1).
Let a1 + a2 + a3 ≠ 3α1, then F(x1,x1) ≡ 0 and the for-
mula (2.21) yields
(∆ – ∆1)F(x1,x2) – ∆1F(x2,x1) = ∆1P(x1) – ∆1P(x2) (3.23)
From the last equality, with a permutation of the
variables we obtain
– ∆1F(x1,x2) + (∆ – ∆1)F(x2,x1) = ∆1P(x2) – ∆1P(x1)
The determinant of the system consisting of the last
two equations is ∆(∆ – 2∆1). If ∆ ≠ 2∆1, the solution of
the last two equations is
F(x1,x2) = ∆1P(x1) – P(x2)
/∆
Then the equality (3.10) gives
f(x1,x2,x3) ≡ 0
Let ∆ = 2∆1, then from (3.23) we obtain
F(x1,x2) – P(x1) = F(x2,x1) – P(x2)
The general solution for the last equation is
F(x1,x2) = P(x1) + G(x1,x2) + G(x2,x1) (3.24)
where P: ℜ → ℜ and G: ℜ2 → ℜ are arbitrary functions
such that
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
216 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
G(x1,x2) = – P(x1)/2
According to the last relation, the equality (3.24)
takes the form
F(x1,x2) = G(x1,x2) + G(x2,x1) – 2G(x1,x1)
and the equality (3.10) becomes
f(x1,x2,x3) = G(x1,x2) + G(x2,x1) – 2G(x1,x2) + G(x2,x3) +
+ G(x3,x2) – 2G(x2,x2) + G(x3,x1) + G(x1,x3) – 2G(x3,x3)
(3.25)
where we have replaced G(x1,x2)∆1 by G(x1,x2).
For
α α α
α α α
α α α
1 2 3
1
2
02 3
3 1
≠
from (3.10) we obtain
(∆ – ∆2)F(x1,x2) – ∆3F(x2,x1) – ∆1F(x1,x1) = 0 (3.26)
First we suppose that ∆ – ∆1 – ∆2 – ∆3 ≠ 0. In this
case, with the substitution x2 = x1, equation (3.26)
reduces to F(x1,x1) ≡ 0. On the basis of the last equality,
the equation (3.26) becomes
(∆ – ∆2)F(x1,x2) – ∆3F(x2,x1) = 0
From the permutation of the variables x1 and x2, from
the above equation it follows that
– ∆3F(x1,x2) + (∆ – ∆2)F(x2,x1) = 0
The system of the last two equations has a nontrivial
solution if and only if the following condition (∆ – ∆2)2
= ∆32 is satisfied.
Let ∆ – ∆2 = ∆3 (≠ 0), then we obtain
F(x1,x2) = G(x1,x2) + G(x2,x1)
where G satisfies G(x1,x1) ≡ 0.
For the case ∆ – ∆2 = – ∆3 (≠ 0) the general solution
is
F(x1,x2) = G(x1,x2) – G(x2,x1)
For ∆ – ∆2 = ∆3 = 0, the unique condition that must
be satisfied by the function F is F(x1,x1) ≡ 0.
The condition (∆ – ∆2)2 ≠ ∆32 gives F(x1,x2) ≡ 0
Next we will pass on to the case ∆ – ∆1 – ∆2 – ∆3 = 0.
Now equation (3.26) can be written as
(∆1 + ∆3)F(x1,x2) – ∆3F(x2,x1) = ∆1R(x1)
where R(x1) = F(x1,x1). By a permutation of the varia-
bles x1 and x2 we obtain
– ∆3F(x1,x2) + (∆1 + ∆3)F(x2,x1) = ∆1R(x2)
The determinant of this system is (∆1 + 2∆3)∆1. If it
is not zero, then
F(x1,x2) = (∆1 + ∆3)R(x1) + ∆3R(x2)
/(∆1 + 2∆3)
From (3.10) it follows that
f(x1,x2,x3) = (∆1
2 + ∆1∆3 + ∆3
2)Q(x1) + (∆1∆2 + ∆1∆3 +
+ ∆2∆3)Q(x2) + ∆3∆Q(x3)
where
Q(x1) = R(x1)/(∆1 + 2∆3)
Let ∆1 = 0, ∆3 ≠ 0. Then
F(x1,x2) = F(x2,x1)
Thus
F(x1,x2) = G(x1,x2) + G(x2,x1)
where G is an arbitrary function ℜ2 → ℜ, and f(x1,x2,x3)
is given by the formula (3.10).
Now we suppose that ∆1 = – 2∆3 ≠ 0. Then
F(x1,x2) + F(x2,x1) = 2R(x1) (3.27)
By a permutation of the variables x1 and x2 we obtain
F(x2,x1) + F(x1,x2) = 2R(x2) (3.28)
From (3.27) and (3.28) we get
R(x1) = R(x2) = c
Thus (3.27) takes on the form
F(x1,x2) – c
 + F(x2,x1) – c
 = 0
which implies that
F(x1,x2) = G(x1,x2) – G(x2,x1) + c
where G: ℜ2 → ℜ is an arbitrary function and c is an
arbitrary real constant.
Now from (2.10) we find
f(x1,x2,x3) = – 2∆3G(x1,x2) – G(x2,x1)
 +
+ (∆ + ∆3)G(x2,x3) – G(x3,x2)
 +
+ ∆3G(x3,x1) – G(x1,x3)
 + c
where c is (another) arbitrary real constant.
In the case ∆1 = ∆3 = 0 equation (3.26) is satisfied for
every function F: ℜ2 → ℜ.
Now we will use the following result.
Lemma 3. 1. Let ∆ ≠ 0. Then the system
∆1 = 0, ∆2 – ∆ = 0, ∆3 = 0 (3.29)
implies α1 = a3, α2 = a1, α3 = a2.
Proof. The system (3.29) can be written in the form
A11(α1 – a3) + A12(α2 – a1) + A13(α3 – a2) = 0
A21(α1 – a3) + A22(α2 – a1) + A23(α3 – a2) = 0 (3.30)
A31(α1 – a3) + A32(α2 – a1) + A33(α3 – a2) = 0
where Aij is the cofactor of the element aij (1 ≤ i, j ≤ 3)
of the determinant ∆. The system (3.30) is a homoge-
neous linear system with respect to α1 – a3, α2 – a1, α3 –
a2. Its determinant is ∆2 ≠ 0, so that it has only the zero
solution.
Thus, the equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) + a3f(x3,x1,x2) =
= a3f(x1,x1,x2) + a1f(x2,x2,x3) + a2f(x3,x3,x1)
has the general solution f(x1,x2,x3) = F(x2,x3).
3° Let ∆ = 0. Then from (3.4) it follows that
a1 + a2 + a3 = 0 or a1 = a2 = a3 ≠ 0.
First we will consider the case a1 = a2 = a3. From
(3.1) and (3.2) we obtain
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Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 217
(α1 – α3)F(x1,x2) + (α2 – α1)F(x2,x3) +
+ (α3 – α2)F(x3,x1) = 0 (3.31)
with the notation f(x1,x1,x2) = F(x1,x2). If α1 = α2 = α3,
then the condition (3.31) is satisfied for every function
F. For the case α1 = α2 = α3 (≠ 0), equation (3.1) takes
the form
a1f(x1,x2,x3) – α1f(x1,x1,x2) + a1f(x2,x3,x1) –
– α1f(x2,x2,x3) + a1f(x3,x1,x2) – α1f(x3,x3,x1) = 0 (3.32)
This quasicyclic equation has the general solution
f(x1,x2,x3) = (α1/a1)F(x1,x2) + U(x1,x2,x3) – U(x2,x3,x1)
(3.33)
with the notation f(x1,x1,x2) = F(x1,x2).
By substitution of (3.33) into (3.32) we obtain
F(x1,x2) – (α1/a1)F(x1,x1) – U(x1,x2,x3) + U(x1,x2,x1)
+ F(x2,x3) – (α1/a1)F(x2,x2) – U(x2,x2,x3) + U(x2,x3,x2)
+ F(x3,x1) – (α1/a1)F(x3,x3) – U(x3,x3,x1) + U(x3,x1,x3) =0
This quasicyclic equation has the general solution
F(x1,x2) = (α1/a1)F(x1,x1) + U(x1,x1,x2) – U(x1,x2,x1) +
+ R(x1) – R(x2)
where R is an arbitrary function ℜ → ℜ.
By using the last equality, for α1 = a1, the equality
(3.33) becomes
f(x1,x1,x2) = U(x1,x2,x3) – U(x2,x3,x1) + U(x1,x1,x2) –
– U(x1,x2,x1) + S(x1) – R(x2) (3.34)
where S: ℜ → ℜ is such that F(x1,x1) = S(x1) – R(x1).
For α1 ≠ a1 it follows from (3.1) that F(x1,x1) = 0.
According to the last identity, the equality (3.33) is
transformed into
f(x1,x2,x3) = U(x1,x2,x3) – U(x2,x3,x1) +
+ (α1/a1)U(x1,x1,x2) – U(x1,x2,x1)
 + R(x1) – R(x2)
Now we will suppose that the parameters αi (1 ≤ i ≤
3) are not all equal. Let α1 ≠ α3. According to the
equality (3.31) for x3 = a (a real constant) we obtain
F(x1,x2) = K(x1) + H(x1) (3.35)
where we used the notations
K(x1) = (α2 – α3)/(α1 – α3)
F(a,x1), H(x2) =
= (α1 – α2)/(α1 – α3)
F(x2,a)
If we substitute F(x1,x2) given by the expression
(3.35) into (3.31), and if we set x1 = u, x2 = x3 = b (a real
constant) and if, on the other hand, we set x1 = x3 = b, x2
= u, we obtain respectively
(α1 – α3)K(u) – K(b)
 + (α3 – α2)H(u) – H(b)
 = 0
(3.36)
(α2 – α1)K(u) – K(b)
 + (α1 – α3)H(u) – H(b)
 = 0
(3.37)
The determinant of this system is
α −α α α
α −α α α
1 3 3 2
1 3
−
−2 1
= (α1 – α2)
2 + (α2 – α3)
2 + (α3 – α1)
2
/2
According to our assumption its value is not 0, then
from (3.36) and (3.37) we find K(u) = K(b) and H(u) =
H(b), hence
F(x1,x2) = m (a real constant) (3.38)
Now the equation (3. 1) becomes
f(x1,x2,x3) – n + f(x2,x3,x1) – n + f(x3,x1,x2) – n = 0 (3.39)
where
n = (α1 + α2 + α3)m/3a1
The general solution of the cyclic functional equation
(3.39) is
f(x1,x2,x3) = p(x1,x2,x3) – p(x2,x3,x1) + n (3.40)
From (3.38) and (3.40) we find
m = F(x1,x2) = p(x1,x1,x2) – p(x1,x2,x1) + n
If we put into the last equality x2 = x1, then m = n.
This is possible if
α1 + α2 + α3 = 3a1 or n = 0
Moreover,
p(x1,x1,x2) – p(x1,x2,x1) = 0 (3.41)
Now we will use the following result.
Lemma 3. 2. Let f(x1,x2,x3) be a function of the form
f(x1,x2,x3) = p(x1,x2,x3) – p(x2,x3,x1)
such that p(x1,x1,x2) = 0. Then
f(x1,x2,x3) = U(x1,x2,x3) – U(x2,x1,x3) – U(x2,x3,x1) –
– U(x1,x3,x2) (3.42)
where U: ℜ3 → ℜ is an arbitrary function.
Proof. Let p(x1,x2,x3) satisfies equation (3.41). We are
looking for p(x1,x2,x3) in the form
p(x1,x2,x3) = k1q(x1,x2,x3) + k2q(x1,x3,x2) + k3q(x2,x1,x3) +
+ k4q(x2,x3,x1) + k5q(x3,x1,x2) + k6q(x3,x2,x1)
where q: ℜ3 → ℜ and ki (1 ≤ i ≤ 6) are real constants.
By a substitution into (3.41) we find
k5 = k1 – k2 + k3, k6 = k4 – k1 + k2
Thus
f(x1,x2,x3) = kq(x2,x1,x3) – q(x1,x2,x3)
 – q(x1,x3,x2) –
– q(x2,x3,x1)
 + ( – k)q(x3,x2,x1) – q(x3,x1,x2)

where k,  are real constants such that k = k3 – k2,  = k4
– k1
If we denote
U(x1,x2,x3) = q(x2,x3,x1) + kq(x3,x1,x2)
we obtain (3.42). Conversely, each function of the form
(2.42) satisfies f(x1,x1,x2) = 0 for arbitrary U: ℜ3 → ℜ.
Moreover, f(x1,x2,x3) satisfies
f(x1,x2,x3) + f(x2,x3,x1) + f(x3,x1,x2) = 0
We note that the representation (3.42) can be ob-
tained just by putting
p(x1,x2,x3) = U(x1,x2,x3) – U(x2,x1,x3)
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218 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
Thus the general solution of the equation (3.1) is in
this case given by
f(x1,x2,x3) = U(x1,x2,x3) – U(x2,x1,x3) –
– U(x2,x3,x1) + U(x1,x3,x2) + n (3.43)
where U is an arbitrary function ℜ3 → ℜ and n is an
arbitrary real constant, n = 0 if α1 + α2 + α3 ≠ 3a1.
Now we will consider the case that a1, a2, a3 are not
all equal. Thus we have
a1 + a2 + a3 = 0 (3.44)
Without any loss of generality, we can assume that a1
≠ a2. Equation (3.1) can be written as
a1 f(x1,x2,x3) – f(x3,x1,x2)
 – a2 f(x3,x1,x2) – f(x2,x3,x1)
 =
= α1F(x1,x2) + α2F(x2,x3) + α3F(x3,x1) (3.45)
where f(x1,x1,x2) = F(x1,x2). Also from (3.2) and (3.3) it
follows that
a1 f(x2,x3,x1) – f(x1,x2,x3)
 – a2 f(x1,x2,x3) – f(x1,x3,x2)
 =
= α1F(x2,x3) + α2F(x3,x1) + α3F(x1,x2) (3.46)
a1 f(x3,x1,x2) – f(x2,x3,x1)
 – a2 f(x2,x3,x1) – f(x1,x2,x3)
 =
= α1F(x3,x1) + α2F(x1,x2) + α3F(x2,x3) (3.47)
By adding (3.45), (3.46) and (3.47) we obtain
(α1 + α2 + α3)F(x1,x2) + F(x2,x3) + F(x3,x1)
 = 0
For α1 + α2 + α3 ≠ 0, the following condition must
be satisfied
F(x1,x2) + F(x2,x3) + F(x3,x1) = 0
This cyclic functional equation has the general solu-
tion
F(x1,x2) = P(x1) – P(x2) (3.48)
where P is an arbitrary function ℜ → ℜ.
From the equation (3.45), (3.46) and (3.47), if we
take into account (3.48), we get
f(x1,x2,x3) + 1/(a1
3 – a2
3)
a1
2α1P(x1) + α2P(x2) +
+ α3P(x3)
 + a2
2α3P(x1) + α1P(x2) + α2P(x3)
 +
+ a1a2α2P(x1) + α3P(x2) + α1P(x3)
 =
= f(x1,x2,x3) + 1/(a1
3 – a2
3)
a1
2α1P(x2) + α2P(x3) +
+ α3P(x1)
 + a2
2α3P(x2) + α1P(x3) + α2P(x1)
 +
+ a1a2α2P(x2) + α3P(x3) + α1P(x1)

The last equation has the general solution
f(x1,x2,x3) + 1/(a1
3 – a2
3)
a1
2α1P(x2) + α2P(x3) +
+ α3P(x1)
 + a2
2α3P(x2) + α1P(x3) + α2P(x1)
 +
+ a1a2α2P(x2) + α3P(x3) + α1P(x1)
 =
= p(x1,x2,x3) + p(x2,x3,x1) + p(x3,x1,x2) (3.49)
where p is an arbitrary function ℜ3 → ℜ.
By virtue of (3.48) f(x1,x2,x3) = P(x1) – P(x2), then
from (3.49) it follows that
P(x1) – P(x2) + 1/(a1
3 – a2
3)
a1
2(α1 + α3)P(x1) +
+ α2P(x2)
 + a2
2(α2 + α3)P(x1) + α1P(x2)
 + a1a2(α1 +
α2)P(x1) + α3P(x2)
 =
= p(x1,x1,x3) + p(x1,x2,x1) + p(x2,x1,x1) (3.50)
For x2 = x1 this equality takes the form
(α1 + α2 + α3)(a1
2 + a2
2 + a1a2)/(a1
3 – a2
3)
P(x1) =
= 3p(x1,x1,x1)
which implies
P(x1) = 3(a1 – a2)/(α1 + α2 + α3)
p(x1,x1,x1)
Now from (3.49) we find the general solution in the
form
f(x1,x2,x3) = p(x1,x2,x3) + p(x2,x3,x1) + p(x3,x1,x2) –
– 3/(a1
2 + a2
3 + a1a2)(α1 + α2 + α3)
a1
2α1p(x2,x2,x2) +
+ α2p(x3,x3,x3) + α3p(x1,x1,x1)
 + a2
2α1p(x3,x3,x3) +
+ α2p(x1,x1,x1) + α3p(x2,x2,x2)
 + a1a2α1p(x1,x1,x1) +
+ α2p(x2,x2,x2) + α3p(x3,x3,x3)
 (3.51)
where p: ℜ3 → ℜ must satisfy the following condition
derived from (3.50)
3(a1 – a2)/(α1 + α2 + α3)
p(x1,x1,x1) – p(x2,x2,x2)
 +
+ 3/(a1
2 + a2
3 + a1a2)(α1 + α2 + α3)
a1
2(α1 +
+ α2)p(x1,x1,x1) + α2p(x2,x2,x2)
 + a2
2(α2 + α3)p(x1,x1,x1)+
+ α1p(x2,x2,x2)
 + a1a2(α1 + α2)p(x1,x1,x1) +
+ α3p(x2,x2,x2)

 = p(x1,x1,x2) + p(x1,x2,x1) + p(x2,x1,x1)
(3.52)
It is easy to see that (3.52) is an equation of the form
p(x1,x1,x2) + p(x1,x2,x1) + p(x2,x1,x1) =
= (3 – γ)p(x1,x1,x1) + γp(x2,x2,x2) (3.53)
where the real constant γ is given by
γ = – 3(a1 – a2)/(α1 + α2 + α3) + 3(a1
2α2 + a2
2α1 +
+ a1a2α3)/(a1
2 + a2
2 + a1a2)(α1 + α2 + α3)
.
Lemma 3. 3. Let f(x1,x2,x3) be a function of the form
f(x1,x2,x3) = p(x1,x2,x3) + p(x2,x3,x1) + p(x3,x1,x2) (3.54)
such that f(x1,x2,x3) = 0. Then
f(x1,x2,x3) = U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) –
– U(x2,x1,x3) – U(x1,x3,x2) –U(x3,x2,x1) (3.55)
where U: ℜ3 → ℜ is an arbitrary function.
Proof. We are looking for a function of the form
p(x1,x2,x3) = k1q(x1,x2,x3) + k2q(x2,x3,x1) + k3q(x3,x1,x3) +
+ k4q(x2,x1,x3) + k5q(x1,x3,x2) + k6q(x3,x2,x1)
where ki (1 ≤ i ≤ 6) are real constants, satisfying
p(x1,x1,x2) + p(x1,x2,x1) + p(x2,x1,x1) = 0 (3.56)
for any function q: ℜ3 → ℜ. By a substitution into
(3.56) we find
k1 + k2 + k3 = k4 + k5 + k6
Thus
f(x1,x2,x3) = (k1 + k2 + k3)q(x1,x2,x3) + q(x2,x3,x1) +
+ q(x3,x1,x2)
 – (k1 + k2 + k3)q(x2,x1,x3) + q(x1,x3,x2) +
+ q(x3,x2,x1)

If we put
U(x1,x2,x3) = (k1 + k2 + k3) q(x1,x2,x3)
we obtain the representation (3.55).
We note that the representation (3.55) can be
obtained from (3.54) by putting
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 219
p(x1,x2,x3) = U(x1,x2,x3) – U(x2,x1,x3)
Let us suppose that p(x1,x2,x3) = S(x1) 
 0. The equa-
tion
p(x1,x1,x2) + p(x1,x2,x1) + p(x2,x1,x1) = (3 – γ)S(x1) + γS(x2)
has a no constant solution of the form
p(x1,x1,x2) = S(x1)
or, more generally,
p(x1,x2,x3) = m1S(x1) + m2S(x2) + (1 – m1 – m2)S(x3)
only if γ = 1. Indeed, we have
(γ – 1)S(x1) – S(x2)
 = 0
On the other hand, any S(x1) ≡ a, where a is a real
constant, satisfies the last equality.
Let us put
p(x1,x2,x3) = Û(x1,x2,x3) + S(x1)
Then Û (x1,x2,x3) satisfies an equation of the form
(3.56) and we have proved this result.
Corollary 3. 4. Let f(x1,x2,x3) be a function of form
(3.54) such that p(x1,x2,x3) satisfies (3.53). Then
f(x1,x2,x3) = U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) –
– U(x2,x1,x3) – U(x1,x3,x2) – U(x3,x1,x2) +
+ S(x1) + S(x2) + S(x3)
where U: ℜ3 → ℜ is an arbitrary function and S is an
arbitrary function ℜ → ℜ for γ = 1, S is equal to a con-
stant a ∈ℜ otherwise.
Thus from (3.51) we find that f(x1,x2,x3) is given by
(3.55) if γ ≠ 1
f(x1,x2,x3) = S(x1) + S(x2) + S(x3) – 3/(a1
2 + a2
2 +
+ a1a2)(α1 + α2 + α3)
a1
2α3S(x1) + α1S(x2) +
+ α2S(x3)
 + a2
2α2S(x1) + α3S(x2) + α1S(x3)
 +
+ a1a2α1S(x1) + α2S(x2) + α3S(x3)
 + U(x1,x2,x3) +
+ U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1).
Now we pass on the case α1 + α2 + α3 = 0. Then from
(3.45), (3.46) and (3.47) we obtain
f(x3,x1,x2) – 1/(a1
3 – a2
3)
a1
2α1F(x3,x1) – α2F(x2,x3)
+
+ a2
2α1F(x1,x2) – α2F(x3,x1)
 + a1a2α1F(x2,x3) –
– α2F(x1,x2)
 =
= f(x1,x2,x3) – 1/(a1
3 – a2
3)
a1
2 α1F(x1,x2) –
– α2F(x3,x1)
 + a2
2α1F(x2,x3) – α2F(x1,x2)
 +
+ a1a2α1F(x3,x1) – α2F(x2,x3)
 (3.57)
The general solution of equation (3.57) is given as
f(x1,x2,x3) = 1/(a1
3 – a2
3)
a1
2α1F(x1,x2) – α2F(x3,x1)
 +
+ a2
2α1F(x2,x3) – α2F(x1,x2)
 + a1a2α1F(x3,x1) –
– α2F(x2,x3)
 + q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2)
(3.58)
where q: ℜ3 → ℜ is an arbitrary function.
From the equality (3.58) we obtain
F(x1,x2) = 1/(a1
3 – a2
3)
a1
2α1F(x1,x1) – α2F(x2,x1)
 –
+ a2
2α1F(x1,x2) – α2F(x1,x1)
 + a1a2α1F(x2,x1) –
– α2F(x1,x2)
 + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
(3.59)
For x2 = x1 (3.59) yields
F(x1,x2) = (α1 – α2)/(a1 – a2)
F(x1,x1) + 3q(x1,x1,x1)
(3.60)
If α1 – α2 = a1 – a2, then q(x1,x1,x1) = 0 and F(x1,x1) =
P(x1), where P is an arbitrary function ℜ → ℜ. Now we
have
1 + a2(α1 – a1)/(a1
2 + a2
2 + a1a2)
F(x1,x2) +
+ a1(α1 – a1)/(a1
2 + a2
2 + a1a2)
F(x2,x1) =
= α1(a1 + a2) + a2
2
P(x1)/(a1
2 + a2
2 + a1a2) +
+ q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) (3.61)
First we consider the particular case, α1 = a1 (then α2
= a2, α3 = a3 = – (a1 + a2))
Now
F(x1,x2) = P(x1) + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
Thus, we find that the functional equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) – (a1 + a2)f(x3,x1,x2) =
= a1f(x1,x1,x2) + a2f(x2,x2,x3) – (a1 + a2)f(x3,x3,x1)
has the general solution
f(x1,x2,x3) = P(x1) + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) +
+ q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2)
where P: ℜ → ℜ is an arbitrary function and q: ℜ3 → ℜ
is an arbitrary function satisfying q(x1,x1,x1) ≡ 0.
Now we consider equation (3.61) in the general case.
With a permutation of the variables x1 and x2 we derive
the equation
a1(α1 – a1)/(a1
2 + a2
2 + a1a2)
F(x1,x2) +
+ 1 + a2(α1 – a1)/(a1
2 + a2
2 + a1a2)
F(x2,x1) =
= α1(a1 + a2) + a2
2
P(x2)/(a1
2 + a2
2 + a1a2) +
+ q(x2,x2,x1) + q(x2,x1,x2) + q(x1,x2,x2) (3.62)
The determinant of the system (3.61), (3.62) is
(a2
2 + a1α1 + a2α1)(2a1
2 + a2
2 – a1α1 + a2α1)/
/(a1
2 + a2
2 + a1a2)
2 (3.63)
If this expression is not 0, then the solution of this
system is
F(x1,x2) = (a1
2 + a2
2 + a2α1)P(x1) – a1(α1 – a1)P(x2)
/
/(2a1
2 + a2
2 – a1α1 + a2α1) + (a1
2 + a2
2 + a1a2)/
/(a2
2 + a1α1 + a2α1)(2a1
2 + a2
2 – a1α1 + a2α1)
 ×
×(a1
2 + a2
2 + a2α1)q(x1,x1,x2) + q(x1,x2,x1) +
+ q(x2,x1,x1)
 – a1(α1 – a1)q(x2,x2,x1) +
+ q(x2,x1,x2) + q(x1,x2,x2)

Now let us suppose that the expression (3.63) is 0.
First let
a2
2 + α 1(a1 + a2) = 0
If a1 + a2 = 0, then a1 = a2 = a3 = 0, which is contra-
diction. Thus
α1 = – a2
2/(a1 + a2), α2 = – a1
2/(a1 + a2)
Now (3.58) takes the form
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
220 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
f(x1,x2,x3) = a1F(x3,x1) + a2F(x2,x3)
/(a1 + a2) +
+ q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2)
while (3.61) becomes
a1/(a1 + a2)
F(x1,x2) – F(x2,x1)
 =
= q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) (3.64)
Equation (3.64) implies
q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) = 0
and
F(x1,x2) – F(x2,x1) = 0
i.e.,
q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) =
= U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1)
where U: ℜ3 → ℜ is an arbitrary function, and
F(x1,x2) = G(x1,x2) + G(x2,x1)
where G: ℜ2 → ℜ is an arbitrary function.
Thus the general solution of the functional equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) – (a1 + a2)f(x3,x1,x2) =
= – a2
2/(a1 + a2)
f(x1,x1,x2) – a1
2/(a1 + a2)
f(x2,x2,x3) –
– (a1
2 + a2
2)/(a1 + a2)
f(x3,x3,x1)
is given by the relation
f(x1,x2,x3) = a1G(x1,x3) + G(x3,x1) + a2G(x2,x3) +
+ G(x3,x2)
/(a1 + a2) + U(x1,x2,x3) + U(x2,x3,x1) +
+ U(x3,x1,x2) – U(x2,x1,x3) – U(x1,x3,x2) – U(x3,x2,x1)
Next we suppose that
2a1
2 + a2
2 – (a1 – a2)α1 = 0
Since a1 ≠ a2, we have
α1 = (2a1
2 + a2
2)/(a1 – a2), α2 = (a1
2 + 2a1a2)/(a1 – a2)
Now (3.58) takes the form
f(x1,x2,x3) = 2a1F(x1,x2) – a1F(x3,x11) – a2F(x2,x3)
/
/(a1 – a2) + q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2)
while (3.61) becomes
a1/(a1 – a2)
F(x1,x2) + F(x2,x1) – 2F(x1,x1)
 =
= q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) (3.65)
Equation (3.65) implies
q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) = 0
and
F(x1,x2) + F(x2,x1) – 2F(x1,x1) = 0
i.e.,
q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) = U(x1,x2,x3) +
+ U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1)
where U: ℜ3 → ℜ is an arbitrary function, and
F(x1,x2) = G(x1,x2) –G(x2,x1) + c
where G: ℜ2 → ℜ is an arbitrary function and c ∈ℜ is
an arbitrary constant.
Thus, the general solution of the functional equation
a1f(x1,x2,x3) + a2f(x2,x3,x1) – (a1 + a2)f(x3,x1,x2) =
= (2a1
2 + a2
2)/(a1 – a2)
 f(x1,x1,x2) +
+ (a1
2 + 2a1a2)/(a1 – a2)
 f(x2,x2,x3) –
– (3a1
2 + 2a1a2 – a2
2)/(a1 – a2)
 f(x3,x3,x1)
is given by the relation
f(x1,x2,x3) = 1/(a1 – a2)
2a1G(x1,x2) – G(x2,x1)
 +
+ a1G(x1,x3) – G(x3,x1)
 – a2G(x2,x3) – G(x3,x2)
 + c+
+ U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1)
Now let (α1 – α2)/(a1 – a2)
 = γ ≠ 1. Then from
(2.60) we find
F(x1,x1) = 3/(1 – γ)
 q(x1,x1,x1) (3.66)
Now we have
1 + a2(α1– a1γ)/(a1
2 + a2
2 + a1a2)
F(x1,x2) +
+ a1(α1 – a1γ)/(a1
2 + a2
2 + a1a2)
F(x2,x1) =
= 3α1(a1 + a2) + a2
2 γ
q(x1,x1,x1)/(a1
2 + a2
2 + a1a2) ×
× (1 – γ) + q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1) (3.67)
By a permutation of the variables x1 and x2 we derive
the equation
a1(α1– a1γ)/(a1
2 + a2
2 + a1a2)
F(x1,x2) +
+ 1 + a2(α1 – a1γ)/(a1
2 + a2
2 + a1a2)
F(x2,x1) =
= 3α1(a1 + a2) + a2
2 γ
q(x2,x2,x2)/(a1
2 + a2
2 + a1a2) ×
× (1 – γ) + q(x2,x2,x1) + q(x2,x1,x2) + q(x1,x2,x2) (3.68)
The determinant of the system (3.67) and (3.68) is
(a1
2 + a1a2)(1 – γ) + a2
2 + (a1 + a2)α1
 ×
×a1
2(1 + γ) + a1a2(1 – γ) + a2
2 – (a1 – a2)α1
/
/(a1
2 + a2
2 + a1a2)
2 (3.69)
If the above expression is not 0, then
F(x1,x2) = 3α1(a1 + a2) + a2
2 γ
/(a1
2 + a1a2)(1 – γ) +
+ a2
2 + (a1 + a2)α1
 a1
2 + a2
2 + a1a2(1 – γ) + a2α1
 ×
× q(x1,x1,x1) – a1(α1 – a1γ)q(x2,x2,x2)/
/a1
2(1 + γ) + a1a2(1 – γ) + a2
2 – (a1 – a2)α1
 +
+ (a1
2 + a2
2 + a1a2)/(a1
2 + a1a2)(1 – γ) + a2
2 +
+ (a1 + a2)α1
 a1
2 + a2
2 + a1a2(1 – γ) + a2α1
 ×
× q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
 –
– a1(α1 – a1γ)q(x2,x2,x1) + q(x2,x1,x2) + q(x1,x2,x2)
/
/a1
2(1 + γ) + a1a2(1 – γ) + a2
2 – (a1 – a2)α1

Now let us suppose that the expression (3.69) is 0.
First let
(a1
2 + a1a2)(1 – γ) + a2
2 + (a1 + a2)α1 = 0
Then
α1 = – a2
2 + (a1
2 + a1a2)(1 – γ)
/(a1 + a2), α2 =
= – a1
2 + (a2
2 + a1a2)(1 – γ)
/(a1 + a2)
Now (3.58) takes the form
f(x1,x2,x3) = (1 – γ)F(x1,x2) + a1F(x3,x1) + a2F(x2,x3)
/
/(a1 + a2) + q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2),
while (3.67) becomes
a1F(x1,x2) – F(x2,x1)
/(a1 + a2) = q(x1,x1,x2) +
+ q(x1,x2,x1) + q(x2,x1,x1) – 3q(x1,x1,x1)
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 221
The last equation implies
q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) = U(x1,x2,x3) +
+ U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) – U(x1,x3,x2) –
– U(x3,x2,x1) + P(x1) + P(x2) + P(x3)
F(x1,x2) = G(x1,x2) + G(x2,x1) – (a1 + a2)P(x1)/a1
where U: ℜ3 → ℜ, G: ℜ2 → ℜ and P: ℜ → ℜ are arbi-
trary functions. The condition (3.66) yields
2G(x1,x1) = (a1 + a2)/a1 + 3/(1 – γ)
P(x1)
Next we suppose that
a1
2(1 + γ) + a1a2(1 – γ) + a2
2 – (a1 – a2)α1 = 0
In this case we have
α1 = a1
2 + a1
2(1 + γ) + a1a2(1 – γ)
/(a1 – a2)
α2 = a1
2 + a2
2(1 – γ) + a1a2(1 + γ)
/(a1 – a2)
Now (3.58) takes the form
f(x1,x2,x3) = a1(1 + γ) + a2(1 – γ)
F(x1,x2) –
– a1F(x3,x1) – a2F(x2,x3)/(a1 – a2) + q(x1,x2,x3) +
+ q(x2,x3,x1) + q(x3,x1,x2),
while (3.67) becomes
a1/(a1 – a2)
F(x1,x2) + F(x2,x1)
 =
= 3/(a1 – a2)
(1 + γ)a1/(1 – γ) + a2
q(x1,x1,x1) +
+ q(x1,x1,x2) + q(x1,x2,x1) + q(x2,x1,x1)
If
3/(a1 – a2)
(1 + γ)a1/(1 – γ) + a2
 ≠ – 1
i.e.,
(2 + γ)a1 + (1 – γ)a2 ≠ 0,
as above we find that
q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) = U(x1,x2,x3) +
+ U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1)
F(x1,x2) = G(x1,x2) – G(x2,x1)
and
f(x1,x2,x3) = a1(1 + γ) + a2(1 – γ)
G(x1,x2) – G(x2,x1)

+ a1G(x1,x3) – G(x3,x1)
 – a2G(x2,x3) – G(x3,x1)
/
/(a1 – a2) + U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) –
– U(x2,x1,x3) – U(x1,x3,x2) – U(x3,x2,x1)
where G: ℜ2 → ℜ and U: ℜ3 → ℜ are arbitrary func-
tions.
If, however
(2 + γ)a1 + (1 – γ)a2 = 0
then
q(x1,x2,x3) + q(x2,x3,x1) + q(x3,x1,x2) =
= U(x1,x2,x3) + U(x2,x3,x1) + U(x3,x1,x2) – U(x2,x1,x3) –
– U(x1,x3,x2) – U(x3,x2,x1) + a1P(x1) + P(x2) + P(x3)

F(x1,x2) = G(x1,x2) – G(x2,x1) + (a1 – a2)P(x1)
where P: ℜ → ℜ and U: ℜ3 → ℜ are arbitrary functions,
and
f(x1,x2,x3) = a1– G(x1,x2) + G(x2,x1) + G(x1,x3) –
– G(x3,x1)
 – a2G(x2,x3) –G(x3,x2)
/(a1 – a2) +
+ (a1 – a2)P(x2) + U(x1,x2,x3) + U(x2,x3,x1) +
+ U(x3,x1,x2) – U(x2,x1,x3) – U(x1,x3,x2) –U(x3,x2,x1)
Example 3. 5. Now we will assume as a meniscus
the relation (3.6). Let assume further as arbitrary its
general solution (3.7) is the function
F(x1,x2,x3) ≡ (x1/ζ1)
2/3 + (x2/ζ2)
2/3 + (x3/ζ3)
2/3 = 1
where ζi (1 ≤ i ≤ 3) are real constants, then the shape
of the meniscus will be given by the expression
f(x1,x2,x3) = (ζ1
–2/3 – ζ3
–2/3)x1
2/3 + (ζ2
–2/3 – ζ1
–2/3)x2
2/3 +
+ (ζ3
–2/3 – ζ2
–2/3)x3
2/3
This shows that the shape of the meniscus changes
cyclically during the mould cycle.
Remark 3. 6. Also, the above results hold for the
vector extension of equation (3.1) of the form
a1f(X1,X2,X3) + a2f(X2,X3,X1) + a3f(X3,X1,X2) =
= α1f(X1,X1,X2) + α2f(X2,X2,X3) + α3f(X3,X3,X1)
f: ℜ3 → ℜ, where Xi = (x1i,x2i,x3i)T are real vectors and
ai, αi (1 ≤ i ≤ 3) are real constants.
4 GENERALIZED RESULTS
As a natural consequence of the previous considered
meniscus equation, we will give the following more
general result.
Theorem 4. 1. The generalized meniscus equation
E(f) ≡ a fi
i
n
=
∑
1
(xi,xi+1,…,xi+n–1) = (4.1)
= a fi
i
n
=
∑
1
(xi,xi,xi+1,…,xi+n–2) (xn+i ≡ xi, n > 1)
where ai, αi (1 ≤ i ≤ n) are real constants, has a
solution if the right-hand side of (4.1) satisfies
(AC + I)Λg(x1,x2,…,xn-1),
g(x2,x3,…,xn),…,g(xn,x1,…,xn-2)

T = O (4.2)
where A = cycl(a1,a2,…,an), Λ = cycl(α1,α2,...,αn),
g(x1,x2,…,xn–1) = f(x1,x1,x2,…,xn–1), C is any non-zero
n×n cyclic matrix with constant real entries satisfying
ACA + A = O, O is the n×n zero matrix and I is the n×n
unit matrix.
If the equality (4.2) holds for some C, then the
general solution of equation (4.1) is given by the
following formula
f(x1,x2,…,xn),f(x2,x3,…,xn,x1),…,f(xn,x1,…,xn-1)

T =
= Bh(x1,x2,…,xn),h(x2,x3,…,xn,x1),…,h(xn,x1,…,xn-1)

T –
– Λg(x1,x2,…,xn-1),g(x2,x3,…,xn),…,g(xn,x1,…,xn-2)

T
(4.3)
where the non-zero n×n cyclic matrix B given by
B = cycl(b1,b2,…,bn)
satisfies the condition
AB = O
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
222 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
and h is an arbitrary real function ℜn → ℜ.
Proof. By a cyclic permutation of the variables in
(4.1) we get
a1 f(x1,x2,…,xn) + a2f(x2,x3,…,xn,x1) + · · · +
+ anf(xn,x1,…,xn-1) = α1g(x1,x2,…,xn-1) + α2g(x2,x3,…,xn) +
+ · · · + αng(xn,x1,…,xn-2)
an f(x1,x2,…,xn) + a1f(x2,x3,…,xn,x1) + · · · +
+ an-1f(xn,x1,…,xn-1) = αng(x1,x2,…,xn-1) + α1g(x2,x3,…,xn) +
+ · · · + αn-1g(xn,x1,…,xn-2)


a2 f(x1,x2,…,xn) + a3f(x2,x3,…,xn,x1) + · · · +
+ a1f(xn,x1,…,xn-1) = α2g(x1,x2,…,xn-1) + α3g(x2,x3,…,xn) +
+ · · · + α1g(xn,x1,…,xn-2)
i.e., in matrix form
AF = ΛG (4.4)
where
F = f(x1,x2,…,xn),f(x2,x3,…,xn,x1),…,f(xn,x1,…,xn-1)

T
and
G = g(x1,x2,…,xn-1),g(x2,x3,…,xn),…,g(xn,x1,…,xn-2)

T
We suppose that equation (4.4) has a solution F and
C satisfies ACA + A = O. Then
(AC + I)ΛG = (AC + I)AF = (ACA + A)F = O
i.e., equation (4.2) must be satisfied. Conversely, let
equation (4.2) hold for a cyclic matrix C. Then – CΛG
is easily seen to be a solution of equation (4.4):
A(– CΛG) = – (AC + I)ΛG + IΛG = IΛG = ΛG
Now let us prove that equality (4.3) gives the general
solution of equation (4.1).
Let f be a solution of equation (4.1), which we will
write in the form
E(f) = L(g) (4.5)
We denote by fh the general solution of the equation
E(f) = 0, and by fp we denote a particular solution of
equation (4.5).
Then f = fh + fp is the general solution of equation
(4.5). Indeed
E(fh + fp) = E(fh) + E(fp) = E(g)
On the other hand, let f be an arbitrary solution of
equation (4.5). Then
E(f – fp) = E(f) – E(fp) = L(g) – L(g) = 0
i.e., f – fp is a solution of the associated homogeneous
equation. So there exists a specialization fh* of the
expression fh such that
f – fp = fh
*, i.e., f = fh
* + fp
Thus fh + fp includes all the solutions of equation
(4.5).
The general solution of the homogeneous equation
E(f) = 0 given in matrix form is BH, where
H = h(x1,x2,…,xn),h(x2,x3,…,xn,x1),…,h(xn,x1,…,xn-1)

T
and a particular solution of the equation E(f) = L(g) in
matrix form is – CΛG, then F = BH – CΛG includes all
the solutions of the nonhomogeneous equation.
On the other hand, every function of the form (4.3)
satisfies the functional equation (4.1).
Remark 4. 2. The same results hold for the vector
extension of equation (4.1) of the form
E(f) ≡ a fi
i
n
=
∑
1
(Xi,Xi+1,…,Xi+n-1) =
= a fi
i
n
=
∑
1
(Xi,Xi,Xi+1,…,Xi+n-2) (Xn+i ≡ Xi, n > 1)
f: ℜn → ℜ, where Xi = (x1i,x2i,…,xni)T are real vectors
and ai, αi (1 ≤ i ≤ n) are real constants.
5 MENISCUS STABILITY
The meniscus stability was considered for the first
time in 15,16, but only according to definition 2.2 for the
solution of the Navier-Stokes equation, including the
pressure gradient. Here, we will give a completely new
matrix approach to the solution of the meniscus stability
problem.
Now we will derive a necessary and sufficient
condition for the stability of the meniscus given by the
quasicyclic functional equation (4.1), i.e., its matrix form
(4.4) using a simple spectral property of compound
matrices.
Let detA ≠ 0, then relation (4.4) takes the form
F = A-1ΛG ≡ SG (5.1)
where S is also a cyclic matrix.
Definition 5. 1. The quasicyclic functional equation
(5.1) is stable if stab(S) < 0.
Proposition 5. 2. For any cyclic matrix S∈ℜ it holds
stab(S) = infµ(S), µ is a LozinskiÇ measure on ℜn
(5.2)
Proof. The relation (5.2) obviously holds for diagoni-
zable matrices in view of
µT(S) = µ(TST
–1) (T is an invertible matrix) (5.3)
and the first two relations in (2.4). Furthermore, the
infimum in (5.2) can be achieved if S is diagonizable.
The general case can be shown based on this observa-
tion, the fact that S can be approximated by diagoni-
zable matrices in ℜ and the continuity of µ(·), which is
implied by the property
µ(A) – µ(B) ≤ A – B
Remark 5. 3. From the above proof it follows that
stab(S) = infµ

(TST –1), T is invertible.
The same relation holds if µ is replaced by µ1.
Corollary 5. 4. Let S∈ℜ. Then stab(S) < 0 ⇔ µ(S) <
0 for some LozinskiÇ measure µ on ℜn.
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226 223
Theorem 5. 5. For stab(S) < 0 it is sufficient and
necessary that stab(S2
) < 0 and (–1)n det(S) > 0.
Proof. Using the spectral property of S2
, the condi-
tion stab(S2
) < 0 implies that at least one eigenvalue of
S can be nonnegative. We may thus suppose that all
eigenvalues are real. It is then simple to see that the
existence of one and only one non-negative eigenvalue is
precluded by the condition (–1)ndet(S) > 0.
Theorem 5.5 and Corollary 5.4 lead to the following
result.
Theorem 5. 6. Suppose that (–1)ndet(S) > 0. Then S
is stable if and only if µ(S2
) < 0 for some LozinskiÇ
measure µ on ℜN, N = n!/2!(n – 2)!.
Theorem 5. 7. If stab(S2
(β)) < 0 for β ∈(a, b), then
(a, b) contains no Hopf bifurcation points of S(β).
Proof. Let β → S(β)∈ℜ be a function that is conti-
nuous for β∈(a, b). A point β0 ∈(a, b) is said to be a
Hopf bifurcation point for S(β) if S(β) is stable for β < β0,
and there exists an eigenvalue λ(β) of S(β) such that λ(β)
> 0, while the rest of the eigenvalues of S(β) are non-
zero for β > β0. From the proof of Theorem 5.5 we see
that stab(S2
) ≤ 0 precludes the existence of a non-nega-
tive eigenvalue of S.
Let S and P be n×n real cyclic matrices. A subspace
Ω ∈ℜ is invariant under S if S(Ω) ⊂ Ω. S is said to be
stable with respect to an invariant subspace Ω if the
restriction of S to Ω, SΩ: Ω → Ω is stable. Let the matrix
P be such that rank P = r (1 < r < n) and
PS = O (5.4)
Then KerP = x∈ℜ, Px = 0 satisfies S(ℜ) ⊂ KerP.
In particular, KerP is an (n–r)-dimensional invariant
space of S. It is of interest to study the stability of S with
respect to KerP when (5.4) holds.
Lemma 5. 8. Let Ω ⊂ ℜ be a subspace such that
S(ℜ) ⊂ Ω and dimΩ = k < n. Then 0 is an eigenvalue of
S, and there exist n – k null eigenvectors that do not
belong to Ω.
Proof. Let ℑ be the quotient space ℜ/Ω. Then ℜ ≅ Ω
⊕ ℑ and S(ℑ) = 0
 since S(ℜ) ⊂ Ω. This establishes the
lemma.
Theorem 5. 9. Suppose that P and S satisfy (5.4) and
rankP = r (1 < r < n). Then for S to be stable with respect
to KerP, it is necessary and sufficient that
1° stab(Sr+2
) < 0
and
2° lim
ε→ +0
sup sign det(εI + S)
 = (–1)n–r
Proof. Let λi (1 ≤ i ≤ n – r) be eigenvalues of SKerP.
By Lemma 5.8, the eigenvalues of S can be written as
λ1, λ2, . . . , λn-r, 0, . . . , 0, (r zeros)
and thus λi +λj, 1 ≤ i < j ≤ n – r ⊂ σ(Sr+2
) by the
spectral property of additive compound matrices dis-
cussed in Section 2. It follows that stab(Sr+2
) < 0
precludes the possibility of more than one non-negative
λi (1 ≤ i ≤ n – r). For ε > 0 sufficiently small
signdet(εI + S)
 = sign(εrλ1 · · · λn-r)
The theorem can be proved using the same arguments
as in the proof of Theorem 5.5.
Remark 5. 10. If r = n in (5.4), then P is of full rank
and hence S = O. If r = n – 1, then KerP is of dimension
1 and thus the eigenvalues of S are λ1 and 0 of multi-
plicity n – 1. From the above proof we know that
Theorem 5.9 still holds in this case, if condition 1° is
replaced by tr(S) < 0.
Corollary 5. 11. Suppose that S and P1 satisfy
P1S = βP1 (5.5)
and rankP1 = r (1 < r < n). Thus S is stable with respect
to KerP1 if and only if the following conditions hold:
1° stab(Sr+2
) < (r + 2)β
and
2° (sign β)r(–1)n–rdet(S) > 0
Proof. Let the matrix P1 be such that rankP1 = r (1 <
r < n) and (5.5) holds for some scalar β ≠ 0. Then KerP1
is an invariant subspace of S. Noting that (5.5) is equi-
valent to P1(S – βI) = O, one can apply Theorem 5.9 to S
– βI and obtain the proof.
6 DISCUSSION
The previous results may be extended in two
different ways.
1° A first possible generalization of equation (3.1) is
the following functional equation
a1f1(x1,x2,x3) + a2f2(x2,x3,x1) + a3f3(x3,x1,x2) =
α1f1(x1,x1,x2) + α2f2(x2,x2,x3) + α3f3(x3,x3,x1)
fi: ℜ3 → ℜ, where ai, αi (1 ≤ i ≤ 3) are real constants.
In other words, it means to extend the representation
of the meniscus with three unknown functions fi (1 ≤ i
≤ 3) instead of by one unknown function f, as was
shown in Section 3. This kind of meniscus representation
is really much better, but this problem is very hard and it
requires a new method of solvability.
If we continue in this way, it will hold for the
generalization of equation (4.1) given by the formula
E(f) ≡ a fi i
i
n
=
∑
1
(xi,xi+1,…,xi+n-1) =
= a fi i
i
n
=
∑
1
(xi,xi,xi+1,…,xi+n-2) (xn+i ≡ xi, n > 1)
fi: ℜn → ℜ, where ai, αi (1 ≤ i ≤ n) are real constants.
This equation is extremely hard and its solution is
unknown up to now.
2° A second generalization of equation (3.1) is the
vector equation
ICE B. RISTESKI: A NEW TOPOLOGY FOR THE TRAJECTORIES OF THE MENISCUS ...
224 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
a1f1(X1,X2,X3) + a2f2(X2,X3,X1) + a3f3(X3,X1,X2) =
= α1f1(X1,X1,X2) + α2f2(X2,X2,X3) + α3f3(X3,X3,X1)
fi: ℜ3 → ℜ, where Xi = (x1i,x2i,x3i)T are real vectors and
ai, αi (1 ≤ i ≤ 3) are real constants.
For equation (4.1) its generalized form is given by
the equation
E(f) ≡ a fi i
i
n
=
∑
1
(Xi,Xi+1,…,Xi+n-1) =
= a fi i
i
n
=
∑
1
(Xi,Xi,Xi+1,…,Xi+n-2) (Xn+i ≡ Xi, n > 1)
fi: ℜn → ℜ, where Xi = (x1i,x2i,…,xni)T are real vectors
and ai, αi (1 ≤ i ≤ n) are real constants.
Really, in this section the considered equations are
more sophisticated than the solved equations in previous
sections, but their solutions are extremely difficult and
up to now unknown to the author. In any case, their
solutions will describe the meniscus form in a way that
will be much closer to reality.
7 CONCLUSION
In this work the analyzed meniscus equation shows
that it is possible to interpret the meniscus shape with a
quasicyclic real functional equation. During continuous
steel casting, the shape of the meniscus changes
according to the mould cycle. The derived results are
appropriate for use in a huge mathematical model for a
description of all the appearances on the meniscus
considering the technical characteristics of the process.
The mathematical results are summarized as follows:
1) The meniscus equation is completely solved in ℜ3,
for all possible cases. The induced topology only
categorizes the trajectories in an orientation space.
2) The extended form of the meniscus equation is deri-
ved too in ℜn by a compact matrix approach,
different from the method used for the solution of the
meniscus equation in ℜ3.
3) The meniscus stability problem is solved by using a
simple spectral property of the compound matrices
and LozinskiÇ measure on ℜn.
In this work the shape of the meniscus during
continuous steel casting is considered for the first time as
a non-fixed characteristic according to the cyclic
operation of the mould. The analysis has shown that it is
more appropriate to use this kind of meniscus modeling
than some approximate form. It is also shown that the
old one-dimensional interpretations of the meniscus may
only be used as an approximation.
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during Continuous Slabs Casting, Inst. Min. & Metall., Skopje 1990
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Univ. Zagreb, 1991
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casting, Metals-Alloys-Technologies 26 (1992), 271–274
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equations, Czechoslovak Math. J. 54 (2004), 1015–1034
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steiner, Univ. Technol., Vienna 2002, 1–16
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Eng. Math. 16 (1982), 209–221
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226 Materiali in tehnologije / Materials and technology 41 (2007) 5, 213–226
L. CIZELJ, I. SIMONOVSKI: MULTISCALE MODELLING OF SHORT CRACKS ...
MULTISCALE MODELLING OF SHORT CRACKS IN
RANDOM POLYCRYSTALLINE AGGREGATES
VE^NIVOJSKO MODELIRANJE KRATKIH RAZPOK V
NAKLJU^NIH VE^KRISTALNIH SKUPKIH
Leon Cizelj, Igor Simonovski
"Jo`ef Stefan" Institute, Reactor Engineering Division, Jamova 39, 1000 Ljubljana, Slovenia
Leon.Cizeljijs.si, Igor.Simonovskiijs.si
Prejem rokopisa – received: 2006-05-17; sprejem za objavo – accepted for publication: 2007-08-16
The identification and explanation of processes potentially responsible for the initiation and development of intergranular cracks
are topics of wide concern. Especially the early phase of the development of cracks seems to be beyond the present
state-of-the-art explanations. An effort was therefore made by the authors to construct a computational model of the crack
growth kinetics at the grain-size scale. The main idea is to divide continuum (e.g., polycrystalline aggregate) into a set of
subcontinua (grains). Random grain structure is modelled using Voronoi-Dirichlet tessellation. Each grain is assumed to be a
monocrystal with random orientation of crystal lattice. Elastic behaviour of grains is assumed to be anisotropic. Crystal
plasticity is used to describe (small to moderate) plastic deformation of monocrystal grains, caused mainly by the strains along
the “incompatible” grain boundaries and at triple points. Finite element method is used to obtain numerical solutions of strain
and stress fields. The analysis is currently limited to two-dimensional models. The paper focuses on the dependence of crack tip
loading (J-integral) on the random orientation of neighbouring grains. The limited number of calculations indicate that the
incompatibility strains, which develop along the boundaries of randomly oriented grains, influence the local stress fields
(J-integrals) at crack tips significantly.
Key words: short cracks, random polycrystalline aggregates, multiscale modelling
Spoznavanju in pojasnjevanju procesov, ki bi lahko povzro~ili nastanek in razvoj medkristalnih razpok, namenjajo raziskovalci
po svetu veliko pozornost. [e najmanj raziskane in pojasnjene so zgodnje faze razvoja razpok. Avtorja sta zato sestavila
ra~unalni{ki model napredovanja razpok na nivoju kristalnih zrn. Klju~na ideja je razdelitev kontinuuma (ve~kristalni skupek)
na mno`ico med sabo povezanih manj{ih kontinuumov (kristalno zrno) z uporabo Voronojevega oz. Dirichletovega mozaika.
Vsako izmed kristalnih zrn nato opi{emo kot monokristal z naklju~no orientirano kristalno re{etko. Predpostavimo anizotropno
elasti~no vedenje, zmerne plasti~ne deformacije pa opi{emo s kristalno plasti~nostjo. Veliko plasti~nih deformacij povzro~ijo `e
nekompatibilnosti specifi~nih deformacij ob kristalnih mejah in v trojnih to~kah. Deformacijska in napetostna polja ra~unsko
ocenimo z metodo kon~nih elementov. Analize so sedaj omejene na ravninske primere. V ~lanku smo se osredinili na odvisnost
vrednosti J-integrala od naklju~ne orientacije okoli{kih kristalnih zrn. Omejeno {tevilo izra~unov nakazuje mo~an vpliv
nekompatibilnih deformacij vzdol` kristalnih mej na porazdelitev napetosti ter specifi~nih deformacij v okolici razpoke, s tem
pa tudi na vrednosti J-integrala.
Klju~ne besede: kratke razpoke, naklju~ni ve~kristalni skupki, ve~nivojsko modeliranje
1 INTRODUCTION
The identification and explanation of processes
potentially responsible for the initiation and development
of intergranular cracks are topics of wide concern.
Despite significant research performed in the past
decades, the root mechanisms of intergranular (stress
corrosion) cracking (IGC, IGSCC) are still not under-
stood completely. Recent research shows that the
intergranular cracking is strongly dominated by the
microstructural features, especially those on the grain
boundaries.
Computational algorithms aiming at modelling and
visualization of the IGSCC initiation and growth on the
grain-size scale have already been proposed 1. Random-
ness of the grain structure and of the crack initiation and
growth processes were assumed. The random crack
growth was simulated with algorithms allowing for crack
branching, coalescence and interference. The method
yielded patterns of cracks with shapes and structure
comparable to those observed in experiments. However,
a number of potentially important microscopic features
(e.g., random orientation and anisotropy of grains, grain
boundary mismatch etc.) were not taken into account. A
convenient approximation with isotropically elastic
continuum was implemented instead, allowing for
simplified but efficient estimation of stress intensity
factors.
In this paper, the dependence of crack tip loading
(J-integral) on the random orientation of neighbouring
grains under anisotropic elasto-plastic material response
is numerically investigated. The simulation framework 2
relies on explicit models of a random grain structure and
finite element solution of the boundary value problem
using standard crystal plasticity models. Similar
approach has been followed by Simonovski et al 3 while
analyzing a transgranular crack emanating from the
surface of a polycrystal. A very detailed study of the
crack tip opening displacements of short crack in mono-
and bi-crystals modelled using standard crystal plasticity
models is available in 4.
Materiali in tehnologije / Materials and technology 41 (2007) 5, 227–230 227
UDK 539.42:548.4 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)227(2007)
The results obtained are discussed and compared
with solutions for homogeneous isotropic and aniso-
tropic plates. The results obtained are important for the
future developments of the already proposed modelling
and visualization of the IGSCC initiation and growth at
the grain-size scale 1.
2 MODEL
The essential features of the proposed multiscale
simulation model are briefly described in this section.
Further details are available in 2.
The random grain structure is modelled as a planar
Voronoi tessellation representing a cell structure
constructed from a Poisson point process by introducing
planar cell walls perpendicular to lines connecting
neighbouring points. This results in a set of convex
polygons embedding the points and their domains of
attraction, which completely fill up the underlying space.
The concept of Voronoi tessellation has recently been
extensively used in materials science, especially to
model random microstructures like aggregates of grains
in polycrystals, patterns of intergranular cracks and
composites.
All tessellations used in this paper were generated
using the code VorTESS 5. Only a subset of Voronoi
tessellations is considered suitable for the finite element
meshing with quadrilaterals 6.
Constitutive modelling. Each grain (as formed ran-
domly by the Voronoi tessellation) is assumed to be
anisotropically elasto-plastic with randomly oriented
crystallographic directions. The utilized crystal plasticity
model assumes that plastic deformation takes place by
simple shear on a specific set of slip planes. A further
constitutive assumption is that the shear rate depends on
the stress only through the Schmid resolved shear stress.
Detailed description of the constitutive models and
algorithmic framework used is given in 7.
Homogenisation. Volume averaging of mesoscopic
strain and stress tensors is used to estimate the effective
macroscopic stress and strain tensors.
The crack tip loading. The loading of the crack tips
is achieved through the prescribed remote macroscopic
biaxial stress field. The J-integrals are then calculated for
each crack tip using the built–in features of ABAQUS 8,
which rely on the Virtual Crack Extension method
combined with the divergence theorem, transforming the
integration domain onto the area enclosed by the chosen
contour.
The scatter caused by the randomly shaped iso-
parametric finite element meshes at crack tips randomly
positioned within Voronoi tessellations has been studied
in isotropic continuum and reported as reasonable (e.g.,
up to about 10 %) elsewhere 9,10. The accuracy of the
J-integral estimates in randomly oriented anisotropically
elastic media has been found reasonable in 10.
3 NUMERICAL EXAMPLE
The planar structure with 101 grains is depicted in
Figure 1 with blue lines representing the intact grain
boundaries. The red line between points A and B
represents cracked grain boundary – a simple straight
inclined crack.
The finite element mesh (isoparametric 8-noded
quadrilaterals with reduced integration) used in sub-
sequent calculations is depicted in Figure 1, too. The
loading of the mesh is prescribed by tensile macrostress
with magnitudes 600 MPa and 300 MPa in directions X
and Y, respectively. The size of the model studied was
significantly smaller than the size of a representative
volume element (RVE), which has been estimated for a
similar non-cracked case to be about 350 grains in elastic
and 800 grains in plastic deformation modes 2. Both
essential types of macroscopic boundary conditions were
therefore simulated: (1) prescribed macroscopic stress
and (2) prescribed macroscopic strain.
L. CIZELJ, I. SIMONOVSKI: MULTISCALE MODELLING OF SHORT CRACKS ...
228 Materiali in tehnologije / Materials and technology 41 (2007) 5, 227–230
/(
)
, /°α
Figure 2: J-integrals and CTOD for an inclined crack in an aniso-
tropic monocrystal (macroscopic equivalent strain of 0.2 %)
Slika 2: Vrednosti J-integrala in CTOD za po{evno razpoko v anizo-
tropnem monokristalu (makroskopska ekvivalentna specifi~na
deformacija 0,2 %)
Figure 1: FE Model of a inclined crack in polycrystal
Slika 1: Mre`a kon~nih elementov s po{evno razpoko v ve~kristalnem
skupku
Two distinct cases are studied: (1) anisotropic con-
tinuum representing a monocrystal and (2) poly-
crystalline aggregate with grains modelled as randomly
oriented anisotropic continua. In both cases, the
influence of grain orientations on the magnitude and
orientation of the crack tip loading are sought under the
assumption of plane strain and both macroscopic
boundary conditions.
Reasonably constant estimates of J-Integral were
obtained for three different contours. The average value
of those three estimates is consistently used in sub-
sequent discussion. The scatter of J-integral in plain
strain is presumably governed by the random orien-
tations of grains in the simulated system. It is therefore
useful to compare such scatter by the variability of
J-integral estimates obtained by assuming an inclined
crack in a homogenous anisotropic plate. For this reason,
all grains in Figure 2 were identically oriented (= mono-
crystal), and material orientations were systematically
varied in increments of 15 degrees. Results are given as
a function of crack orientation. Additionally, the J inte-
gral estimates obtained from scaled Crack Tip Opening
and Crack Tip Sliding Displacements (CTOD and
CTSD, resp.) in a similar case 3 are also plotted for
comparison. Qualitative agreement between J and
CTOD&CTSD estimated is deemed reasonable. It
should however be noted, that 3 analyzed a surface crack,
which might well explain the quantitative differences in
the response.
The scatter of J-integral in a polycrystal was studied
using 30 realizations of random grain orientations within
the fixed grain structure (Figure 3). The results are
presented for macrostress and macrostrain boundary
conditions. Two data points (Tip B and Tip A) are
plotted for each realization of random grain orientations
and each type of boundary conditions. A rather large
difference between J integrals in both crack tips
(extremes indicated by green arrows) is found in some
cases. This significantly exceeds the finite element
numerical error and therefore clearly indicates existence
of a preferential crack growth direction as opposed to the
well-known symmetric behaviour of such crack in
isotopic conditions. The observed scatter seems to be
generally consistent with the angular variations of
J-integrals in monocrystal.
The total strain of about 0.2 % indicates that the
calculations were performed in the neighbourhood of the
macroscopic yield strength, with only a subset of grains
experiencing plastic deformation. This particular choice
is expected to maximize the scatter of J-integrals due to
highly scattered microscopic stress and strain fields.
4 CONCLUSIONS
The influence of randomly oriented anisotropic
elasto-plastic grains on the microscopic stress fields at
crack tips is studied numerically in this paper using
Voronoi tessellation and finite element method. The
limited number of calculations indicates that the incom-
patibility strains, which develop along the boundaries of
randomly oriented grains, influence the local stress fields
(J-integrals) at crack tips significantly.
Results clearly indicate significant difference
between the J-integrals at both crack tips. This supports
existence of a preferential crack growth direction as
opposed to the well-known symmetric behaviour of the
same crack in isotopic conditions. It is also a clear
indication that mode II represents a significant part of
the total crack tip loading. Purely elastic estimates of J
integral at the onset of global yielding (0.2% strain) may
be more than one order of magnitude lower than those
calculated with account for the incompatibility strains
along the grain boundaries. The influence of the macro-
scopic boundary conditions seems to be pronounced at
the onset of global plastification.
The results obtained are especially important for the
future developments of the modelling and visualization
of the initiation and growth of intergranular and trans-
granular cracks on the grain-size scale.
5 REFERENCES
1 Cizelj, Leon, Riesch-Oppermann, Heinz. Modeling the early
development of secondary side stress corrosion cracks in steam
generator tubes using incomplete random tessellation. Nuclear
Engineering and Design 212 (2001), 21–29
2 Kova~, Marko. Influence of microstructure on development of large
deformations in reactor pressure vessel steel. Dissertation. University
of Ljubljana, Slovenia, 2004
3 Simonovski, Igor, Nilsson, K.-F., Cizelj, L. Crack tip displacements
of microstructurally small cracks in 316L steel and their dependance
on crystallographic orientations of grains, Fatigue and Fracture of
Eng. Materials and Structures 30 (2007), 463–478
4 Potirniche, G.P. Finite element modeling of crack tip plastic
anisotropy with application to small fatigue cracks and textured
aluminum alloys. Dissertation. Mississippi State University,
Mississippi, USA, 2003
5 Riesch-Oppermann, Heinz. VorTess, Generation of 2-D random
Poisson-Voronoi Mosaics as Framework for the Micromechanical
L. CIZELJ, I. SIMONOVSKI: MULTISCALE MODELLING OF SHORT CRACKS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 227–230 229
/(
)
Figure 3: J-integrals and CTOD for an inclined crack in a polycrystal
(macroscopic equivalent strain of 0.2 %)
Slika 3: Vrednosti J-integrala in CTOD za po{evno razpoko v
ve~kristalnem skupku (makroskopska ekvivalentna specifi~na
deformacija 0,2 %)
Modelling of Polycristalline Materials. Karlsruhe, Germany: For-
schungszentrum Karlsruhe; Report FZKA 6325, 1999
6 Weyer, Stefan; Fröhlich, Andreas; Riesch-Oppermann, Heinz;
Cizelj, Leon, Kova~, Marko. Automatic Finite Element Meshing of
Planar Voronoi Tessellations. Engineering Fracture Mechanics 69
(2002), 954–958
7 Huang, Yonggang. A User-material Subroutine Incorporating Single
Crystal Plasticity in the ABAQUS Finite Element Program.
Cambridge, Massachussets: Harvard University; MECH-178, 1991
8 Hibbit, Karlsson & Sorensen Inc. ABAQUS/Standard User’s
Manual, Version 5.8. Pawtucket, R.I., USA: Hibbit, Karlsson &
Sorensen Inc., 1998
9 Kova~, Marko and Cizelj, Leon. Numerical Analysis of Interacting
Cracks in Biaxial Stress Field. Proc of Int Conf Nuclear Eergy in
Central Europe; Portoro`, Slovenia. 1999. 259–266
10 Cizelj Leon, Kov{e, Igor. Short intergranular cracks between
randomly oriented anisotropically elastic grains. 4th CNS Interna-
tional Steam Generator Conference, May 5–8, 2002, Toronto,
Ontario, Canada. Proceedings. Canadian Nuclear Society, 2002
L. CIZELJ, I. SIMONOVSKI: MULTISCALE MODELLING OF SHORT CRACKS ...
230 Materiali in tehnologije / Materials and technology 41 (2007) 5, 227–230
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
CHANGES TO THE FRACTURE BEHAVIOUR OF
MEDIUM-ALLOYED LEDEBURITIC TOOL STEEL
AFTER PLASMA NITRIDING
SPREMEMBE V NA^INU PRELOMA SREDNJE LEGIRANEGA
LEDEBURITNEGA JEKLA ZARADI PLAZEMSKEGA NITRIRANJA
Jur~i Peter1, Franti{ek Hnilica2, Jií Cejp2
1ECOSOND, s. r. o., K
í`ová 1018, 150 21 Prague 5, Czech Republic
2Czech Technical University, Faculty of Mechanical Engineering, Karlovo nám. 2, 121 35 Prague 2, Czech Republic
jurciecosond.cz
Prejem rokopisa – received: 2006-09-19; sprejem za objavo – accepted for publication: 2007-07-17
Three-point test specimens made from VANADIS 4 Extra cold-work steel were heat treated using two basic regimes and a
different hardness was obtained in each case. The cross-section of the specimens was 10 mm × 10 mm. Plasma nitriding was
carried out using various combinations of temperature, processing time and atmosphere. Fracture-toughness tests using the
method of static three-point bending showed the dominant role of the presence of a nitrided layer on both the bending strength
and the fracture mechanism. Only if the material was not plasma nitrided did the role of the austenitizing temperature become
clear, and in this case the higher the temperature, the lower the bending strength. The initiation and the propagation of the
fracture were low-energy ductile for the steel that was hardened and tempered. The presence of the plasma-nitrided region on
the surface changed the initiation as well as the propagation mechanism to that of cleavage. The thickness of the cleavage region
increased as the nitrided region became thicker, which additionally lowered the bending strength.
Key words: Vanadis 4 cold work steel, heat treatment, plasma nitriding, three-point bending strength, fracture surface
Trito~kovni preizku{anci iz jekla Vanadis 4 Ekstra za hladna orodja so bili toplotno obdelani na dva osnovna na~ina na razli~no
trdoto. Prerez preizku{ancev je bil 10 mm × 10 mm. Nitriranje v plazmi je bilo izvr{eno z razli~no kombinacijo temperature,
procesiranja in atmosfere. Preizkusi `ilavosti loma po metodi trito~kovnega upogiba so pokazali dominantno vlogo nitrirane
plasti na upogibno trdnost in na mehanizem preloma, le pri jeklu, ki ni bilo nitrirano, je pri{el do izraza vpliv temperature: ~im
vi{ja je bila temperatura, tem ni`ja je bila upogibna trdnost. Za~etek in propagacija preloma sta bila duktilna-maloenergijska pri
kaljenem in popu{~enem jeklu. Zaradi nitrirane plasti sta se spremenila za~etek in propagacija razpoke v cepljenje. Debelina
cepilne plasti je bila ve~ja pri ve~ji debelini nitrirane plasti, kar je dodatno zmanj{alo upogibno trdnost.
Klju~ne besede: jeklo Vanadis za hladna orodja, toplotna obdelava, nitriranje, trito~kovni upogib, povr{ina preloma
1 GENERAL REMARKS
The ledeburitic steels made via the powder
metallurgy (P/M) technique have a considerably finer
and much more isotropic microstructure than materials
with the same chemical composition produced by the
conventional ingot-fabrication route. The favourable
structural parameters are reflected in the fracture
toughness, which is several times greater than that of
conventionally produced steels. To improve the surface
hardness and to increase the wear resistance, the steels
are nitrided, PVD- or CVD-layered or duplex-coated.
The occurrence of surface layers formed by various
diffusion processes affects the mechanical properties,
markedly improving the wear resistance, hardness, and
in many cases also the fatigue strength; however, these
layers also lower the fracture toughness. Nevertheless, it
is very important to determine exactly the extent of the
lowering of the fracture toughness, since this property is
a very important parameter for the end-user of the tools.
The powder metallurgy of rapidly solidified particles,
which is a common name for the production of the group
of materials with an excellent combination of micro-
structure and properties, is a rapidly expanding area in
metallurgy. Many newly developed materials are intro-
duced to industry every year. For these materials, the
understanding of their behaviour under the condition of
surface layering is of essential importance. In this paper,
the results of an investigation of fracture behaviour for
the newly developed cold-work steel Vanadis 4 Extra
processed with plasma nitriding are presented and
discussed.
2 EXPERIMENTAL
Specimens of the steel Vanadis 4 Extra (1.37 % C,
0.43 % Si, 0.38 % Mn, 4.66 % Cr, 3.47 % Mo, 3.65 %
V, 0.08 % Cu, Fe bal.) were heat treated (hardened and
tempered) in a vacuum furnace to different hardnesses,
Table 1. Next, the specimens were plasma nitrided in a
RUBIG – Micropuls plasma furnace using various
combinations of temperature and dwell time (Table 1).
The fracture toughness was determined with a
three-point bending test, with a distance between the
supports of 80 mm. The specimens were loaded at the
central point with a loading speed of 1 mm/min up to
fracture.
Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236 231
UDK 539.42:669.14.018.252:621.785 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)231(2007)
The nitrided layers were investigated using light
microscopy (the thickness of the diffusion layer),
microhardness tests (depth profiles of the microhardness,
Nht*), a WDX analyser (concentration depth profiles),
X-ray diffraction (phase constitution of the surface).
Scanning electron microscopy was used for the
examination of the surface of the fractures.
Table 1: Nitriding of the specimens
Tabela 1: Nitriranje preizku{ancev
Specimen
set
Hardness
HRC
Plasma nitriding
1–4 57
470 °C/30 min/N2:H2 = 1:3,
500 °C/60 min/N2:H2 = 1:3,
530 °C/120 min/N2:H2 = 1:3,
470 °C/30 min + 470 °C/75 min,
N2:H2 = 1:10
5–8 60
470 °C/30 min/N2:H2 = 1:3,
500 °C/60 min/N2:H2 = 1:3,
530 °C/120 min/N2:H2 = 1:3,
470 °C/30 min + 470 °C/75 min,
N2:H2 = 1:10
3 RESULTS AND THEIR DISCUSSION
The microstructure of the substrate steel after
quenching and tempering to a hardness of HRC 57 is
shown in Figure 1. It consists of a martensitic matrix
and fine (several microns) globular carbide particles
(Figure 1). The microstructure of the steel processed to a
hardness of 60 HRC is similar to that with the hardness
of HRC 57 (Figure 2).
The nitrided region differs from the substrate
strongly in terms of etching sensitivity due to the nitride
precipitates formed in the near-surface layer; this is
typical for all nitrided ledeburitic steels. For the steel
processed at a low temperature, the nitrided layer is free
of a compound sub-layer (Figure 3). For the materials
processed at a higher temperature and/or for a longer
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
232 Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236
Figure 4: Microstructure of the steel heat treated to HRC 57 and
nitrided at 530 °C for 120 min
Slika 4: Mikrostruktura jekla, ki je bilo toplotno obdelano na HRC 57
in nitrirano 120 min pri 530 °C
Figure 1: Microstructure of the steel heat treated to HRC 57
Slika 1: Mikrostruktura jekla, ki je bilo toplotno obdelano na trdoto
HRC 57
Figure 2: Microstructure of the steel heat treated to HRC 60
Slika 2: Mikrostruktura jekla, ki je bilo toplotno obdelano na trdoto
HRC 60
Figure 3: Microstructure of the steel heat treated to HRC 57 and
nitrided at 470 °C for 30 min
Slika 3: Mikrostruktura jekla, ki je bilo toplotno obdelano na HRC 57
in nitrirano 30 min pri 470 °C
time, a compound "white" layer is also obtained (Figure
4).
The X-ray diffraction spectra show no presence of a
compound layer on the surface of the specimens
processed at 470 °C for 30 min. The surface micro-
structure consists of martensite and 13 % is the -nitride
(Figure 5). On the other hand, up to 70 % of the
-nitride was found in the case of the specimen
processed at 530 °C for 120 min. This definitely
indicates the presence of a compound layer with a
thickness of several µm (Figure 6).
The input of nitrogen into the surface induces a
considerable surface hardness increase. The hardness is
also increased below the surface and the thickness of the
region with elevated hardness is related to the diffusion
depth of nitrogen. The initial hardness of the material
does not have any substantial effect on the surface
hardness, but influences slightly the depth of the nitrided
region, according to the criterion: core hardness HV0.05 =
50 (Figures 7 and 8).
Figure 9 shows how the bending strength changes
when the core hardness, nitriding temperature and the
processing dwell time are increased. It is evident that the
austenitising temperature, resulting in a different core
hardness, is a relevant factor influencing the three-point
bending strength only when the steel is not nitrided. In
the nitrided steel, the presence of the nitrided layer by
itself lowers the bending strength considerably and the
austenitizing temperature does not play a significant role.
The bending strength is decreased if the thickness of the
nitrided region is increased. It is also important that the
diffusion annealing in a nitrogen-poor atmosphere (the
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236 233
,
/M
P
a
F
Figure 9: Bending strength as a function of nitriding parameters and
core hardness
Slika 9: Upogibna trdnost pri razli~nih parametrih nitriranja in trdoti
jekla
2 /°n
n/
s
Figure 6: X-ray patterns of the steel, nitrided at 530 °C for 120 min
Slika 6: Rentgenski spekter jekla, ki je bilo nitrirano 120 min pri 530
°C
,
Figure 7: Hardness depth profiles of the nitrided steel with a core
hardness of HRC 57
Slika 7: Globinski profil trdote nitriranega jekla s trdoto jekla HRC 572 /°n
n/
s
Figure 5: X-ray patterns of the steel, nitrided at 470 °C for 30 min
Slika 5: Rentgenski spekter jekla, ki je bilo nitrirano 30 min pri 470
°C
,
Figure 8: Hardness depth profiles of the nitrided steel with a core
hardness of HRC 60
Slika 8: Globinski profil trdote nitriranega jekla s trdoto jekla HRC 60
last two columns) does not lead to an improvement in the
bending strength.
The bending strength of the non-nitrided Vanadis 4
Extra steel is higher than that of Vanadis 6 and
comparable with that of M2-type steel 5. The nature of
the difference was not explained so far; however, it can
be assumed that Vanadis 4 Extra differs from Vanadis 6
in the molybdenum content and that molybdenum nitride
particles can affect the bending strength in an
undesirable way. Nevertheless, the M2-type steel also
contains molybdenum and the bending strength of the
nitrided material remained much higher. Further and
more detailed investigations are needed to make a more
reliable conclusion about the cause of the change in the
bending strength after nitriding.
The fractographical analysis was designed to
investigate the fracture initiation and propagation for
specimens with and without a nitrided region of different
thickness.
In all of the specimens the fracture is initiated on the
tensile strained side, in several centres, and propagated
in the specimen (Figure 10). The crack propagation is
different for the non-nitrided and nitrided samples. In the
case of the non-nitrided material, the propagation of the
crack occurs with the de-cohesion at the carbide-matrix
interface and the fracture surface exhibits a shallow
dimpled morphology (Figure 11). The crack propagation
does not consume a large amount of energy, since the
dimples are relatively flat and the plastically deformed
volume of steel is not large. For this reason, this type of
fracture is low-energy transcrystalline. Similarly, as for
steel Vanadis 6, a lower austenitizing temperature did
not change the mechanism of the failure and only some
secondary cracks were observed at the surface 5.
The steel austenitized at a lower temperature had a
higher fracture toughness because of its smaller grain
size. The austenite grain size increased with the
temperature and the products of the austenite
decomposition (like martensite) were coarser, too. These
phenomena are well known as the limiting ones for the
fracture toughness and can explain the obtained results
of the three-point bending strength.
The mechanism of fracture initiation in the case of
the nitrided specimens differs a great deal from that of
the non-nitrided specimens. The fracture clearly exhibits
the characteristics of transcrystalline cleavage (Figure
12) with the thickness of the cleavage layer
corresponding to that of the nitrided region. At higher
magnification, small steps are visible on the cleavage
facets, indicating the microcracks’ propagation at
different levels of the same lattice plane. The
investigations on various ledeburitic steels showed that
the microstructure of the nitrided steel consisted of
martensitic platelets containing nitrogen, and ultra-fine
nitride particles 6. Coarser nitride particles are broken
during the propagation of the crack and can act as nuclei
for the crack re-initiation. In the SEM micrograph in
Figure 14, the case of a brittle particle with spokewise
cracks propagating in the surrounding area is shown. The
steps on the cleavage facets can be related to the platelet
shape of martensite. Figure 15 shows that in the core
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
234 Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236
Figure 10: Fracture surface of un-nitrided specimen, processed to
HRC 57
Slika 10: Prelomna povr{ina preizku{anca s trdoto HRC 57, ki ni bil
nitriran
Figure 12: Fracture surface of specimen nitrided at 530 °C for 120
min
Slika 12: Prelomna povr{ina preizku{anca, ki je bil nitriran 120 min
pri 530 °C
Figure 11: Fracture surface of the specimen in Figure 10
Slika 11: Prelomna povr{ina preizku{anca s slike 10, detail
material, the fracture propagates again according to the
trancrystalline low-energy ductile mechanism with
de-cohesion at the carbide-matrix interface and a small
plastic deformation.
The lowering of the bending strength after the plasma
nitriding is due to the fact that cleavage crack propa-
gation requires only a negligible plastic deformation. All
of the energy input into the material is spent only for the
formation of two new surfaces. This is a different, when
compared to the non-nitrided material, where a low
plastic deformation occurs throughout the specimens
and, as a consequence, the three-point strength was
considerably higher. The lowering of the fracture
toughness at an increased nitriding temperature and/or
time can be explained by the fact that the area of
cleavage of the total cross-sectional area is greater due to
the thickness of the nitrided layer.
Based on the results presented in this work as well as
in the papers published previously 5,6 it seems that the
lowering of the bending strength, and fracture toughness
in general, due to the occurrence of a nitrided region of
the surface, is a systematic phenomenon and cannot be
avoided completely. On the other hand, the nitriding
brings several beneficial effects to the materials and
components, such as an increase in the fatigue lifetime of
the specimens and tools, and an improvement in the wear
resistance, corrosion resistance, adhesion of thin PVD
layers, etc. Therefore, the nitriding will be required from
industrial producers and/or users of tools. It is, therefore,
necessary to minimize the lowering of the fracture
toughness, through the optimization of the nitriding
process.
4 CONCLUDING REMARKS
– In the case of the non-nitrided steel, the austenitizing
temperature has an important influence on the
fracture behaviour. The three-point bending strength
decreases as the austenitizing temperature increases
because of the grain coarsening at the higher
austenitizing temperature.
– The main mechanism of the fracture initiation is the
nucleation of dimples at the carbide-matrix interface
in the case of non-nitrided specimens. The fracture
propagation is ductile and low energy.
– The presence of the plasma-nitrided layer at the
surface lowers significantly the bending strength.
The thicker is the nitrided layer the lower is the
fracture toughness, since the cleavage region, where
a small amount of energy is spent for the crack
propagation, in greater with a thicker nitrided region.
– The lowering of the bending strength for the steel
Vanadis 4 Extra is more remarkable than that for the
steels Vanadis 6 and M2, processed in the same
nitriding conditions.
– Transcrystalline cleavage was found to be the main
mechanism of crack propagation in the case of
nitrided layers. The thickness of the cleavage
regions corresponds well with the thickness of the
nitrided regions determined by metallographic
methods.
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236 235
Figure 15: Fracture surface of the specimen in Figure 12, core steel
Slika 15: Prelomna povr{ina preizku{anca s slike 12, jedro
preizku{anca
Figure 14: Fracture surface of the specimen in Figure 12, detail of
cleavage facets
Slika 14: Prelomna povr{ina preizku{anca s slike 12, detail s
cepilnimi facetami
Figure 13: Fracture surface of the specimen in Figure 12
Slika 13: Prelomna povr{ina preizku{anca s slike 12, detail
ACKNOWLEDGEMENTS
The authors wish to thank the Ministry of Education
and Youth of the Czech Republic for the financial
support for the solution of the Project Eureka E!3437
PROSURFMET.
5 LITERATURE
1 Jur~i, P., Suchánek, J., Stola
, P.: In.: Proceedings of the 5th ASM
Heat Treatment and Surface Engineering Conference in Europe, 7–9
June 2000, Gothenburg, Sweden, 197
2 Jur~i, P., Suchánek, J., Stola
, P., Hnilica, F., Hrubý, V.: In.:
Proceedings of the European PM 2001 Congress, October 22–24,
2001, Nice, France, 303–308
3 Musilová, A., Jur~i, P.: Acta Metallurgica Slovaca, 7 (2001), 1,
Special Issue METALLOGRAPHY 01, Gabriel Janák, 25–27 April
2001, Stará Lesná, Slovak republic, 265–268
4 Jur~i, P., Hnilica, F.: Powder Metallurgy Progress, 3 (2003)1, 10–19
P. JUR^I ET AL.: CHANGES TO THE FRACTURE BEHAVIOUR OF MEDIUM-ALLOYED ...
236 Materiali in tehnologije / Materials and technology 41 (2007) 5, 231–236
T. KOSMA^ ET AL.: THE FRACTURE AND FATIGUE OF SURFACE-TREATED TETRAGONAL ZIRCONIA ...
THE FRACTURE AND FATIGUE OF
SURFACE-TREATED TETRAGONAL ZIRCONIA (Y-TZP)
DENTAL CERAMICS
PRELOM IN UTRUJENOST POVR[INSKO OBDELANE
TETRAGONALNE (Y-TZP) DENTALNE KERAMIKE
Toma` Kosma~1, ^edomir Oblak2, Peter Jevnikar2
1Jo`ef Stefan Institute, Jamova 39, 1001 Ljubljana, Slovenia;
2Faculty of Medicine, University of Ljubljana, Vrazov trg 2, 1101 Ljubljana, Slovenija
tomaz.kosmacijs.si
Prejem rokopisa – received: 2006-05-17; sprejem za objavo – accepted for publication: 2007-06-26
The effects of dental grinding and sandblasting on the biaxial flexural strength of Y-TZP ceramics containing the mass fraction
of 3 % yttria were evaluated. Dental grinding at high rotation speed lowers the mean strength under static loading and the
survival rate under cyclic loading. Sandblasting, in contrast, may provide a powerful tool for surface strengthening also
resulting in a substantially higher survival rate under cyclic loading. Fractographic examination of ground specimens revealed
that failure originated from radial cracks extending up to 50 µm from the grinding groves into the bulk of the material. However,
no evidence of grinding-induced surface cracks could be obtained by SEM analysis of the ground samples, prepared by a
standard bonded-interface technique. Sandblasting, in contrast, introduces lateral cracks, which are not detrimental to the
strength of Y-TZP ceramics. The "medical-grade" Y-TZP ceramics also containing 0.25 % of dispersed alumina used in this
work exhibited full stability under hydrothermal conditions.
Key words: tetragonal zirconia; dental grinding; sandblasting; fracture origin; fatigue
Ocenjen je bil vpliv zobnih bru{enj in peskanja na dvoosno upogibno trdnost keramike Y-TZP z molskim dele`em itrijevega
oksida 3 %. Zobno bru{enje pri veliki hitrosti vrtenja zmanj{a trdnost pri stati~ni obremenitvi in trajnostno dobo pri cikli~ni
obremenitvi. Nasprotno pa je peskanje lahko u~inkovit na~in za utrditev povr{ine, ki pomembno pove~a tudi trajnostno dobo pri
cikli~ni obremenitvi. Fraktografsko opazovanje bru{enih preizku{ancev je pokazalo, da se je prelom za~el iz radialnih razpok v
globino do 50 µm iz dna brusilnih `lebov v notranjost preizku{anca, ~eprav pri SEM pregledu ni bila odkrita nobena povr{inska
brusilna razpoka na bru{enih preizku{ancih, pripravljenih po standardni povr{insko vezani tehniki. Peskanje nasprotno od
bru{enja ustvari lateralne razpoke, ki ne vplivajo na trdnost keramike Y-TZP. Tetragonalna (Y-TZP) keramika za medicinske
namene, ki vsebuje tudi 0,25 % dispergiranega aluminijevega oksida in je bila uporabljena v tem delu, je bila popolnoma
stabilna v hidrotermalnih razmerah.
Klju~ne besede: tetragonalni cirkonijev oksid, zobno bru{enje, peskanje, za~etek razpoke, utrujenost
1 INTRODUCTION
Yttria partially stabilized tetragonal zirconia (Y-TZP)
has become increasingly popular as an alternative
high-toughness core material in dental restorations
because of its biocompatibility, acceptable aesthetics and
attractive mechanical properties. Compared to other
dental ceramics, the superior strength, fracture toughness
and damage tolerance of Y-TZP are due to a stress-
induced transformation toughening mechanism operating
in this particular class of ceramics 1. Y-TZP is currently
used as a core material in full-ceramic crowns and
bridges, implant superstructures, orthodontic brackets
and root dental posts 2–5. Like most technical ceramics,
zirconia dental restorations are produced by dry- or
wet-shaping of ceramic green bodies which are than
sintered to high density. For the material’s selection and
microstructural design the following two criteria should
be taken into consideration: the damage tolerance upon
mechanical surface treatment and the aging behavior in
an aqueous environment. Dental grinding is involved in
reshaping and the final adjustment of the prosthetic
work, whereas sandblasting is commonly used to im-
prove the bond between the luting agent and the
prosthetic work. Because Y-TZP ceramics exhibit a
stress-induced transformation, the surface of the
mechanically treated prosthetic work is expected to be
transformed into the monoclinic form, i.e. constrained,
and also damaged. Under clinical conditions, where
dental restorations are exposed to thermal and mecha-
nical cycling in a chemically active aqueous environ-
ment over long periods, these grinding- and
impact-induced surface flaws may grow to become stress
intensifiers, facilitating fracture at lower levels of
applied stress. Furthermore, with prolonged time under
clinical conditions the metastable tetragonal zirconia
may start transforming spontaneously into the mono-
clinic structure 6. This transformation is diffusion-
controlled and is accompanied by extensive micro-
cracking, which ultimately leads to strength degradation
7. Therefore, extensive research work was undertaken to
evaluate the effects of mechanical surface treatment and
aging on the strength and reliability of various Y-TZP
ceramics.
Materiali in tehnologije / Materials and technology 41 (2007) 5, 237–241 237
UDK 539.42:691.49:546.831 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)237(2007)
In our previous studies 8,9 we have shown that dental
grinding using a coarse-grit diamond burr at a high
rotation speed lowers the mean strength and reliability,
whereas sandblasting improves the mean strength, at the
expense of somewhat lower reliability. The fine-grained
materials exhibited higher strength after sintering, but
they were less damage tolerant upon grinding than
tougher, coarse-grained materials. Standard grade
3Y-TZP ceramics were more susceptible to low-tempe-
rature degradation than a special, corrosion resistant
3Y-TZP grade also containing a small amount of
dispersed alumina. Besides, no grain-size dependence of
the diffusion-controlled transformation was observed
with this material. Based on these results, coarse-grained
zirconia containing a small amount of alumina was
suggested for dental applications.
Here we report on the fracture and fatigue of
surface-treated tetragonal zirconia (Y-TZP) dental
ceramics. Fracture mechanics was used to calculate the
effective length of mechanically induced surface flaws
acting as the stress concentrators, relative to the depth of
the stress-induced surface compressive layer which
contributes to strengthening. The results were verified by
a conventional fractographic examination as well as by
SEM analysis of surface-treated samples, which were
prepared by a standard bonded-interface technique.
2 EXPERIMENTAL WORK
Disc-shaped specimens ((15.5 ± 0.03) mm in dia-
meter ad (1.5 ± 0.03) mm thick) were fabricated from a
commercially available ready-to-press Y-TZP powder
(TZ-3YSB-E, Tosoh, Japan) containing the mass fraction
of yttria 3 % in the solid solution and a 0.25 % alumina
addition to suppress the t-m transformation during aging,
by uniaxial dry pressing and pressureless sintering in air
for 4 h at 1450 °C and 1550 °C, respectively.
After firing, the top surface of the specimens was
submitted to a different surface treatment. A coarse grit
(150 µm) and a fine grit (50 µm) diamond burr were
chosen for dry and wet surface grinding, in order to
simulate clinical conditions. The grinding load of about
100 g was exerted by finger pressure, the grinding speed
was 150,000 r/min. For sandblasting, discs were
mounted in a sample holder at a distance of 30 mm from
the tip of the sandblaster unit, equipped with a nozzle of
5 mm in diameter. Samples were sandblasted for 15 s
with 110 µm fused alumina particles at 4 bar. Before and
after each surface treatment the samples were analyzed
by XRD, using CuKα radiation. The relative amount of
transformed monoclinic zirconia on the specimens’
surfaces was determined according to the method of
Garvie and Nicholson 10. The thickness of the trans-
formed surface layer of surface-treated samples was
calculated using the x-ray determination method 11.
Although this method yields conservative values, it can
be used to compare the influence of various surface
treatments on the thickness of the surface compressive
layer.
Aging of pristine and mechanically treated materials
in an aqueous environment was performed under iso-
thermal conditions at 140 °C for 24 h. After aging,
specimens were analysed by XRD for phase compo-
sition.
Biaxial flexural strength measurements were
performed according to ISO 6872 at a loading rate of 1
mm/min. Surface-treated specimens were fractured with
the surface treated side under tension. The load to failure
was recorded for each disc and the flexural strength was
calculated using the equations of Wachtman et al.12. The
variability of the flexural strength values was analyzed
using the two-parameter Weibull distribution function.
Cyclic loading experiments were performed using an
Instron Ltd, Model 8871 machine. The load varied from
50 N to 850 N at a freqency of 15 Hz. After 106 cycles
the specimens were "statically" loaded to fracture. For
specimens which failed before one milion cycles The
number of cycles to failure was registered.
After biaxial flexural strength measurements the
fracture surfaces were examined by SEM. The existence
of grinding- and sandblasting-induced sub-surface flows
was evidenced by SEM analysis of polished interfaces
perpendicular to the ground and sandblasted surface,
respectively. For this examination, specimens were
prepared using a standard bonded-interface technique, as
described elsewhere 13.
3 RESULTS AND DISCUSSION
The main characteristics of the sintered materials are
listed in Table 1. The relative density of sintered
specimens exceeded 99 % of the theoretical value and
they were 100 % tetragonal. An SEM micrograph of the
sintered material, showing equiaxed grains with the
mean size of 0.57 µm, is represented in Figure 1.
Table 1: Sintering conditions and main characteristics of sintered
ceramics
Tabela 1: Pogoji sintranja in osnovne zna~ilnosti sintrane keramike
Sintering
conditions
Mean grain
size
d/µm
Flexural
strength
MPa (SD)
KIc
MPa m1/2 (SD)
1450 °C/4 h 0.51 1080 (75) 5.08 (0.10)
1550 °C/4 h 0.59 990 (111) 5.18 (0.12)
Tosoh, Tokyo, Japan
During dental grinding, tens of µm of material were
removed by a single pass as the burr was moved back
and forth across the surface and the process was always
accompanied by extensive sparking. During sandblasting
about 60 µm of material was uniformly removed but
sparks were not observed during this operation.
Microscopic examination of the ground and sandblasted
samples revealed that in both cases the materials surface
T. KOSMA^ ET AL.: THE FRACTURE AND FATIGUE OF SURFACE-TREATED TETRAGONAL ZIRCONIA ...
238 Materiali in tehnologije / Materials and technology 41 (2007) 5, 237–241
was in part plastically deformed (Figure 2). The depth of
the intersecting grinding grooves and parallel grit
scratches, representing the most characteristic feature of
the ground surface morphology, varied with the diamond
grit size. The eroded surface was wrinkled with sharp,
randomly oriented scores, and surface pits were readily
observed on an otherwise plain sandblasted surface. In
spite of high stresses during grinding the amount of
transformed zirconia on the ground surfaces was almost
negligible, and so was the transformed zone depth, as
calculated from the relative amounts of the monoclinic
phase. It is assumed that during grinding the locally
developed temperatures exceeded the m->t trans-
formation temperature and the reverse transformation
occurred. Higher amounts of the monoclinic zirconia,
about 15–17%, were detected on sandblasted samples,
which yielded the transformed-zone depth values
ranging from 0,3 µm to 0,5 µm. It is interesting to note
that these values roughly correspond to the mean grain
size of the sintered ceramics.
The mean values of biaxial flexural strength and the
respective standard deviations are graphically repre-
sented in Figure 3. Dental grinding evidently lowered
the mean strength, whereas sandblasting provided a
powerful tool for strengthening. The counteracting effect
of dental grinding and sandblasting on flexural strength
can be explained by considering two competing factors
influencing the strength of surface treated Y-TZP
ceramics: residual surface compressive stresses, which
contribute to strengthening, and mechanically induced
surface flaws, which cause strength degradation 8.
Since almost no monoclinic zirconia was detected on
the ground specimens, the contribution of the grinding-
induced strengthening must have been negligible,
regardless of grinding conditions used and the material
tested. The strength of ground materials is thus mainly
T. KOSMA^ ET AL.: THE FRACTURE AND FATIGUE OF SURFACE-TREATED TETRAGONAL ZIRCONIA ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 237–241 239
Figure 3: Mean biaxial flexural strength values for as-sintered and
surface treated Y-TZP ceramics. 1 – As sintered, 2 – Dry ground (50
µm diamond burr), 3 – Dry ground (150 µm diamond burr), 4 –
Sandblasted. Error bars represent one SD from the mean.
Slika 3: Povpre~na dvoosna upogibna trdnost sintrane in povr{insko
obdelave keramike Y-TZP. 1 – sintrano, 2 – suho bru{eno (diamantni
brus 50 µm), 3 – suho bru{eno (diamantni brus 150 µm), 4 – peskano.
Obmo~je raztrosa prikazuje eno SD od povpre~ja.
Figure 2: SEM micrographs showing Y-TZP surface morphology
after: A) dry grinding using 150 µm diamond burr and B) sandblasting
Slika 2: SEM-posnetka, ki prikazujeta morfologijo povr{ine keramike
Y-TZP po: A) suhem bru{enju z diamantnim brusom 150 µm in B) po
peskanju
Figure 1: SEM micrograph showing the microstructure of sintered
tetragonal zirconia
Slika 1: SEM-posnetek, ki prikazuje mikrostrukturo sintranega
cirkonijevega oksida
determined by the critical defect size to initiate failure,
which can be estimated using the Griffith strength
relation 14
δ
ϕf
IC
cr
= ⋅1
K
c
(1)
where δf is the fracture stress, ϕ is a geometric constant
(= 2/π1/2 for surface line cracks), KIC is the fracture
toughness and ccr is the critical defect size to initiate
failure. Calculated ccr values for sintered coarse-
grained specimens before and after dry grinding using
150 µm and 50 µm grit burr were 17.3, 31.0 µm and 24.1
µm, respectively. Since Eq. (1) does not take into
account any of the residual surface stresses that may
exist in the material, the calculated ccr values should be
regarded as the effective length of strength-controlling
defects, which would result in an equivalent strength of
the material without any residual surface stresses.
Surface grinding increases the effective critical defect
size, presumably by generating radial surface cracks. A
fractographic examination of the ground specimens
indeed revealed that failure originated from radial
cracks extending several tens of µm (up to 50 µm) from
the grinding grooves into the bulk of the material
(Figure 4). However, no evidence of grinding-induced
surface cracking could be obtained by SEM
examination of a polished interface perpendicular to the
ground surface (Figure 5). This observation is in
agreement with recently published results by Xu et al 15,
who reported on a noticeable grit size dependence of
strength degradation upon machining of Y-TZP, but
they could not find any evidence of grinding-induced
surface cracking. Since radial cracks, which are readily
seen in fracture surfaces, were not formed during
grinding, they must have been initiated and extended
from a grinding groove during loading until they
reached the critical length for failure initiation.
Subcritical crack growth from a grinding groove during
cyclic loading resulted in the lowest survival rate during
fatigue experiments, whereas the strength of "survived"
specimens was nearly the same as that of the material
which was not subjected to cyclic loading.
In contrast to grinding, sandblasting is capable of
transforming a larger amount of zirconia in the surface
of Y-TZP ceramics indicating lower temperatures during
this operation. Surface flaws, which are introduced by
sandblasting, do not seem to be strength determining,
otherwise the strength of the material would have been
reduced instead of being increased. Since lateral crack
chipping is the most prevalent mechanism involved in
the erosive wear of ceramics, lateral cracks could be
expected in these samples, which was later confirmed by
T. KOSMA^ ET AL.: THE FRACTURE AND FATIGUE OF SURFACE-TREATED TETRAGONAL ZIRCONIA ...
240 Materiali in tehnologije / Materials and technology 41 (2007) 5, 237–241
Figure 6: SEM micrograph of a polished interphase perpendicular to
the sandblasted surface of a fine-grained Y-TZP, showing lateral crack
chipping
Slika 6: SEM-posnetek polirane povr{ine, pravokotne na peskano
ploskev fino zrnate Y-TZP, ki prikazuje lateralno lu{~ilno razpoko
Figure 4: SEM micrograph of the fracture surface of fine grained dry
ground Y-TZP
Slika 4: SEM-posnetek prelomne povr{ine fino suho bru{ene Y-TZP
Figure 5: SEM micrograph of a polished interphase perpendicular to
the ground (150 µm grit) of a fine grained Y-TZP
Slika 5: SEM-posnetek polirane povr{ine, pravokotne na bru{eno
(zrno 150 µm) fino zrnato Y-TZP
microscopic examination using a bonded-interface
technique (Figure 6). Fractographic examination of
sandblasted samples confirmed that the failure of these
samples was initiated from a lateral crack linked to
subsurface cracks (Figure 7). It seems that sandblasting
introduces surface flaws, which are not detrimental to
the strength of Y-TZP ceramics statically loaded to
fracture. However, under clinical conditions these impact
flaws may grow to become stress intensifiers, causing
accidental failure at lower levels of applied stress.
After autoclaving at 140 °C for 24 h, traces of the
monoclinic zirconia were identified on the surface of
sintered specimens, but the strength degradation has not
yet occurred. The same observation was made with the
ground and sandblasted specimens.
4 CONCLUSIONS
The surface grinding using a coarse-grit diamond
burr at a high rotation speed lowers the mean strength of
tetragonal zirconia Y-TZP ceramics, whereas sand-
blasting provides a powerful method for surface
strengthening. The counteracting effect of dental
grinding and sandblasting was explained in terms of two
competing factors influencing the strength of surface
treated 3Y-TZP ceramics: residual surface compressive
stresses, which contribute to strengthening, and
mechanically induced surface flaws, which cause
strength degradation.
5 REFERENCES
1 E. C. Subbarao: Adv. Ceram., 3 (1981), 1–24
2 O. Keith, R. P. Kusy, J. Q. Whitley: Am. J. Orthod. Dentofacial.
Orthop., 106 (1994), 605–614
3 K. H. Meyenberg, H. Lüthy, P. Schärer: J. Esthet. Dent. 7 (1995),
73–80
4 A. Wohlwend, S. Studer, P. Schärer: Quintessence. Dent. Technol., 1
(1997), 63–74
5 R. Luthardt, V. Herold, O. Sandkuhl, B. Reitz, J. P. Knaak, E. Lenz:
Dtsch. Zahnarztl. Z., 53 (1998), 280–285
6 T. Sato., M. Shimada: J. Am. Ceram. Soc. 68 (1985) 68, 356–359
7 D. J. Kim: J. Euro. Ceram. Soc., 17 (1997) 17, 897–903
8 T. Kosma~, ^. Oblak, P. Jevnikar, N. Funduk, L. Marion: Dent.
Mater. 15 (1999), 426–433
9 T. Kosma~, ^. Oblak, P. Jevnikar, N. Funduk, L. Marion: J. Biomed.
Materi. Res., 53 (2000), 304–313
10 R. C. Garvie, P. S. Nicholson: J. Am. Ceram. Soc., 55 (1972),
303–305
11 T. Kosma~, R. Wagner, N. Claussen: J. Am. Ceram. Soc., 64 (1981),
C72–C73
12 J. B. Wachtman, W. Capps, J. Mandel: J. Mater. Sci., 7 (1972),
188–194
13 F. Guiberteau, N. P. Padture, B. R. Lawn: J. Am. Ceram. Soc., 77
(1994), 1825–1831
14 B. R. Lawn: Fracture of brittle solids, 2nd ed. Cambridge, Cambridge
University Press, UK, 1993
15 H. K. K. Xu, S. Jahanmir, L. K. Ives: Mach. Sci. Tech., 1 (1997),
49–66
T. KOSMA^ ET AL.: THE FRACTURE AND FATIGUE OF SURFACE-TREATED TETRAGONAL ZIRCONIA ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 237–241 241
Figure 7: SEM micrograph of a fracture surface of fine-grained dry
ground and sandblasted Y-TZP. Failure originated from a 50 µm deep
surface pit
Slika 7: SEM-posnetek prelomne povr{ine finozrnate suho bru{ene in
peskane Y-TZP. Za~etek preloma je v 50 µm globoki povr{inski
zajedi.

F. ZUPANI^ ET AL.: POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z
ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
SURFACE OF Cu-Sn-Zn-Pb ALLOY IRRADIATED WITH
ULTRAVIOLET NITROGEN LASER
Franc Zupani~1, Tonica Bon~ina1, Davor Pipi}2, Vi{nja Hen~ - Bartoli}2
1 Univerza v Mariboru, Fakulteta za strojni{tvo, Smetanova 17, SI-2000 Maribor, Slovenija
2 Sveu~ili{te u Zagrebu, Fakultet elektrotehnike i ra~unarstva, Zavod za primijenjenu fiziku, Unska 3, 10 000 Zagreb, Hrva{ka
franc.zupanicuni-mb.si
Prejem rokopisa – received: 2007-05-07; sprejem za objavo – accepted for publication: 2007-07-09
Povr{ino zlitine Cu-Sn-Zn-Pb smo obsevali z laserskimi impulzi du{ikovega laserja (valovna dol`ina 337 nm). Pri tem sta se
spremenila tako topografija povr{ine kot tudi mikrostruktura pod njo. Ker je s klasi~nimi metalografskimi metodami zelo te`ko
primerno pripraviti obsevano povr{ino za mikroskopsko opazovanje, smo kot temeljno orodje za metalografsko preiskavo
uporabili fokusirani ionski curek (FIB), kajti FIB lahko odstranjuje material na specifi~nih mestih v mikro- in nanoobmo~ju in
odkrije mikrostrukturo brez prej{nje metalografske priprave. To nam je omogo~ilo, da smo raziskali vpliv obsevanja z laserjem
na spremembo oblike povr{ine, ugotovili profil kraterjev ter mikrostrukturne spremembe v obmo~ju toplotnega vpliva.
Klju~ne besede: laserska ablacija, bakrova zlitina, fokusirani ionski curek (FIB), mikrostruktura, topografija povr{ine
Surface of a Cu-Sn-Zn-Pb alloy was irradiated by ultraviolet nitrogen laser pulses (wavelength 337 nm). As a result both surface
topography and microstructure beneath the surface changed. Since it is very difficult to adequately prepare the damaged regions
for microscopical observations using classical metallographic methods, we used a focussed ion beam (FIB) as the main tool for
microstructural characterisation. Namely, FIB can remove material at specific sites in micro- and nanoregions and reveal
microstructure without any previous metallographic preparation. This allowed us to investigate the influence of laser pulses on
change of surface topography and subsurface microstructure.
Key words: laser ablation, copper alloy, focussed ion beam, microstructure, surface topography
1 UVOD
Obdelava povr{in kovinskih gradiv z laserjem se
hitro razvija. Laserski `arki krajevno segrejejo povr{ino
materiala na zelo visoko temperaturo in u~inkujejo do
globine 10–100 µm. V odvisnosti od energije laserski
`arki segrevajo, talijo ali uparjajo snov oziroma ustvar-
jajo plazmo. Trajanje energijskega impulza je lahko 1 ns
ali manj. Kasnej{e ohlajanje lahko vodi do ponovnega
strjevanja z drobnozrnato mikrostrukturo, v jeklih se
lahko pojavi tudi premena avstenit/martenzit. Pri
nekaterih zlitinah je lahko ohlajanje dovolj hitro, da se
tvori steklasta faza.1
Pri laserski ablaciji uparjamo snov s povr{ine mate-
riala. Postopek med drugim uporabljamo za kemijsko
analizo 2 ter za nana{anje tankih prevlek 3. (V Slovarju
slovenskega knji`nega jezika 4 pojem laserska ablacija ni
opredeljen, najbolj soroden pomen besede "ablacija" je
definiran v geologiji: odna{anje sipkega zemeljskega
materiala z de`jem, odplakovanjem.)
Fokusirani ionski curek (FIB), ki navadno uporablja
galijeve ione, ima premer od 5 nm do nekaj mikro-
metrov. Ko deluje kot mikroskop, je njegova lo~ljivost
nekoliko manj{a, kot je lo~ljivost vrsti~nega elektron-
skega mikroskopa, vendar ima bistveno bolj{i orienta-
cijski kontrast. Z njim lahko odvzemamo ali nana{amo
material na izbranih mestih z natan~nostjo vsaj 100 nm.
Ta zna~ilnost omogo~a, da se uporablja v najrazli~nej{e
namene, od popravila elektronskih vezij, preko
3D-mikroskopije, do izdelave najrazli~nej{ih 3D-objek-
tov v nano- in mikrometrskem obmo~ju. Kombinacija
fokusiranega ionskega curka in vrsti~nega elektronskega
mikroskopa bistveno izbolj{a zmogljivosti obeh. 5,6
Glavni cilj tega dela je prikazati uporabnost foku-
siranega ionskega curka pri opredelitvi vpliva obsevanja
z laserskimi `arki na spremembo topografije povr{ine in
mikrostrukture pod povr{ino.
2 INTERAKCIJA LASERJA S POVR[INAMI
KOVIN
Pri laserski ablaciji s pulzirajo~o ultravijoli~no
svetlobo potekajo {tevilni kompleksni procesi 7. Kadar
kratek laserski pulz osvetli povr{ino kovine, lasersko
energijo takoj absorbirajo prosti elektroni, pri procesu, ki
je nasproten zavornemu sevanju. Absorbirana energija se
takoj spremeni v gibanje mre`e v obliki elektronskih in
fononskih nihanj v obdobju nekaj pikosekund, kar
povzro~i segrevanje povr{ine. Porazdelitev temperature
v materialu po laserskem impulzu lahko izra~unamo z
ena~bo za prenos toplote. Ko obsevalna doza prese`e
ablacijski prag, se snov na osvetljeni povr{ini najprej
stali, nato pa za~ne izparevati. S tem za~ne povr{ina
oddajati delce – poteka ablacija kovinske tar~e. Blizu
ablacijskega praga je para tako razred~ena, da lahko
Materiali in tehnologije / Materials and technology 41 (2007) 5, 243–247 243
UDK 669.35:620.179.1 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)243(2007)
zanemarimo njeno interakcijo z lasersko svetlobo. Pri
ve~jih dozah je prehod snovi iz pregrete povr{ine zelo
hiter (tudi z eksplozivnim izparevanjem), tako da nastane
plinski oblak z veliko gostoto, v katerem je del delcev
tudi ioniziran. Pojavi se tudi interakcija med oblakom in
laserskimi `arki, v kateri se porabi ve~ina intenzitete
laserskega `arka. Tako se dele` energije, ki dose`e
kovinsko povr{ino, mo~no zmanj{a. Po drugi strani
absorbirana energija segreva plinski oblak in inducira
nastanek plazme. Plazma mo~no absorbira laserske `arke
z inverznim zavornim sevanjem in lo~i laserski impulz
od povr{ine; plazma dejansko {~iti povr{ino, tako da le
malo laserskega `arka dose`e povr{ino kovine. Poleg
tega plazma lo~i zrak od ablacijske povr{ine in omeji
povr{insko reakcijo, ~eprav je temperatura povr{ine
dovolj visoka. Tako ima pove~anje doze majhen vpliv na
kemijsko sestavo v kraterju zaradi za{~itnega u~inka
plazme (sen~enja).
Pri du{ikovem laserju je navadno gostota energije v
gori{~ni to~ki neenakomerna, kar povzro~i na o`ar~eni
povr{ini velike temperaturne gradiente 8. V sredi{~u
laserske to~ke se material upari in tudi ionizira ter ima
veliko te`njo, da se {iri v vzdol`ni in pre~ni smeri.
Sosednja obmo~ja, ki so se samo stalila, so zato
izpostavljena velikemu tlaku v pre~ni smeri, ki ga
povzro~a plinski oblak med {irjenjem. Rezultat je
po{kodba povr{ine; nastane krater z dvignjenim robom,
na sosednjo povr{ino pa lahko izvr`e kapljice teko~e
snovi. Pri pulzirajo~em delovanju laserja se obsevano
obmo~je izmeni~no segreva in ohlaja, kar vodi do
spremembe povr{ine kot tudi mikrostrukture pod njo.
3 EKSPERIMENTALNO DELO
Za preiskavo smo uporabili vzorec rde~e litine
(Cu-Sn-Zn-Pb), ki je imel naslednjo kemijsko sestavo:
6,12 % Sn, 5,75 % Zn, 3,06 % Pb, 0,52 % Ni, 0,31 % Fe,
drugo Cu.
Povr{ino vzorca smo obsevali s kratkimi impulzi
(trajanje 6 ns, frekvenca 1 Hz) ultravijoli~ne svetlobe
du{ikovega laserja, ki je imela valovno dol`ino 337,1
nm. Povpre~na energija laserskega impulza je bila
3 × 10–3 J, obsevano obmo~je na vzorcu pa je bilo veliko
0,75 mm2. Obsevali smo tri obmo~ja, in to z desetimi,
dvajsetimi in stotimi impulzi. Energijska gostota ni bila
enaka po celotni obsevani povr{ini, kar je zna~ilno za to
vrsto laserja 8. S spektroskopskimi meritvami (monokro-
mator SPEX) smo ugotovili, da je bila elektronska tem-
peratura plazme neposredno nad kraterjem ≈ 14 000 K.
Povr{ino obsevanega obmo~ja smo opazovali z
vrsti~nima elektronskima mikroskopoma Quanta 200 3D
in Sirion 400 NC (oba Fei Company). Za mikrokemi~no
analizo EDS smo uporabili sistem INCA 350 (Oxford
Analytical). Glavno delo je potekalo s fokusiranim
ionskim curkom (FIB), ki je sestavni del mikroskopa
Quanta 200 3D. Z elektronsko mikroskopijo smo najprej
poiskali obmo~ja, ki so bila obsevana z lasersko
svetlobo. Nato smo vzorec nagnili za 52°, tako da je
ionski curek padal na povr{ino pod kotom 90°. Za ionsko
mikroskopijo smo uporabili majhne toke ionov (navadno
10 pA), medtem ko smo za grobo odvzemanje materiala
uporabili tokove 3–20 nA, za srednje grobo odvzemanje
tokove 0,5–1 nA in za glajenje prerezanih povr{in
0,1–0,5 nA.
4 REZULTATI IN DISKUSIJA
Mikrostrukturo zlitine Cu-Sn-Zn-Pb sestavljata trdna
raztopina na osnovi bakra (αCu) in evtekti~ni oto~ki (αCu
+ βPb). Svinec je namre~ malo topen v bakru in se
razme{a `e v teko~em stanju 9, medtem ko sta Sn in Zn
precej bolj topna 10,11 in se substitucijsko vgradita v trdno
raztopino αCu. Povr{ina zlitine je bila pred obdelavo z
laserjem stru`ena, sledovi so delno opazni na desni strani
slike 1. Zato je bila plast pod povr{ino mo~no
deformirana, le na povr{ini je bila tanka plast po vsej
verjetnosti dinami~no rekristaliziranih enakoosnih
kristalnih zrn. Slika 1 prikazuje povr{ino, kjer je bila
vro~a to~ka laserske svetlobe. Pri obsevanju se je talina
segrela nad vreli{~e, zato je na tistem mestu nastal krater
(K). Njegov rob (R) je dvignjen nad povr{ino, vidni so
u~inki hitrega segrevanja in ohlajanja. Z ve~anjem
{tevila impulzov so postajali kraterji ve~ji in globlji. Po
desetih impulzih je bila velikost kraterja 40 µm × 20 µm,
globina 15 µm; po dvajsetih impulzih: 42 µm × 20 µm,
globina 15 µm ter po stotih impulzih: 50 µm × 40 µm,
globina 30 µm.
V okolici kraterja so na nekaterih mestih vidne
kapljice (D), ki jih je raz{irjajo~ plinski oblak v vro~i
to~ki razpr{il naokoli. V drugih obmo~jih se je povr{ina
segrela le nad temperaturo likvidus, nastala je kapilarna
valovitost (KV), katere valovna dol`ina je okoli 5 µm.
F. ZUPANI^ ET AL.: POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
244 Materiali in tehnologije / Materials and technology 41 (2007) 5, 243–247
K – krater (crater), R – rob kraterja (crater edge), KV – kapilarna
valovitost (capilary vawes), D – kapljica (droplet)
Slika 1: Krater po dvajsetih laserskih impulzih (SEM, sekundarni
elektroni)
Figure 1: Crater after twenty laser pulses (SEM, secondary electrons)
Dodatne informacije na tem mestu smo dobili, ko smo s
FIB naredili pre~ni rez. Mikroposnetek (slika 2) razkriva
reliefnost povr{ine; relativna vi{inska razlika med vrhovi
in dolinami je nekaj mikrometrov. Na njih so {e manj{e
izbokline, ki merijo v vi{ino nekaj desetink mikrometra,
med njimi pa je povr{ina gladka. Mikroposnetek tudi
razkriva, da je povr{ina prekrita s pribli`no 100 nm
debelo sivo plastjo (C), z njo pa so napolnjene tudi
povr{inske vdolbine. Analiza EDS je pokazala, da je v
glavnem iz ogljika. Njeno navzo~nosti si lahko
pojasnimo le s tem, da je bila povr{ina pred obdelavo
kontaminirana s snovjo, ki vsebuje ogljik. Pod povr{ino
je okoli 8 µm debela plast usmerjenih kristalnih zrn αCu,
ki v pre~ni smeri merijo od nekaj desetink mikrometra
do enega mikrometra. (Razdalje v vodoravni smeri lahko
dobimo neposredno z merjenjem na osnovi ~rtice,
medtem ko moramo v navpi~ni smeri vsako razdaljo
deliti s cos 52°, kajti ionski curek je bil nagnjen za 52°
proti povr{ini pre~nega prereza.) Najverjetnej{a razlaga
za njihovo navzo~nost je, da se je ≈ 8 µm debela zunanja
plast zlitine stalila, pri strjevanju pa so kristalna zrna
rasla v smeri pravokotno na povr{ino zaradi hitrega
odvoda toplote v nasprotni smeri. V tej plasti sta opazna
tudi dva oto~ka evtektika (αCu + βPb), ki sta ozna~ena s P.
Poudariti moramo, da kontrast med kristalnimi zrni αCu
izvira iz razli~ne kristalografske orientacije kristalnih zrn
glede na smer ionskega curka. Velja namre~, da so
kristalna zrna, ki imajo smeri z majhnimi vrednostmi
Millerjevih indeksov vzporedne z ionskim curkom,
temna, druga pa so svetlej{a. Razlog za to je, da ioni v
smereh z majhnimi Millerjevimi indeksi zaradi kanalske-
ga pojava prodrejo globoko v material, pri tem pa na
povr{ini inducirajo le malo sekundarnih elektronov 6.
Pod to plastjo je obmo~je, ki ima enakomeren meglen
videz; to je gotovo za~etna hladno deformirana mikro-
struktura, ki se pri obsevanju ni spremenila.
Slika 3 prikazuje pre~ni prerez kraterja, ki je nastal
po obsevanju z desetimi laserskimi impulzi. Pre~ni
prerez poteka skozi ravnino, v kateri je krater najdalj{i in
najgloblji. Dobro je viden zelo razgiban relief povr{ine v
okolici kraterja. Sam krater meri v dol`ino pribli`no 40
µm in sega okoli 15 µm pod za~etno povr{ino. Na robu
kraterja je del materiala izvr`en nad za~etno povr{ino –
to je bil material, ki je bil staljen in hitro ohlajen. Slika
3a je posnetek z odbitimi elektroni, ki dajejo Z-kontrast;
obmo~ja z ve~jim vrstnim {tevilom Z so svetlej{a. Na
F. ZUPANI^ ET AL.: POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
Materiali in tehnologije / Materials and technology 41 (2007) 5, 243–247 245
P – evtektik αCu + βPb (eutectic αCu + βPb), C – snov, bogata z
ogljikom (carbon-rich substance), KV – kapilarna valovitost (capilary
vawes), ZP – zgornja povr{ina (upper surface), PP – pre~ni prerez FIB
(FIB cross-section)
Slika 2: Mikrostruktura na mestu, obsevanem z manj{o energijsko
gostoto (slika s sekundarnimi elektroni, ki so jih inducirali ioni,
{tevilo laserskim impulzov: 20)
Figure 2: Microstructure of the alloy at a site irradiated with lower
energy density (secondary electrons induced by ion beam, number of
pulses 20)
K – krater (crater), KV – kapilarna valovitost (capilary vawes), ZP –
zgornja povr{ina (upper surface), PP – pre~ni prerez FIB (FIB cross-
section), P – evtektik αCu + βPb (eutectic αCu + βPb)
Slika 3: Mikroposnetka obmo~ja na mestu z najve~jo energijsko
gostoto pri {tevilu laserskih impulzov N = 10 po rezanju s FIB: a)
slika z odbitimi elektroni, b) slika s sekundarnimi elektroni, ki so jih
inducirali ioni.
Figure 3: Micrographs of the site irradiated with the high-energy
density after 10 laser pulses. a) secondary electron image, b) FIB-
induced secondary electron image.
sliki lahko opazimo obmo~ja s tremi razli~nimi odtenki.
Zelo svetla obmo~ja so bogata s svincem (P), na temnih
mestih zunaj kraterja ter tudi na povr{ini kraterja je snov
bogata z ogljikom ter sivkasta obmo~ja zlitinske osnove.
Slika s sekundarnimi elektroni, ki so jih inducirali ioni
(slika 3b), nima tako dobrega faznega kontrasta, so pa
zelo poudarjene topolo{ke zna~ilnosti, delno pa se `e
razkrije mikrostruktura v pre~nem prerezu. Mikro-
struktura je spremenjena pribli`no do 5 mikrometrov pod
dnom kraterja, kjer so opazni trakovi kristalnih zrn, pod
tem obmo~jem pa v mikrostrukturi ne opazimo posebnih
zna~ilnosti, razen por, ki so ve~inoma v stiku z obmo~ji,
ki so bogata s svincem.
Krater in njegova okolica sta bila po stotih impulzih
`e v ve~jem obsegu prekrita s snovjo, bogato z ogljikom
(oznaka C, slika 4a). Slika 4b prikazuje dno kraterja.
Vidno je, da staljena plast ni bistveno debelej{a kot na
drugih mestih (8-10 µm). Novonastala kristalna zrna so
razli~nih velikosti; nekatera merijo le nekaj desetink
mikrometra, druga pa tudi do 2 µm. Kristalna zrna so
ve~inoma enakoosna, kar verjetno pomeni, da se je po
ve~kratnem obsevanju segrela tudi podlaga, zato se
odvod toplote v smeri pravokotno na povr{ino zmanj{a.
Pod plastjo enakoosnih kristalnih zrn je prav tako
enakomerno sivo, deformirano obmo~je. Morda se je
dodatno deformiralo tudi zaradi udarnega delovanja
plinske faze, ki nastane pri obsevanju z laserjem. Da bi
to dokazali, bi bilo treba spremeniti za~etno stanje
povr{ine. Najbolj smiselno se zdi, da bi z `arjenjem
dosegli povsod enako velika enakoosna kristalna zrna
αCu, da bi lahko po obdelavi z laserjem la`e ugotovili,
kaj se je dogajalo med obsevanjem.
5 SKLEPI
Neenakomerna energijska gostota ultravijoli~ne
laserske svetlobe du{ikovega laserja je povzro~ila
spremembo povr{inske topografije in mikrostrukture pod
povr{ino. Zaradi povr{inske kontaminacije je bilo
obmo~je, obsevano z laserjem, prekrito s tanko plastjo
snovi, bogate z ogljikom. Pod njo je okoli 5–10 µm
debela staljena in hitro strjena plast, v njej pa so v
odvisnosti od {tevila impulzov ter energijske gostote
usmerjena in/ali enakoosna kristalna zrna. V vro~i to~ki
laserskega `arka je nastal krater, ki je v odvisnosti od
{tevila impulzov segal v globino 10–20 µm; staljen in
izvr`en material pa je tvoril rob kraterja, ki se je dvigal
nekaj mikrometrov nad za~etno povr{ino.
Fokusirani ionski curek se je pri raziskavi topografije
povr{ine in mikrostrukture zlitine Cu-Sn-Zn-Pb v
obmo~ju, ki je bilo obsevano s kratkimi impulzi
du{ikovega laserja, pokazal kot zelo primerno orodje za
odvzemanje materiala in metalografsko preiskavo. Poleg
tega je mogo~e na pre~nih prerezih, izdelanih s FIB-om,
izvesti tudi analizo EDS, s katero dobimo {e dodatne
informacije o kemijski sestavi pod prosto povr{ino
raziskanega materiala.
Zahvala
Avtorji se zahvaljujejo doc. dr. Lidiji ]urkovi}
(Sveu~ili{te u Zagrebu, Fakultet za strojarstvo i bro-
dogradnju) za kemijsko analizo preiskovane zlitine.
6 LITERATURA
1 R. E. Smallman: Modern physical metallurgy and materials engi-
neering : science, process, applications, 6th ed., Oxford, Butterworth
Heinemann, 1999
2 O. V. Borisov, X. L. Mao, A. Fernandez, M. Caetano, R. E. Russo:
Spectrochimica Acta Part B 54 (1999), 1351–1365
F. ZUPANI^ ET AL.: POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
246 Materiali in tehnologije / Materials and technology 41 (2007) 5, 243–247
K – krater (crater), KV – kapilarna valovitost (capilary vawes), ZP –
zgornja povr{ina (upper surface), PP – pre~ni prerez FIB (FIB
cross-section), L – pretaljena plast (melted and resolidified layer), C –
snov bogata z ogljikom (carbon-rich substance)
Slika 4: Mikroposnetka obmo~ja na mestu z najve~jo energijsko
gostoto pri {tevilu laserskih impulzov N = 100: a) slika s sekundarnimi
elektroni (pogled od zgoraj), b) pre~ni prerez (sekundarni elektroni, ki
so jih inducirali ioni).
Figure 4: Micrographs of a site irradiated with the highest energy
density after 100 laser pulses. a) secondary electron image (top view),
b) FIB-induced secondary electron image (FIB cross-section).
3 N. Patel, G. Guella, A. Kale, A. Miotello, B. Patton, C. Zanchetta, L.
Mirenghi, P. Rotolo: Applied Catalysis A: General 323 (2007) 18–24
4 Slovar slovenskega knji`nega jezika, Ljubljana DZS, 1994, str. 2
5 F. Zupani~: Vakuumist 26 (2006), 4–9
6 J. Orloff, M. Utlant, L. Swanson: High Resolution Focused Ion
Beams, FIB and Its Applications, Kluwer Academic/Plenum
Publishers, New York, 2003
7 D. W. Zeng, K. C. Yung, C. S. Xie: Applied Surface Science 217
(2003), 170–180
8 Z. Andrei}, V. Hen~ - Bartoli}, D. Gracin, M. Stubi~ar: Applied
Surface Science 136 (1998), 73–80
9 D. J. Chakrabarti, D. E. Laughlin (Cu-Pb) in T. B. Massalski (Ed.),
Binary Alloy Phase Diagrams, Second Edition, ASM International,
1990, 1452–1454
10 N. Saunders, A. P. Miodownik (Cu-Sn) in T. B. Massalski (Ed.),
Binary Alloy Phase Diagrams, Second Edition, ASM International,
1990, 1481–1483
11 A. P. Miodownik (Cu-Zn), in: T. B. Massalski (Ed.), Binary Alloy
Phase Diagrams, Second Edition, ASM International, 1990,
1508–1510
F. ZUPANI^ ET AL.: POVR[INA ZLITINE Cu-Sn-Zn-Pb PO OBSEVANJU Z ULTRAVIJOLI^NIM DU[IKOVIM LASERJEM
Materiali in tehnologije / Materials and technology 41 (2007) 5, 243–247 247

L. GUSHA ET AL.: A PRELIMINARY S-N CURVE FOR THE TYPICAL STIFFENED-PLATE PANELS ...
A PRELIMINARY S-N CURVE FOR THE TYPICAL
STIFFENED-PLATE PANELS OF SHIPBUILDING
STRUCTURES
PRELIMINARNA KRIVULJA S-N ZA TOGE PLO[^ATE PANELE
ZA LADJEDELNI[KE STRUKTURE
Luljeta Gusha1, Skender Lufi2, Marenglen Gjonaj2
1Technological University "I. Q. Vlora", Marine Faculty, Naval Engineering Department, Lagja: Pavaresia, Rruga: Sadik Zataj, Vlora-Albania
2Politechnical University of Tirana, Mechanical Faculty, Mechanical Department, Sheshi: Nene Tereza, Tirana-Albania
gushaaul.com.al
Prejem rokopisa – received: 2006-05-17; sprejem za objavo – accepted for publication: 2007-07-10
This paper presents the results of a preliminary study focused on the structural behavior of typical stiffened plate panels used for
shipbuilding structures and their fatigue strength under a lateral load. The investigated panels are thin plates, welded with
longitudinal bulb stiffeners through alternate welding seams. This makes the panel a composite structural element with a
complex strength behavior. The aim of the research was to obtain data about the failure conditions of the panels. Testing covers
the bending tests carried out on the real-size panels of shipbuilding structures. A reliable definition of a fatigue design curve
was not possible due to the limited number of specimens, although a tentative S-N curve was drawn on the basis of the test data.
Key words: shipbuilding panels, fatigue, real-size testing, S-N data
^lanek predstavlja rezultate preliminarne {tudije ciljane na strukturno vedenje tipi~nih togih ladijskih panelov in utrujenostno
trdnost pri bo~ni obremenitvi. Paneli so tanke plo{~e zvarjene z podol`nimi rebri za prepre~enje izbo~enja z alternativnimi
spoji. Taki paneli so kompozitni strukturni elementi s kompleksnim trdnostnim vedenjem. Cilj raziskave je bil opredeliti
podatke o pogojih za nastanek preloma panelov. Preizkusi so obsegali upogib panelov realne velikosti za ladijske strukture.
Zanesljiva opredelitev krivulje S-N ni bila mogo~a zaradi omejenega {tevila preizkusnih panelov. Zato je bila dolo~ena le
poizkusna krivulja S-N na podlagi rezultatov preizkusov.
Klju~ne besede: ladjedelni{ki paneli, utrujenost, preizku{anje v realni velikosti, podatki S-N
1 INTRODUCTION
Stiffened plate panels are the basic structural
components of a ship’s structure. Fatigue constitutes a
major source of local damage in ships and other marine
structures, since the most important loading on the
structure, the wave-inducted loading, consists of large
numbers of load cycles of alternating sign. The
prevention of fatigue failure in ship structures is strongly
dependent on proper attention to the design and
fabrication of structural details. Much of the quantitative
information on fatigue obtained by experiments and S-N
fatigue design curves is drawn on the basis of test data.
With the test results, and based on Wöhler’s diagram,
a tentative attempt is made to construct an S-N curve for
the stiffened panels, where the thin plates are welded
with longitudinal bulb stiffeners through alternate
welding seams.
According to IIW documents, more than 15
specimens are in general necessary to establish the
fatigue limit and more than 25 for the S-N curve, using
static analysis methods (e.g., the staircase method) 8.
2 EXPERIMENTAL
The experimental measurements were made at the
DINAV of the Naval Structural Laboratory of Università
degli Studi di Genova.
2.1 Data on the panel and model description
As a model for the experimental test, a stiffened plate
panel of real size, simply supported, was considered. The
effects of stress and initial distortion were not
considered.
Panel-type stiffeners were welded in the span
between the transversal T-beams, using alternate
welding seams with a length of 50 mm and a step of 200
mm. It is worth pointing out that at a 50-mm interval the
stiffeners are welded to the plate, alternatively on one or
the other. The panels were built according to standard
fabrication practice using semi-automatic arc welding.
The welding parameters are as follows:
Wire: FRO Fluxofil 19, d = 1 mm
Voltage: 23/24 V, Current: 140/150 A
Welding speed: 50 cm/min
Throat: 3.5 mm
Materiali in tehnologije / Materials and technology 41 (2007) 5, 249–253 249
UDK 539.44 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 41(5)249(2007)
Table 1: Geometrical characteristics of the panel
Tabela 1: Geometrijske zna~ilnosti panelov
Plate dimensions (1800 × 2600 × 5) mm
Stiffeners(HP80X6) Ix = 39.0 cm
4 Wmin = 8.15 cm
3
Effective plate width
included (s = 500 mm)
Ix = 155 cm
4 Wmin = 21 cm
3
Transversal beams (180 × 90 × 5 × 8) mm(180 × 5; 90 × 8) mm
Reference standard: MM-042F-331
Welding on unpainted surfaces
Table 2: Characteristics of materials
Tabela 2: Lastnosti materiala
Yield stress y = 355 N/mm
2
Yield load (s = 500 mm) Py = 49.7 MPa
Young’s modulus Ex = 2·10
5 MPa
Shear modulus Gxy = 0.793·10
5 MPa
Poisson’s modulus 
 = 0.33
Density of the material d = 7.9·10–5 kg/mm3
All the panels were manufactured from the same
material. The panel is modeled as shown in Figure 1,
and is considered to be supported on two T-beams along
both its longest sides (2600 mm) and along two other
free sides (Figure 2).
The assumed failure criteria are:
• A crack propagation over the section of the bulb,
• A number of cycles N = 1.0 × 106
2.2 Procedure and experimental tests
The bending was achieved with a transversal beam
along the whole panel width. The loading beam had an
"I" section with two large flanks of size (1800 × 200)
mm and a thickness of 20 mm 1,12. Its inertia moment is
large enough to ensure a constant distribution of the load
along the beam. Between the flat and the panel surface, a
thick rubber strip was interposed during the tests in order
to prevent damage to the surface. The strip had a width
of 200 mm and this should be regarded as the area of the
200 MPa load application. The load jack is hinged to a
frame and acts vertically downward, as shown in
Figures 2 and 3. It was selected on the basis of the
predicted limit load of 10 t. A load cell was interposed
between the jack and the loading beam 3,12. Seven linear
strain gauges for each panel and one load cell (200 MPa
max. load), as in Figure 3, were applied for each test.
Two rows of linear strain gauges were placed near the
loading "I" beam (Figure 3). These strain gauges
measure the strain in the longitudinal direction of the
bulb after each loading sequence. The panels were tested
in the range of about 0.7 yield stress with a sinusoidal
pulsating load. The maximum stress during the fatigue
test did not exceed the yield stress 7,8,9,10. The load and
strain were continuously monitored. The tests were
monitored with strain-gauge measurements and visual
inspections. The signals were stored in the data files in
millivolts and converted into the real physical quantities
by means of a calibration (Figure 4). The displacement
transducers were placed in their position and then
calibrated "in situ" with mechanical gauges.
The acquisition program was run for every test at a
sampling rate of 1.4–1.5 Hz 1,8 for a predetermined
period of time. For each acquisition the maximum,
minimum, mean and range values were evaluated and
subsequently converted into stress and the acting force.
The range versus cycles and the maximum range
versus cycle plots are presented in Figure 5 and 6.
L. GUSHA ET AL.: A PRELIMINARY S-N CURVE FOR THE TYPICAL STIFFENED-PLATE PANELS ...
250 Materiali in tehnologije / Materials and technology 41 (2007) 5, 249–253
Figure 1: Sketch (a) and isometric (b) view of the tested panel
Slika 1: Skica (a) in izometri~en (b) pogled preizkusnega panela
Figure 2: View of the testing device
Slika 2: Preizkusna naprava
a)
b)
2.3 Results and discussion
The results are shown in graphical form in Figures 5,
6 and 7. The panel was loaded with a pulsating
sinusoidal loading wave with a range of about 41 MPa,
at a mean level of about 25 MPa for 106 cycles (stress
ratio R 
 0.1). No cracks were found.
The range load was then increased up to 52 MPa, at a
mean level of 33 MPa (stress range R 
 0.1) for 1.35 ×
106 cycles. Then another 106 cycles were applied at 46.5
MPa, at the same mean level. No cracks were detected
with a visual examination and strain-gauges signal
analysis.
At about 1.45 × 106 cycles a crack started from the
head of the central bulb stiffener about the span middle
and propagated up to the plating, while a second crack
started in the west side bulb stiffener in the same
position and propagated up to the stiffener web. From
the analysis of the gauge measurements, it is concluded
that the crack propagated for about 104 cycles.
The boundary conditions of the panel are considered
as simply supported, the load is applied in the center of
the panel and the panel is in a positive bending moment
condition with the maximum of the bending moment in
the center of the panel.
L. GUSHA ET AL.: A PRELIMINARY S-N CURVE FOR THE TYPICAL STIFFENED-PLATE PANELS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 249–253 251
Figure 8: Composite cross-section of beam
Slika 8: Pre~ni prerez rebra
Figure 5:. Test measurements plot: range vs. cycles.
Slika 5: Rezultati meritev obremenitev v odvisnosti od {tevila
amplitud
Figure 4: Gauges and load-cell measurements at the end of the test
after about 1.45 × 106 cycles
Slika 4: Doze in meritve obremenitvenih celic pri koncu preizkusa pri
1,45 ⋅ 106 amplitudah
Figure 3: View of the position of the strain gauges
Slika 3: Polo`aji merilnih doz
Figure 6: Test measurements plot: max vs. cycles
Slika 6: Rezultati meritev obremenitev v odvisnosti od {tevila am-
plitud
Figure 7: Cracks in the central stiffener, both sides and the lateral
stiffener (after panel dismantling)
Slika 7: Razpok v centralnem rebru, obe strani in bo~na utrditev (po
demonta`i panela)
According to the elastic beam theory, in this case, the
neutral axis is near the plate, as in Figure 8, so the
maximum stress is achieved at the head of the bulb.
In the preliminary assessment using the finite-
element method, for these panels 11 and in static loading,
three critical areas of the panels’ collapse were
identified:
a) The region including the plate area between the
stiffeners (in compression);
b) The region including the plate area around the
stiffeners (in compression);
c) The region including the head area of the stiffeners
(in tension)
The specific stress-strain situation in each of these
regions defines the type of collapse that can occur in
them. The expected collapse in the region (a) is generally
of a static nature, while in the regions (b) and (c) it is
generally of a fatigue type. In this case, since the region
(b) is in compression, the region (c) is expected to
collapse in fatigue 2,4,10,11. The finite-element analysis
showed the region (c) as the hottest stress area between
the (b) and (c) regions.
3 PRELIMINARY S-N CURVE
Due to the limited number of specimens it was not
possible to obtain a curve. For this reason, a preliminary
S-N curve was drawn on the basis of the test data.
According to the IIW documents, more than 15 speci-
mens are necessary to establish a reliable fatigue limit
and more than 25 for the S-N curve, using static analysis
methods (e.g., the staircase method) 8. To construct the
S-N curve, we relied on Wohler’s curve. The shape of
this curve is shown in Figure 9 1,2,3,4,5,6,10.
Where:
• S is the upper point, static resistance r vs. 103 cycles.
• G is the point of the fatigue limit a vs. 2 × 106
cycles (or an endurance limit).
• In the lg σa – lg N diagram the curve slope is a con-
stant k.
• The point A is the point of the conventional refe-
rence, NA, σA
N Nk A
kσ σa A= (1)
k
lg
N
N
lg
=
10
10
A
a
A
σ
σ
(2)
A typical value of k = 3 is given in several references
1,2,3,13.
In our case, since the load is a sinusoidal pulsing
load: r ≈ 0.1, ∆σE ≅ σy = 355 N/mm2 (see HSS ∆σE =
330 –360 N/mm2), where ∆σE = σa∞, and σy is the stress
at the S point. The coordinates of the point G are 1.5·105
and 330, and the panel is loaded with 42 MPa at 106
cycles and with 52 MPa at 0.45·106 cycles.
From the De Saint Venant relation, σ =
M y
I
b
xx
, it is
deducted:
At 106 cycles, P = 42 MPa, A = 278.8 N/mm2 and at
0.45·106 cycles, P = 52 MPa, T = 278.8 N/mm2.
The proof of the service fatigue strength on the basis
of the damage is the accumulation rule according to
Miner:
D =
N
N
j
fjj
n
=
∑
1
(3)
D  Dper where Dper = 0.5 – 1.0 (4)
where D is the total damage, Dper is the permissible total
damage, Nj is the number of cycles on level j, Nfj is the
number of cycles of failure on the level j according to
the allowed stress S–N curve, j is the number of the
stress level and n is the total number of stress levels .
• Let us now deduce the value of k:
1 1
1
2
2
= +
N
N
N
Nf f
(5)
N Nf
k
f
k
1 1 2 2σ σ= (6)
Starting from the conventional point S(103, y) with
σy = 355 N/mm2 it is possible to write:
L. GUSHA ET AL.: A PRELIMINARY S-N CURVE FOR THE TYPICAL STIFFENED-PLATE PANELS ...
252 Materiali in tehnologije / Materials and technology 41 (2007) 5, 249–253
Figure 10: Wöhler curve for the panels
Slika 10: Wöhlerjeva krivulja za panele
Figure 9: Typical S-N curve for a mild steel
Slika 9: Tipi~na krivulja S-N za konstrukcijsko jeklo
10 3
2
1
1
2
2
σ
σ
σ
σ
y
k k
N N
⎛
⎝
⎜
⎞
⎠
⎟ =
⎛
⎝
⎜
⎞
⎠
⎟ + (7)
where, 1 = A = 278.8 N/mm
2, 2 = T = 345.2 N/mm
2
From the three mentioned equations a tentative value
of k = 10 is deduced. This is in agreement with
NAFEMS 3, because the crack propagated in the bulb
head, without weld seams and the material behaves as if
it is unwelded 2.
From equations 5,6 Nf1, Nf2 we deduced:
N
N
f
f
1
2
4810000
568091
=
=
⎧
⎨
⎩
In this step it is necessary to verify that a = E,
assuming a known number of cycles corresponding to
the endurance limit a = E at 2·106.
From the relation N Nk kσ σa A= we can write:
N Nf
k
f
k
1 σ σ1 E E= ⇒ σE = 166.006 N/mm2
Finally, it is possible to construct the preliminary
S–N curve for the panel shown in Figure 10.
The S-N curve with these characteristic points, based
on the damage-accumulation rule according to Miner
and the curve of Wohler, has the following coordinates.
A(Nf1, 1) = A(4810000, 278.8)
T(Nf2, 2) = T(568091 ,345.2)
G(NE, σE) = G(2·10
6, 166.006)
S(103, Y) = S(10
3, 166.006)
4 CONCLUSIONS
From the experimental point of view, it is evident
that the points A and T are situated in the safety zone.
The value of the curve’s slope is k = 10, and it can
increased up to k = 14 (the results can be in a scatter
band with the value of "k" from 10 up to 14).
Future results will permit a further revision and
implementation with the aim to obtain good agreement
between the Wöhler curve and the experimental data.
5 REFERENCES
1 G. Sines, J. L. Waisman, Metal fatigue, London, (1959)
2 S. J. Maddox, Fatigue strength of welded structure, England, 1991
3 B. P. Atzori, Introduzione alla progettazione a fatica, NNAMFES,
Italy, 2001
4 O. Hughes, Ship structural design, Sydney, 1983
5 O. Hughes, J. B. Caldwell, Marine structures selected topics,
examples and problems, Sydney, 1991
6 J. J. Jensen, Load and global response of ships, Denmark, 2000
7 I. S. O. Standarted proposal. Fatigue testing of welded components,
France, 1996
8 H. P. Lieurade, I. Huther, IIW Fatigue Testing Standard and Effect
of Quality and Weld Improvement Methods WRC Proceeding IIW,
1996
9 International Institute of Welding, IIW Recomendation on large scale
fatigue testing of welded components, Germany, 1999
10 D. Radaj, C. M. Sonsino, Fatigue assessment of welded joints by
local approaches, England, 1998
11 L. Gusha, S. Lufi, Strain-stress analysis of stiffened plate panels of
shipbuilding structures, ACTA Universitatis Pontica Euxinus,
Rumania 5 (2005), 2
12 L. Gusha, Typical stiffened plate panels in shipbuilding – ultimate
and fatigue strength. Universita degli Studi di Genova DINAV,
Genova (Internal Report 03CRI04), 2003
13 M. Gjonaj, F. Pejani, Detale Makinash. Kritere Kryesore te Lloga-
ritjes se Detaleve te Makinave, Tirana, Albania, 1987
L. GUSHA ET AL.: A PRELIMINARY S-N CURVE FOR THE TYPICAL STIFFENED-PLATE PANELS ...
Materiali in tehnologije / Materials and technology 41 (2007) 5, 249–253 253