VSEBINA – CONTENTS
PREGLEDNI ^LANKI – REVIEW ARTICLES
Surface modification of materials using an extremely non-equilibrium oxygen plasma
Modifikacija povr{ine materialov z izrazito neravnovesno kisikovo plazmo
M. Mozeti~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Middle triassic dry-land phases in Southern Slovenia
Srednjetriasne kopenske faze v ju`ni Sloveniji
S. Dozet, M. Godec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
IZVIRNI ZNANSTVENI ^LANKI – ORIGINAL SCIENTIFIC ARTICLES
Enhanced fatigue analysis – incorporating downstream manufacturing processes
Izbolj{ana utrujenostna analiza – vklju~itev izdelavnih procesov
W. Eichlseder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A new complex vector method for balancing chemical equations
Nova kompleksna metoda za uravnote`enje kemi~nih ena~b
I. B. Risteski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Application of Grey relation analysis (GRA) and Taguchi method for the parametric optimization of friction stir welding
(FSW) process
Uporaba Greyjeve analize (GRA) in Taguchijeve metode za parametri~no optimizacijo varjenja z vrtilno-tornim procesom (FSW)
H. Aydin, A. Bayram, U. Esme, Y. Kazancoglu, O. Guven . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
Phase relations in the Pt–Ag17.5–Si4.5 ternary alloy
Fazna odvisnost v ternarni zlitini Pt–Ag17.5–Si4.5
G. Klan~nik, P. Mrvar, J. Medved . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A new solution of the harmonic functions in the theory of elasticity
Nekatere zna~ilnosti harmoni~nih funkcij v teoriji o elastni~nosti
V. V. Chygyryns’kyy, V. G. Shevchenko, I. Mamuzic, S. B. Belikov. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Influence of vacuum processing on the content of some elements in molten metal
Vpliv vakuumskega procesiranja na vsebnost nekaterih elementov v kovinskih talinah
D. Pihura, D. Mujagi} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
STROKOVNI ^LANKI – PROFESSIONAL ARTICLES
Influence of 
-ferrite on the fatigue resistance of blade materials
Vpliv 
-ferita na utrujenostno trdnost jekel za lopatice
P. Hájková, J. Janovec, J. Siegl, B. Smola, M. Vyroubalová, D. Tùmová . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
18. MEDNARODNA KONFERENCA O MATERIALIH IN TEHNOLOGIJAH, 15. – 17. november, 2010, Portoro`, Slovenija
18th INTERNATIONAL CONFERENCE ON MATERIALS AND TECHNOLOGY, 15–17 November, 2010, Portoro`, Slovenia . . . . . . 233
ISSN 1580-2949
UDK 669+666+678+53 MTAEC9, 44(4)163–232(2010)
MATER.
TEHNOL.
LETNIK
VOLUME 44
[TEV.
NO. 4
STR.
P. 163–232
LJUBLJANA
SLOVENIJA
JULY–AUG.
2010

M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
SURFACE MODIFICATION OF MATERIALS USING AN
EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
MODIFIKACIJA POVR[INE MATERIALOV Z IZRAZITO
NERAVNOVESNO KISIKOVO PLAZMO
Miran Mozeti~
Odsek za tehnologijo povr{in in optoelektroniko, Institut "Jo`ef Stefan", Jamova cesta 39, 1000 Ljubljana, Slovenija,
miran.mozetic@ijs.si
Prejem rokopisa – received: 2010-01-04; sprejem za objavo – accepted for publication: 2010-02-05
Several technological processes based on the interaction of extremely non-equilibrium oxygen plasma are described. A plasma
with a low kinetic temperature of heavy particles and an extremely high density of neutral oxygen atoms is created in glass
plasma reactors by inductively coupled radio-frequency (RF) discharges. The density of the charged particles is kept low at
around 1016 m–3, while the density of the neutral oxygen atoms may exceed a value of 1 × 1022 m–3. The neutral oxygen atoms
are chemically active and readily react with solid materials, even at low temperatures. The interaction of the oxygen atoms with
metal samples often causes a rapid nucleation of metal oxide isles and the spontaneous growth of one- or two-dimensional
structures, such as nanowires and nanobelts. These "nanofeatures" are often monocrystalline. The exposure of different
carbon-rich materials to an oxygen plasma is applied for the functionalization with oxygen functional groups as well as for the
controlled oxidation of various materials. The superhydrophilicity of several polymers and composites can be achieved. The
technique is suitable for the destruction of bacteria and thus the sterilization of delicate materials.
Keywords: plasma, oxygen, metal oxide nanoparticles, polymer functionalization, sterilization
Opisani so nekateri tehnolo{ki postopki za obdelavo materialov z izrazito neravnovesno kisikovo plazmo. Stanje plinske plazme
z nizko kineti~no energijo te`kih delcev in zelo veliko gostoto nevtralnih kisikovih atomov se vzpostavi v induktivno sklopljeni
radiofrekven~ni plinski razelektritvi v steklenem plazemskem reaktorju. Gostota nabitih delcev v tak{ni plazmi je razmeroma
majhna in navadno ne presega vrednosti 1 × 1016 m–3, medtem ko lahko gostota nevtralnih kisikovih atomov prese`e vrednost
1 × 1022 m–3. Nevtralni kisikovi atomi so kemijsko izredno aktivni in reagirajo s povr{ino trdnih snovi `e pri nizki temperaturi.
Interakcija kisikovih atomov s povr{ino kovinskih materialov pogosto povzro~i bliskovito nukleacijo oto~kov kovinskega
oksida, iz katerih spontano rastejo eno- ali dvodimenzionalne strukture, kot so nano`ice in nanotrakovi. Nanomateriali so
pogosto monokristalini~ni. Materiali, ki vsebujejo ogljik, prav tako reagirajo z atomskim kisikom. @e majhna doza atomov
povzro~i modifikacijo povr{ine tovrstnih materialov z nastankom kisikovih funkcionalnih skupin. Tako obdelani polimerni
materiali so v~asih superhidrofilni. Podalj{ana izpostava organskih materialov kisikovim atomom vodi k postopni oksidaciji in s
tem po~asnemu jedkanju materialov. Pojav izkori{~amo pri sterilizaciji ob~utljivih organskih materialov, ki ne prenesejo
klasi~nih postopkov.
Klju~ne besede: plazma, kisik, nanodelci kovinskega oksida, funkcionalizacija polimerov, sterilizacija
1 INTRODUCTION
The increasing demands on the quality as well as the
miniaturization of products facilitate the development of
new technological processes. These new technologies
should ensure the good quality of products, low produc-
tion costs and should be environmentally acceptable.
Classical approaches to the treatment of materials have
been almost exhausted and industry is looking for new
ideas. The majority of breakthrough technologies that
have appeared in recent years are based on the applica-
tion of non-equilibrium environments, especially non-
equilibrium gaseous plasma.1 A variety of gases are used
to create a plasma with suitable properties1. A plasma of
particular interest is created with oxygen1. Molecular ox-
ygen is passed through an electrical discharge. The fast
electrons then excite the neutral molecules with the orig-
inal Maxwell-Boltzmann (MB) distribution causing a
dramatic shift to the non-equilibrium state. Apart from
neutral molecules in low ro-vibrational states, substantial
amounts of highly excited molecules, neutral atoms and
ionized molecules and atoms are created.2–3 While the
distribution of neutral molecules over translational states
is preserved close to MB due to the poor kinetic energy
exchange between fast electrons and slow neutrals dur-
ing elastic collisions, it is not true for internal energy dis-
tribution. Two excited states of molecular oxygen are
metastable (at the excitation energy of approximately
1 eV and 2 eV) and their concentration is far above the
values calculated using MB distribution at room temper-
ature.4 Furthermore, the dissociation degree is far from
the equilibrium value and the same applies for the ion-
ization fractions. Different temperatures that are suitable
for a description of such a non-equilibrium state of oxy-
gen depend on the particular conditions (discharge and
plasma parameters).5 The largest shift from MB distribu-
tion is often observed by passing the oxygen through the
radio frequency of microwave discharges created in glass
plasma reactors. At the neutral gas kinetic temperature
close to room temperature, the other temperatures (T) are
as follows: vibrational temperature 500–1000 K, dissoci-
ation temperature 100 000 K or even higher, ionization
temperature 10 000–50 000 K, electron temperature
Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171 165
UDK 533 ISSN 1580-2949
Review article/Pregledni ~lanek MTAEC9, 44(4)165(2010)
15 000–50 000 K.6 Such a non-equilibrium gas often acts
as an ordinary gas at room or slightly elevated tempera-
ture in terms of the heat dissipation on solid materials,
while it acts as an extremely hot gas (well over 10,000
K) in terms of chemical activity. Obviously, heavily
non-equilibrium oxygen is used for the rapid chemical
modification of solid materials at low temperature.7
2 DETERMINATION OF THE KEY PLASMA
PARAMETERS
Plasma is a partially ionized gas, and the most obvi-
ous plasma parameters are the density of the charged
particles (free electrons and positive ions) and their tem-
peratures (or the kinetic energy distribution functions in
the case the particles that are away from the MB distribu-
tion). Apart from these, another couple of parameters are
often stated, i.e., the plasma potential (Vs) and the Debye
length (
D). In many cases, both are calculated with
equations:8
V V
kT
e
m
ms f
e
e
− =
− +
2 20
ln (1)
and
 s
e
e 0
=
kT
N e2
(2)
In equations (1) and (2), Vs is the plasma potential
(often called the space potential), Vf is the floating poten-
tial, e0 is the elementary charge, k is the Boltzmann con-
stant, Te is the electron temperature, m+ is the positive
ion mass, me is the electron mass, Ne is the electron den-
sity, and 0 is the vacuum permittivity. These two equa-
tions are useful only as long as some requirements are
fulfilled. Obviously, one requirement is the MB distribu-
tion of electrons. The set of parameters Te, Ne, Vs, 
D is
useful for describing the electrical properties of the
plasma, but tell little about its chemical reactivity. Apart
from these classical parameters, many others are neces-
sary for a decent characterization of the processing
plasma, and they include the rotational and vibrational
temperature of the neutral and charged molecules, the
density of the neutral atoms, the density of the meta-
stable molecules and atoms, and the ionization fractions
of the molecules and atoms. In a plasma of electro-
negative gases, the density of the negatively charged
heavy particles is not negligible, either. Complete plasma
characterization is obviously a difficult task, so numer-
ous methods have been developed and applied more or
less successfully. Among them, electrical, magnetic and
catalytic probes9–18 are worth mentioning, along with
more complicated techniques, such as a variety of optical
emission19–21 and absorption techniques,22 the propaga-
tion of electromagnetic waves,6 and chemical titrations.23
A proper determination of the parameters is crucial for a
calculation of the fluxes of the reactive plasma particles
on the surface of the samples as well as the kinetic en-
ergy of the charged particles interacting with plasma fac-
ing materials. Although modern plasma laboratories are
well equipped with several different tools, plasma is of-
ten characterized insufficiently.
3 SINTHESIS OF METAL OXIDE
NANOPARTICLES
A novel technology based on the application of
non-equilibrium oxygen plasma is the synthesis of large
quantities of metal oxide nanoparticles.24–32 Metallic foils
are exposed to oxygen plasma with a dissociation frac-
tion of the order of 10 % (corresponding to a dissociation
temperature of 22 500 K).24 Neutral oxygen atoms are,
by far, more reactive than their parent molecules and
interact with the surface of solid materials both chemi-
cally and physically. Chemical reactivity causes the
formation of metal oxides, while the physical interaction
is predominantly demonstrated by the accommodation
(relaxation) of metastable molecules and the hetero-
geneous surface recombination of neutral oxygen
atoms.33–34 Furthermore, the bombardment of the metal
surface with positive ions from the plasma is observed as
well as the neutralization of charged particles. All the
chemical and physical reactions are heavily exothermic,
causing a localized disturbance of the surface atoms
from their (quasi)equilibrium positions. The surface
atoms become extremely mobile, so a disturbance of the
crystalline structure occurs.35–37 The extremely high
mobility of the surface atoms allows for the creation of
(meta)stable chains of metal oxides, stretching perpendi-
cularly from the surface plain, as shown by molecular
dynamics simulations.38 An example of such a chain is
shown in Figure 1. The chains act as nuclei for the
growing of nanofeatures from the surface of the metal
samples. As long as the mobility of the surface atoms is
much higher in the preferred direction than in other
orientations, the growth is one-dimensional, resulting in
M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
166 Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171
Figure 1: The results of a molecular dynamics simulation of the
interaction of oxygen atoms added to the surface of nickel foil. The
figure shows the formation of NiO chains. Dark and white particles
stand for the O and Ni atoms, respectively.
Slika 1: Rezultati simulacije molekulske dinamike interakcije
kisikovih atomov na povr{ini nikljeve folije. Slika prikazuje tvorjenje
verig NiO. Temni krogci prikazujejo kisikove atome, beli pa nikljeve.
long nanowires rather than other forms. Figure 2
represents a SEM image of such nanowires, growing
from the surface of niobium foil exposed to an extremely
large flux of oxygen atoms of close to 1 × 1024 m–2 s–1.24
It is assumed that the nanowires are monocrystalline
Nb2O5, the most stable form of niobium oxide. In the
case of a lower dissociation fraction (lower O flux at
about 3 × 1023 m–2 s–1), other features grow from the
surface of the metal foils. Figure 3 shows nanobelts of
rectangular shape growing from the surface of the
niobium foils. Fluxes below about 1 × 1023 m–2 s–1 do not
produce such interesting nanoparticles. Namely, at a low
flux, the surface mobility of the atoms is too low to allow
for the rapid growth of one- or two-dimensional
structures, so the oxidation mechanism is close to that
characteristic for a thermal (equilibrium) treatment.
Similar results are obtained with many other metals,
including Fe, V, Mo, Cr.29
4 SURFACE ACTIVATION OF POLYMER
MATERIALS
A popular application of oxygen plasma is the sur-
face functionalization of organic materials.39–56 Polymer
materials are usually hydrophobic with a fairly low sur-
face energy. This hydrophobicity is usually the conse-
quence of a poor concentration of polar functional
groups on the surface of the polymers. In thermal equi-
librium the concentration of the functional groups on the
surface is the same as the bulk concentration. The sur-
face properties of polymer materials are changed by the
incorporation of different functional groups on the very
surface.39–40,50–51,54–55 Obviously, the new surface func-
tional groups are not very stable and tend to decay spon-
taneously.47 Several techniques for the functionalization
of organic materials have appeared. Since the presence
of foreign functional groups on the surface is not thermo-
dynamically favorable, the obvious technique is the ap-
plication of a non-equilibrium gaseous plasma. As men-
tioned above, non-equilibrium oxygen plasma with the
kinetic temperature close to room temperature acts as an
almost perfect medium for the surface functionalization
with oxygen functional groups. Its extreme chemical re-
activity allows for rapid functionalization at minimal
thermal loading.43 The applicability of this technology is
demonstrated for the case of industrial LDPE (low-den-
sity polyethylene) polymer foil used for packaging (Fig-
ures 4 and 5).57 The polymer is exposed to oxygen
plasma at almost room neutral gas kinetic temperature, a
dissociation fraction of about 10 % and a rather low ion-
ization fraction of 1 × 10–6 (corresponding to the ioniza-
tion temperature of about 12 000 K). The material heat-
ing due to the surface neutralization of charged particles
is therefore negligible. Furthermore, the Debye length in
such plasma is much larger than the mean free path of
oxygen molecules or atoms. Therefore, the kinetic en-
ergy that the ions gain passing the potential difference
between space and floating potentials (see equation (1))
is effectively distributed to neutral molecules or atoms
before colliding with the polymer surface. The polymer
material is therefore the subject of extremely poor heat-
ing, but high chemical modifications. Assuming an infi-
nite planar geometry, the flux of ions (j+) and atoms (ja)
M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171 167
Figure 2: Nanowires of niobium pentoxide stretching from the surface
of a niobium foil during exposure to an extremely non-equilibrium
oxygen plasma with the following parameters: a neutral gas kinetic
temperature of about 500 K, an electron temperature of 20 000 K, an
ionization fraction of about 1 × 10–6 and a dissociation fraction of
about 13 %.
Slika 2: Nano`i~ke niobijevega pentoksida, rasto~e na povr{ini
niobijeve folije med izpostavo neravnovesni kisikovi plazmi z
naslednjimi parametri: temperatura nevtralnega plina okoli 500 K,
temperatura elektronov 20 000 K, dele` ionizacije okoli 1 × 10–6 in
dele` disociacije okoli 13 %
Figure 3: Rectangular nanoparticles of niobium pentoxide growing
from the surface of niobium foil during exposure to highly non-
equilibrium oxygen plasma with the following parameters: a neutral
gas kinetic temperature of about 500 K, an electron temperature of 20
000 K, an ionization fraction of about 8 × 10–7 and a dissociation
fraction of about 4 %.
Slika 3: Pravokotni nanodelci niobijevega pentoksida, rasto~i na
povr{ini niobijeve folije med izpostavo neravnovesni kisikovi plazmi z
naslednjimi parametri: temperatura nevtralnega plina okoli 500 K,
temperatura elektronov 20 000 K, dele` ionizacije okoli 8 × 10–7 in
dele` disociacije okoli 4 %.
onto the surface of the polymer does not depend on
sheath properties, and is as follows:58
j+ = ¼ n+ <v+> (3)
ja = ¼ na <va> (4)
Here, n+ is the density of positively charged ions,
<v+> is the average absolute random velocity of posi-
tively charged ions assuming an MB distribution, na is
the neutral atom density, and <va> is the average absolute
random velocity of neutral oxygen atoms, assuming an
MB distribution.
There are few reasons for the acceleration of heavy
charged particles (molecular and atomic ions) in
high-frequency discharges. Namely, as the frequency of
the electrical field approaches a few MHz, the local elec-
tromagnetic oscillations become too fast to be followed
by heavy particles. Only the electrons can be heated (can
gain kinetic energy) in radio-frequency fields. The distri-
bution of heavy charged particles over translational states
is thus the same as those for neutral particles – MB close
to room temperature. The average velocity of the posi-
tively charged ions in equation (3) is therefore the same
as for the neutral particles at room temperature (T): <v+>
= 440 m/s for molecular ions and 630 m/s for atomic
ions. The average velocity was calculated from equation
v kT m= 8 / π , where k is the Boltzmann constant and m
is the ion mass.58 Since the density of the molecules in
the plasma is an order of magnitude larger than the den-
sity of the atoms, and since the dissociative ionization is
energetically less favorable than the ordinary ionization
of the molecules, the contribution of atomic ions is easily
neglected and <v+> can be approximated to the value
characteristic for molecules, i.e., 440 m/s. The resultant
flux of ions (j+) on the polymer surface exposed to
plasma is thus:58
j+ = ¼ n+ <v+> = ¼ 3 × 10
16 m–3 × 440 m/s =
= 3.3 × 1018 m–2s–1 (5)
Ions are recombined on the surfaces of all solid mate-
rials at a very high rate, close to 100 %. The thermal load
per unit area caused by ion recombination is:42
P+ = j+ Wi = 3.3 × 10
18 m–2s–1 × 12 × 1.6 × 10–19 J =
= 6 W m–2 (6)
Here, P+ is the power dissipated by the oxygen ion
recombination per unit area and Wi is the ionization en-
ergy of the oxygen molecules, i.e., about 12 eV. The
power P+ is extremely low, so heating by ions is always
negligible, as long as the ionization fraction is that low.
The upper calculation is valid for the case when oxygen
molecular ions are found in the ground state. In practice,
this is never completely true, since there are always some
excited ions in oxygen plasma. Fortunately, the density
of ions with a high excitation energy in plasma with a
low ionization fraction and a rather low electron temper-
ature is usually negligible, so the upper calculation is a
good approximation. The positive ions are accelerated in
the sheath between the plasma and the surface (accord-
ing to equation (1)) so the heating by ions is larger than
the value given by (6). The maximum heating is obvi-
ously in the case of collision-less sheath when the ions
gain a kinetic energy of e0 (Vs – Vf)8. As explained above,
however, the ions lose their kinetic energy in the gas
phase as long as the Debye length is much larger than the
mean free path of the molecules.
The heating of polymer materials by the recombina-
tion of neutral atoms is often also negligible. The ther-
mal load per unit area due to this effect is calculated tak-
ing into account equation (4):59
M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
168 Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171
Figure 5. High-resolution C1s XPS peak for industrial LDPE polymer
foil treated undert same conditions as in Figure 4
Slika 5: Visoko lo~ljivi spekter XPS vrha C1s industrijske polimerne
folije LDPE, obdelane pri istih pogojih kot na sliki 4
Figure 4: The XPS (X-ray photoelectron spectroscopy) survey spectra
of untreated and plasma-treated industrial LDPE polymer foil. The
oxygen concentration in the surface film of the polymer is increased to
23 % after treatment with plasma with the following parameters: a
neutral gas kinetic temperature of about 400 K, an electron tempe-
rature of 20 000 K, an ionization fraction of about 1 × 10–6 and a
dissociation fraction of about 10 %. The treatment time was 3 s.
Slika 4: Pregledni spekter XPS (rentgenske fotoelektronske spektro-
skopije) neobdelane in plazemsko obdelane polimerne folije LDPE.
Koncentracija kisika na povr{ini polimera je narasla na 23 % po
obdelavi s plazmo z naslednjimi parametri: temperatura nevtralnega
plina okoli 400 K, temperatura elektronov 20 000 K, dele` ionizacije
okoli 1 × 10–6 in dele` disociacije okoli 10 %. ^as obdelave je bil 3 s.
Pa = 1/4 na <va>  WD/2 = 1/4 × 6 × 10
20 m–3 ×
× 630 m s–1 × 10–4 × 5.2 × 1.6 × 10–19 J / 2 = 4 W m–2 (7)
Here,  is the recombination coefficient and WD is the
dissociation energy, i.e., WD = 5.2 eV. The recombination
coefficient for flat polymer materials is low, often of the
order of 10–4. Taking into account this value, the power
dissipated on the polymer surface due to the recombina-
tion of atoms is approximately Pa = 4 W m–2. The total
contributions of the neutral atom recombination (Pa) and
the neutralization of ions (P+) are P = Pa + P+ and are
therefore about 10 W m–2. A polymer foil with a thick-
ness d of 10 µm is therefore heated at:58
dT/dt = P/( cp d) = 10 W m
–2 / (103 kg m–3 ×
× 1 kJ kg–1 K–1 × 10 × 10–6 m) = 1 K/s (8)
This value, of course, depends on the properties of a
particular polymer (the density  and the thermal capac-
ity cp) and is only a rough estimation. It depends also on
the surface roughness (rough materials are heated at a
faster rate) and the thickness (thicker materials are
heated at lower rates). Since the flux of atoms on the sur-
face is enormous (in the above calculation the value of
1 × 1023 m–2 s–1 was taken into account), the surface of the
polymer is saturated with oxygen functional groups in
much less than a second, so far before any substantial
heating of the material occurs.
5 STERILIZATION OF DELICATE MATERIALS
Since plasma treatment does not heat the materials
much, oxygen plasma is used for the sterilization of
delicate biocompatible materials.60–65 Namely, oxygen
plasma interacts with bacteria already at room tempe-
rature, causing the destruction of the bacterial cell wall
and, therefore, sterilization. The sterilization effect is
illustrated by Figures 6 and 7. Figure 6 is an SEM
image of the live bacterium Bacillus Stearothermophilus.
The bacteria were deposited on substrates and treated
with oxygen plasma with a neutral gas kinetic tempera-
ture of about 400 K, an electron temperature of 20 000
K, an ionization fraction of about 3 × 10–6 and a
dissociation fraction of about 10 %.62 The effect of the
plasma treatment is well illustrated by Figure 7. The
treatment time was 55 s. The reactive particles from the
oxygen plasma (mainly neutral oxygen atoms) readily
reacted with the bacteria. The capsule protecting the live
bacteria disappeared and the bacterial cell wall is badly
damaged. What remains from the bacteria after the
plasma treatment is just the remnants of the bacterial
cytoplasm. Obviously, some parts of the cytoplasm do
not react with the plasma particles as much as the
bacterial cell wall, let alone the capsule. The type of
bacteria was not chosen accidentally. Bacteria Bacillus
Stearothermophilus is famous for its resistance to
prolonged heating at elevated temperature.66 Even a half
an hour treatment at 100 °C is not sufficient to ensure the
sterility of objects contaminated with this type of
bacteria. Plasma sterilization is therefore a promising
method for the destruction of bacteria on biocompatible
objects that do not stand standard autoclaving in humid
air at 130 °C. Since the bacterium presented in Figure 7
is very badly damaged it can be concluded that plasma
treatment times shorter than a minute are sufficient for
the sterilization by extremely non equilibrium oxygen
plasma.
6 CONCLUSIONS
Some applications of extremely non-equilibrium oxy-
gen plasma were presented. An oxygen plasma with a
high degree of dissociation is created in a radio-fre-
quency discharge. The neutral gas kinetic temperature re-
mains close to room temperature, so the thermal loads on
substrates are minimal. The chemical reactivity of such
plasma, on the other hand, is extremely high. Such
plasma is suitable for the modification of both inorganic
and organic materials. The exposure of metal foils leads
to the spontaneous growth of metal oxide nanoparticles.
M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171 169
Figure 7: SEM image of Bacillus Stearothermophilus after treatment
with plasma for 55 s. The plasma parameters were the same as for the
treatment of the polymer foil (as in Figure 4).
Slika 7: SEM slika bakterije Bacillus Stearothermophilus po obdelavi
v plazmi 55 s. Parametri plazme so bili isti kot pri obdelavi polimerne
folije (kot na sliki 4).
Figure 6: SEM image of Bacillus Stearothermophilus before plasma
treatment
Slika 6: SEM slika bakterije Bacillus Stearothermophilus pred pla-
zemsko obdelavo
As long as the flux of atoms on the surface of the sam-
ples is very large, an oxide film will grow in the form of
long nanowires. At moderate flux, the oxide is often in
the form of rectangular nanoparticles. The treatment of
polymer materials with an oxygen plasma leads to a sur-
face functionalization with oxygen-rich functional
groups. Even a brief exposure results in a saturation of
the surface with polar functional groups, causing a dra-
matic increase in the surface energy and thus the
wettability. Treatment with oxygen plasma also causes
the destruction of bacteria and therefore represents a
suitable medium for the sterilization of delicate biocom-
patible objects.
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lus_stearothermophilus
M: MOZETI^: SURFACE MODIFICATION OF MATERIALS USING AN EXTREMELY NON-EQUILIBRIUM OXYGEN PLASMA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 165–171 171

S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN
SLOVENIA
SREDNJETRIASNE KOPENSKE FAZE V JU@NI SLOVENIJI
Stevo Dozet1, Matja` Godec2
1Geological Survey of Slovenia, Dimi~eva ulica 14, 1000 Ljubljana, Slovenia
2Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana, Slovenia
matjaz.godec@imt.si
Prejem rokopisa – received: 2009-11-20; sprejem za objavo – accepted for publication: 2010-03-18
In the Middle Triassic lithostratigraphic sequence of southern Slovenia and neighbouring Gorski Kotar, Croatia, three dry-land
phases have been recognized: 1 – between the Lower and Middle Triassic, 2 – in between the Anisian stratigraphic sequence,
and 3 – between the Anisian and Ladinian beds. Traces of the Anisian volcanism have also been discovered. The dry-land
episodes of the non-deposition and volcanic activity are ascribed to synsedimentary tectonic movements of the Alpine tectonic
cycle. However, new data enable a new approach to the paleogeographical evolution of the area. In the Triassic "Dolomite"
Complex, lenses of kaolinite deposits occur, representing an important correlation horizon for the basal Ladinian beds. They
consist chiefly of kaolinite and quartz admixed in smaller quantities by hematite, goethite, nontronite, gibbsite, groutite and
montmorillonite. The kaolinite deposits were formed in a marsh with weathering of the pyroclastic materials.
Key words: carbonate, clastic and volcanic rocks, dry-land phases, Middle Triassic, External Dinarides, Slovenia, SEM, EDS,
XRD
V srednjetriasnem litostratigrafskem zaporedju ju`ne Slovenije in sosednjega Gorskega Kotarja (Hrva{ka) so ugotovljene tri
okopnitve, in sicer: 1 – med spodnjim in srednjim triasom, 2 – v aniziju ter 3 – med anizijem in ladinijem. Odkriti so tudi
sledovi anizijskega vulkanizma. Okopnitve in vulkanizem povezujemo s so~asnimi tektonskimi premiki Alpidskega tektonskega
cikla. Novi podatki omogo~ajo nov pogled na paleogeografski razvoj obravnavanega ozemlja. V triasnem dolomitnem
kompleksu se pojavljajo le~e kaolinitnih usedlin, ki so pomemben korelacijski horizont za bazalne ladinijske plasti. Sestojijo v
glavnem iz kaolinita in kremena, katerima so v manj{ih koli~inah prime{ani hematit, goethit, nontronit, gibbsit, groutit in
montomorilonit. Kaolinitne usedline so nastale v mo~virju pri razpadu piroklasti~anega materiala.
Klju~ne besede: karbonatne, klasti~ne in vulkanske kamnine, kopenske faze, srednji trias, Zunanji Dinaridi, Slovenija, SEM,
EDS, XRD
1 INTRODUCTION
The territory that is in question, covers the southern
Slovenia and the extreme northern parts of Gorski Kotar,
which belongs to neighbouring Croatia. In the
framework of the elaboration of the Basic Geologic Map
on the scale of 1 : 100 000 the systematic regional
geologic researches included among other Ko~evje and
Gorski Kotar 1 areas, White Carniola (Bela krajina) 2 and
central Slovenia 3, that lie on the topographic Maps
Grosuplje (No 45), Ribnica (No 54), Ko~evje (No 55)
and ^rnomelj (No 56) on the scale of 1 : 25 000. In the
last ten years the Geological Survey of Slovenia
performed a detailed geological mapping for the
elaboration of the Geologic Maps of Slovenia on the
scales 1 : 25 000 and 1 : 50 000. The Map Sheet
Grosuplje 1 : 25 000 consists of 25 sections on the scale
of 1 : 10 000 embracing about 675 km2. Also elaborated
and digitalized is the Map Sheet Grosuplje on the scale 1
: 25 000. On the Map Sheet Grosuplje 1 : 25 000 lies
among other Blo{ka planota that was the central part of
our researches.
In southern Slovenia the dry-land phases and
interruptions in sedimentation appear in different, and
mostly in two, lithostratigraphic sequences: 1) in the
variegated heterogeneous Triassic (Schytian-Carnian)
Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 173
UDK 552.54(497.4):537.533.35 ISSN 1580-2949
Review article/Pregledni ~lanek MTAEC9, 44(4)173(2010)
Figure 1: Location sketch map of the study area
Slika 1: Polo`ajna skica obravnavanega ozemlja
chiefly clastic sedimentary succession of Ko~evje,
Carniola (Bela krajina) and Gorski Kotar, and 2) in the
monotonous, light Middle Triassic "Dolomite" Complex
on Bloke plateau.
The purpose of this work was to find answers and
evidence for the following questions: What is the real
age of the "Dolomite" Complex and its parts? Was the
sedimentation of the "Dolomite" Complex continuous or
interrupted? Are in the Triassic "Dolomite" Complex of
Bloke plateau also present the Anisian, Ladinian and
Cordevolian beds? How many interruptions occurred in
the complex, and how long did they last? Finally, the aim
of this paper is also to explain the particularities of the
paleogeographical evolution of the study area during the
Triassic period.
The listed problems required detailed stratigraphic
investigations of the Middle Triassic "Dolomite"
Complex and adjacent beds in the field and additional
systematic laboratory examinations (Figure 1).
From the geographical point of view the investigated
area belongs to the Lower and Inner Carniola
(Dolenjska, Notranjska) 17,21, while, geologically and
geotectonically, however, to the Slovenian and Dinaric
Carbonate Platforms and the Lower and Inner Carniola
Blocks 4,6,19.
2 EXPERIMENTAL PROCEDURES
In the past, geological researches in Slovenia were
focused on systematic regional and detailed geological
mappings of the considered area for the elaboration of
the Basic Geological Map SFRY on the scale of 1 : 100
000 on the Map Sheet Ribnica (OGK I – RI). In the past
ten years a detailed geological mapping for the
Geological Map of Slovenia 1 : 50 000 on the Map Sheet
Grosuplje 1 : 25 000 (GKS-GR) was achieved. It was
accompanied by systematic laboratory examinations (a
determination of the macro- and microfossils, XRD,
geochemical and sediment-petrographic analyses and
SEM/EDS analyses). The main mapping methods for the
elaboration of the OGK I-RI were the mapping of all the
exposures and the tracing of contacts, whereas the main
methods for the elaboration of the GKS-GR were the
mapping of motorway traces as well as stratimetric
measurements and sedimentologic research of well-
exposed representative cross-sections (stratimetric
profiling).
Carbonate rocks were arranged according to FOLK’s
23 and DUNHAM’s 24 classifications, whereas the clastic
sediments were defined regarding FOLK’s 23 and
PETTIJOHN’s 25 classifications.
For the detailed analysis six selected rock samples
were taken from three different regions in wider Bloke
area, two from Puharje, two from Kaplanovo and two
from @gajnarji (Table 1). The elemental analyses of all
six samples were performed using a field-emission
scanning electron microscope JEOL JSM 6500F
equipped with an EDS analyzer. Due to their very soft
nature and the high crumbling tendency the samples
were only carefully ground with 500 SiC paper. The
accelerating voltage of the primary electron beam was 15
kV and the probe current was 0.2 nA.
Table 1: Designations of six selected samples that were taken at
different places close to the Bloke region
Tabela 1: Oznake {estih izbranih vzorcev, ki so bili odvzeti na
razli~nih mestih blizu podro~ja Bloke
Sample
designation
Name of the sample by the place where it
was taken
1 Puharje 1
2 Puharje 2
3 Kaplanovo 1
4 Kaplanovo 2
5 @gajnarji 1
6 @gajnarji 2
With the aim to obtain information about the
minerals present in the rocks, the phase analyses were
performed using Philips Analytical X-ray diffraction
(XRD) using CuK1 radiation (0.15406 nm) in the 2
range from 0 to 70°.
The fraction of each mineral present in the six
analysed samples was calculated from the XRD
measurements and verified with EDS. The fraction of
each mineral is expressed in approximate integer values,
rounded up to 10 % or 5 %.
3 RESULTS
3.1 Geology
For the explanation of the paleogeographic
development of the southern part of the Slovenian
territory in the Middle Triassic period it is useful to
recall the geological column of the clasticly developed
Ko~evje region and on the other side of the calcareously
developed Bloke plateau (Blo{ka planota).
3.1.1 Geological column of the Ko~evje region
The geological column of the Ko~evje region consists
of two parts: the lower clastic part and the upper
carbonate part. The lower part is composed of the
carbonate clastic Carboniferous, Permian and Triassic
systems and the upper part involves the Upper Triassic
series as well as the Jurassic and Cretaceous systems.
In the Kaptol-Mo{enik cross-section south of
Ko~evje 26 the clastic formation of the Younger Paleozoic
beds is subdivided into three lithostratigraphic units,
which are in analogy with similar developments in
Gorski Kotar and Ortnek 20 attributed to the Lower
Permian (Trogkofelian). A relatively shallow sedimen-
tation area, where the Lower Permian clastic rocks
originated, began in the time of the Saalian movements
to rise, so that the Ko~evje and Gorski Kotar areas were
in the Middle and Upper Permian dry land where in that
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
174 Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183
time no significant folding occurred. During a slow
uplifting a relatively flat mainland originated, built of
pretty clayey micaceous sediments with interbeds of
compact quartz sandstones and conglomerates. The dry
land was the product of the activity of the final strokes of the
Hercynian orogeny. The elevation of the considered area
was accompanied by a fault tectonics in Gorski Kotar;
however, with a weak magmatic activity as well 25.
The conclusions related to the dry land could also be
deduced to the stratigraphic gap of the Scythian beds
lying discordantly upon the Lower Permian bed. For the
tectonic-erosional character of this boundary speaks also
about 20-cm-thick limonite crust in the Gorski Kotar
region 26–29 on the boundary between the Younger
Paleozoic and Triassic sediments, that is a product of the
strong oxidizing conditions. The elevation of the
considered territory that began on the boundary between
the Lower and Middle Permian in the time of the Saalian
phase, continued uninterruptedly into the Pfälzian
phase, with which the Hercynian tectogenetic cycle in
the Alps was ended. The Hercynian orogeny in the
Dinarides area was followed by a common descending
and afterwards the transgression of the Lower Triassic
sea, which is according to U. Premru 16 closely
connected with the extinguishing Herzynian orogenetic
cycle. In the Triassic sea in the Ko~evje, Gorski Kotar
and Bela Krajina areas, a pretty variegated clastic-
carbonate succession of sediments originated (Figure 2),
of which the lower part is of Scythian and the upper one
of Carnian age 10–13,30,31. They are distinguished between
themselves according to lithology, mineral composition,
heavy minerals, geochemical composition, cement,
diagenesis, source material, environment, Energy-index
and climate.
In the Scythian part of the clastic-carbonate
succession oolitic and sandy dolomite prevail, and there
is also less of the sandy dolomitic marlstone, claystone,
micaceous sandstone and intraformational conglomerate.
The Carnian (Julian and Tuvalian) beds are composed of
red dolomitic marlstone containing interbeds of
yellowish, greenish grey and grey dolomite as well as
greenish grey claystone. Between heavy minerals
turmaline prevails in the Scythian beds, and zircon in the
Carnian beds.
The Scythian and Carnian sediments originated in a
shallow sea and semiarid climate. The mainland in the
background of the sedimentary basin was low. Toward
the end of the Lower Triassic epoch the Ko~evje region
began slowly to uplift, as consequence of the Old Slove-
nian phase 32. The elevation of the terrain is evidenced
by interbeds of breccias and conglomerates in the
Triassic sedimentary succession in the Ko~evje region
9,32, the Gorski Kotar region 33–35 and the Lower Carniola
(Dolenjska) region 34–37.
The continuous shallowing of the already relatively
shallow Lower Triassic sea brought on a differentiation
of the sedimentary area and finally originated the
positive paleostructure comprising the Ko~evje region,
Bela krajina and Gorski Kotar 22. The new dry land was
gently anticlinally folded 38, and the folding was locally
accompanied with weak radial tectonics. These events
caused a reduction and later a complete interruption of
the sedimentation in this part of the Dinarides, that lasted
until the transgression of the Carnian (Julian) sea. The
emersion in the Ko~evje region lasted through the whole
Anisian, Ladinian and Cordevolian epochs. With respect
to the existence of dry land it is concluded with reference
to the absence of the Middle Triassic and Lower Carnian
(Cordevolian) rocks in this area, too.
3.1.2 "Dolomite" Complex on Bloke plateau
On the northeastern border of the Bloke plateau the
Middle Triassic sedimentary succession occurs in a
carbonate development, in which dolomite greatly
prevails; therefore, it was denominated as the "Dolomite"
Complex (Figure 3). In this Complex also more or less
dolomitized limestones and even pure limestones occur,
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 175
Figure 2: Correlation of the Scythian-Carnian carbonate-clastic
succession of Ko~evje, Gorski Kotar and Lower Carniola
Slika 2: Primerjava skitsko-karnijskega karbonatno-klasti~nega
zaporedja Ko~evske, Gorskega Kotarja in Dolenjske
that are due to the numerous dasycladal mostly
rock-forming algae of the Diplopora genus, known under
the name Diplopora Limestones and occurring in the
complex above all in the uppermost part. In the Basic
Geologic Map of SFRY on the scale 1 : 100 000 BUSER
3,39 ascribed the considered the "Dolomite" Complex to
the Cordevolian substage, that is considered in Slovenia
for several ten years already as belonging to the Lower
Carnian substage. The dolomite or carbonate develop-
ment of the Triassic beds also occurs in other parts of
Slovenia and it is ascribed to its dissimilar age, from
Anisian to Norian and Rhaetian 3,5,14,15,41–44.
3.1.3 Stratigraphic position
In the considered area the "Dolomite" Complex
occurs mostly to an incomplete extent, since its
uppermost or lowermost or both parts are tectonically cut
off or eroded. In the valley of the Black Ravine (^rni
potok) at Karlovica is exposed the footwall of the
"Dolomite" Complex, represented by a several ten-
meters-thick stratigraphic sequence of rosy, grey, platy
dolomitic limestone, in which in the [krilje and Podgozd
areas the gastropod Natiria costata was found 3, that in
Slovenia occurs in the uppermost Scythian beds. The
conodont analysis of the samples from the beds at
Podgozd was not positive 45.The hanging wall of the
"Dolomite" Complex and their contacts are exposed on
Veliki Vrh, where concordantly upon the "Dolomite"
Complex is located a red pelitic and poorly oolitic
iron-rich bauxite representing the basal horizon of the
Julian beds. Concordantly upon the 30-meters-thick band
of Carnian (Julian and Tuvalian) beds rests a dolomite in
the Lofer development belonging to the Main Dolomite
Formation. The "Dolomite" Complex thus lies con-
cordantly between the Schythian and Julian-Tuvalian
beds and represents, supposing that the sedimentation
was uninterrupted, the Anisian, Ladinian (Fassanian and
Langobardian) and Lower Carnian (Cordevolian) beds.
3.1.4 Lithology
The stratigraphic sequence of carbonate rocks
denominated as the "Dolomite" Complex, originated on
the Slovenian and Dinaric Carbonate Platforms, is not a
homogeneous lithologic unit, since in some sections it
includes, beside the dolomite, also, a strongly
dolomitized limestone and bedded biostromal Diplopora
Limestone. Also there, where the "Dolomite" Complex is
built of pure dolomite, exists a certain heterogeneousity
of the carbonate sequence as with regard to
sedimentologic properties in the considered area, the
lower and upper parts can be distinguished.
In the about 120-meters-thick interval of the lower
part of stratigraphic sequence of the "Dolomite"
Complex alternate light-grey to very-light-grey bedded
fenestral, laminated and very fine-grained dolomites.
The fenestral dolomite is a dolomite with numerous
more or less parallel shrinkage pores. They may be an
open space in the rock or completely or partly filled with
secondarily introduced sediment or cement. The
laminated dolomite is rare. In the rock the lamination is
not produced by stromatolitic crusts, but by numerous,
with stratification parallel, shrinkage pores filled with
sparry calcite, limonite or sediments. The fine-grained
dolomite (microdolosparite) does not show any porosity,
whereas, the porosity of the laminated dolomite consists
mostly of horizontal pores. The described carbonate
sediments originated in a shallow lagune in an intertidal
and supratidal environment. The types of carbonate rocks
show that the Energy-index of the sea was low.
In the middle and upper part of the "Dolomite"
Complex there is an about 350-to-400-meters-thick
sedimentary succession built chiefly of massive and, here
and there, bedded dolomite. The dolomite of the middle
and upper part of the "Dolomite" Complex is mostly
very light grey to almost white and sometimes it is bluish
white, also the stratification of this sediment is rare.
Almost always it is coarse-grained, respectively very
coarse-crystallized, which speaks for its origin at a late
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
176 Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183
Figure 3: Geologic column of the "Dolomite Complex" with its
footwall and hanging wall beds
Slika 3: Geolo{ki stolpec "Dolomitnega" kompleksa s talninskimi in
krovninskimi plastmi
diagenesis of the Diplopora Limestone. Usually, it is
quite porous. The pores occur mostly in places that in
former times were dasycladal algae. The pores and fine
cracks are filled as a rule with fine dolomite crystals. The
porosity of the rock is intragranular, sometimes,
however, it is represented by isolated pores and fine
cracks. The massive dolomite is compact and it is usually
crumbling because of the coarse-grained texture. During
weathering it disintegrates into a dolomitic sand,
mylonite and ultramylonite.
The Diplopora Limestone is bedded and light grey,
medium grey to grey or rosy. According to the texture it
belongs to biosparite, bointrasparite, biolithite and rarely
biomicrite. Commonly, it contains on the bed surfaces
numerous round, oval and oblong sections of dasycladal
algae of the genus Diplopora, that are mostly
rock-forming. Diploporas lived on vast grasses on the
bottom of the Cordevolian shallow and warm sea. More
or less well-preserved Diplopora remains can also be
found in the dolomite. Algae are better preserved in the
limestones than in the dolomites. On the northern
Borderland of the Bloke plateau we found in the
Diplopora Limestone also an about 7.5-meters-thick
interbed of bedded medium-grey stromatolitic limestone.
The Diplopora Limestone passes laterally and vertically
into the more or less dolomitized limestone or dolomitic
limestone. In southern Slovenia in some places we come
across larger complexes of dolomitic limestones. It is
interesting and important that the dasycladacean are
well-preserved in the dolomitic limestone as well. On the
other hand, well-preserved dasycladacean remains are
very rare due to diagenesis in the coarse-grained
dolomite.
3.1.5 Tectonics
Our researches showed that in the Middle Triassic
period there were three dry-land phases:
The first phase began already at the boundary
between the Scythian and Anisian (Ko~evje area and
wider surrounding), the second one took place in the
middle of the Anisian stage (Bloke plateau and environs)
and the third phase was discovered somewhere in the
middle part of the "Dolomite" Complex representing the
boundary between the Anisian and Ladinian stage.
The first epeirogenic phase, denominated as Ko~evje
epeirogenic phase 9 is the largest phase especially with
respect to the time of its lasting (from the end of the
Scythian to the Julian). Besides the Ko~evje region it
also affected the White Carniola and Gorski Kotar area
as well as some places in the Lower Carniola northwest
of Ko~evje. Unequal strongly intensified epeirogenic
activity formed a positive structure with the centre in the
Ko~evje area. Upon the gently convexed antiform no
sedimentation occurred. Evidence for the dry land is the
absence of Anisian, Ladinian and Cordevolian rocks in
the Ko~evje region; however, primary exposures of the
above-enumerated beds were not found in the White
Carniola and Gorski Kotar as well. Anisian, Ladinian
and Cordevolian rocks are also not found in the Triassic
cross-sections of the Ko~evje, White Carniola and
Gorski Kotar regions. The Triassic columns consist of
variegated (prevalently red) clastic sediments and more
ore less frequent interbeds of dolomites and dolomitic
marlstones. The lower part of the many-coloured
clastic-carbonate succession belongs to Scythian; the
upper one, however, to the Carnian. This is evidenced by
macro- and microfossils as well as with petrographic-
sedimentologic methods, heavy metals analysis and with
XRD methods 28,30,35. Since on the boundary between the
Scythian and Carnian sequences there are no Anisian,
Ladinian and Cordevolian rocks, and because the
Carnian sedimentary succession starts nowhere with a
thicker horizon of basal breccia or conglomerate, we
believe that orogenetic forces and fault tectonics in that
time were not present and that the newly formed dry land
was flat and built of fine- and very fine-grained
sediments.
Traces of the second Middle Triassic dry-land phase
are exposed on the top of the lower third of the
"Dolomite" Complex in the upper part of the sequence of
light bedded fenestral dolomite, which is, according to
its concordant position upon the rosy grey, thick-plated
dolomitic limestone containing the gastropod Natiria
costata and with reference to sedimentologic characte-
ristics, attributed to the Anisian. The main evidence for
this dry land phase, that was the shortest and the weakest
at all three, is a 0.75-m- to 1.5-m-thick lense-like
interbed of pale greenish kaolinite. The darker greenish
tuff, however, speaks for volcanic activity 18 in the
Anisian epoch. Otherwise, the Anisian volcanic rocks are
more wide-spread in the Croatian territory 46.
The third Middle Triassic dry-land phase is
discovered in the "Dolomite" Complex of Bloke plateau
on the boundary between the underlying light fenestral
dolomite, to which we ascribed the Anisian age, and the
overlying very light grey to almost white massive
coarse-grained porous dolomite, that in spots lies under
the medium-grey to rosy bedded biostromal Diplopora
Limestone, which is attributed in Slovenia to Cordevolian
and Lower Carnian. The evidence for the Middle Triassic
phase is based on finding 0.5- to 1.25-thick lense-like
interbeds of kaolinite, bauxite and iron crust on the
boundary between the Anisian light bedded fenestral
dolomite and the Carnian-Ladinian white coarse-grained
massive dolomite.
3.1.6 Paleogeography
The relatively shallow sedimentary environment in
which the Lower Permian clastic rocks were deposited,
began at the end of the Lower Permian during Saalian
movements to elevate so that the Ko~evje and Gorski
Kotar area was in the Upper Permian a dry land. In this
area in that time no important folding occurred. During
the slow uplift a relatively flat dry land originated, built
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 177
of clayey sediments with interbeds of sandstones and
conglomerates. This dry land was a consequence of the
activity of the last strokes of the Hercynian orogeny 27,28.
After the last strokes of this orogeny (Pfälzian phase)
the transgression of the Scythian sea takes place. The
Scythian sediments originated in a very shallow sea, in
between the intertidal band with erosional activity of the
sea currents and waves on half-consolidated sediments of
the sea bottom, that were mixed with a fine-grained
dolomitic matrix. The Scythian intraformational conglo-
merate speaks, above all, for a shallow sea that from time
to time flew off and was followed by drying out
(desiccation) and cracking of the half consolidated mud
material, of which fragments were afterwards rounded
by sea currents. The material supply from the land was
pretty considerable.
Wave ripple marks and cross-stratification in the
Scythian sediments point to a turbulent shallow sea. The
dry land in the vicinity of the sedimentary area had to be
low and flat, since from this dry land came only fine and
very fine rock material.
The absence of coarse-grained and non-sorted rock
material speaks for the conclusion, that in the vicinity of
the considered area in the Middle Triassic there were no
mountains or higher elevated dry land. After the
deposition of the Scythian sediments, the Ko~evje region
was elevated and a gently convexed antiform originated,
upon which there was no sedimentation until the
transgression of the Carnian sea.
The similar Scythian sedimentation on the Slovenian
Carbonate Platform also originated on the Bloke plateau
and in its surroundings, with a distinction that it was
more calcareous.
In the Anisian epoch the carbonate platform in
southern Slovenia became larger and more stable. The
rock material supply from the land was insignificant.
Monotonous sedimentation of light dolomites point at a
stable shelf and sedimentologic properties of the
carbonate rocks indicate a neritic sedimentation in the
tidal area. In the middle of Anisian epoch the intensified
epeirogenic movements assisted the forming of
short-lasting local dry land with traces of volcanism,
bauxite and kaolinite were preserved on the surface.
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
178 Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183
Figure 4: BE images of all six samples performed on an area 1.17 mm × 0.88 mm. The chemical inhomogeneity is shown by the lighter and
darker areas in the micrographs
Slika 4: Slike, dobljene z odbitimi elektroni vseh {estih vzorcev, posnete na podro~ju velikosti 1,17 mm × 0,88 mm. Kemijska nehomogenost se
ka`e v porazdelitvi temnej{ih in svetlej{ih podro~ij na sliki
Greater activity of the Anisian volcanoes was recorded in
the Croatian territory 46. After the short-lasting dry land,
that can be temporally correlated with the Monte Negro
orogenic phase, repeatedly followed a neritic sedimen-
tation on the carbonate platform, equal as before to the
short-lasting interruption of sedimentation.
On the boundary between the Anisian and Ladinian
stage followed a new, still more intense dry-land phase,
which is evidenced by here and there preserved lenses of
bauxite, kaolinite and iron crusts. In the whole area of
southern Slovenia originated then on the Dinaric
Platform carbonate sediments in a shallow sea. In the
shallow and warm sea grew calcareous algae, forming
numerous submarine lawns. With their coalcareous
skeletons they contributed most to the origin of several
hundreds of meters thick limestone sequence. Later the
Diplopora Limestone changed at late diagenesis in part
or completely into a coarse-grained (crystallized)
dolomite 47.
The part of the "Dolomite " Complex, originated after
the Anisian/Ladinian dry-land phase, the Slovenian
geologists attributed to the Lower Carnian respectively to
Cordevolian. In S. Buser’s, and A. Ramov{’s opinions 7
the algal species Diplopora annulata Schafhäutl
occurring in southern Slovenia only in the Cordevolian
beds, represents a guide fossil for this area. In our
opinion, it was formed in the Langobardian and
Cordevolian epoch.
The dry-land phases were detected and evidenced by
the following signs and materials: erosion surfaces,
barite-filled pockets, continental breccias and conglo-
merates, paleokarstic phenomena, red residual sedi-
ments, kaolinite deposits, as well as iron and kaolinite
crusts.
Upon the size and thickness of the lense-like
kaolinite deposits, that range normally from 4 to 9
meters and amount in spots even to 17.5 meters,
influenced also the morphology of the solid rock.
Namely, the kaolinite deposits were not formed or
accumulated upon the elevated parts of the paleorelief.
The paleorelief served only as a trap for the considered
deposits 50. The kaolinite deposits start in rare places
with a thin layer of basal calcareous breccio-conglo-
merate with a kaolinite matrix.
The kaolinite deposits originated most probably from
a pyroclastic material that was deposited into a marsh
and there changed into the kaolinite 51–53.
3.2 Laboratory examinations
3.2.1 SEM/EDS analysis
Semi-quantitative EDS analyses of all six selected
samples were obtained. BE images of the investigated
samples are shown Figure 4. The rocks are not homo-
geneous and consist of areas with different chemical
compositions. The higher intensity of the backscattered
electrons represents a lighter colour in the BE image;
therefore, it can be concluded that such areas consist of
elements having higher atomic numbers. The overall
elemental analyses of all six samples are shown in Table 2.
All the samples have approximately mass fraction 60 %
of oxygen, the exception being sample 4, where the
content of oxygen is less than half, most probably due to
the higher amount of iron. Silicon and aluminium are the
next two most represented elements. In sample 1 the
mass fraction of aluminium is almost 26 %, while in
sample 2, 5 and 6 the content of aluminium is close to 17
% and in other samples it is of 8 %. The higher content
of silicon is in sample 3 (27 %) followed by sample 4
and 5 (approximately 19 %) and there is near 10 % in
samples 1 and 4. Iron gives the colour of the rocks; the
higher content of iron and their oxides give a more
reddish colour to the samples. The highest content of
iron, of approximately 28 %, was found in sample 4, it
was in sample 2 at 4.7 %, in samples 1 and 5 at 2.7 %, in
sample 3 at 1.6 % and in sample 6 at 0.4 %. Some of the
samples also contain a minor content of titanium,
magnesium and some other elements presented in Table
1. The EDS spectra of all the examined samples are
presented in Figure 5. In each sample a few spot
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 179
Table 2: Chemical composition of six samples determined with EDS on a field of view 1,17 mm × 0.88 mm; mass fractions w/%, mole fractions
x/%
Table 2: Kemi~na sestava {estih vzorcev dolo~ena z EDS-analizo na vidnem polju 1,17 mm × 0,88 mm; masni dele`i w/%, molski dele`i x/%
O Mg Al Si K Ca Ti Fe
Sample 1
w/% 60.0 – 25.7 10.9 – – 0.6 2.7
x/% 72.8 – 18.5 7.5 – – 0.3 0.9
Sample 2
w/% 59.5 – 17.1 18.6 – – – 4.7
x/% 72.9 – 12.4 13.0 – – – 1.7
Sample 3
w/% 57.9 2.3 8.3 27.0 2.2 0.6 – 1.6
x/% 71.2 1.9 6.0 18.9 1.1 0.3 – 0.6
Sample 4
w/% 51.9 – 8.4 9.4 – – 2.0 28.3
x/% 73.1 – 7.0 7.6 – – 0.9 11.4
Sample 5
w/% 60.5 0.3 17.4 18.7 – – 0.4 2.7
x/% 73.3 0.2 12.5 12.9 – – 0.2 0.9
Sample 6
w/% 61.2 – 17.9 20.0 – – 0.5 0.4
x/% 73.3 – 12.7 13.6 – – 0.2 0.1
analysis were performed, which clearly demonstrate the
scattering of individual elements over the sample,
especially iron, titanium and also silicon as well as
aluminium, while oxygen is almost the same in all the
examined samples. The results of the chemical compo-
sition of the individual small areas of sample 2 are
shown in Figure 6 and Table 3.
Table 3: Chemical composition of individual small areas of sample 2
determined with EDS; mass fractions w/%, mole fractions x/%
Table 3: Kemi~na sestava posameznih manj{ih podro~ji vzorca 2
dolo~ena z EDS-analizo; masni dele`i w/%, molski dele`i x/%
O Al Si Ti Fe
Spectrum 1
w/% 49.5 10.4 11.2 1.2 27.7
x/% 70.4 8.8 9.0 0.5 11.3
Spectrum 2
wt.% 54.5 14.1 15.2 0.8 15.4
x/% 71.5 11.0 11.4 0.3 5.8
Spectrum 3
w/% 60.8 17.6 18.7 0.2 2.7
x/% 73.5 12.6 12.9 0.1 0.9
Spectrum 4
w/% 58.5 16.1 17.1 0.4 7.9
x/% 73.1 7.0 7.6 0.9 11.4
Spectrum 5
w/% 54.5 13.4 14.4 0.7 17.0
x/% 72.0 10.5 10.8 0.3 6.4
3.2.2 XRD analysis
Figure 7 to 12 shows the XRD spectra of six samples
with labeled constituting minerals. The XRD results
correspond very well with the EDS elemental analyses
taking into account that there are some elements that
cannot be analyzed, e.g., hydrogen, and that light
elemental analysis is not very precise when using EDS.
This analysis is difficult because of their low photon
energy, what may lead to high absorption in the
specimen and in the components of the spectrometer 48.
Sample 1 is dark-red oolitic pisolitic bauxite or boehmite
bauxite with some minor minerals such as hematite,
kaolinite, nantronite and berthierine (Figure 7). The
share of the boehmite phase is at 60 %, while the share
of other minerals is approximately 10 % for each.
Sample 2 is red pelitic bauxite or iron kaolinite (Figure
8). The ratio of the kaolinite mineral to the hematite is 8
to 2. Sample 3 is a quartz sandstone with a tuff
admixture (Figure 9). The content of quartz is 80 %,
while in the other two phases it is 10 % for each. Sample
4 is an iron-kaolinite-rich phase (Figure 10) where the
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
180 Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183
Figure 6: (a) BE image of sample 2 with marked areas of EDS
analysis; (b) five EDS spectra performed on different areas
Slika 6: (a) Slika odbitih elektronov vzorca 2 z ozna~enimi mesti EDS
analize; (b) pet EDS spektrov, posnetih na razli~nih podro~jih
Figure 5: EDS spectra of all six samples taken for an area of 1.17 mm
× 0.88 mm
Slika 5: EDS spektri {estih vzorcev posneti na podro~ju 1,17 mm ×
0,88 mm
Figure 7: XRD-spectrum of sample 1 with labelled constituted mine-
rals of analysed rock
Slika 7: XRD-spekter vzorca 1 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
ratio of the goethite and kaolinite are 6 to 4. Sample 5 is
kaolinite with the addition of iron in the goethite phase
(Figure 11). The content of kaolinite in sample 5 is 70
%, while goethite and nontronite are represented each as
10 %, and gibbsite and groutite as 5 % each. Sample 6 is
pure kaolinite (Figure 12).
4 CONCLUSIONS
In Southern Slovenia – which in the Scythian and
Anisian epoch belonged to the Slovenian Carbonate
Platform including the whole Dinarides and from the
Upper Cordevolian to the end of the Mesozoic period to
the Dinaric Carbonate Platform – two Triassic sedi-
mentary developments can be distinguished: the clastic
and the carbonate. The clastic development occurs in the
Ko~evje and Bela krajina regions, while the carbonate
appears in the Bloke plateau and in some other places in
Slovenia. In the last three years we focused our research
work on the Bloke plateau area, to the "Dolomite"
Complex in particular.
With systematic regional and detailed geological and
laboratory researches we came to the following con-
clusions:
1. The "Dolomite" Complex on the Bloke plateau is not
a homogenous lithologic unit, but it consists of two,
and in some places even of three or four, rock
sequences.
2. In the Ko~evje region, there was no sedimentation in
the Anisian, Ladinian and Cordevolian epochs, since
there are no Middle Triassic and Lower Carnian
(Cordevolian) rocks exposed at the surface, and
because these rocks do not crop out anywhere in the
White Carniola and Gorski Kotar, as well.
3. The sedimentation in the "Dolomite" Complex of the
Bloke plateau was not continuous; it was interrupted
two times for a certain time: in the middle of the
S. DOZET, M. GODEC: MIDDLE TRIASSIC DRY-LAND PHASES IN SOUTHERN SLOVENIA
Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 181
Figure 12: XRD-spectrum of sample 6 with labelled constituted
minerals of analysed rock
Slika 12: XRD-spekter vzorca 6 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
Figure 10: XRD-spectrum of sample 4 with labelled constituted
minerals of analysed rock
Slika 10: XRD-spekter vzorca 4 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
Figure 9: XRD-spectrum of sample 3 with labelled constituted
minerals of analysed rock
Slika 9: XRD-spekter vzorca 3 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
Figure 8: XRD-spectrum of sample 2 with labelled constituted
minerals of analysed rock
Slika 8: XRD-spekter vzorca 2 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
Figure 11: XRD-spectrum of sample 5 with labelled constituted
minerals of analysed rock
Slika 11: XRD-spekter vzorca 5 z ozna~enimi minerali, ki sestavljajo
analizirane kamne
Anisian epoch and on the boundary between the
Anisian and Ladinian stage.
4. About S. Buser and A. Ramov{ statement 7, i.e., that
in southern Slovenia the dasycladal species Diplo-
pora annulata Schafhäutl occurs only in the Corde-
volian beds, we came to the conclusion that it was so
only there, where the Ladinian is clasticly developed.
In other places in Slovenia, however, we have also
the Middle Triassic carbonate development ("Dolo-
mite" Complex), where according to our data, the
beds containing diploporas extend into the Ladinian
stage as well. The considered dry-land phase started
probably already in the topmost Anisian and lasted
for some time into the Ladinian epoch (probably in
the whole Fassanian). Afterwards, started the
sedimentation of the Diplopora Limestone of the
coarse-grained dolomites, that fill the Ladinian and
Cordevolian interval of the Triassic lithostratigraphic
column 15–17,49.
5. In the Anisian epoch volcanoes very probably also
erupted.
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Materiali in tehnologije / Materials and technology 44 (2010) 4, 173–183 183

W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
ENHANCED FATIGUE ANALYSIS – INCORPORATING
DOWNSTREAM MANUFACTURING PROCESSES
IZBOLJ[ANA UTRUJENOSTNA ANALIZA – VKLJU^ITEV
IZDELAVNIH PROCESOV
Wilfried Eichlseder
University of Leobenm, Franz-Josef-strasse 18, A-8700 Leoben, Austria
wilfried.eichlseder@unileoben.ac.at
Prejem rokopisa – received: 2009-08-19; sprejem za objavo – accepted for publication: 2010-02-17
Most failures of engineering components can be traced back to cyclic loads which cause fatigue of the material. These cyclic
loads can be either constant or variable. This influence has to be taken into account in the fatigue analysis, the aim of which is to
assure a minimum required lifetime of a component without breakdown. Simultaneously, a lightweight design is intended by
reducing the size of the component. The strength of materials under cyclic loading is essentially lower than those under static
loading. However, the cyclic strength is of more importance in practical applications, as more components fail by cyclic loading
than by static loads. Regrettably, the strength of cyclic loaded components was discovered relatively late and the first systematic
investigations were performed in the 19th century. Cyclic loadings result from mechanic or thermal service loads, which can be
superimposed to static loads. When looking at a truck, the chassis is loaded by the dead load of the vehicle. Due to loading and
unloading procedures the load is changed into a quasi-static way. Additional cyclic loads emerge during driving operation in
addition to the static loads due to braking, acceleration, cornering, or physical conditions just like bumps on the roadway or air
drag. Additional loading is also due to resonance vibrations of engine, gearbox, fuel tank, or the spare wheel. These cyclic loads
lead to fatigue of material hence overstraining them to a crack. These cracks propagate and finally lead to failure of the
component.
Key words: fatigue life, complex structures, simulation, manufacturing processs, finite elements
Izra~un trajnostne dobe za geomerti~no kompleksne strukture zahteva poznanje lokalne utrujenosti materiala. Po navadi
raziskava lokalnega utrujenostnega vedenja komponent ni mogo~a, ker je treba za {tudij vplivnih parametrov preveliko ~asa.
Zato je potrebna metoda za izra~un trajnostne dobe z upo{tevanjem lokalnih vplivov. Lokalna utrujenostna trajnostna doba je
odvisna od zarez, vrste obremenitve, temperature, vi{ine in sekvence obremenitve itd., mo~an je tudi vpliv procesa izdelave. V
tem ~lanku so obravnavani procesi izdelave in njihov vpliv na mikrostrukturo in na utrujenostno trajnostno dobo. Posebej so
raziskani vplivi litja, oblikovanja, varjenja in povr{inske obdelave. Razli~ni procesi so bili simulirani, da bi bil dose`en lokalni
u~inek, kot so ~as strjevanja, hitrost deformacije in rezidualne napetosti. Pripravljeni so bili preizku{anci z opredeljenimi
procesnimi parametri, njihova utrujenostna trajnostna doba pa je bila dolo~ena eksperimentalno. Ti rezultati so bili podlaga za
napoved utrujenostne trajnostne dobe geometri~no kompleksnih oblik na podlagi rezultatov analize s kon~nimi elementi, ki je
upo{tevala lokalne napetosti, gradiente napetosti in proces izdelave. Simulacija procesa izdelave in razumevanje vpliva lokalnih
razmer na utrujenostno trajnostno dobo omogo~a dobro simulacijo zapiranja verige od procesa izdelave do napovedi te dobe.
Klju~ne besede: utrujenostna trajnostna doba, kompleksne strukture, simulacija, procesi izdelave, kon~ni elementi
1 FATIGUE ANALYSIS
The science of fatigue analysis deals with the fatigue
of components under cyclic loads as in their regular ser-
vice life. The main goal is to obtain spatial dimensions
of the components in a way that they will serve the mini-
mum lifetime required without failure, with the focus
also lying on weight reduction by using reduced amount
of material. Basically dimensioning of components can
be done in two ways.
Dimensioning by testing
In this case the expected loads are applied on proto-
types. When the required fatigue life is not achieved, a
modified prototype is built and tested again. This devel-
opment cycle continues until the required specifications
are reached. The weak points in components or complete
construction could be easily detected as against over-
sized regions with no visible damage.
Dimensioning by simulation
Based on stresses, knowledge of material behaviour
and the loading spectrum, a lifetime calculation is
performed. At spots that do not achieve the required
lifetime, the geometry or the material is changed or
reinforced. Whereas less stressed spots provide scope for
a reduction in weight by means of material reduction.
The latter has the advantage that it can be applied
very early in the engineering phase and is also less time
and cost intensive. However, in contrast to testing, the
accuracy of simulation is lesser. Hence, in the initial
phase of mass product development, simulations are
performed and in the final stages, developed components
are tested to evaluate their fatigue resistance.
1.1 Damage accumulation
In real life applications, dynamic loads have variable
amplitudes instead of constant amplitudes that too with
different frequencies and stochastic sequences. The dam-
Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192 185
UDK 620.178.3:539.42 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)185(2010)
age under cyclic load is based on the formation and mo-
tion of crystalline defects in the material which has not
yet been measured in a quantitative way. Hence, the
knowledge of an engineer depends on empirical models.
In the last few decades numerous hypothesis have been
established. One of the oldest with distinction from oth-
ers and also most often used is that of Palmgren and
Miner. This model, like the other models, concentrates
on describing the two following observations:
a) There exists an additional damage Di due to a number
of load cycles ni with a stress amplitude of i, which
is caused by the amplitude itself and the foregoing
loadcycles and loads.
b) The accumulation of the single damage leads to a
global damage D, at a critical damage parameter Dc
failure occurs.
1.1.1 Palmgren Miner Method
The most famous rule to define the damage accumu-
lation was already described by Palmgren in 1924 and
Miner in 1945. Thereby, the service life at single con-
stant amplitude is considered as the base for calculating
the service life at variable amplitudes. The damage D1
due to on single load cycle is given by,
D
N i
1
1
= (1)
where Ni is the number of load cycles to fracture of the
S-N curve at a load level ai.
When, at the load level ai, the load cycles appear
with a frequency ni, the damage Di at that level is given
by:
D
n
N
i
i
1 = (2)
By summing up the damage at each load cycle for a
specific load spectrum, one gets the total damage
D
n
N
i
ii
=
=
∑
1
(3)
wherein i is the spectrum level, ni the number of load
cycles on the ith level and ai the corresponding load
amplitude. Experiments show that components fail
below or above the damage sum of 1 whereas, compo-
nents are to fail, as per definition, when the damage sum
reaches a critical value of 1:
D = 1,0 (4)
2 EXTENDING THE SIMULATION CHAIN –
INTEGRATION OF THE MANUFACTURING
PROCESS
For lifetime evaluation of complex geometric compo-
nents, concepts based on local stresses or strains are re-
quired (Figure 1). For investigating local stresses, finite
element method (FEM) has been well established. For
lifetime calculation, knowledge of local fatigue strength
of material, described by the S-N curves of component
and, the service conditions are needed, but these are not
identical with the data of fatigue tests performed on
idealised specimens. In an ideal case the strength of a
component is determined by tests and can be used for
lifetime calculation. Since no components exist in the
construction phase, S-N curves are obtained by simula-
tion.
These influences can either lead to an increase or de-
crease in stiffness and when occurring synchronous,
these effects may boost or weaken each other. The exper-
imental investigation of these effects on fatigue strength
at a full scale is quite time consuming and also cost in-
tensive whereby making it impractical. Hence, these tests
are performed very selectively. The fatigue strength char-
acteristics are not uniform all over the component and
the S-N curve of the whole component is obtained by en-
compassing all the local S-N curves. The task is to trans-
fer the S-N curve of the specimen with respect to the in-
fluencing factors into an S-N curve of the component.
In the following consideration, a few models for de-
scribing the influence of the production process are ex-
plained. These models facilitate lifetime calculation
based on the results of finite element calculations.
2.1 Notches and load type
Notches, zones of bending stress, or components
exposed to irregular force flow show irregular characte-
ristics of the stress level. These irregularities can be
derived from the stresses which gives the stress gradient
 or with reference to local stress, the relative stress
gradient * (Figure 2):


*
max
= ⎛⎝
⎜ ⎞
⎠
⎟1 d
dx
(5)
As the stress gradient can be calculated easily with
the help of finite element results, it is obvious to use it
for the assessment of stresses and to describe the influ-
ences on the S-N curve in the case of notches as well as
under tension/compression and bending loading.
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
186 Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192
Figure 1: Lifetime prediction based on local stresses and strains
Slika 1: Napoved trajnostne dobe na podlagi lokalnih napetosti in
deformacij
2.1.1 Relative Stress Gradient Concept (RSG)
Notches are characterized by an increase in stress and
formation of stress gradient within the component.
Further, beams subjected to bending loading show a
stress gradient. In both cases the material exhibits a
locally higher fatigue stress limit than those under pure
tensile loading on un-notched specimens which is a
result of the support effect (Figure 3).
Two different fatigue limit parameters form the base
for describing the influence of the stress gradient on the
fatigue limit of any component:
– The fatigue limit zdw of an un-notched specimen
exposed to tension-compression load with the
relative stress gradient * = 0 and
– The fatigue limit bw of a bending specimen with
thickness b and a relative stress gradient * = 2/b.
To describe the fatigue limit of components with ar-
bitrary stress gradients one has to interpolate or extrapo-
late these values. Experience shows that the relationship
between fatigue limit and stress gradient is not linearly
proportional, with increasing stress gradient the growth
of the fatigue limit is reduced (Figure 4).
To describe this relationship, an exponential curve is
chosen, which is characterised by the exponent KD:
 



D
K
/b
= + −
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟zdw*
bw
zdw
D
1 1
2
*
(6)
  D n= ⋅zdw (7)
For the fatigue life calculation with the help of
S-N-curve according to (8) two additional parameters are
necessary; the number of cycles at the fatigue limit ND
and the slope k of the S-N-curve.
N N D
D
= ⋅
⎛
⎝
⎜
⎞
⎠
⎟


a
for  a D≥ (8)
Generally speaking, with an increase in the sharpness
of a notch the slope k gets steeper, while the number of
cycles at the fatigue limit ND decreases. To describe the
trend of ND and k of the S-N-curves, the corresponding
minimum and maximum values at low and high gradient
(*) from test results are defined.
For the description of these values between these lim-
its, exponential functions are chosen:
The values lg (NDmin) and lg (NDmax) describe the
lowest and highest number of cycles at fatigue limit,
while kmin and kmax refer to the minimum and maximum
slope of the S-N-curve versus the stress gradient. The
dependence of ND and k as a function of n is described
by the exponents Kn and Kk, respectively.
lg lg
lg lg
n
( ) ( )
( ) ( )
min
max min
N N
N N
nD D
D D
K= +
−

(9)
k K
K K
n K
= +
−
min
max min

n
(10)
The equations (6) to (10) define the S-N curve for a
component with notches and irregular stress distribution
and can further be used for lifetime calculation of com-
plex geometric components. This allows for an efficient
analysis of S-N curves and an estimation of lifetime of
finite element structures with more than a hundred thou-
sand degrees of freedom in every node of the structure.
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192 187
Figire 4: Fatigue limit at 107 load cycles depending on stress gradient
Slika 4: Meja utrujenosti pri 107 obremenilnih ciklih v odvisnosti od
gradienta napetosti
Figure 3: Stress gradient in notch for tension-compression and
bending
Slika 3: Gradient napetosti za razteg-tlak in upogib
Figure 2: Stress gradient in notch root
Slika 2: Gradient napetosti na dnu zareze
2.2 Statistical size effect
The statistical size effect supposes that the flaws in a
component of large volume are more than those in a
smaller component owing to to statistical reasons.
Hence, bigger components show a higher probability for
the occurance of critical local stress due to a flaw. This
could probably lead to a crack which would grow and
eventually lead to the failure of the component. For eval-
uation of the statistical size effect the stress integral for
the high loaded volume (11) or the high loaded area (12)
was proposed:
V V
m
V



=
⎛
⎝
⎜
⎞
⎠
⎟∫
max
d (11)
A A
m
A



=
⎛
⎝
⎜
⎞
⎠
⎟∫
max
d (12)
max represents the notch root stress,  is the stress of a
volume or area element, m represents the Weibull
exponent which characterizes the homogeneity of the
material. For easier numerical calculation of highest
loaded volume it is proposed in 3 that the volume V90 %
which is loaded by at least 90 % of the maximum stress
should be taken into account for evaluation. For
calculation of a correction factor for fatigue strength the
ratio between highest loaded volume of a reference
specimen Vref to the highest loaded volume V of the
component is built.
n
V
V
m
st
ref=
⎛
⎝
⎜ ⎞
⎠
⎟
1
(13)
n
A
A
m
st
ref=
⎛
⎝
⎜ ⎞
⎠
⎟
1
(14)
2.3 Technological influences on forged components
Forging at different tool speeds primarily influences
the microstructural properties of the material. In addi-
tion, a change in mechanical properties of the component
like fatigue strength behaviour is also noticed, which is
caused by a difference in grain size or grain size distribu-
tion.
The influences on the fatigue limit are not uniform
over the complete structure but dependent on local pa-
rameters like:
– Local degree of deformation
– Local deformation rate
– Forging temperature
– Grain Size (mean value, variation)
– Inclusions in the microstructure
– Segregation direction
In Figure 5 the basic principle of a forging process is
shown, while Figure 6 depicts the distribution of degree
of deformation of a component calculated by simulation. 4.
The S-N curve for influence of degree of deformation
on fatigue strength of 16MnCr4 is depicted in Figure 7.
188 Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
Figure 7: Influence of degree of deformation on performance of
fatigue limit
Slika 7: Vpliv stopnje deformacije na velikost utrujenostne trdnosti
Figure 5: Principle chain of production process
Slika 5: Veriga postopkov procesa izdelave
Figure 6: Simulation of degree of deformation
Slika 6: Simulacija stopnje deformacije
Rotating bending fatigue tests have been performed on
un-notched specimens with local degrees of deformation
 = 0,  = 2.16 and  = 3. The fatigue limit shows a
clear dependence on local degree of deformation. 4.
The number of cycles at fatigue limit nearly stays
constant while the slope k of the S-N curve steepens with
lowering fatigue strength. The reason for the decrease of
strength is explained by metallographic analysis. Forged
specimens show greater variation of grain size. This is
ascribed to grain growth during the heating phase prior
to the forging process. Hence, the influence of a chosen
forging temperature on fatigue strength is a result of
change in grain size and grain size distribution.
2.4 Influence of heat treatment on fatigue strength
Simulation of the cooling during the annealing
process of a component is a big benefit for investigating
the influence of heat treatment on fatigue strength. The
temperature distribution during cooling calculated by
simulation is shown in Figure 8. When combining local
temperatures in dependence of time with continuous
TTT curves, the composition of local microstructure can
be estimated and hence the local strength calculated 4
(Figure 9).
In the example shown below, the microstructure of
the heat treated component shows a higher amount of
martensite on the surface than 10 mm below the surface,
which corresponds to the machining allowance.
Hence, on the surface of the machined component a
considerably different microstructure with different
strength characteristics exists. The influence of micro-
structural difference on the fatigue strength is shown in
the S-N curve in Figure 10. For the investigated material
the fatigue strength shows an increase of 7 % when
tempered at 540 °C instead of 620 °C.
2.5 Technological influence on cast aluminum compo-
nents
The technological influences, like casting process
and refinement of microstructure, have an important ef-
fect on the fatigue behaviour of cast aluminum alloys.
Furthermore, the local dendrite arm spacing (DAS) and,
the distribution and size of pores can change the lifetime
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192 189
Figure 10: Influence of tempering temperature on fatigue strength
Slika 10: Vpliv temperature popu{~anja na trdnost pri utrujenosti
Figure 9: Continuous TTT curve of 42CrMo4
Slika 9: Kontinuirni TTT-diagram za 42CrMo4
Figure 8: FE Simulation of annealing
Slika 8: Simulacija `arjenja s kon~nimi elementi
Figure 11: Definition of DAS 5
Slika 11: Definicija DAS 5
behaviour of the components. DAS and porosity are
mainly affected by the freezing rate, which depends on
the casting process, wall thickness, etc. Additionally, ox-
ides and impurities affect fatigue strength.
Generally a dendrite grows from a single nucleus,
which could be foreign particles or fragments of other
grains as shown in Figure 11. The dendrite arm spacing
is only affected by the local freezing rate or solidification
time, whereas the grain size especially depends on the
number of nuclei in the melt. Therefore a correlation
between grain size and freezing rate cannot be shown.
The grain of a cast material can be built up by differ-
ent dendrites of same origin. While grain size depends
on number of apparent nuclei, DAS is only affected by
local freezing rate and respective solidification time. In
this way no correlation can be found between grain size
and solidification time.
2.5.1 Die casting
For dimensioning dynamically loaded components
with regard to fatigue lifetime, the knowledge of the lo-
cal S-N-curve is necessary. These local S-N-curves, de-
termined by the material, are essentially influenced by
component specific effects, such as type of loading, size,
stress gradient, temperature and the production process.
In the case of cast components the technological influ-
ences can be described by DAS. DAS is not constantly
spread within the component. Its distribution depends on
the conditions under which the local solidification oc-
curs. As models for the simulation of solidification are
known and available local DAS can be determined in a
very early stage of the development process. Based on
the S-N-curves obtained by simulation and the local
stress concept, which takes into account the local stress
gradient, fatigue lifetime of geometrically complex, cast
components can be calculated. 1
For the die casting process DAS is a good parameter
for characterizing the microstructure with respect to fa-
tigue strength. Beside DAS, size and distribution of po-
rosity also influence fatigue strength of cast aluminium
alloys. Simulation of the freezing process allows the cal-
culation of DAS with respect to local freezing rate. The
S-N curves in Figure 12 show the strong influence of
DAS on fatigue strength of cast aluminium alloys. Speci-
men with DAS of 30 µm show a higher fatigue strength
than the ones with 70 µm, additionally the slope of the
S-N curve increases with greater DAS, and the number
of cycles at fatigue limit decreases 6.
2.5.2 High-Pressure die casting
While for die casting DAS is very important for fa-
tigue strength, gas porosity plays a main role for high
pressure die casting due to turbulent flow conditions
which can’t be avoided completely.
In 7 geometry and distribution of pores has been in-
vestigated using computer tomography (CT). CT analy-
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
190 Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192
Figure 13: Results of computer tomography: a – Overview of tomo-
graphed cube; b – pore pressed by dendrites; c – meshed pore.
Slika 13: Rezultati ra~unalni{ke tomografije: a – pogled na tomogra-
firano kocko, b – pore, ki jih stiskajo dendriti, c – zamre`ene pore
Figure 12: S-N curves for specimens with different DAS 6
Slika12: S-N-krivulje za razli~en DAS6
sis shows that the shape of gas pores obviously differs
from a sphere as shown in Figure 13.
By finite element calculations, the impact of gas
pores on the stress state in specimen has been examined.
Local stress elevations which result out of irregular
shape and notches show values in a range from 1,8 to
2,9. The larger the size of a pore, the stronger are the
dendrites pressed into the surface, thus resulting in
sharper notches.
Figure 14 shows the correlation between local stress
elevation of approximately spherical gas pores and the
pore diameter for tomographed pores with a range in di-
ameter from 50 µm to 350 µm.
2.5.3 Calculation of pore distribution in high pressure
die casting components
In 8 for calculation of pore distribution in a high pres-
sure die casting component the use of the so called statis-
tical porosity model is suggested. For a defined tempera-
ture range, post pressure and hold pressure, the
corresponding Weibull distribution of porosity in the
component is calculated by (15)
k c p c p c T c p cWeibull solid solid rel solid= + − + −( ) (1
2
2 3 4
2
5 p c
d c p c T c p c
solid
Weibull solid rel solid
+
= − + −
6
7 8 9 10
)
( ) ( )
(15)
Exact position of a pore in a component is not pre-
dictable, but distribution of porosity within a defined do-
main is. The porosity distribution provides information
regarding the percentage of elements within a domain,
defined by the temperature distribution during the freez-
ing process.
Pore diameter is an essential input in the fracture me-
chanics material model described in (8) in order to calcu-
late the pore size dependence of fatigue strength. Hence,
with the help of (16) the local porosity in the component
has to be converted to a single pore of an equivalent di-
ameter.
[ ]
d
A
equ
element Porosity=
4 100% /
π
(16)
wherein, Aelement corresponds to the area of the element,
which originally was used for porosity calculation.
Multiplying porosity with element area gives the area of
the pore within the element. The area equivalent dia-
meter dequ can be calculated using this equation. If two
small pores exist in an element in the original
calculation the recalculation will result into one big pore
which will further lead to conservative results.
Calculated pore distribution in comparison to real pore
distribution is presented in Figure 15 and an obvious
and pretty good correlation between both is observed.
The calculated pore distribution depends on post
pressure and shows a different pore distribution for each
calculation cycle as it relies on a random function. This
reflects reality in a good way, as components even if they
are of the same charge never show exactly the same pore
distribution. But in a statistic point of view, the results
can be compared by looking at the density and size of
pores.
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192 191
Figure 15: Pore distribution in real component (computertomography)
(a), calculated pore distribution (b)
Slika 15: Porazdelitev por v realnem elementu (ra~unalni{ka
tomografija) (a), izra~unana porazdelitev por (b)
Figure 14: FEM calculation for tomographed spherical gas pores with
varying diameters 7
Slika 14: FEM-izra~un za tomografirane sferi~ne plinske pore z
razli~nim premerom7
Figure 16: Distribution of safety against cyclic failure in the
component
Slika 16: Porazdelitev varnosti priti cikli~ni obremenitvi komponente
a)
b)
2.5.4 Calculation of safety against cyclic failure
A fracture mechanic model defines the correlation
between fatigue strength and pore size 8. With this
model, the local load capacity depending on the pore size
can be calculated in each element. The load in the com-
ponent is further calculated by FEM. Local safety for
each element is due to the correlation between local load
capacity and local load. The distribution of safety against
cyclic failure for the above shown reference component
is depicted in Figure 16.
3 CONCLUSION
For further optimization of components with the fo-
cus on higher utilisation of materials beside the knowl-
edge of local stresses also knowledge of local strengths
is needed. The later shows distinction on many different
influences
– Kind of loading (Push/Pull, Bending , Torsion)
– Geometry and size
– Temperature
– Mean Stress
– Surface Layer (Surface topography, residual stress,
microstructure, hardness)
– Load sequence
– Production process, Casting, Deforming, Cutting or
Welding
– etc.
All these influences lead to higher or lower strength
of the component. When occurring simultaneously the
effects can either strengthen or attenuate each other. The
experimental investigation of all these effects on fatigue
strength in its whole sum is due to time and cost factors
nearly impossible and can only be performed punctual,
additional simulation is needed for calculation.
On the basis of stress calculations with FEM and
simulation of casting and deforming process the
dimensioning process of components was shown. These
examples show how interdisciplinary work enhances the
significance of strength calculations.
4 REFERENCES
1 Bargel H. J., Schulze G.: Werkstoffkunde, Springer-Verlag Berlin
Heidelberg, 2005
2 Böhm J.: Zur Vorhersage von Dauerschwingfestigkeiten ungekerbter
und gekerbter Bauteile unter Berücksichtigung des statistischen
Größeneinflusses. Dissertation, TU München, 1980
3 Sonsino, C. M.: Zur Bewertung des Schwingfestigkeitsverhaltens
von Bauteilen mit Hilfe örtlicher Beanspruchungen. Konstruktion 45
(1993)
4 Fröschl J.: Fatigue effects of forged components: Technological ef-
fects and multiaxial fatigue, Dissertation, Montanuniversität Leoben,
2006
5 Altenpohl, D.: Aluminium von innen. Aluminium-Verlag, Düssel-
dorf, 1994
6 Minichmayr, R., Eichlseder, W.: Lebensdauerberechnung von
Gussbauteilen unter Berücksichtigung des lokalen Dendritenarmab-
standes und der Porosität, Gießerei, (2003) 5
7 Powazka D.: Einfluss der Porosität auf die Betriebsfestigkeit von
Al-Druckgussbauteilen, Dissertation, Montanuniversität Leoben,
2009
8 Oberwinkler Christian: Virtuelle betriebsfeste Auslegung von Alumi-
nium-Druckgussbauteilen, Dissertation, Montanuniversität Leoben,
2009
W. EICHLSEDER: ENHANCED FATIGUE ANALYSIS ...
192 Materiali in tehnologije / Materials and technology 44 (2010) 4, 185–192
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
A NEW COMPLEX VECTOR METHOD FOR
BALANCING CHEMICAL EQUATIONS
NOVA KOMPLEKSNA METODA ZA URAVNOTE@ENJE KEMI^NIH
ENA^B
Ice B. Risteski
2 Milepost Place # 606, Toronto, Ontario, Canada M4H 1C7
ice@scientist.com
»Just as tensor analysis is the proper language of kinematics,
so linear algebra is the proper language of stoichiometry.«
Aris, R.; Mah, R. H. S.; Ind. Eng. Chem. Fundam. 2 (1963), 90.
Prejem rokopisa – received: 2009-10-22; sprejem za objavo – accepted for publication: 2010-01-21
In this article, the author discovers a paradox of balancing chemical equations. The many counterexamples illustrate that the
considered procedure of balancing chemical equations given in the paper1 is inconsistent. A new complex vector method for
paradox resolution is given too.
Keywords: paradox, balancing, chemical equations, vector space.
V ~lanku avtor opisuje paradoks pri uravnote`enju kemijskih reakcij. Ve~ primerov dokazuje, da je procedura uravnote`enja
kemijskih reakcij v viru1 inkonsistentna. Predstavljena je nova kompleksna vektorska metoda za re{itev paradoksa.
Klju~ne besede: paradoks, uravnote`enje, kemijske reakcije, vektorski prostor
1 INTRODUCTION
Balancing chemical equations is a basic matter of
chemistry, if not one of its most important issues, and it
plays a main role in its foundation. Indeed, it is a subtle
question which deserves considerable attention.
Also, this topic was a magnet for a number of
researchers around the world, because the general
problem of balancing chemical equations was considered
as one of the hardest problems in chemistry, as well as in
mathematics.
In chemical literature there are a great number of
papers which consider the problem of balancing
chemical equations in different chemical ways, but all of
them offer only some particular procedures for balancing
simple chemical equations. Very often, these chemical
procedures generate absurd results, because most of
them are founded on presumed chemical principles, but
not on genuine exact principles. And these presumed
chemical principles generate paradoxes! The balancing
chemical equations does not depend on chemical
principles, it is a mathematical operation which is
founded on true mathematical principles.
There is a number of paradoxes in chemistry about
balancing chemical equations, and these will be
systematized and studied in a special article of higher
level by the same author which will appear in the near
future. In this article only one paradox in balancing
chemical equations, discovered in1 is considered.
The author would like to emphasize very clearly, that
balancing chemical equations is not chemistry; it is just
linear algebra. Although it is true, that it is not
chemistry, it is very important for chemistry! In this
particular case the following question comes up: If
balancing chemical equations is not chemistry, then why
is it considered in chemistry? Or maybe more important
is this question: If it is linear algebra, then it should be
studied in mathematics, so why bother chemists?
The author will address the above questions assuming
that: the problem of balancing chemical equations was a
multidisciplinary subject and for its solution both
mathematicians and chemists are needed. The job of
chemists is to perform reactions, while balancing their
equations is a job for mathematicians. Mathematics, as a
servant to other sciences, is a problem solver, but
chemistry is a result user.
The skepticism about balancing chemical equations
by chemical principles appeared a long time ago. Let’s
quote the opinions of three chemists.
In 1926 the American chemist Simons2 wrote: The
balancing of an equation is a mathematical process and
independent of chemistry. The order of steps in the
process is as essential as the order of steps in long
division and the process is much simpler than is ordinary
assumed. Seventy-one years later, the Dutch chemist Ten
Hoor3, made a similar statement: Balancing the equation
of the reaction is a matter of mathematics only.
Now is the right place to paraphrase the criticism of
the American chemist Herndon4: The major changes that
have taken place over seven decades are substitutions of
the terms šchange in oxidation number’ and šalgebraic
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 193
UDK 54 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)193(2010)
method’ for the terms švalence change’ and šmethod of
undetermined coefficients’, respectively. This assertion
shows very clearly the picture of chemists’ contributions
to balancing chemical equations by chemical principles.
Probably expressing his satisfaction with the chemists’
contributions to balancing chemical equations, Herndon,
the editor of the Journal of Chemical Education4, decided
that further discussion of equation balancing will not
appear in the Journal unless it adds something substan-
tively new to what has already appeared.
In view of the above assertions this question arises:
Are there in chemistry šchemical principles’ capable of
offering solutions of the general problem of balancing
chemical equations? Maybe more interesting is the
question: What are šchemical principles’ – a rhetorical
sophism of chemists or their hopes?
More interesting for us is to give an answer to both
questions from a scientific view point. Perhaps, the more
appropriate short answer to the first question is:
šChemical principles’ are not defined entities in
chemistry, and so this term does not have any meaning.
They are not capable to provide solutions of the general
problem of balancing chemical equations, because they
are founded on an intuitive basis and they represent only
a main generator for paradoxes. However, the necessary
and sufficient conditions for a complete solution of the
general problem of balancing chemical equations lie
outside of chemistry, and we must look for them in an
amalgamated theory of n-dimensional vector spaces,
linear algebra, abstract algebra and topology. It is a
very hard problem of the highest level in chemistry and
mathematics, which must be considered only on a
scientific basis. Like this should look the answer to the
first question.
The answer to the second question will be described
in this way: Actually, šchemical principles’ are a remnant
of an old traditional approach in chemistry, when
chemists were busy with the verification of their results
obtained in chemical experiments. It is true, that until
second half of 20th century there was no mathematical
method for balancing chemical equations in chemistry,
other than Bottomley’s algebraic method5. Chemists
balanced simple particular chemical equations using
only Johnson’s change in oxidation number procedure6,
Simons – Waldbauer – Thrun’s partial reactions
procedure2,7 and other slightly different modifications
derived from them. The šchemical principles’ were an
assumption of traditional chemists, who thought before
the appearance of Jones’ problem8, that the solution of
the general problem of balancing chemical equations is
possible by use of chemical procedures. But, practice
showed that the solution of the century old problem is
possible only by using contemporary sophisticated
mathematical methods.
These questions require a deeper elaboration than
given here, and it will be a main concern of the author in
his future research.
2 PRELIMINARIES
The exact statements about balancing chemical
equations agree with the following well-known results.
Theorem 1. Every chemical reaction can be reduced
in a matrix equation Ax = 0, where A is a reaction
matrix, x is a column-vector of the unknown coefficients
and 0 is a null column-vector.
Proof. The proof of this theorem immediately follows
from9.
Remark 2. The coefficients satisfy three basic prin-
ciples (corresponding to a closed input-output static
model10,11)
• the law of conservation of atoms,
• the law of conservation of mass, and
• the time-independence of the reaction.
Theorem 3. Every chemical reaction can be pre-
sented as a matrix Diophantine equation Ax = By, where
A and B are matrices of reactants and products, respec-
tively, and x and y are column-vectors of unknown
coefficients. The proof of this theorem is given in.12,13
This theorem is actually the Jones’8 problem founded
by virtue of Crocker’s procedure for balancing chemical
equations14, and from this problem the formalization of
chemistry began. This problem is a milestone in
chemistry and mathematics as well, and just it opened
the door in chemistry to enter a new mathematical
freshness. Jones8 with his problem transferred the
general problem of balancing chemical equations from
the field of chemistry into the field of mathematics and
opened way to solve this problem with mathematical
methods founded on principles of linear algebra.
Before the appearance of this problem, the approach
of balancing chemical equations was intuitive and useful
only for some elementary chemical equations. Now,
some well-known results from the theory of complex
n-dimensional vector spaces15,16, for resolution of
balancing chemical equations will be introduced. Here,
by C is denoted the set of complex numbers and by Cn is
denoted the Euclidian n-dimensional vector space with
complex entries.
Definition 4. A vector space over the field C consists
of a set V of objects called vectors for which the axioms
for vector addition hold
(A1) If u, v  V, then (u + v)  V,
(A2) u + v = v + u, u, v  V,
(A3) u + (v + w) = (u + v) + w, u, v, w  V,
(A4) u + 0 = u = 0 + u, u  V,
(A5) – u + u = 0 = u + (– u), u  V,
and the axioms for scalar multiplication
(S1) If u  V, then au  V, a  C,
(S2) a(u + v) = au + av, u, v  V 	 a  C,
(S3) (a + b)u = au + bu, u  V 	 a,b  C,
(S4) a(bu) = (abu), u  V 	 a,b  C,
(S5) 1u = u, u  V.
194 Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Remark 5. The content of axioms (A1) and (S1) is
described with thes assertion that V is closed under
vector addition and scalar multiplication. The element 0
in axiom A4 is called the zero vector.
Definition 6. If V is a vector space over the field C, a
subset U of V is called a subspace of V if U itself is a
vector space over C, where U uses the vector addition
and scalar multiplication of V.
Definition 7. Let V be a vector space over the field
C, and let vi  V (1 
 i 
 n). Any vector in V of the form
v = a1v1 + a2v2 + … + anvn, where ai  C, (1 
 i 
 n) is
called a linear combination of vi, (1 
 i 
 n).
Definition 8. The vectors v1, v2, …, vn are said to
span or generate V or are said to form a spanning set of
V if V = span{v1, v2, …, vn}. Alternatively, vi  V (1 
 i

 n) span V, if for every vector v  V there exist scalars
ai  C (1 
 i 
 n) such that
v = a1v1 + a2v2 + … + anvn,
i. e., v is a linear combination of
a1v1 + a2v2 + … + anvn.
Remark 9. If V = span{v1, v2, …, vn}, then each
vector v  V can be written as a linear combination of
the vectors v1, v2, …, vn. Spanning sets have the property
that each vector in V has exactly one representation as a
linear combinations of these vectors.
Definition 10. Let V be a vector space over a field C.
The vectors vi  V (1 
 i 
 n) are said to be linearly
independent over C, or simply independent, if it satisfies
the following condition: if
s1v1 + s2v2 + … + snvn = 0,
then
s1 = s2 = … = sn = 0.
Otherwise, the vectors that are not linearly indepen-
dent, are said to be linearly dependent, or simply depen-
dent.
Remark 11. The trivial linear combination of the
vectors vi, (1 
 i 
 n) is the one with every coefficient
zero
0v1 + 0v2 + … + 0vn.
Definition 12. A set of vectors {e1, e2, …, en} is
called a basis of V if it satisfies the following two
conditions
1° e1, e2, …, en are linearly independent,
2° V = span{e1, e2, …, en}.
Definition 13. A vector space V is said to be of finite
dimension n or to be n-dimensional, written dim V = n, if
V contains a basis with n elements.
Definition 14. The vector space {0} is defined to
have dimension 0.
3 PARADOX APPEARANCE
Chemistry as other natural sciences is not immune of
paradoxes. Unlike other natural sciences, in chemistry
paradoxes appeared some time later, and it has only two,
while in other sciences many such contradictions are
met. These paradoxes are well-known Levinthal’s
paradox17: The length of time in which a protein chain
finds its folded state is many orders of magnitude shorter
than it would be if it freely searched all possible
configurations, and Structure-Activity Relationship
(SAR) paradox18: Exceptions to the principle that a small
change in a molecule causes a small change in its
chemical behavior are frequently profound.
However, these paradoxes are not alone and there are
more, but now another will be mentioned, which appears
in the balancing of chemical equations.
In the paper1, the so-called formal balance numbers
(FBN) are introduced, like this: Formal balance numbers
are an aid that may grossly facilitate the problem of
balancing complex redox equations. They may be chosen
as being equal to the traditional values of oxidation
numbers, but not necessarily. An inspection of the redox
equation may suggest the optimal values that are to be
assigned to formal balance numbers. In most cases,
these optimal values ensure that only two elements will
šchange their state’ (i. e. the values of the formal balance
numbers), allowing the use of the oxidation number
technique for balancing equations, in its simplest form.
Just like for oxidation numbers, the algebraic sum of the
formal balance numbers in a molecule/neutral unit is 0,
while in an ion it is equal to its charge (the sum rule).
Promptly, it was detected that the procedure given in1
boils down to using of well-known unconventional
oxidation numbers, which previously were advocated by
Tóth19 and Ludwig20.
Consider this sentence from previous definition: They
may be chosen as being equal to the traditional values of
oxidation numbers, but not necessarily. It is a paradox! If
the formal balance numbers can be the same as oxidation
numbers or not, then the whole definition is illogical.
This definition represents only a contradictory premise,
which does not have any correlation with balancing che-
mical equations. The above definition does not speak
anything about balancing chemical reactions in a chemi-
cal sense of the word, or their solution in a mathematical
sense. In order for a chemical equation to be balanced
the first necessary and sufficient condition is its solvabi-
lity, but the above definition is far from it.
The so-called formal balance numbers, which actu-
ally are the same as the well-known oxidation numbers,
are not any criterion for balancing chemical equations.
If the chemical equation is not formalized, then it
generates only paradoxes. To support this assertion some
ordinary counterexamples will be given.
1° Balancing chemical equations by using of change
in oxidation number procedure has a limited usage! It
holds only for some simple equations. Is it possible to
determine valence of elements in some complex organic
molecules as they are C2952H4664N812O832S8Fe4,
C738H1166N812O203S2Fe, and so on?
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 195
Not yet! For instance, one may show the power-
lessness of that procedure by the following two counter-
examples.
Example 1.
2993 C2952H4664N816O832S8Fe4
+ 9300568 CsClO3 + 408724 Li3PO4
 11972 Li4Fe(CN)6 + 8763504CO2
+ 2370456CsNO3 + 1178284 LiHCl2
+ 408724 H3PO4 + 6944000 Cs0.998Cl
+ 23944 H2S + 5753504 H2O.
What is the valence of C, N, S and Fe?
Example 2.
249000 C738H1166N812O203S2Fe
+ 1098133767 Pb(NO3)2
+ 46812000 HSiCl3 + 7716178 P2O5
+ 23148534 CoCl2  46812000 SiCl3.989
+ 7716178 Co3(PO4)2 + 124500 H2SO4
+ 549616500 Pb1.998O3 + 124500 Fe2(SO4)3
+ 15313500 C12H22O11 + 2398455534 NO2.
The same previous question holds for this counter-
example too. This means, the solution of the above
equations is based on very sophisticated matrix
methods21–23, but in no case on the change in oxidation
number procedure!
As consequence of that, same holds for so-called
formal balance numbers. That procedure is useless to
balance complex chemical equations as it does not give
effects for balancing chemical equations. Still, there are
many causes for arguing why that elementary procedure
is useless, but the above mentioned two counterexamples
are enough to show that it is ineffective.
2° Consider this simple reaction
Example 3.
x1 NaNO2 + x2 FeSO4 + x3 H2SO4
 x4 NaHSO4 + x5 Fe2(SO4)3 + x6 NO.
In the above chemical reaction N and Fe change the
valence, but chemical equation is impossible! Thus, is
the procedure of formal balance numbers useful for
balancing all chemical equations? Obviously, not! Since
it does not have the capability to detect if some chemical
equation is possible or not.
3° Now, one more counterexample will be given,
where that particular procedure of formal balance
numbers is impossible.
For instance, consider this chemical reaction
Example 4.
128147987406 FeSO4
+ 27736706439 PrTlTe3
+ 286573109604 H3PO4
 128404797000 Fe0.998(H2PO4)2·H2O
+ 13945051000 Tl1.989(SO3)3
+ 13889187000 Pr1.997(SO4)3
+ 83210119317 TeO2 + 14881757802 P2O3
+ 44645273406 H2S.
Is it possible to balance the above chemical equation
by the procedure of formal balance numbers?
Not yet! Since it is not possible to determine the
valence of Fe, Tl and Pr. Then, on what basis the author
of the paper1 states, that his procedure (there called
method) is proposed for fast and easy balancing of
complex redox equations? On top of all, he states that the
procedure (there called method) is probably the fastest of
all possible methods! Really, a very modest statement is
offered by the author, which is wrong, not only because
of today’s current balancing methods view point, but also
from an earlier view point, when that procedure was
published.
Is it a method when somebody can find on every step
counterexamples? Obviously, the answer is negative! It is
merely a picture of the old chemical traditionalism. Or,
maybe it is a lonely case lost in the newest progressive
and contemporary mathematical generalism!
4 PARADOX RESOLUTION BY A NEW
COMPLEX VECTOR METHOD
With the purpose of solution of the paradox, in this
section a new complex vector method of balancing
chemical equations will be developed. This method is
founded on the theory of n-dimensional complex vector
spaces.
Theorem 15. Suppose that chemical equation
x1v1 + x2v2 + … + xnvn = 0, (1)
where vi (1 
 i 
 n) are the molecules and xi (1 
 i 

n) are unknown coefficients is a vector space V over the
field C spanned by the vectors of the molecules vi (1 
 i

 n). If any set of m vectors of the molecules in V is
linearly independent, then m 
 n.
Proof. Let be V = span{v1, v2, …, vn}. We must show
that every set {u1, u2, …, um} of vectors in V with m > n
fails to be linearly independent. This is accomplished by
showing that numbers x1, x2, …, xm can be found, not all
zero, such that
x1u1 + x2u2 + … + xmum = 0.
Since V is spanned by the vectors v1, v2, …, vn, each
vector uj can be expressed as a linear combination of vi
uj = a1jv1 + a2jv2 + … + anjvn.
Substituting these expressions into the preceding
equation gives
0 = x a a xj
j
m
ij
i
n
j ij
j
m
j i
i
n
= = ==
∑ ∑ ∑∑⎛⎝⎜
⎞
⎠
⎟ =
⎛
⎝
⎜
⎞
⎠
⎟
1 1 11
v v .
This is certainly the case if each coefficient of vi is
zero, i. e., if
a xij j
j
m
=
∑
1
= 0, (1 
 i 
 n).
This is a system of n equations with m variables x1,
x2, …, xm, and because m > n, it has a nontrivial solution.
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
196 Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203
This is what we wanted. Now we shall prove the follow-
ing results.
Theorem 16. Let U be a subset of a vector space V of
the chemical equation (1) over the field C. Then U is a
subspace of V if and only if it satisfies the following
conditions
0  U, where 0 is the zero vector of V, (2)
If u1, u2  U, then (u1 + u2)  U, (3)
If u  U, then au  U, a  C. (4)
Proof. If U is a subspace of V of the chemical
equation (1), it is clear by axioms (A1) and (S1) that the
sum of two vectors in U is again in U and that any scalar
multiple of a vector in U is again in U. In other words, U
is closed under the vector addition and scalar multipli-
cation of V. The converse is also true, i. e., if U is closed
under these operations, then all the other axioms are
automatically satisfied. For instance, axiom (A2) asserts
that holds u1 + u2 = u2 + u1, u1, u2  U. This is clear
because the equation is already true in V, and U uses the
same addition as V. Similarly, axioms (A3), (S2), (S3), (S4)
and (S5) hold automatically in U, because they are true in
V. All that remains is to verify axioms (A4) and (A5).
If (2), (3) and (4) hold, then axiom (A4) follows from
(2) and axiom (A5) follows from (4), because – u = (–1)u
lies in U, u  U. Hence, U is a subspace by the above
discussion. Conversely, if U is a subspace, it is closed
under addition and scalar multiplication and this gives
(3) and (4). If z denotes the zero vector of U, then z = 0z
in U. But, 0z = 0 in V, so 0 = z lies in U. This proves (2).
Remark 17. If U is a subspace of V of the chemical
equation (1) over the field, then the above proof shows
that U and V share the same zero vector. Also, if u  U,
then – u = (–1)u  U, i. e., the negative of a vector in U
is the same as its negative in V.
Proposition 18. If V is any vector space of the che-
mical equation (1) over the field C, then {0} and V are
subspaces of V.
Proof. U = V clearly satisfies the conditions of the
Theorem 16. As to U = {0}, it satisfies the conditions
because 0 + 0 = 0 and a0 = 0, a  C.
Remark 19. The vector space {0} is called the zero
subspace of V of the chemical equation (1) over the field
C. Since all zero subspaces look alike, we speak of the
zero vector space and denote it by 0. It is the unique
vector space containing just one vector.
Proposition 20. If v is a vector of some molecule in a
vector space V of the chemical equation (1) over the field
C, then the set Cv = {av, a  C} of all scalar
multiplies of v is a subspace of V.
Proof. Since 0 = 0v, it is clear that 0 lies in Cv. Given
two vectors av and bv in Cv, their sum av + bv = (a + b)v
is also a scalar multiple of v and so lies in Cv. Therefore
Cv is closed under addition. Finally, given av, r(av) =
(ra)v lies in Cv, so Cv is closed under scalar multipli-
cation. Now, if we take into account the Theorem 16,
immediately follows the statement of the proposition.
Theorem 21. Let U = span{v1, v2, …, vn} in a vector
space V of the chemical equation (1) over the field C.
Then,
U is a subspace of V containing each of vi (1 
 i 

n), (5)
U is the smallest subspace containing these vectors in
the sense that any subspace of V that contains each of vi
(1 
 i 
 n), must contain U. (6)
Proof. First we shall proof (5). Clearly
0 = 0v1 + 0v2 + … + 0vn
belongs to U. If
v = a1v1 + a2v2 + … + anvn
and
w = b1v1 + b2v2 + … + bnvn
are two members of U and a  U, then
v + w
= (a1 + b1)v1 + (a2 + b2)v2 + … + (an + bn)vn,
av = (aa1)v1 + (aa2)v2 + … + (aan)vn,
so both v + w and av lie in U. Hence, U is a subspace of
V. It contains each of vi (1 
 i 
 n). For instance,
v2 = 0v1 + 1v2 + 0v3 + … + 0vn.
This proves (5).
Now, we shall prove (6). Let W be subspace of V that
contains each of vi, (1 
 i 
 n). Since W is closed under
scalar multiplication, each of aivi (1 
 i 
 n) lies in W
for any choice of ai (1 
 i 
 n) in C. But, then aivi (1 
 i

 n) lies in W, because W is closed under addition. This
means that W contains every member of U, which proves
(6).
Theorem 22. The intersection of any number of
subspaces of a vector space V of the chemical equation
(1) over the field C is a subspace of V.
Proof. Let {Wi: i  I} be a collection of subspaces of
V and let W =  (Wi: i  I). Since each Wi is a subspace,
then 0  Wi, i  I. Thus 0  W. Assume u, v  W.
Then, u, v  Wi, i  I. Since each Wi is a subspace,
then (au + bv)  Wi, i  I. Therefore (au + bv)  W.
Thus W is a subspace of V of the chemical equation (1).
Theorem 23. The union W1 
 W2 of subspaces of a
vector space V of the chemical equation (1) over the field
C need not be a subspace of V.
Proof. Let V = C2 and let W1 = {(a, 0): a  C} and
W2 = {(0, b): b  C}. That is, W1 is the x-axis and W2 is
the y-axis in C2. Then W1 and W2 are subspaces of V of
the chemical equation (1). Let u = (1, 0) and v = (0, 1).
Then the vectors u and v both belong to the union W1 

W2, but u + v = (1, 1) does not belong to W1 
 W2. Thus
W1 
 W2 is not a subspace of V.
Theorem 24. The homogeneous system of linear
equations obtained from the chemical equation (1), in n
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 197
unknowns x1, x2, …, xn over the field C has a solution set
W, which is a subspace of the vector space Cn.
Proof. The system is equivalent to the matrix
equation Ax = 0. Since A0 = 0, the zero vector 0 
 W.
Assume u and v are vectors in W, i. e., u and v are
solutions of the matrix equation Ax = 0. Then Au = 0 and
Av = 0. Therefore, a, b 
 C, we have A(au + bv) = aAu
+ bAv = a0 + b0 = 0 + 0 = 0. Hence au + bv is a solution
of the matrix equation Ax = 0, i. e., au + bv 
 W. Thus
W is a subspace of Cn.
Theorem 25. If S is a subset of the vector space V of
the chemical equation (1) over the field C, then
1° the set span{S} is a subspace of V of the chemical
equation (1) over the field C which contains S.
2° span{S}  W, if W is any subspace of V of the
chemical equation (1) over the field C containing S.
Proof. 1°. If S = , then span{S} = {0}, which is a
subspace of V containing the empty set . Now assume
S  . If v 
 S, then 1v = v 
 span{S}, therefore S is a
subset of span{S}. Also, span{S}   because S  .
Now assume v, w 
 span{S}; say
v = a1v1 + … + amvm
and
v = b1w1 + … + bnwn,
where vi, wj 
 S and ai, bj are scalars.
Then
v + w = a1v1 + … + amvm + b1w1 + … + bnwn
and for any scalar k,
kv = k(a1v1 + … + amvm) = ka1v1 + … + kamvm belong to
span{S} because each is a linear combination of vectors
in S. Thus span{S} is a subspace of V of the chemical
equation (1) over the field C which contains S.
2°. If S = , then any subspace W contains S, and
span{S} = {0} is contained in W. Now assume S  
and assume vi 
 S  W (1  i  m). Then all multiples
aivi 
 W (1  i  m) where ai 
 C, and therefore the
sum (a1v1 + … + amvm) 
 W. That is, W contains all
linear combinations of elements of S. Consequently,
span{S}  W, as claimed.
Proposition 26. If W is a subspace of V of the che-
mical equation (1) over the field C, then span{W} = W.
Proof. Since W is a subspace of V of the chemical
equation (1) over the field C, W is closed under linear
combinations. Hence, span{W}  W. But W  span{W}.
Both inclusions yield span{W} = W.
Proposition 27. If S is a subspace of V of the che-
mical equation (1) over the field C, then span{span{S}}
= span{S}.
Proof. Since span{S} is a subspace of V, the above
Proposition 26 implies that span{span{S}} = span{S}.
Proposition 28. If S and T are subsets of a vector space
V of the chemical equation (1) over the field C, such that
S  T, then span{S}  span{T}.
Proof. Assume v 
 span{S}. Then
v = a1u1 + … + arur,
where ai 
 C, (1  i  r) and ui 
 S (1  i  r). But S
 T, therefore every ui 
 T (1  i  r). Thus v 

span{T}. Accordingly, span{S}  span{T}.
Proposition 29. The span{S} is the intersection of all
the subspaces of a vector space V of the chemical
equation (1) over the field C which contains S.
Proof. Let {Wi} be the collection of all subspaces of a
vector space V of the chemical equation (1) containing S,
and let W =  Wi. Since each Wi is a subspace of V, the
set W is a subspace of V. Also, since each Wi contains S,
the intersection W contains S. Hence span{S}  W. On
the other hand, span{S} is a subspace of V containing S.
So span{S} = Wk for some k. Then W  Wk = span{S}.
Both inclusions give span{S} = W.
Proposition 30. If span{S} = span{S 	 {0}}, then
one may delete the zero vector from any spanning set.
Proof. By Proposition 28, span{S}  span{S 	 {0}}.
Assume v 
 span{S 	 {0}}, say
v = a1u1 + … + anun + b⋅0
where ai, b 
 C (1  i  n) and ui 
 S (1  i  n).
Then v = a1u1 + … + anun, and so v 
 span{S}. Thus
span{S 	 {0}}  span{S}. Both inclusions give span{S}
= span{S 	 {0}}.
Proposition 31. If the vectors vi 
 V (1  i  n)
span a vector space V of the chemical equation (1) over
the field C, then for any vector w 
 V, the vectors w, vi
(1  i  n) span V.
Proof. Let v 
 V. Since the vi (1  i  n) span V,
there exist scalars ai (1  i  n) such that v = a1v1 + … +
anvn + 0w. Thus w, vi (1  i  n) span V.
Proposition 32. If vi (1  i  n) span a vector space
V of the chemical equation (1) over the field C, and for k
> 1, the vector vk is a linear combination of the pre-
ceding vectors vi (1  i  k – 1), then vi without vk span
V, i. e., span{v1, v2, …, vk–1, vk+1, …, vn} = V.
Proof. Let v 
 V. Since the vi (1  i  n) span V,
there exist scalars ai (1  i  n) such that
v = a1v1 + … + anvn.
Since vk is a linear combination of vi (1  i  k – 1),
there exist scalars bi (1  i  k – 1) such that
vk = b1v1 + … + ak–1vk–1.
Thus
v = a1v1 + … + akvk + … + anvn
= a1v1 + … + ak(b1v1 + … + bk–1vk–1) + … + anvn
= (a1 + akb1)v1 + … + (ak–1 + akbk–1)vk–1
+ ak+1vk+1 + … + anvn.
Therefore,
span{v1, v2, …, vk–1, vk+1, …, vn } = V.
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198 Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203
Proposition 33. If Wi, (1  i  k) are subspaces of a
vector space V of the chemical equation (1) over the field
C, for which W1  W2  …  Wk and
W = W1 	 W2 	 … 	 Wk,
then W is a subspace of V.
Proof. The zero vector 0 
 W1, hence 0 
 W.
Assume u, v 
 W. Then, there exist j1 and j2 such that u

 Wj1 and v 
 Wj2. Let j = max(j1, j2). Then Wj1  Wj
and Wj2  W, and so u, v 
 Wj. But Wj is a subspace.
Therefore (u + v) 
 Wj and for any scalar s the multiple
su 
 Wj. Since Wj  W, we have (u + v), su 
 W. Thus
W is a subspace of V.
Proposition 34. If Wi (1  i  k) are subspaces of a
vector space V of the chemical equation (1) over the field
C and Si (1  i  k) span Wi (1  i  k), then S = S1 	
S2 	 … 	 Sk spans W.
Proof. Let v 
 W. Then there exists j such that v 

Wj. Then v 
 span{Sj}  span{S}. Therefore W 
span{S}. But S  W and W is a subspace. Hence
span{S}  W. Both inclusions give span{S} = W, i. e., S
spans W.
Theorem 35. Let {v1, v2, …, vn} be a linearly inde-
pendent set of vectors in a vector space V of the chemical
equation (1) over the field C, then the following condi-
tions
1° {v, v1, v2, …, vn} is a linearly independent set,
2° v does not lie in {v1, v2, …, vn},
are equivalent for a vector v in V.
Proof. Assume 1° is true and assume, if possible, that
v lies in span{v1, v2, …, vn}, say,
v = a1v1 + … + anvn.
Then
v – a1v1 – … – anvn = 0
is a nontrivial linear combination, contrary to 1°. So 1°
implies 2°. Conversely, assume that 2° holds and assume
that
av + a1v1 + … + anvn = 0.
If a  0, then
v = (–a1/a)v1 + … + (–an/a)vn,
contrary to 2°. So a = 0 and
a1v1 + … + anvn = 0.
This implies that
a1 = … = an = 0,
because the set {v1, v2, …, vn} is linearly independent.
By this is proved that 2° implies 1°.
Proposition 36. Let {v1, v2, …, vn} be linearly
independent in a vector space V of the chemical equation
(1) over the field C, then {a1v1, a2v2, …, anvn}, such that
the numbers ai (1  i  n) are all nonzero, is also
linearly independent.
Proof. Suppose a linear combination of the new set
vanishes
s1(a1v1) + s2(a2v2) + … + sn(anvn) = 0,
where si (1  i  n) lie in C.
Then
s1a1 = s2a2 = … = snan = 0
by the linear independence of {v1, v2, …, vn}. The fact
that each ai  0 (1  i  n) now implies that s1 = s2 =
… = sn = 0.
Proposition 37. No linearly independent set of vec-
tors of molecules can contain the zero vector.
Proof. The set {0, v1, v2, …, vn} cannot be linearly
independent, because
10 + 0v1 + … + 0vn = 0,
is a nontrivial linear combination that vanishes.
Theorem 38. A set {v1, v2, …, vn} of vectors of mole-
cules in a vector space V of the chemical equation (1)
over the field C is linearly dependent if and only if some
vi is a linear combination of the others.
Proof. Assume that {v1, v2, …, vn} is linearly depen-
dent. Then, some nontrivial linear combination vanishes,
i. e.,
a1v1 + a2v2 + … + anvn = 0,
where some coefficient is not zero. Suppose a1  0.
Then
v1 = (–a2/a1)v2 + … + (–an/a1)vn,
gives v1 as a linear combination of the others. In gene-
ral, if ai  0, then a similar argument expresses vi as
linear combination of the others.
Conversely, suppose one of the vectors is a linear
combination of the others, i. e.,
v1 = a2v2 + … + anvn.
Then, the nontrivial linear combination 1v1 – a2v2 –
… – anvn equals zero, so the set {v1, v2, …, vn} is not
linearly independent, i. e., it is linearly dependent. A
similar argument works if any vi (1  i  n) is a linear
combinations of the others.
Theorem 39. Let V  0 be a vector space of the che-
mical equation (1) over the field C, then
1° each set of linearly independent vectors is a part
of a basis of V,
2° each spanning set V contains a basis of V,
3° V has a basis and dim V  n.
Proof. 1° Really, if V is a vector space that is spanned
by a finite number of vectors, we claim that any linearly
independent subset S = {v1, v2, …, vk} of V is contained
in a basis of V. This is certainly true if V = span{S}
because then S is itself a basis of V. Otherwise, choose
vk+1 outside span{S}. Then S1 = {v1, v2, …, vk, vk+1} is
linearly independent by Theorem 35. If V = span{S1},
then S1 is the desired basis containing S. If not, choose
vk+2 outside span{v1, v2, …, vk, vk+1} so that S2 = {v1, v2,
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 199
…, vk, vk+1, vk+2} is linearly independent. Continue this
process. Either a basis is reached at some stage or, if not,
arbitrary large independent sets are found in V. But this
later possibility cannot occur by the Theorem 15 because
V is spanned by a finite number of vectors.
2° Let V = span{v1, v2, …, vm}, where (as V  0) we
may assume that each vi  0. If {v1, v2, …, vm} is linearly
independent, it is itself a basis and we are finished. If
not, then according to the Theorem 38, one of these
vectors lies in the span of the others. Relabeling if
necessary, it is assumed that v1 lies in span{v2, …, vm} so
that V = span{v2, …, vm}. Now repeat the argument. If
the set {v2, …, vm} is linearly independent, we are
finished. If not, we have (after possible relabeling) V =
span{v3, …, vm}. Continue this process and if a basis is
encountered at some stage, we are finished. If not, we
ultimately reach V = span{vm}. But then {vm} is a basis
because vm  0 (V  0).
3° V has a spanning set of n vectors, one of which is
nonzero because V  0. Hence 3° follows from 2°.
Corollary 40. A nonzero vector space V of the
chemical equation (1) over the field C is finite dimen-
sional only if it can be spanned by finitely many vectors.
Theorem 41. Let V be a vector space of the chemical
equation (1) over the field C and dim V = n > 0, then
1° no set of more than n vectors in V can be linearly
independent,
2° no set of fewer than n vectors can span V.
Proof. V can be spanned by n vectors (any basis) so
1° restates the Theorem 15. But the n basis vectors are
also linearly independent, so no spanning set can have
fewer than n vectors, again by Theorem 15. This gives
2°.
Theorem 42. Let V be a vector space of the chemical
equation (1) over the field C and dim V = n > 0, then
1° any set of n linearly independent vectors in V is a
basis (that is, it necessarily spans V),
2° any spanning set of n nonzero vectors in V is a
basis (that is, they are necessarily linearly independent).
Proof. 1° If the n independent vectors do not span V,
they are a part of a basis of more than n vectors by pro-
perty 1° of the Theorem 39. This contradicts Theorem
41.
2° If the n vectors in a spanning set are not linearly
independent, they contain a basis of fewer than n vectors
by property 2° of Theorem 39, contradicting Theorem
41.
Theorem 43. Let V be a vector space of dimension n
of the chemical equation (1) over the field C and let U
and W denote subspaces of V, then
1° U is finite dimensional and dim U  n,
2° any basis of U is a part of a basis for V,
3° if U  W and dim U = dimW, then U = W.
Proof. 1° If U = 0, dimU = 0 by Definition 14. So
assume U  0 and choose u1  0 in U. If U = span{u1},
then dimU = 1. If U  span{u1}, choose u2 in U outside
span{u1}. Then {u1, u2}, is linearly independent by
Theorem 35. If U = span{u1, u2}, then dimU = 2. If not,
repeat the process to find u3 in U such that {u1, u2, u3} is
linearly independent and continue in this way. The pro-
cess must terminate because the space V (having dimen-
sion n) cannot contain more than n independent vectors.
Therefore U has a basis of at most n vectors, proving 1°.
2° This follows from 1° and Theorem 39.
3° Let dimU = dimW = m. Then any basis {u1, u2, …,
um} of U is an independent set of m vectors in W and so
is a basis of W by Theorem 42. In particular, {u1, u2, …,
um} spans W so, because it also spans U, W = span{u1,
u2, …, um} = U. By this is proved 3°.
Proposition 44. If U and W are subspaces of a vector
space V of the chemical equation (1) over the field C,
then U + W is a subspace of V.
Proof. Since U and W are subspaces, 0 
 U and 0 

W. Hence 0 = 0 + 0 
 U + W. Assume v, v’ 
 U + W.
Then there exist u, u’ 
 U and v, v’ 
 W such that v = u
+ w and v’ = u’ + w’. Since U and W are subspaces, u +
u’ 
 U and w + w’ 
 W and for any scalar k, ku 
 U and
kw 
 W. Accordingly, v + v’ = (u + w) + (u’ + w’) = (u +
u’) + (w + w’) 
 U + W and for any scalar k, kv = k(u +
w) = ku + kw 
 U + W. Thus U + W is a subspace of V.
Proposition 45. If U and W are subspaces of a vector
space V of the chemical equation (1) over the field C,
then U and W are contained in U + W.
Proof. Let u 
 U. By hypothesis W is a subspace of
V and so 0 
 W. Hence u = u + 0 
 U + W. Accordingly,
U is contained in U + W. Similarly, W is contained in U +
W. By this the proof is finished.
Proposition 46. If U and W are subspaces of a vector
space V of the chemical equation (1) over the field C,
then U + W is the smallest subspace of V containing U
and V, i. e., U + W = span{U, W}.
Proof. Since U + W is a subspace of V containing
both U and W, it must also contain the linear span of U
and W, i. e., span{U, W}  U + W.
On the other hand, if v 
 U + W, then v = u + w = 1u
+ 1w, where u 
 U and w 
 W. Hence, v is a linear
combination of elements in U 	 W and so belongs to
span{U, W}. Therefore, U + W  span{U, W}. Both
inclusions give us the required result.
Proposition 47. If W is a subspace of a vector space
V of the chemical equation (4. 2) over the field C, then W
+ W = W.
Proof. Since W is a subspace of V, we have that W is
closed under vector addition. Therefore, W + W  W. By
Proposition 45, W  W + W. Thus, W + W = W. By this
the proof is finished.
Proposition 48. If U and W are subspaces of a vector
space V of the chemical equation (1) over the field C,
such that U = span{S} and W = span{T}, then U + W =
span{S 	 T}.
Proof. Since S  U  U + W and T  W  U + W,
we have S 	 T  U + W. Hence span{S 	 T}  U + W.
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
200 Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203
Now assume v 
 U + W. Then v = u + W, where u 
 U
and w 
 W. Since U = span{S} and W = span{T}, u =
a1u1 + … + arur and w = b1w1 + … + bsws, where ai, bj 

C, uj 
 S, and wi 
 T. Then v = u + w = a1u1 + … + arur
+ b1w1 + … + bsws. Thus, U + W  span{S 	 T}. Both
inclusions yield U + W = span{S 	 T}.
Proposition 49. If U and W are subspaces of a vector
space V of the chemical equation (1) over the field C,
then V = U + W if every v 
 V can be written in the form
v = u + w, where u 
 U and w 
 W.
Proof. Assume, for any v 
 V, we have v = u + w
where u 
 U and w 
 W. Then v 
 U + W and so V  U
+ W. Since U and V are subspaces of V, we have U + W
 V. Both inclusions imply V = U + W.
By this, in whole is given the skeleton of the complex
vector method.
5 AN APPLICATION OF THE MAIN RESULTS
In this section the above complex vector method will
be applied on some chemical equations for their balanc-
ing. All chemical equations balanced here appear for the
first time in professional literature and they are chosen
with an intention to avoid all well-known to date che-
mical equations which were repeated many times in the
chemical journals for explanation of certain particular
techniques for balancing of some chemical equations
using only atoms with integer oxidation numbers.
1° First, we shall consider the case when the chemi-
cal reaction is infeasible.
Example 5. Consider this chemical reaction
z1 K4Fe(CN)6 + z2 K2S2O3

 z3 CO2 + z4 K2SO4 + z5 NO2 + z6 FeS,
zi 
 C, (1  i  6).
From the following scheme
v 1
=
K
4F
e(
C
N
) 6
v 2
=
K
2S
2O
3
v 3
=
C
O
2
v 4
=
K
2S
O
4
v 5
=
N
O
2
v 6
=
F
eS
K 4 2 0 2 0 0
Fe 1 0 0 0 0 1
C 6 0 1 0 0 0
N 6 0 0 0 1 0
S 0 2 0 1 0 1
O 0 3 2 4 2 0
follows this vector equation
z1v1 + z2v2 = z3v3 + z4v4 + z5v5 + z6v6,
i. e.,
z1 (4, 1, 6, 6, 0, 0)
T + z2 (2, 0, 0, 0, 2, 3)
T
= z3 (0, 0, 1, 0, 0, 2)
T + z4 (2, 0, 0, 0, 1, 4)
T
+ z5 (0, 0, 0, 1, 0, 2)
T + z6 (0, 1, 0, 0, 1, 0)
T,
or
(4z1 + 2z2, z1, 6z1, 6z1, 2z2, 3z2)
T
= (2z4, z6, z3, z5, z4 + z6, 2z3 + 4z4 + 2z5)
T.
From the system of linear equations
4z1 + 2z2 = 2z4,
z1 = z6,
6z1 = z3,
6z1 = z5,
2z2 = z4 + z6,
3z2 = 2z3 + 4z4 + 2z5,
one obtains the contradictions z2 = 3z1 and z2 = 44z1/3,
that means that the system is inconsistent. According to
Definition 8, the vectors v1, v2, v3, v4, v5 and v6 of the
molecules of the chemical reaction (5. 1) do not gene-
rate a vector space V, and according to the Definition 10
they are linearly independent, i. e., we have only a
trivial solution zi = 0, (1  i  6), that means that the
chemical reaction is infeasible.
2° Next, we shall consider the case when the chemi-
cal reaction is feasible and it has a unique solution.
This type of chemical equations really is the most
appropriate for study the process of balancing chemical
equations, because it gives an excellent opportunity for
application of the group theory.
At once, we would like to emphasize here, that by
application of groups theory one may determine Sylow
subgroups, conjugacy classes of maximal subgroups,
proper normal subgroups, and so one. The main reason
why we confined ourselves to the next group of calcu-
lations is limitation of the size of the work.
Example 6. Consider this chemical reaction
z1 Fe2(SO4)3 + z2 PrTlTe3 + z3 H3PO4

 z4 Fe0.996(H2PO4)2·H2O + z5 Tl1.987(SO3)3
+ z6 Pr1.998(SO4)3 + z7 Te2O3
+ z8 P2O5 + z9 H2S, zi 
 C, (1  i  9).
According to the scheme given below
v 1
=
F
e 2
(S
O
4)
3
v 2
=
P
rT
lT
e 3
v 3
=
H
3P
O
4
v 4
=
F
e 0
.9
96
(H
2P
O
4)
2·
H
2O
v 5
=
T
l 1
.9
87
(S
O
3)
3
v 6
=
P
r 1
.9
98
(S
O
4)
3
v 7
=
Te
2O
3
v 8
=
P
2O
5
v 9
=
H
2S
Fe 2 0 0 0.996 0 0 0 0 0
S 3 0 0 0 3 3 0 0 1
O 12 0 4 9 9 12 3 5 0
Pr 0 1 0 0 0 1.998 0 0 0
Tl 0 1 0 0 1.987 0 0 0 0
Te 0 3 0 0 0 0 2 0 0
H 0 0 3 6 0 0 0 0 2
P 0 0 1 2 0 0 0 2 0
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 201
one obtains the following vector equation
z1v1 + z2v2 + z3v3 = z4v4 + z5v5 + z6v6
+ z7v7 + z8v8 + z9v9,
i. e.,
z1 (2,3,12,0,0,0,0,0)T + z2 (0,0,0,1,1,3,0,0)T
+ z3 (0,0,4,0,0,0,3,1)T
= z4 (0.996,0,9,0,0,0,6,2)T
+ z5 (0,3,9,0,1.987,0,0,0)T
+ z6 (0,3,12,1.998,0,0,0,0)T
+ z7 (0,0,3,0,0,2,0,0)T + z8 (0,0,5,0,0,0,0,2)T
+ z9 (0,1,0,0,0,0,2,0)T,
from where follows this system of linear equations
2z1 = 0.996z4,
3z1 = 3z5 + 3z6 + z9,
12z1 + 4z3 = 9z4 + 9z5 + 12z6 + 3z7 + 5z8,
z2 = 1.998z6,
z2 = 1.987z5,
3z2 = 2z7,
3z3 = 6z4 + 2z9,
z3 = 2z4 + 2z8,
From the last system one obtains the required solu-
tion. This show that the vectors vi (1 
 i 
 9) generate a
vector space V and they are linearly dependent. The
balanced reaction has this form
17839883133 Fe2(SO4)3
+ 12843034110 PrTlTe3
+ 81542933266 H3PO4
 35823058500 Fe0.996(H2PO4)2·H2O
+ 6463530000 Tl1.987(SO3)3
+ 6427945000 Pr1.998(SO4)3
+ 19264551165 Te2O3
+ 4948408133 P2O5 + 14845224399 H2S.
3° Next, the case when the chemical reaction is non-
unique will be considered, i. e., when it has infinite
number of solutions.
Example 7. Consider double reduction of titanium
dioxide with carbon and chlorine given by the reaction
z1 TiO2 + z2 C + z3 Cl2  z4 TiCl4
+ z5 CO + z6 CO2.
This reaction plays an important role in theory of
metallurgical processes, especially in processes of direct
reduction of metal oxides. Sure, that this reaction is not
unique, and there are many other reactions of that kind,
which may to be used for analysis of this particular case.
From the following scheme
v 1
=
T
iO
2
v 2
=
C
v 3
=
C
l 2
v 4
=
T
iC
l 4
v 5
=
C
O
v 6
=
C
O
2
Ti 1 0 0 1 0 0
O 2 0 0 0 1 2
C 0 1 0 0 1 1
Cl 0 0 2 4 0 0
follows this vector equation
z1v1 + z2v2 + z3v3 = z4v4 + z5v5 + z6v6,
i. e.,
(z1, 2z1, 0, 0)
T + (0, 0, z2, 0)
T
+ (0, 0, 0, 2z3)
T = (z4, 0, 0, 4z4)
T
+ (0, z5, z5, 0)
T + (0, 2z6, z6, 0)
T,
or
(z1, 2z1, z2, 2z3)
T
= (z4, z5 + 2z6, z5 + z6, 4z4)
T,
i. e., immediately follows this system of linear equations
z1 = z4,
2z1 = z5 + 2z6,
z2 = z5 + z6,
2z3 = 4z4,
which general solution is z3 = 2z1, z4 = z1, z5 = -2z1 +
2z2, z6 = 2z1 – z2, where z1 and z2 are arbitrary complex
numbers.
Now, balanced general chemical reaction has this
form
z1 TiO2 + z2 C + 2z1 Cl2  z1 TiCl4
+ (- 2z1 + 2z2) CO + (2z1 – z2) CO2,
(z1, z2  C).
According to the Definition 8, the vectors v1, v2, v3,
v4, v5 and v6 of the molecules of the chemical reaction
generate infinite number of vector spaces V∞, and
according to the Definition 10 they are linearly depen-
dent, i. e., we have an infinite number of solutions (z1, z2,
2z1, z1, – 2z1 + 2z2, 2z1 – z2), (z1, z2  C), that means
that the this chemical reaction is non-unique.
6 DISCUSSION
In his previous work22, the author announced a
cleaning of chemistry from barren intuitionism and its
substitution by an elegant formalism from one side, and
substitution of the old chemical traditionalism by a new
mathematical generalism, from other side. This
announcement is realized in this work that gives a new
contribution to the theory as well as practice of balancing
chemical equations.
The complex vector method of balancing chemical
equations augmented the research field in chemistry and
made obsolete the old traditional approach, and gave
reliable results for paradox resolution.
By this work will begin consideration of paradoxes in
chemistry as a serious object, and it will increase re-
searchers’ carefulness to avoid appearance of paradoxes.
7 CONCLUSION
The new complex mathematical method of balancing
chemical equations, which was used for the solution of a
paradox is farewell to the chemical tradition, which still
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
202 Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203
respects the composers of general chemistry textbooks,
that affirm that chemistry is a science which studies the
structure of substances, how they react when combined
or in contact, and how they behave under different
conditions. These subjects include a great part of the
matter to which chemistry was applied.
In this work the foundation of chemistry is enriched
by one more new topic, and a contribution to a new
formalization of chemistry founded by virtue of a new
complex vector method of balancing chemical equations
is offered. This work opens doors for the next research in
chemistry for diagnostic of paradoxes and their resolu-
tion. It will accelerate the newest mathematical research
in chemistry and it will surmount the barriers hampering
the development of chemistry.
This work is a critical survey that requires changes of
chemical thinking. Hence, it must be distinguished from
the uncritical penetration, in which chemistry itself is
developed sometimes.
ACKNOWLEDGEMENT
Author would like to thank Prof. Dr Valery C. Cova-
chev from the Bulgarian Academy of Sciences and to the
Dutch chemist Marten J. Ten Hoor for reading the article
and for their remarks.
In addition, author would like to thank Prof. Dr Franc
Vodopivec from the University of Ljubljana for his
helpful suggestions.
8 REFERENCES
1 Petru{evski, V. M.; Glas. hem. tehnol. Makedonija 17 (1998), 141
2 Simons, J. H.; J. Chem. Educ. 3 (1926), 1305
3 Ten Hoor, M. J.; J. Chem. Educ. 74 (1997), 1367
4 Herndon, W. C.; J. Chem. Educ. 74 (1997), 1359
5 Bottomley, J.; Chem. News 37 (1878), 110
6 Johnson, O. C.; Chem. News 42 (1880), 51
7 Waldbauer, L. J.; Thrun, W. E.; J. Chem. Educ. 3 (1926), 1430
8 Jones, M.; SIAM Rev. 13 (1971), 571
9 Alberty, R. A.; J. Chem. Educ. 68 (1991), 984
10 Zeggeren, V. F.; Storey, S. H.; The Computation of Chemical
Equlibria, Cambridge Univ. Press: London, 1970
11 Smith, W. R.; Missen, R. W.; Chemical Reaction Equilibrium Analy-
sis: Theory and Algorithms, Wiley: New York, 1982
12 Risteski, I. B.; SIAM Problems & Solutions, 2007, 1–10
13 Risteski, I. B.; J. Korean Chem. Soc. 52 (2008), 223
14 Crocker, R.; J. Chem. Educ. 45 (1968), 731
15 Halmos, P.; Finite Dimensional Vector Spaces, Springer Vlg: Berlin,
1998
16 Kurepa, S.; Kona~no dimezionalni vektorski prostori i primjene, 5-to
izd., Sveu~ilisna naklada Liber: Zagreb, 1990
17 Levinthal, C.; J. Chim. Phys. 65 (1968), 44
18 Patani, G. A.; LaVoie, E. J.; Chem. Rev. 96 (1996), 3147
19 Tóth, Z.; J. Chem. Educ. 74 (1997), 1270
20 Ludwig, O.; J. Chem. Educ. 74 (1997), 1270
21 Risteski, I. B.; Internat. J. Math. Manuscripts 1 (2007), 180
22 Risteski, I. B.; Bol. Soc. Quím. Méx. 2 (2008), 104
23 Risteski, I. B.; J. Chin. Chem. Soc. 56 (2009), 65
ICE B. RISTESKI: A NEW COMPLEX VECTOR METHOD FOR BALANCING CHEMICAL EQUATIONS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 193–203 203

H. AYDIN ET AL.: APPLICATION OF GREY RELATION ANALYSIS (GRA) ...
APPLICATION OF GREY RELATION ANALYSIS (GRA)
AND TAGUCHI METHOD FOR THE PARAMETRIC
OPTIMIZATION OF FRICTION STIR WELDING (FSW)
PROCESS
UPORABA GREYJEVE ANALIZE (GRA) IN TAGUCHIJEVE
METODE ZA PARAMETRI^NO OPTIMIZACIJO VARJENJA Z
VRTILNO-TORNIM PROCESOM (FSW)
Hakan Aydin1, Ali Bayram2, Ugur Esme3*, Yigit Kazancoglu4, Onur Guven5
1,2Uludag University, Faculty of Engineering and Architecture, Department of Mechanical Engineering, 16059, Gorukle-Bursa/Turkey
3Mersin University Tarsus Technical Education Faculty Department of Mechanical Education, 33480, Tarsus-Mersin/Turkey
4Izmir University of Economics, Department of Business Administration, 35330, Balcova-Izmir/Turkey
5Mersin University, Engineering Faculty, Department of Mechanical Engineering, 33400, Mersin/Turkey
uguresme@gmail.com
Prejem rokopisa – received: 2010-02-15; sprejem za objavo – accepted for publication: 2010-02-25
This study focused on the multi-response optimization of friction stir welding (FSW) process for an optimal parametric
combination to yield favorable tensile strength and elongation using the Taguchi based Grey relational analysis (GRA). The
objective functions have been selected in relation to parameters of FSW parameters; rotating speed, welding speed and tool
shoulder diameter. The experiments were planned using Taguchi’s L8 orthogonal array. Multi-response optimization was applied
using Grey relation analysis and Taguchi approach to solve the problem. The significance of the factors on overall quality
characteristics of the welding process has also been evaluated quantitatively by the analysis of variance (ANOVA) method.
Optimal results have been verified through confirmation experiments. This study has also showed the application feasibility of
the Grey relation analysis in combination with Taguchi technique for continuous improvement in welding quality.
Keywords: Friction stir welding, Grey relation analysis, Taguchi method, optimization
Cilj raziskave je bila ve~odgovorna optimizacija procesa varjenja z vrtilnim trenjem (FSW) za kombinacijo parametrov za
dosego ugodnih raztr`ne trdnosti in raztezka z uporabo Taguchi-Greyjeve racionalne analize. Primerne funkcije so bile izbrane v
povezavi s FSW-parametri: hitrost vrtenja, hitrost varjenja in premer ramen orodja. Preizkusi so bili izvr{eni z uporabo
Taguchijeve ortogonalne mre`e L8. Odgovori so bili optimizirani z uporabo Greyjeve analize s Taguchijevim pribli`kom. Pomen
dejavnikov splo{nih zna~ilnosti procesa varjenja je bil kvantitativno analiziran z analizo variance (ANOVA). Optimalni rezultati
so bili preverjeni s preizkusi. Raziskava je tudi pokazala uporabnost Greyjeve analize v povezavi s Taguchijevo tehniko za stalno
izbolj{anje tehnike varjenja.
Klju~ne besede: vrtilno torno varjenje, Greyjeva analiza odvisnosti, Taguchijeva metoda, optimizacija
1 INTRODUCTION
In today’s manufacturing world, quality is of vital
importance. Quality can be defined as the degree of cus-
tomer’s satisfaction as provided by the procured product.
The product quality depends on the desired requirements
gained in the product that suits its functional require-
ments in various areas of application.1
In the field of welding, weld quality mainly depends
on the welding type, mechanical properties of the weld
metal and heat affected zone (HAZ), which in turn is in-
fluenced by metallurgical characteristics and chemical
compositions of the weld.1 Moreover, these mechani-
cal-metallurgical features of the weldment directly re-
lated to welding process parameters. In other words,
weld quality depends on welding process parameters.1
The welding of aluminum and its alloys has always
represented a great challenge for designers and technolo-
gists.2 Friction stir welding (FSW) is a welding tech-
nique, patented in 1991 by TWI.3,4
As a solid-state process, FSW can avoid the forma-
tion of solidification cracking and porosity associated
with fusion (FSW) welding processes and significantly
improve the weld properties of aluminum alloys.4,5 As il-
lustrated in Figure 1, this technique involves a non-con-
Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211 205
UDK 621.791:669.715 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)205(2010)
Figure 1: Schematic representation of FSW
Slika 1: Shemati~na predstavitev FSW
sumable, cylindrical, rotating tool (usually hardened
steel) which moves between the seam of two butted
plates and stirs them together.2–6 The effect of friction stir
welding on the material is both on heat flow and plastic
strain. The heat is generated by friction between the tool
shoulder and the top of the sheets.
When compared to traditional welding techniques,
FSW strongly reduces the presence of distortions and
residual stresses.8–11 The FSW process is a solid state
process and therefore a solidification structure is absent
in the weld. A detailed description of the FSW process is
present in the literature.7–16 The process can be easily
monitored and replicated. In addition, it does not
produce any major safety hazards, such as fume or
radiation.17 This process is used to bond metals without
fusion or filler materials.18 FSW of aluminum has several
advantages over fusion welding processes. Problems
arising from fusion welding of aluminum alloys, such as
solidification cracking, liquation cracking and porosity,
are eliminated with FSW, due to its solid-state
nature.17–23
The Taguchi method is very popular for solving
optimization problems in the field of production
engineering.24,25 The method utilizes a well-balanced
experimental design (allows a limited number of experi-
mental runs) called orthogonal array design, and signal-
to-noise ratio (S/N ratio), which serve the objective
function to be optimized (maximized) within experi-
mental domain.24 However, traditional Taguchi method
cannot solve multi-objective optimization problem. To
overcome this, the Taguchi method coupled with Grey
relational analysis has a wide area of application in
manufacturing processes. This approach can solve
multi-response optimization problem simultaneously.26,27
Planning the experiments through the Taguchi
orthogonal array has been used quite successfully in
process optimization by Chen and Chen,28 Fung and
Kang,29 Tang et al,30 Vijian and Arunachalam31 as well as
Zhang et al.32 Therefore, this study applied a Taguchi L8
orthogonal array to plan the experiments on FSW
welding process. Three controlling factors including
rotating speed (w), welding speed (V) and shoulder
diameter (d) were selected. The Grey relational analysis
is then applied to examine how the welding process
factors influence the tensile strength (TS) and percent
elongation (e). An optimal parameter combination was
then obtained. Through analyzing the Grey relational
grade matrix, the most influential factors for individual
quality targets of FSW welding process can be identified.
Additionally, the analysis of variance (ANOVA) was also
utilized to examine the most significant factors for the
tensile strength and elongation in FSW welding process.
2 GREY RELATIONAL ANALYSIS (GRA)
2.1 Data Preprocessing
In Grey relational analysis, experimental data i.e.,
measured features of quality characteristics are first nor-
malized ranging from zero to one. This process is known
as Grey relational generation. Next, based on normalized
experimental data, Grey relational coefficient is calcu-
lated to represent the correlation between the desired and
actual experimental data. Then overall Grey relational
grade is determined by averaging the Grey relational co-
efficient corresponding to selected responses.26 The over-
all performance characteristic of the multiple response
process depends on the calculated Grey relational grade.
This approach converts a multiple response process opti-
mization problem into a single response optimization sit-
uation with the objective function is overall Grey rela-
tional grade. The optimal parametric combination is then
evaluated which would result highest Grey relational
grade. The optimal factor setting for maximizing overall
Grey relational grade can be performed by Taguchi
method.26,33
In Grey relational generation, the normalized E corre-
sponding to the smaller-the-better (SB) criterion which
can be expressed as:
x k
y k y k
y k y ki
i i
i i
( )
max ( ) ( )
max ( ) min ( )
=
−
−
(1)
TS should follow the larger-the-better (LB) criterion,
which can be expressed as:
x k
y k y k
y k y ki
i i
i i
( )
( ) min ( )
max ( ) min ( )
=
−
−
(2)
where xi(k) is the value after the Grey relational genera-
tion, min yi(k) is the smallest value of yi(k) for the kth re-
sponse, and max yi(k) is the largest value of yi(k) for the
kth response.26 An ideal sequence is x0(k) (k = 1, 2,
3......, 8 for the responses. The definition of Grey rela-
tional grade in the course of Grey relational analysis is
to reveal the degree of relation between the 16 se-
quences [x0(k) and xi(k), i = 1, 2, 3.......,8]. The Grey re-
lational coefficient # can be calculated as:



i i
k
k
( )
( )
min max
max
=
−
+
∆ ∆
∆ ∆0
(3)
where ∆ 0 0i ix k x k= −( ) ( ) = difference of the absolute
value x0(k) and xi(k);  is the distinguishing coefficient
0    1; ∆ min
min min ( ) ( )= ∀ ∈ ∀ −j i k x k x kj0 = the
smallest value of 0i; and
∆ max
max max ( ) ( )= ∀ ∈ ∀ −j i k x k x kj0 = largest value of
0i. After averaging the Grey relational coefficients, the
Grey relational grade i can be computed as:
g
n
ki i
k
n
=
=
∑1
1

 ( ) (4)
where n is the number of process responses. The higher
value of Grey relational grade corresponds to intense
relational degree between the reference sequence x0(k)
and the given sequence xi(k). The reference sequence
x0(k) represents the best process sequence; therefore,
H. AYDIN ET AL.: APPLICATION OF GREY RELATION ANALYSIS (GRA) ...
206 Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211
higher Grey relational grade means that the corre-
sponding parameter combination is closer to the
optimal. The mean response for the Grey relational
grade with its grand mean and the main effect plot of
Grey relational grade are very important because
optimal process condition can be evaluated from this
plot.26
3 EXPERIMENTAL PROCEDURE AND TEST
RESULTS
3.1 Experimental Details
AA1050-H22 aluminum alloy material was used as a
workpiece material with the thickness of 4 mm. The
workpieces were machined out in 360 mm lengths and
200 mm widths. The mechanical properties and percent
composition of workpiece material is listed in Table 1.
1.2367 (X38CrMoV5-3) hardened and threaded (left
screw with 0.8 mm pitch) pins with the shoulder diame-
ters of 15 mm and 20 mm were used as welding tools.
The dimensions of the welding tools are shown in Fig-
ure 2.
The pre-machined aluminum plates were fixed rig-
idly on the table of the vertical semiautomatic milling
machine for lap joint of FSW as shown in Figure 3.
The rotating tool was fixed to the spindle of the mill-
ing machine and then the spindle of the milling machine
was adjusted at an angle of 2–3° away from the spindle
travel path. To generate the required pre-frictional heat-
ing, the shoulder of the rotating tool was held in its ini-
Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211 207
H. AYDIN ET AL.: APPLICATION OF GREY RELATION ANALYSIS (GRA) ...
Table 1: Chemical and mechanical properties of AA1050 aluminum alloy
Tabela 1: Kemi~na sestava in mehanske lastnosti zlitine AA1050
Chemical
composition
w/%
Al Mg Si Mn Zn Fe Ti Sn
Balance 0.007 0.18 0.05 0.033 0.30 0.009 0.182
Mechanical
properties
Yield strength (MPa) Tensile strength (MPa) Elongation (%) Vickers Hardness (HV)
155 175 4 50
Table 2: Process parameters and their limits
Tabela 2: Parametri in limite procesa
Parameters Notation Unit
Levels of factors
1 2 3 4
Rotating speed w r/min 740* 1070 1520 2140
Welding speed V mm/min 80* 224 – –
Shoulder diameter d mm 15* 20 – –
*Initial factor settings
Table 3: Orthogonal array L8 of the experimental runs and results
Tabela 3: Ortogonalna mre`a L8 eksperimentalnih varkov in rezultatov
Run no Experimental results
w V d
TS/
MPa
E/%
Fracture location
HAZ: Heat affected zone
TMAZ: Thermo-mechanically affected zone
NZ: Nugget zone
BM: Base metal
1 1 1 1 93 14.8 The interface between HAZ and TMAZ onthe retreating side
2 1 2 2 65 5.50 NZ
3 2 1 1 90 17.3 BM
4 2 2 2 89 13.5 HAZ on the retreating side
5 3 1 2 92 18.3 BM
6 3 2 1 93 14.5 HAZ on the advancing side
7 4 1 2 94 19.1 BM
8 4 2 1 92 14.1 HAZ on the advancing side
Figure 2: Dimensions of the welding tools used in the experiments: a)
20 mm shoulder diameter, b) 15 mm shoulder diameter
Slika 2: Dimenzije pri preizkusih uporabljenih varilnih orodij: a) pre-
mer ramena 20 mm, b) premer ramena 15 mm
tial position for 30 s rubbing with the surface of the
workpiece.
Figure 4 shows the dimensions of the tensile test
specimens prepared according to TS138 EN10002-1
standard. The tensile tests were carried out at a room
temperature and crosshead speed of 10 mm/min using
using a ZWICK Z-050 tensile testing machine. Each ten-
sile test is the average of four specimens cut from the
same joint.
3.2 Process Parameters and Test Results
In full factorial design, the number of experimental
runs exponentially increases as the number of factors as
well as their level increases. This results huge experi-
mentation cost and considerable time.26 So, in order to
compromise these two adverse factors and to search the
optimal process condition through a limited number of
experimental runs, Taguchi’s L8 orthogonal array consist-
ing of 8 sets of data has been selected to optimize the
multiple performance characteristics of FSW. Experi-
ments have been conducted with the process parameters
given in Table 2, to obtain butt welding on AA1050-H22
aluminum 4 mm thickness with (360 × 200) mm dimen-
sions by FSW welding process.
Table 3 shows the selected design matrix based on
Taguchi L8 orthogonal array consisting of 8 sets of coded
conditions and the experimental results for the responses
of TS and E. All these data have been utilized for analy-
sis and evaluation of optimal parameter combination re-
quired to achieve desired quality weld within the experi-
mental domain.
4 PARAMETRIC OPTIMIZATION OF FSW
PROCESS
4.1 Evaluation of Optimal Process Condition
First, by using Eqs. (1) and (2), experimental data
have been normalized to obtain Grey relational genera-
tion.26 The normalized data and 
0i for each of the re-
sponses have been furnished in Table 4 and Table 5 re-
spectively. For TS larger-the-better (LB) and for E
smaller-the-better (SB) criterion has been selected.
Table 4: Grey relational generation of each performance characteris-
tics
Tabela 4: Generiranje Greyjeve odvisnosti za zna~ilnosti vsakega
preizkusa
Run no
TS E
Larger-the-better Smaller-the-better
Ideal sequence 1 1
1 0.966 0.463
2 0.000 1.000
3 0.862 0.132
4 0.828 0.412
5 0.931 0.059
6 0.966 0.338
7 1.000 0.000
8 0.931 0.368
Table 5: Evaluation of 
0i for each of the responses
Tabela 5: Ocena 
0i za vsak odgovor
Run no Ra HV
Ideal sequence 1 1
1 0.034 0.537
2 1.000 0.000
3 0.138 0.868
4 0.172 0.588
5 0.069 0.941
6 0.034 0.662
7 0.000 1.000
8 0.069 0.632
Table 6 shows the calculated Grey relational
coefficients (with the weights of TS = 0.7 and E = 0.3)
of each performance characteristic using Eq. (3).
Table 6: Grey relational coefficient of each performance characte-
ristics (TS = 0.7, E = 0.3)
Tabela 6: Greyjevi koeficienti odvisnosti za zna~ilnosti vsakega
preizkusa (TS = 0.7, E = 0.3)
Run no TS E
Ideal sequence 1 1
1 0.953 0.359
2 0.412 1.000
3 0.829 0.564
4 0.795 0.741
5 0.906 0.531
6 0.951 0.685
7 1.000 0.507
8 1.000 0.706
208 Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211
H. AYDIN ET AL.: APPLICATION OF GREY RELATION ANALYSIS (GRA) ...
Figure 3: FSW applications on conventional vertical milling machine;
1 milling head, 2 welding tool, 3 aluminum plates, 4 Steel backing
plate, 5 clamping setup, 6 machine table
Slika 3: Uporaba FSW na pokon~nem vrtalnem stroju; 1 vrtalna glava,
2 varilno orodje, 3 aluminijevi plo{~i, 4 jeklena oporna plo{~a, 5 pri-
jemno orodje, 6 delovna miza stroja
Figure 4: Dimensions of tensile test specimens
Slika 4: Dimenzije raztr`nega preizku{anca
The Grey relational coefficients, given in Table 7, for
each response have been accumulated by using Eq. (4) to
evaluate Grey relational grade, which is the overall
representative of all the features of FSW quality. Thus,
the multi-criteria optimization problem has been
transformed into a single equivalent objective function
optimization problem using the combination of Taguchi
approach and Grey relational analyses. Higher is the
value of Grey relational grade, the corresponding factor
combination is said to be close to the optimal26.
Table 7: Grey relational grade
Tabela 7: Stopnje Greyjeve odvisnosti
Run no Grey relationalgrade Rank
1 0.7747 6
2 0.5882 8
3 0.7493 7
4 0.7787 5
5 0.7936 4
6 0.8710 2
7 0.8520 3
8 0.9119 1
Table 8 shows the S/N ratio based on the larger-
the-better criterion for overall Grey relational grade
calculated by using Eq. (5).
S N
n y ii
n
/ lg= −
⎡
⎣⎢
⎤
⎦⎥=
∑10 1 12
1
(5)
where n is the number of measurements, and yi is the
measured characteristic value.
Graphical representation of S/N ratio for overall Grey
relational grade is shown in Figure 5. The dashed line is
the value of the total mean of the S/N ratio.
As indicated in Figure 5, the optimal condition for
the FSW of aluminum alloy becomes w4V1d1. Table 9
shows the mean Grey relational grade ratio for each level
of the process parameters.
Table 8: S/N ratio for overall Grey relational grade
Tabela 8: Razmerje S/N za splo{no Greyjevo stopnjo
Run no S/N
1 –2.22
2 –4.61
3 –2.51
4 –2.17
5 –2.01
6 –1.20
7 –1.39
8 –0.80
Table 9: Response table for the mean Grey relational grade
Tabela 9: Odgovori za povpre~no stopnjo po Greyu
Factors
Grey relational grade
Level 1 Level 2 Level 3 Level 4 Max-Min
w 0.68 0.76 0.83 0.88 0.20
V 0.79 0.78 – – 0.01
d 0.83 0.75 – – 0.08
Total mean Grey relational grade = 0.79
4.2 Analysis of Variance (ANOVA)
The purpose of the analysis of variance (ANOVA) is
to investigate which welding parameters significantly af-
fect the performance characteristic.26,33,34 This is accom-
plished by separating the total variability of the grey re-
lational grades, which is measured by the sum of the
squared deviations from the total mean of the grey rela-
tional grade, into contributions by each welding parame-
ters and the error.26,34 Thus
SS SS SST F e= + (6)
where
SS j m
j
p
T = −
=
∑ ( )
 
 2
1
(7)
and
SST Total sum of squared deviations about the mean

j Mean response for jth experiment

m Grand mean of the response
p Number of experiments in the orthogonal array
SSF Sum of squared deviations due to each factor
SSe Sum of squared deviations due to error
In addition, the F test was used to determine which
welding parameters have a significant effect on the
H. AYDIN ET AL.: APPLICATION OF GREY RELATION ANALYSIS (GRA) ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211 209
Figure 5: S/N ratio plot for the overall Grey relational grade
Slika 5: Razmerje S/N za splo{no Greyjevo odvisnost
Table 10: ANOVA results of FSW process
Tabela 10: ANOVA-rezultati FSW-procesa
Parameter Degree ofFreedom
Sum of
Square
Mean
Square F
Contribu-
tion (%)
w 3 0.050 0.020 0.88 65.61
V 1 0.015 0.002 0.62 19.68
D 1 0.010 0.010 0.47 13.12
Error 1 0.0012 0.010 1.58
Total 6 0.0762 100
performance characteristic. Usually, the change of the
welding parameter has a significant effect on the
performance characteristic when the F value is large.
ANOVA for overall Grey relational grade is shown in
Table 10.
4.3 Confirmation Test
After evaluating the optimal parameter settings, the
next step is to predict and verify the enhancement of
quality characteristics using the optimal parametric com-
bination. The estimated Grey relational grade 
 using the
optimal level of the design parameters can be calculated
as:

 ( )
 
 
= + −
=
∑m j m
i
o
1
(8)
where 
m is the total mean Grey relational grade, 
i is
the mean Grey relational grade at the optimal level, and
o is the number of the main design parameters that
affect the quality characteristics.26 Table 11 indicates
the comparison of the predicted tensile strength and
elongation with that of actual by using the optimal
welding conditions. Good agreement between the actual
and predicted results has been observed (improvement
in overall Grey relational grade was found to be as
0.20).
Table 11: Results of confirmation test
Tabela 11: Rezultati preizkusov preverjanja
Initial factor
settings
Optimal process condition
Prediction Experiment
Factor levels w1V1d1 W4V1d1 W4V1d1
TS 93 96
E 14.8 12.3
S/N ratio of overall
Grey relational
grade
–2.22 –0.58 –1.80
Overall Grey
relational grade 0.72 0.89 0.92
Improvement in Grey relational grade = 0.20
In Taguchi method, the only performance feature is
the overall Grey relational grade; and the aim should be
to search a parameter setting that can achieve highest
overall Grey relational grade.26,33 The Grey relational
grade is the representative of all individual performance
characteristics. In the present study, objective functions
have been selected in relation to parameters of tensile
strength and elongation. The weight calculations were
done by using Analytic Hierarchy Process (AHP) and the
weights were found to be as 0.70 and 0.30 for the re-
sponses of tensile strength and elongation respectively.
The results showed that using optimal parameter set-
ting (w4V1d1) caused lower elongation with higher tensile
strength.
5 CONCLUSION
Taguchi method is a very effective tool for process
optimization under limited number of experimental runs.
Essential requirements for all types of welding processes
are higher tensile strength with lower elongation. This
study has concentrated on the application of Taguchi
method coupled with Grey relation analysis for solving
multi criteria optimization problem in the field of friction
stir welding process. Experimental results have shown
that tensile strength and elongation of welded
AA1050-H22 aluminum alloy are greatly improved by
using Grey based Taguchi method.
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Materiali in tehnologije / Materials and technology 44 (2010) 4, 205–211 211

G. KLAN^NIK ET AL.: PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY ALLOY
PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY
ALLOY
FAZNA ODVISNOST V TERNARNI ZLITINI Pt–Ag17.5–Si4.5
Grega Klan~nik, Primo` Mrvar, Jo`ef Medved
University of Ljubljana, Faculty of Natural Science and Engineering, Department of Materials and Metallurgy, A{ker~eva 12,
SI-1000 Ljubljana, Slovenia
grega.klancnik@ntf.uni-lj.si
Prejem rokopisa – received: 2009-12-01; sprejem za objavo – accepted for publication: 2010-04-22
An investigation of the Pt–Ag17.5–Si4.5 ternary alloy was carried out. Thermodynamic data on platinum systems are scarce and
are mostly based on binary systems. No accurate thermodynamic study of the Pt–Ag–Si ternary system has been published so
far. Thermodynamic calculations were used as an indication of the possible phase composition for the investigated
platinum-based alloy. The results are based on differential scanning calorimetry (DSC) tests, scanning electron microscopy with
energy-dispersive X-ray spectroscopy (SEM – EDS) and X-ray diffraction measurements (XRD). The analyses revealed the
presence of two phases that were found after homogenization at 900 °C: a high-temperature phase, Pt2Si, and a Ag (Pt) solid
solution.
Keywords: Pt–Ag–Si alloys, thermal analysis, thermodynamics
V tem prispevku je opisana raziskava, ki je osredinjena na ternarno zlitino Pt-Ag17.5–Si4.5. Za zlitinske sisteme, ki so osnovani
na platini, so termodinamski podatki skopi in temeljijo predvsem na binarnih faznih sistemih. Nobena termodinamska {tudija {e
ni bila opravljena na ternarnem sistemu Pt–Ag–Si. Za izbiro ustrezne kemijske sestave smo uporabili termodinamski izra~un.
Prvi rezultati so bili dobljeni z diferen~no vrsti~no kalorimetrijo (DSC), rastrskim elektronskim mikroskopom z energijsko
disperzivno rentgensko spektroskopijo (SEM – EDS) in rentgensko analizo (XRD). Analize so potrdile obstoj dveh binarnih faz,
dobljeni po homogenizaciji na 900 °C. Najdeni fazi sta visokotemperaturna faza Pt2Si in trdna raztopina na osnovi srebra Ag
(Pt).
Klju~ne besede: Pt-Ag-Si zlitine, termi~na analiza, termodinamika
1 INTRODUCTION
In this study we have investigated an alloy from the
ternary Pt–Ag–Si system with the aim to determine the
phase composition of the low-melting platinum-based
alloy. Using thermodynamic equilibrium calculations of
the phase diagram, an indication of a suitable chemical
composition for the alloy can be made. From the analysis
of the ternary alloy the investigated new information on
the extension of the binary phases into the ternary system
was expected. Differential scanning calorimetry, scan-
ning electron microscopy with EDS analyses, and X-ray
diffraction measurements were used for the investigation.
2 THEORY
The binary boundary systems of the ternary
Pt–Ag–Si system according to the literature are shown in
Figure 1.2,3 The Pt–Si binary system is rather complex
with several intermetallic phases (Pt3Si, Pt12Si5, Pt2Si,
Pt6Si5 and PtSi). The phases of the binary system Pt–Si
are presented in Table 1. The Ag–Pt binary system is a
Materiali in tehnologije / Materials and technology 44 (2010) 4, 213–217 213
Table 1: The binary phases taken from the Pt–Si system
Tabela 1: Faze v binarnem sistemu Pt–Si
Phase Space group Structure type
Lattice parameter /nm
Ref
a b c
Pt3Si C12/m1 Pt3Ge 0.7724(2) 0.7767(2) 0.5390(2) 5
HT-Pt3Si Pnma Fe3C 0.5579 0.7697 0.5520 6
Pt12Si5 I4/m Ni12P5 0.9607 0.9607 0.5542 6
HT-Pt12Si5* P4/nZ / 0.13404 0.13404 0.5451 7
Pt2Si I4/mmm ThH2 0.3933 0.3933 0.5910 8
HT-Pt2Si P6-2M Fe2P 0.6440 0.6440 0.3573 8
PtSi Pbnm FeB 0.5932(1) 0.5595(1) 0.3603(1) 9
PtSi Pnma MnP 0.5595 0.3603 0.5932 9
Pt6Si5* P121/m1 / 0.15308 0.348 0.6120 10
Pt2Si3 P63/mmc Sc2O2S 0.3841(1) 0.3841 1.1924(5) 11
* – no data present, HT – high-temperature phase
UDK 536:543.57:546.92 57 28 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)213(2010)
peritectic system with several reactions in the solid state.
The Ag–Pt phase diagram assessed by Karakaya and
Thompson3 identified a superstructure at low tempera-
ture and a miscibility gap between the silver and the pla-
tinum-rich corner.1 The intermediate phases are known
as  (PtAg),  and ’ (Pt3Ag). Other phases, such as ’,
’’ (PtAg3) and ’, are also possible. Invariant reactions
other than the peritectic are uncertain, because of the
lack of conclusive studies. The phase diagram of the
Si–Ag binary system is a eutectic type, without inter-
mediate or intermetallic phases.
3 EXPERIMENTAL
The chemical composition of the alloy was selected
on the basis of the results of thermodynamic calculations
of the liquidus surface for the isothermal section of the
ternary system Pt–Ag–Si using Thermo Calc 4 for
Windows (TCW 4) with the SSOL4 database.
The composition of the alloy is presented in Table 2.
The alloy was melted in a high-purity Al2O3 crucible
in an argon atmosphere from elements of high-purity
grade (99.99). The melting was performed with a high-
temperature furnace, where the platinum was heated up
to 1800 °C. The temperature control was established
with a type-D thermocouple (W- 3 % Re – W- 25 % Re).
After adding silver and silicon, the melt was mixed to
achieve homogeneity. The alloy was then re-melted and
slowly cooled to room temperature. The melting crucible
inside the high-temperature furnace is presented in
Figure 2.
The alloy was investigated with thermal analyses
(STA – 449 Jupiter, Netzsch) and X-ray diffraction
(XRD) patterns. These X-ray diffraction patterns were
recorded at room temperature using Cu K1 and K2
radiation with a Bruker X8 APEX II CCD diffracto-
meter. The diffraction patterns were compared with those
G. KLAN^NIK ET AL.: PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY ALLOY
214 Materiali in tehnologije / Materials and technology 44 (2010) 4, 213–217
Table 2: Chemical composition of the investigated alloy
Tabela 2: Kemi~na sestava preiskovane zlitine
Sample
mass fraction (w/%) mole fraction (x /%) mass, m/g
Pt Ag Si Pt Ag Si Pt Ag Si
1 77.952 17.522 4.526 55.312 22.436 22.277 1.957 0.439 0.113
Figure 1: Phase diagrams of the binary systems: Pt–Si2 (a), Ag–Pt3
(b) and Si–Ag4 (c)
Slika 1: Fazni diagrami binarnih sistemov: Pt–Si2 (a), Ag–Pt3 (b) and
Si–Ag4 (c)
Figure 2: High-temperature furnace and position of the type-D
thermocouple
Slika 2: Visokotemperaturna pe~ in polo`aj termoelementa tipa D
generated for known compounds and used to calculate
the refined unit-cell parameters. The second recording of
the X-ray diffraction patterns, using the same radiation,
for the aged sample was carried out using PANalytical
X-pert PRO MPD for samples with flat surfaces.
The maximum temperature for the STA 449 Jupiter –
Netzsch device was 1550 °C for the as-cast sample, with
a 20 K/min heating rate up to 1200 °C for the homoge-
nized sample. The measurements were performed in an
inert atmosphere of argon, applying the empty crucible
as a reference. The heating program for the homogenized
sample was as follows: heating at 25 K/min to the
homogenization temperature (900 °C), holding for 30
min, and then further heating at 5 K/min up to 1200 °C.
The microstructure was examined on polished speci-
mens using a Jeol 5610 scanning electron microscope
(SEM) equipped for energy-dispersive X-ray spectro-
scopy (EDS). Several EDS point analyses were obtained
for each phase.
4 RESULTS
4.1 Thermodynamic calculation
The calculation of the phase diagrams with TCW4 is
an important tool for the design of materials and it
significantly decreases the amount of experimental work
required. The purpose of the thermodynamic calculations
was to predict the liquid region inside the ternary
Pt–Ag–Si system. The last liquid region served for a
determination of the nominal alloy composition with a
predominating platinum component (with a minimum
content of the mass fraction 50 %). Below a temperature
of 1089 °C the thermodynamic calculation predicts the
crossing of the liquidus surfaces in the ternary system.
Figure 3 presents the isothermal section of Pt–Ag–Si at
1089 °C, which is the lowest liquidus temperature for an
alloy with a content of platinum and the mass fraction of
Ag 40 % and 6 % of Si. The composition of the inve-
stigated alloy is inside the region of this composition.
4.2 Differential scanning calorimetry (DSC)
The aim of the DSC analyses was to determine the
characteristic temperatures of possible reactions in the
alloy in the temperature interval between 25 °C and 1550
°C. The DSC heating curve is presented in Figure 4a.
Two peaks were found on the DSC heating curve in
Figure 4a, with a total heat of fusion of 89.147 J/g,
which is the contribution of two different main reactions.
Figure 4a represents the melting point at 964 °C with a
heat of fusion of 5.887 J/g, with an additional reaction at
969.2 °C. The second main reaction is found at 1044.3
°C, with a heat of fusion of 83.26 J/g. The dislocated
first peak away from the second one is assumed to be the
result of a minute content of the first melting phase and a
distinct temperature difference between the first and
second reactions. The presence of a minute content of
the first melting phase is also discernable from the values
of the enthalpy of fusion (Figure 4a). Two peaks were
also found on the DSC cooling curve in Figure 4b,
which are also the contribution of two different main
reactions, with a total heat of solidification of 102.726
J/g. The liquidus temperature determined from the
cooling curve is at 1019.9 °C, and the solidification
occurs with a high undercooling, with a solidification
enthalpy of 97.45 J/g for the first reaction. With further
cooling to 832 °C an exothermic peak with a small
initiation at 853.2 °C and an enthalpy of solidification of
5.276 J/g was found. The high undercooling is discussed
in relation to the theoretical liquidus determined from the
heating curve. High undercooling usually indicates a
lack of nucleation sites inside the melt at the theoretical
solidification temperature determined during the heating
G. KLAN^NIK ET AL.: PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY ALLOY
Materiali in tehnologije / Materials and technology 44 (2010) 4, 213–217 215
Figure 4: DSC analyses of the alloy: heating (a) and cooling curve (b)
Slika 4: DSC analiza zlitine: segrevalna (a) in ohlajevalna krivulja (b)
Figure 3: Isothermal section of the Pt–Ag–Si ternary system: liquid
region at 1089 °C and crossing of the liquidus surfaces
Slika 3: Izotermni prerez ternarnega sistema Pt–Ag–Si: podro~je
taline pri 1089 °C in prerez liquidusnih ploskev
of the same alloy. The nucleation of the liquid during the
melting is facilitated by the large number of micro-
structural defects, such as grain boundaries, especially
segregating impurities, and makes the determination of
the liquidus more reliable.
The homogenization temperature is selected below
the first reaction. For the alloy homogenized at 900 °C,
the heating curve between 900 °C and 1200 °C is shown
in Figure 5. It shows a melting point at 961.8 °C. A
reaction was also recorded at 969.1 °C, and the second
strong effect was recorded at 1032 °C. The liquidus
temperature determined from the DSC heating curve was
1065.7 °C.
4.3 Energy-dispersive X-ray spectroscopy (EDS)
The points of EDS analyses are presented in Figure
6. The results for the homogenized sample, gathered in
Table 3, show a possible intermetal phase matrix based
on platinum and silicon, which has been observed
already.2 The mole fraction of platinum in solution in the
silver phase is about 2 %. Small deviations in the
platinum content inside the silver-based phase are in
relation to the reaction at 969.1 °C (spectra 1 and 2 in
Figure 6). It is confirmed later in this paper that silicon
only appears in the Pt–Si system (Figure 7c).
Table 3: EDS analyses
Tabela 3: EDS analiza
Specter Element mole fractionx / %
mass fraction
w / %
1 AgPt
97.51
2.49
95.58
4.41
2 AgPt
97.55
2.44
95.66
4.33
3 AgPt
98.43
1.56
97.19
2.80
4
Si
Ag
Pt
26.21
2.57
71.21
4.93
1.86
93.19
4.4 X-ray diffraction phase analysis (XRD)
The XRD analyses of the homogenized sample con-
firmed the presence of a small content of a silver-based
phase and a larger amount of HT-Pt2Si (high-tempe-
rature) metastable phase (Figure 7b,c). The recorded
G. KLAN^NIK ET AL.: PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY ALLOY
216 Materiali in tehnologije / Materials and technology 44 (2010) 4, 213–217
Figure 6: SEM micrograph of the microstructure after homogeniza-
tion at 900 °C
Slika 6: SEM-mikroskopska slika mikrostrukture po homogenizaciji
na 900 °C
Figure 5: The DSC heating curve of the homogenized sample
Slika 5: DSC-segrevalna krivulja homogeniziranega vzorca
Figure 7: X-ray powder diffraction pattern: experimental pattern (a),
silver-based phase (b), HT–Pt2Si phase (c) and diffraction pattern of
the aged sample at room temperature (d)
Slika 7: Posnetki rentgenska pra{kovne difrakcije: eksperimentalni
posnetek (a), srebrova faza (b), faza HT–Pt2Si (c) in difrakcijski
posnetek staranega vzorca na sobni temperaturi (d)
X-ray diffraction pattern of the alloy is shown in Figure
7a. The X-ray diffraction pattern recorded again after 2
months of aging at room temperature for possible
changes in the HT Pt2Si phase revealed the presence of a
high-temperature phase HT-Pt2Si and a silver-based
phase (Figure 7d).
5 DISSCUSSION
After homogenization at 900 °C the melting point
was at 961.8 °C, and near to the melting point of silver.
The EDS analyses confirmed the presence of the silver
phase with a small content of platinum. The EDS ana-
lyses also revealed a phase with a high content of
platinum. The X-ray diffraction patterns after homoge-
nization at 900°C revealed the existence of two phases: a
silver-based phase and a HT-Pt2Si phase. The determined
lattice constant for the silver-based phase is
0.408364(18) nm. The lattice parameter for silver is at
0.40857 nm, with an Fm-3m space group. The deter-
mined HT–Pt2Si phase has a lattice parameter a =
0.6460310(45) nm and b = 0.3577385(45) nm with a
P6-2m space group. The literature, shows that the
HT–Pt2Si or the –Pt2Si is a congruent phase with the
following lattice parameters: a = 0.6436 nm and c =
0.3569 nm or a = 0.6440 nm and c = 0.3573 nm. Thus,
similar values to those in our investigation. From Figure
8 it can be concluded that the nominal alloy composition
in mole fractions (55.36 % Pt, 22.43 % Ag, 22.27 % Si)
is really a two-phase region of the silver-based phase and
the HT-Pt2Si phase inside the Gibbs concentration
triangle between the silver-rich corner and the binary
Pt2Si phase.
6 CONCLUSIONS
After homogenization at 900 °C the Pt–Ag17.5–Si4.5
ternary alloy was characterized by the presence of two
compounds: the HT–Pt2Si phase and a silver-based
phase. It can be concluded from the X-ray diffraction
patterns, the EDS and the DSC that the alloy with mass
fractions 17.5 % of silver and 4.5 % of silicon has a
melting point at 961.8 °C as a result of the melting of the
silver-based phase. The effect at 1032 °C, determined
from the DSC heating curve, is related to the melting of
the HT-Pt2Si phase and the liquidus temperature of the
alloy is at 1065.5 °C.
7 LITERATURE
1 M. H. F. Suiter, C. Colinet, A. Pasturel, Ab initio calculation of the
phase stability in Au-Pd and Ag-Pt alloys, Physical review, B 73
(2006) 17, 174204
2 L. E. Tanner, H. Okamoto, The Pt-Si system (Platinum-Silicon),
Phase Diagram Evaluation; Journal of Phase Equilibria, 12 (1991) 5,
571
3 I. Karakaya, W. T,Thompson, The Ag-Pt system, Royal military
college of Canada, 8 (1987) 4, 334–430
4 ASM handbook, Alloy phase diagrams, ASM international, 3 (1992),
410
5 A. Gribanov, A. Grytstiv, E. Royanian, P. Rogl, E. Bauer, G. Giester,
Y. Seropegin, On the system cerium-platinum-silicon; Journal of
solid state chemistry; 181 (2008), 2964–2975
6 R. Phal Ram, S. Bhan, On the constitution of platinum-silicon alloys;
Zeitschrift fuer Metallkunde, 69 (1978), 524–529
7 W. Gold, K. Schubert, Kristallstruktur von Pt12Si5, Zeitschrift fuer
Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie,
128 (1969), 406–413
8 R. Gohle, K. Schubert, Zum aufbau des systems platin-silizium; Zeit-
schrift fuer metallkunde, 55 (1978), 503–511
9 H. Pfeisterer, K. Schubert, Neuen phasen vom Mn P (B-31) – typ;
Zeitschrift fuer metallkunde, 41 (1950), 358–367
10 R. Gohle, K. Schubert, Journal of physics and Chemistry of Solids,
46 (1985), 631–641
11 B. Y. Tsaur, J. W. Mayer, K. N. Tu, Ion beam induced metastable
Pt2Si3 phase, Journal of Applied Physics, 51 (1980), 5326–5333
G. KLAN^NIK ET AL.: PHASE RELATIONS IN THE Pt–Ag17.5–Si4.5 TERNARY ALLOY
Materiali in tehnologije / Materials and technology 44 (2010) 4, 213–217 217
Figure 8: Isothermal section of the Ag–Pt–Si system at 900°C
Slika 8: Izotermni prerez sistema Ag–Pt–Si pri 900 °C

V. CHYGYRYNS’KYY ET AL.: A NEW SOLUTION OF THE HARMONIC FUNCTIONS ...
A NEW SOLUTION OF THE HARMONIC FUNCTIONS IN
THE THEORY OF ELASTICITY
NEKATERE ZNA^ILNOSTI HARMONI^NIH FUNKCIJ V TEORIJI
O ELASTNI^NOSTI
Valeryy Victorovich Chygyryns’kyy1, Vladimir Grigorovich Shevchenko1,
Ilija Mamuzic2, Sergey Borisevich Belikov1
1Zaporozhskyy National Technical University, Str. Zukovsky 64, Zapori``'a, Ukraine
2University of Zagreb, Faculty of Mettalurgy Sisak, Str. A. N. heroja 3, Sisak, Croatia
mamuzic@simet.hr
Prejem rokopisa – received: 2008-11-21; sprejem za objavo – accepted for publication: 2009-08-17
A new approach to the solution of a plane problem of the theory of elasticity with the use of two harmonic functions with a
Cauchy-Riemann analytical link is developed. The analysis of the harmonic functions shows that some allow a new approach to
the solution of problems of the theory of elasticity. For the solution of linear differential equations a fundamental substitution is
used, written in the general form (x,y) = y = C · exp , with  = (x,y) as a function of the strain centre.
The transformations are explained with the properties of harmonic functions, where the Cauchy-Riemann relations can be used.
The considered variants extend the possibilities for solutions and, if necessary, to obtain suitable functions for predetermined
tasks. The new method is universal and can be effectively used when the fields of stresses and strains are described with
trigonometric expressions.
Key words: theory of elasticity, harmonic functions, Cauchy-Riemann expressions.
Razvit je bil nov na~in re{evanja ravninskega problema teorije elasti~nosti s Cauchy-Riemanovo analiti~no zvezo. Analiza
harmoni~nih funkcij poka`e, da nekatere omogo~ajo nov na~in za re{itve problemov iz teorije elasti~nosti. Za re{itev linearnih
diferencialnih ena~b se uporablja temeljna substitucija, zapisana v splo{ni obliki z (x,y) = y = C · exp , z  = (x,y) kot
funkcijo sredi{~a deformacije.
Transformacije smo razlo`ili z lastnostmi harmoni~nih funkcij, pri katerih je dovoljena uporaba Cauchy-Riemanovih povezav.
Upo{tevane variante raz{irjajo mo`nost re{itev in, ~e je potrebno, omogo~ijo, da dobimo re{itve za vnaprej na~rtovano uporabo.
Nova metoda je univerzalna in se lahko u~inkovito uporabi, ~e so polja napetosti in deformacij opisana s trigonometri~nimi
odvisnostmi.
Klju~ne besede: teorija elesti~nosti, harmoni~ne funkcije, Cauchy-Riemanovi izrazi
1 INTRODUCTION AND FORMULATION OF
THE TASK
The analysis of the peculiarities of the harmonic
functions shows that some of them allow new
approaches to the solution of problems of the theory of
elasticity. Let us consider a plane problem of this theory.
We have a set of equilibrium equations1.
∂
∂
∂
∂
 x xy
x y
+ = 0;
∂
∂
∂
∂
 xy y
x y
+ = 0 (1)
The equation of joint strains
∇ + =2 0( ) x y (2)
The stresses’ boundary conditions

 
  n =
−
⋅ ⋅
x y
xy2
2 2sin cos (3)
Applying these expressions, the harmonic law of the
distribution of contact stresses is determined2, which
formally coincides with that in3:
  n = − ⋅ −( , ) sin( )x y AF 2
where (x,y) is the coordinate function of the strain
centre; A is the constant determining the elastic state of
a deformable medium; F is the coordinate function
characterizing the allocation of contact shearing
stresses;  is the slope angle of an element.
In place of equations (1) and (2), the biharmonic
equation (4) can be applied:
∇ = +
⋅
+4
4
4
4
2 2
4
42
  ∂
∂
∂
∂ ∂
∂
∂x x y y
(4)
with  as a stress function.
The expression fulfils the boundary conditions (3)
 xy x y A= ⋅( , ) sin( )F (5)
The stress difference in (3) is determined with
  x y x y A− = ⋅ ⋅2 ( , ) cos( )F (6)
2 SOLUTION OF THE TASK
The fundamental substitution is often used during the
solution of linear differential equations4, which can be
written in the following general form
(x,y) =  = C · exp  (7)
Materiali in tehnologije / Materials and technology 44 (2010) 4, 219–222 219
UDK 539.3:534.1 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)219(2010)
with  = (x,y) being the unknown coordinate function
of the strain centre.
Let us examine the harmonic functions AF and .
The analytical link between them is admitted by the
Cauchy-Riemann expressions 4,5
 x yA= ± F  y xA= ± F (8)
After the derivation of equation (5), consideration of
equation (7) and substitution in the equilibrium equa-
tions we obtain
∂
∂

  
x
yx
C A+ ⋅ ⋅ ⋅exp sin( F
+ ⋅ ⋅ ⋅C A Ay   F exp cos(
∂
∂

  
y
xy
C A+ ⋅ ⋅ ⋅exp sin( F
+ ⋅ ⋅ ⋅C A Ax   F Fexp cos(
with y, AFx as the partial derivatives of the appropriate
functions of the coordinates y and x. Passing from one
variable to the other with the help of (8), we obtain,
after integration and simplifications, the normal and
shearing stresses
    x C A f y C= ⋅ ⋅exp cos( F
    y C A f x C= ⋅ ⋅exp cos( F (9)
  xy C A= ⋅ ⋅exp sin( F
Substituting f(y) = f(x) = 0 in (9), we obtain the
relation (6) that fulfils the boundary conditions for
equation (3).
Considering (9), the sum of the stresses is
 x y C+ = 2
and the equation of joint strains (2) is automatically
fulfilled. It is interesting that during the evaluation of
the Laplacian for each value C A  ⋅ ⋅exp cos( F and the
substitution (8) the identity 0 0≡ is obtained. Using this
peculiarity, the sum of stresses can be expressed as a
product of the functions
    ' exp cos(= + = ⋅ ⋅ ⋅x y n C AF (10)
with n as the number that defines the influences of
hydrostatic pressure on the medium of the stressed state
in the strain zone.
By substituting (10) in (2) we obtain
[ ]∇ + = ∇ ⋅ ⋅ ⋅ =     x y n C A) exp cos( F
[{= ⋅ ⋅ +n C A Axx x y x y      exp ) )F F
]+ ⋅     yy y x y xA A AF F F) ) cos(
[ ] }− + + + ⋅2 2 A A A A Ax y y xx yyF F F F Fsin( ) (11)
It is clear from the analysis of the differential equa-
tion (11), that it turns to identity under the condition of
 x yA= ± F  y xA= ± F
This is the relation (8), which was introduced as an
assumption during the solution of the equilibrium
equations. Differentiating further, we obtain
 xx yxA= ± F  yy xyA= ± F
 xy yyA= ± F  yx xxA= ± F
The last relations convert equation (11) into identity.
The last expressions show that the indicated functions
are harmonic, i.e.
 xx yy+ = 0 A Axx yyF F+ = 0
It is remarkable that the operators of the trigono-
metrical functions are equal to zero. This peculiarity
shows that the function (10) also fulfils the biharmonic
equation (4). Considering equation (10), the Laplace of
the equation has the form
[ ]∇ = ∇ ⋅ ⋅ =    C Aexp cos( F
= ⋅ ⋅
+
+
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟
⎡
⎣
⎢ ⋅C
A
A
xx x y
yy y x
 
  
  
exp c
2 2
2 2
F
F
os(AF 
−
⋅ + +
+ +
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎤
⎦
⎥ =
2 2
0
 x x y y
xx yy
A A
A A
A
F F
F F
Fsin( )
Let us introduce the symbolisms
L x y L A Axx x y yy y x( , ) = = +     
2 2 2 2F F
M x y M A A A Ax x y y xx yy( , ) = = ⋅ ⋅ + ⋅2 2   F F F F
Then, with consideration of the symbolisms, the
accurate form of the Laplace equation is obtained
    ( , ) cos( sin(x y L A M A= = ⋅ ⋅F F
If the factors in the trigonometrical functions are
equal to zero, also the operators L = M = 0. For a more
integrated analysis let us write a Laplacian for the
function (x,y)
∇ = +
⋅
+ =4
4
4
4
2 2
4
42
  ∂
∂
∂
∂ ∂
∂
∂x x y y
( )= ∇ = +⎛
⎝
⎜
⎞
⎠
⎟ ⋅ ⋅ ⋅ =4
2
2
2
2  
∂
∂
∂
∂x y
L A M Acos( sin(F F
= − ⋅ ⋅ − ⋅ + − ⋅ −(L L A M A M A L L Axx x x x xx yy yF F F F
  
 M A M A A L Ay y yy x x⋅ − ⋅ ⋅ ⋅ +F F F F) cos(
 L A M M A L A L Axx xx x y y yy⋅ + − ⋅ + ⋅ + ⋅ +F F F F
2
+ − ⋅ ⋅M M A Ayy y yF F
2 ) sin( 
It is expected that the partial derivatives from zero
functions are equal to zero, thus, ∇ ≡4  . Let us write
the partial derivatives of separate operators and track the
mechanism of the turning into identity of the harmonic
functions
L A Ax xxx yyx x xx y yx    + + − +) (2 2 F F
+ −( )2 2 y yx x xxA AF F
Following (8), we have
 x yA= − F  y xA= F
 xx yxA= − F  yy xyA= F
 xy yyA= − F  yx xxA= F
 xxx yxxA= − F  yyy xyyA= F
V. CHYGYRYNS’KYY ET AL.: A NEW SOLUTION OF THE HARMONIC FUNCTIONS ...
220 Materiali in tehnologije / Materials and technology 44 (2010) 4, 219–222
 xyy yyyA= − F  yxx xxxA= F (12)
Let us substitute (12) in the operator Lx.
[ ]L
xx xx yy x xx x xx
= − + − +
∂
∂
( ) ( )( )     2 2
[ ]+ − ≡2 2 0   y yx y yx( )( )
We have obtained the identity, "quod erat demon-
strandum".
The same approaches take place during the evalu-
ation of the operator Lxx. Let us write it as
L A Axx xxxx yyxx xx xx yx yx= − + − +( ) ( )   2 2 F F
+ − − − −( ) ( )2 2 2 2   x xxx y yxx xx xx yx yxA A A AF F F F
− −( )2 2A Ax xxx y yxxF F  
Substituting (12) into the expression for Lxx and
factoring out a flexion on x from the first brackets, after
conversion the identity 0 0≡ is obtained.
Thus, the operators
L L M M L L M M L Mxx yy xx yy x y x y= = = = = = = = = = 0
demonstrate that the function (10) fulfils the biharmonic
equation (4) and it can be used for the evaluation of the
components of the stress tensor. It is necessary to ensure
that the field of stresses and the stress function are
described, as a matter of fact, with identical expressions
(9) and (10) linked analytically6 with


x y
=
∂
∂
2
2 

y x
=
∂
∂
2
2 

xy y x
=
∂
∂ ∂
2
Such schemes of transformations are explained with
the properties of harmonic functions where the Cauchy-
Riemann relations can be applied. The considered
variants allow us to extend the possibility of solutions
and, if necessary, to obtain suitable functions for the
development of a predetermined result.
Let us return to the expressions for the stress tensor
components and consider the equilibrium equations in
the components of the stress deviator. Let us introduce
the symbols
  x x f y C
' ' ( )= − − −
  y y f x C
' ' ( )= − − − (13)
with  being the mean stress.
Considering (13), the equilibrium equation (1), can
be rewritten in the form6
∂
∂
∂
∂
2 2
0
 x xy
x y
' '
+ =
∂
∂
∂
∂
 xy y
x y
+ =
' '
0
By analogy with (9) and with integration and
simplifications we obtain the stress tensor components
    x C A f y C= ⋅ ⋅exp cos(
    y C A f x C= − ⋅ ⋅exp cos(
  xy C A= ⋅ ⋅exp sin( (14)
With:  x yA= ± F ,  y xA= ± F
It follows from expressions (14) that their deviator
part for the normal stresses C A  ⋅ ⋅exp cos( F coin-
cides with the shifting part  in (10). Considering (13),
(6) is fulfilled and the boundary conditions (3) are
satisfied.
The outcome (14) can be generalized. The analytical
link of the functions with the opposite signs is obtained
in relations (8) and different signs of an index in an
exponential curve result can be obtained. Therefore, the
index of an exponential curve in a solution will be not
unique. The exponential function can be written in the
form of a sum with the use of the hyperbolic cosine or
sine in the general form
[ ]      x C ch C sh A f y C= ⋅ ± ⋅( ( cos( F
[ ]      y C ch C sh A f x C= − ⋅ ± ⋅( ( cos( F
[ ]    xy C ch C sh A= + ⋅ ± ⋅( ( sin( F (15)
In these expressions it is assumed that the arguments
of the trigonometric and exponential functions can be
represented in the form of a series of harmonic functions
interlinked with the Cauchy-Riemann relations.
3 COMPARISON TO OTHER SOLUTIONS
The solutions of a plane problem with the help of a
trigonometric series are often used. For example, the
following combination of functions is often met3:
[ ]       = ⋅ ⋅ ⋅ ⋅ + ⋅ − ⋅sin( exp( exp(x C y C y (16)
Let us ascertain whether the Cauchy-Riemann
relation exists between the arguments of trigonometric
and exponential functions
A xF = ⋅  = ± ⋅y A xF = 
A yF =   y = ±  x = 0
The obtained relations take place for the functions
  x yA= = F   y xA= ± = ±F
The peculiarity of these solutions is that they are
common and do not contradict known partial solutions.
The arguments AF and  are harmonic functions of the
coordinates x and y. They can be rather complicated and
cannot be determined from the linear dependences for
one coordinate.
Let us analyze the possibilities of the solution (14).
The elementary variant of a harmonic function of two
variables is A A x yF = ⋅ ⋅ . Applying the relations (8), it is
written as
 = ⋅ −
1
2
2 2A x y( )
Thus,
 x yA A x A x= = ⋅ = ⋅ F  y xA A y A y= ± = ± ⋅ = ⋅F
Each function fulfils the Laplace equations.
V. CHYGYRYNS’KYY ET AL.: A NEW SOLUTION OF THE HARMONIC FUNCTIONS ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 219–222 221
4 CONCLUSION
A new approach to the solution of a plane problem of
the theory of elasticity based on the use of two harmonic
functions with the Cauchy-Riemann analytical link is
developed. The new method is universal and can be
effectively used when the fields of stresses and strains
are described with trigonometric expressions.
5 REFERENCES
1 Bezuchov N. I. Osnovy teoriy uprugosty, plastichnosty i polzuchesty
(Basics of the theory of elasticity, plasticity and creep). Vysshaja
shkola, 1968, 498 s
2 Malinin N. N. Prikladnaja teorija plastichnosty i polzuchesty
(Practical theory of plasticity and creep), Mashinostroenie, 1975, 399
s
3 Chigirins’kyy V. V., Mazur V. L., Legotkin G.I., Slepynin A. G. and
al. Proizvodstvo vysokoeffektivnogo metalloprokata (High efficient
production of rolling stock), Dnepropetrovsk, PBA «Dnepro-VAl»,
2006, 261 s
4 Tihonov A. N., Samarskiy A. A. Uravnenija matematicheskoy fiziki
(Equations of mathematical physics), Nauka, 1977, 735 s
5 V. V. Chygyryns’kyy, I. Mamuzic, G. V. Bergeman. Analysis of the
State of Stress of a Medium under Conditions of Inhomogeneous
Plastic Flow, Metalurgija, 43 (2004), 87–93
6 Nikiforov S. N. Teorija uprugosty i plastichnosty (Theory of ela-
sticity and plasticity), Gosudarstvennoe izdatelstvo literatury po
stroitelstvu i architecture, 1958, 283 s
V. CHYGYRYNS’KYY ET AL.: A NEW SOLUTION OF THE HARMONIC FUNCTIONS ...
222 Materiali in tehnologije / Materials and technology 44 (2010) 4, 219–222
D. PIHURA, D. MUJAGI]: INFLUENCE OF VACUUM PROCESSING ON THE CONTENT ...
INFLUENCE OF VACUUM PROCESSING ON THE
CONTENT OF SOME ELEMENTS IN MOLTEN METAL
VPLIV VAKUUMSKEGA PROCESIRANJA NA VSEBNOST
NEKATERIH ELEMENTOV V KOVINSKIH TALINAH
Dervi{ Pihura, Dervi{ Mujagi}
University of Zenica, Metallurgical institute "Kemal Kapetanovi}" Zenica, Travni}ka c. 7, 72000 Zenica, Bosnia and Herzegovina
dervis.pihura@gmail.com
Prejem rokopisa – received: 2008-09-22; sprejem za objavo – accepted for publication: 2010-03-15
Residual and trace elements can affect physical, chemical and mechanical properties, especially those of high-alloy materials
and steels. With a treatment of the molten metal in a vacuum-induction furnace and an electron-beam furnace the content of
residual elements is diminished, depending on the treatment time and the vacuum pressure.
Key words: vacuum treatment, trace elements, electron-beam furnace, vacuum-induction furnace
Rezidualni elementi in elementi v sledovih vplivajo na kemi~ne in mehanske lastnosti, posebno pri visokolegiranih zlitinah in
jeklih. Pri obdelavi taline v vakuumski indukcijski pe~i in v pe~i z elektronskim curkom se vsebnost rezidualnih elementov
zmanj{a v odvisnosti od trajanja obdelave in od tlaka.
Klju~ne besede: vakuumska obdelava, elementi v sledovih, pe~ z elektronskim curkom, vakuumska indukcijska pe~
1 INTRODUCTION
A vacuum is a powerful tool for improving the
physical and chemical properties of ferrous and
nonferrous alloys with vacuum melting, re-melting and
treatment. Depending on the vacuum process used, the
physical and chemical properties of the alloys can be
improved. For this reason, it is important to select the
vacuum treatment process, keeping in mind the required
change in the properties of the alloy.
2 VACUUM PROCESSES
Vacuum melting is applied for the melting of raw
materials in coreless induction furnaces, and is
particularly suited to the melting and casting of different
high-alloyed materials because it enables an exact
control of the temperature and the chemical compo-
sition.1–6 In this way, the physical characteristics and
properties of the melted alloys can be tailored. The
charged components are melted in vacuum or a
protective atmosphere (Ar, N or H) and then vacuumed
with a partial pressure of down to 10–4 mbar. Since the
alloy may contain elements with a high vapour pressure,
by selecting the proper vacuum partial pressure the
chemical composition of the alloy must be kept in mind.
With a vacuum treatment, the alloy is refined and its
physical, chemical and mechanical properties as well as
cleanliness may be changed. It is possible, in a vacuum
furnace, to also treat the alloy with vacuum de-oxidation
using appropriate additions or, through the effect of the
vacuum itself, to decrease the content of the gasses N2
and H2 as well as to change the content of elements such
as Sb, As, Bi, Sn, Cu, etc. By decreasing the content of
some elements, non-metallic inclusions and gases,
physical and mechanical properties that are better suited
to a particular application are obtained.
Arc furnaces and electron-beam furnaces are mostly
used for the re-melting of previously produced alloys
applying a non-vacuum process. The best, which must be
used for such an operation, are the electric arc furnace
and electron-beam furnace, which is of particular interest
for the aim of this paper. The electron-beam vacuum
furnace is applied on the pilot and industrial scales for
ingots up to 20 t. In this furnace, droplets of the molten
alloy, or electrode, solidify rapidly in a water-cooled
copper mould. The heat for melting is obtained with 3 to
12 electron beams targeted on the re-melted electrode.
Depending on the energy consumption, the local
electrode temperature may be increased to a few
thousand degrees and, in this way, the conditions for
evaporation of the high-partial-pressure components of
the alloys are achieved. The high local temperature
makes it difficult to measure the exact temperature of the
molten drops. The evaporation may be considered as one
of the advantages of the re-melting process, especially as
the melt droplets form in a vacuum of different grades
down to 1 · 10–6 mbar. In this way, the molten metal is
exposed to vacuum for a sufficient time and the extrac-
tion of the gases and the evaporation of high-vapour-
pressure components is ensured. After re-melting, the
alloy density and cleanliness are improved and the
physical properties are substantially improved. The
re-melted alloys can then be used for high-temperature
and high-stress applications, as well as for the parts of jet
engines.
Materiali in tehnologije / Materials and technology 44 (2010) 4, 223–226 223
UDK 669.18 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 44(4)223(2010)
A good technology for the vacuum treatment of
molten metal is the circulating or RH process that is
applied only on an industrial scale. In this process the
molten metal is lifted from the container in the vacuum
chamber, processed for the determined time and re-flown
in the melting furnace. As part of the process the surface
of the alloy surface in direct contact with the vacuum is
increased and the rate and the extent of extraction is also
increased. This process is also suited to the highest
quality of steel grades.
3 EFFECT OF VACUUM TREATMENT
In a vacuum some of the physical properties of a
molten metal are changed. Dissolved gases behave in
accordance with Sieverts’ and Nernst’s laws and their
extraction becomes possible. Nitrogen and hydrogen are
extracted directly, while the extraction of oxygen occurs
through vacuum de-oxidation with carbon and the
extraction of CO.
Specific interactions between the elements present in
the melt may help to displace some elements from the
solution in the molten mother metal. The interaction
coefficients are a measure of the mutual bonding energy
of the dissolved X and Y atoms relative to the X-Fe and
Y-Fe bonding. For dilute solutions, the bonding energies
are independent of each other and additive, whereas in
more concentrated solutions, Z atoms may affect the
X-Y bonding. For multi-components solutions, the total
activity coefficient for each element will depend on the
mutual effect of each component. For a diluted solution
the interaction can be described with a simple relation.
In the extraction process, the effect of the partial
pressure of elements in the melt is of essential import-
ance. For a number of elements, the effect of temperature
on the partial pressure is shown in Figure 1. Generally,
elements with a higher partial pressure are extracted
more quickly from the melt; however, the evaporation
also depends on the geometry, the vacuuming time and
the ferro-static pressure in the reacting part of the melt.
At the melting temperature of iron, the vapour pressure
of the impurities and the alloying elements decreases in
the order Ba > Pb > Sn > Mn > Si > Fe = 0.84 > 0.45 >
0.45 > 0.035 > 8 ·10–4 > 2 · 10–4 bar.5,6
4 RESULTS
In this work the results of investigations of the effect
of the vacuum treatment of several alloys – low carbon
iron and steel, alloys with about 60 % Ni and 18 % Cr,
with 50 % Ni, steel with 18 % Cr and 10 % Ni and a
low-alloy steel – in a vacuum-induction furnace and an
electron-beam furnace are presented. In Figure 1 the
dependence of the partial vapour pressure on temperature
is shown for different elements, as well as for elements
found as impurities in the investigated alloys.
In Figures 2 and 3 the effect of electron-beam
melting on the content of some elements are shown –
impurities in low-carbon iron, low-carbon steel, nickel or
nickel chromium alloy types such as perm-alloy, Cr/Ni –
18/8 or 18/10, Cr/Ni – 18/60.
Figure 2 presents the results of the content change of
tin, arsenic, copper and antimony, while in Figure 3 the
effects of the melting rate (kg/min) and the power ((kW
h)/kg) are shown for arsenic and different groups of
alloys. The content of all the investigated elements is
diminished by maintaining the melt in vacuum; however,
the extent and the kinetics of evaporation are different
for the different alloys.
For the three elements with greater evaporation rates,
two phases are seen on the kinetics curve: an initial
phase of rapid evaporation down to a determined critical
content and a phase with a much lower evaporation rate.
For antimony the evaporation rate is 5.5 times lower in
the second than in the first phase. The shape of the
kinetics curves in Figure 2 is related to the activity in the
elements in solution in the molten iron. It is assumed that
the evaporation rate is greater down to the equilibrium
224 Materiali in tehnologije / Materials and technology 44 (2010) 4, 223–226
D. PIHURA, D. MUJAGI]: INFLUENCE OF VACUUM PROCESSING ON THE CONTENT ...
Figure 2: Decrease of some residual elements during electron-beam
melting: Sn (), As (
), Cu (
) and Sb (x) in low-carbon iron
Slika 2: Zmanj{anje vsebnosti nekaterih elementov pri taljenju z
elektronskim curkom: Sn (), As (
), Cu (
) in Sb (x)
Figure 1: Dependence of the vapour or partial pressure for different
elements on the temperature5,6
Slika 1: Odvisnost parcialnega parnega tlaka od temperature za raz-
li~ne elemente5,6
activity, which depends on the bath’s chemical compo-
sition and the temperature.
The evaporation of antimony was the largest, but it is
also large for copper. It is low for arsenic, while it is
scarcely discernible for tin. In Figure 3 the effect of the
melting power and rate is shown for arsenic for the
groups of melts of low-carbon iron (1), steel (2),
perm-alloy (6), Cr/Ni – 18/8 (3) or 18/10 (4) and Cr/Ni –
18/60 (5). It indicates that an average decease of arsenic
of 22 % for different groups of materials depends on the
content of the alloying elements and on the influence of
their physical properties on each material. The depen-
dences are based on our own experimental findings6 and
reference.5 The comparison of melting characteristics
indicates a simple relation between the melting rate and
the power consumption for different materials for the
same average loss of arsenic.
The effect of melting in the vacuum induction
furnace is, in principle, similar to the electron-beam
furnace, with the higher evaporation rate for antimony
and the lowest for tin (Figure 4). The evaporation rate
increases with the increasing content of the element in
the alloy and it is quite different for the same content of
different elements in the alloy. Compared with the
evaporation rate of tin, the evaporation rate is 2.5 times
greater for arsenic, five times greater for copper and 10
times greater for antimony. The difference is very
probably related to the difference of the activity of the
elements in solution in the molten metal and their partial
pressure and depends on the interaction coefficient of
some element also, as the activity of the impurities in
molten metal is affected by the presence of other
components. To describe this phenomenon it is
convenient to maintain the same standard state as with
the binary melts, and to introduce an activity coefficient
(fx), which describes the effect of the third component.
This coefficient depends on the interaction coefficients,
which differ greatly,5 and is the product of the interaction
coefficients and the content of the third component in the
melt.
It is assumed that with a vacuum treatment on the
industrial scale the relative differences in the evaporation
of elements would be similar to those presented in this
work, while the absolute rate of evaporation and the
change of element contents would depend for all
elements on the conditions specific to the processing,
especially the presence of the slag layer on the surface of
the molten alloy.
The evaporation rates of sulphur, silicon and
phosphorus by melting in an electron-beam vacuum
furnace is low and are virtually unaffected by the melting
rate, while the evaporation of manganese is large and
again independent on the melting rate (Figure 5). Again,
the difference is very probably related to the activity of
the elements in solution in the iron bath. With respect to
the loss of elements during other methods of vacuum
processing of iron alloys, for phosphorus, sulphur,
silicon and manganese a similar conclusion seems to be
justified, as was earlier suggested for tin, arsenic, copper
and antimony. It should also be considered that the slag
could have a positive effect on the change of the content
D. PIHURA, D. MUJAGI]: INFLUENCE OF VACUUM PROCESSING ON THE CONTENT ...
Materiali in tehnologije / Materials and technology 44 (2010) 4, 223–226 225
Figure 5: Decrease of some elements during melting in an electron-
beam vacuum furnace: P (), Si (
), S (
) and Mn (×) depending on
the melting rate of Cr/Ni – 18/8 (3), 18/10 (4) and perm-alloy6
Slika 5: Zmanj{anje vsebnosti nekaterih elementov pri taljenju v pe~i
z elektronskim curkom; P (), Si (
), Cu (
) in Mn (×) v odvisnosti
od hitrosti taljenja Cr/Ni – 18/8 (3), 18/10 (3) in perm. zlitine6
Figure 3: Loss of 22 % of arsenic by electron-beam melting for
different group of melts and different melting rate MIZ (*) 6, As (
)4
& lit. (
)5; low-carbon iron (1), steel (2), Cr/Ni – 18/8 (3) or 18/10
(4), Cr/Ni – 18/60 (5), perm-alloy (6) elemente
Slika 3: Izguba 22 % arzena pri taljenju v pe~i z elektronskim curkom
za razli~ne skupine zlitin pri razli~ni hitrosti taljenja; MIZ (*) 6, As
(
)4, ref.5 (
). Malooglji~no `elezo (1), jeklo (2), Cr/Ni 18-8 (3 in
18-10(4), Cr/Ni 18/60 (5), perm. zlitina (6)
Figure 4: Decrease of some residual elements during the melting of
low-carbon steels and alloys1,2,6 in a vacuum induction furnace: Sn
(), As (
), Cu (
) and Sb (×)
Slika 4: Zmanj{anje vsebnosti nekaterih rezidualnih elementov pri
taljenju malooglji~nih jekel in zlitin1,2,6 v vakuumski indukcijski pe~i:
Sn (), As (
), Cu (
) in Mn(×)
of phosphorus and particularly sulphur in the melted iron
alloy.
In vacuum, the gases are extracted from the metal
bath according to Sieverts’ law. The degassing intensity
also depends on the metal bath temperature and the
chemical composition, which determine the coefficients
of interaction. This is confirmed by the data in Table 1,
showing that the degassing constant is different for
alloys with different chemical compositions and tem-
peratures.
Table 1: Constant nitrogen-degassing rate (KN ·10–4)1,2,6
Tabela 1: Konstanta hitrosti degazacije du{ika (KN · 10–4)1,2,6
Tempera-
ture T/°C
Material
Cr/Ni –
18/8
Cr/Ni –
18/60
Perm
alloy LAS
Cr/Ni –
18/10
1450 0.41 2.51 2.91 – –
1500 1.11 1.82 – 1.41 0.81
1550 1.45 1.12 1.09 1.35 1.39
1600 1.82 0.75 0.81 1.21 1.83
5 CONCLUSION
The experimental data presented show the loss of
elements as a result of evaporation in vacuum is very
different; it is below the detection limits for some
elements, while it is significant for other elements
present in the iron bath. The evaporation rate is
particularly high for antimony, copper and manganese,
while it is virtually null for tin and low sulphur,
phosphorus and silicon. The findings show that the
evaporation rate depends on the interaction coefficient of
the elements in solution in the metal bath and the
temperature. When extrapolating the experimental
findings to other vacuum processes for iron alloys, it
should be kept in mind that the evaporation process also
depends on the geometry of the vacuum exposure of the
melt, and remembering that the presence of the slag on
the metal bath surface would significantly affect the
evaporation loss in two ways: by preventing a contact
between the bath surface and the atmosphere and by a
direct chemical reaction of the slag with some elements
in solution in the molten bath, especially sulphur and
phosphorus.
6 REFERENCES
1 Pihura D. at al. Promjena sadr`aja du{ika u vakuumskoj pe} (Change
of nitrogen content in the vacuum furnace), Zenica, 2008
2 Pihura D. at al. Osvajanje proizvodnje legura tipa nimonic i nitronic
u vakuumskoj pe}i (Manufacturing of nimonic and nitronic type
alloys in a vacuum furnace), Zenica, 2007
3 Pihura D. Kazanska metalurgija (Ladle metallurgy) – VI-VIII dio;
Met. Inst., Zenica, 1986/9
4 Pihura D. Vacuum Melting Technology Monogr., UNIDO, Vienna,
1980
5 Winkler O, Bakhish Vacuum metallurgy, Elsevier, London 1971
6 Pihura D. Vakuumske tehnike i tehnologije (Vacuum technique and
technology), Met. Inst., Zenica, 1979
D. PIHURA, D. MUJAGI]: INFLUENCE OF VACUUM PROCESSING ON THE CONTENT ...
226 Materiali in tehnologije / Materials and technology 44 (2010) 4, 223–226
P. HÁJKOVÁ ET AL.: INFLUENCE OF 
-FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
INFLUENCE OF 
-FERRITE ON THE FATIGUE
RESISTANCE OF BLADE MATERIALS
VPLIV 
-FERITA NA UTRUJENOSTNO TRDNOST JEKEL ZA
LOPATICE
Pavlína Hájková1, Jiøí Janovec1, Jan Siegl2, Bohumil Smola3, Michaela Vyroubalová1,
Daniela Tùmová1
1CTU in Prague – Faculty of Mechanical Engineering, Karlovo nám. 13, 121 35 Praha 2, Czech Republic
2CTU in Prague – Faculty of Nuclear Sciences and Physical Engineering, Trojanova 13, 121 35 Praha 2, Czech Republic
3Charles University in Prague – Faculty of Mathematics and Physics, KE Karlovu 5, 121 35 Praha 2, Czech Republic
pavlina.hajkova@fs.evut.cz
Prejem rokopisa – received: 2009-11-16; sprejem za objavo – accepted for publication: 2010-02-21
The low-pressure parts of a TG 1000 MW turbine, for the third wheels, are made from modified 12 % Cr martensitic steel AK1
TD.9, and the fourth, are made from X2CrNiMo13-4 steel. The fracture of the third turbine wheel at the point of the connection
part of the blade has a high-cycle fatigue character with multiple initiation sites. An analysis of the microstructure has proved
the influence of both the 
-ferrite content and the arrangement on the initiation point of the fatigue process. Also, an inspection
of the fatigue crack’s kinetics in relationship to the operating mode of the power plant has been carried out.
Key words: low-pressure turbine, steel, fatigue fracture, microstructure, 
 ferrite
Nizkotla~ni deli turbine TG 1000 MW iz tretjega venca so iz modificiranega 12-odstotnega Cr martenzitnega jekla AK TF.9, iz
~etrtega venca pa iz jekla X2CrNiMo 13-4. Prelom tretjega turbinskega venca v to~ki vpetja ima zna~aj visoke cikli~ne
utrujenosti z mnogimi za~etnimi mesti. Analiza mikrostrukture je pokazala vpliv vsebine in porazdelitve 
-ferita na za~etno
to~ko procesa utrujenosti. Ugotovljena je bila tudi povezava kinetike utrujenostne razpoke z na~inom dela turbine.
Klju~ne besede: nizkotla~na turbina, jeklo, utrujenostna razpoka, mikrostruktura, 
-ferit
1 INTRODUCTION
In October 2008 the blade of the third turbine wheel
(3TW) placed on the third low-pressure part (3LP) of the
TG1 broke. Consequently, the broken blade damaged the
turbine wheels numbers 3 and 4 on the LP1, LP2 and
LP3 (see Figure 1). The break in the blade occurred near
the connection part (Figure 2) during the blade’s
operation. Cracks in the connection-part regions within
the turbine wheels of other low-pressure parts were
discovered with defectoscopic analysis. The damaged
connection part is shown in Figure 2. The chemical
composition, mechanical properties, fractography and
time-dependent estimation of the crack propagation of
the removed blades were tested. The scheme of the
damaged blades within the whole turbine vessel is shown
in Figure 3.
2 EXPERIMENTAL PROCEDURE
2.1 Materials
The investigated blades (see Table 1) were drop
forged, and a new steel, 1.4939, was investigated as a
possible substitute material for the 3.TW.
Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232 227
UDK 621.039.568:669.14.018:539.42 ISSN 1580-2949
Professional article/Strokovni ~lanek MTAEC9, 44(4)227(2010)
Figure 2: Damage to the connection part
Slika 2: Po{kodba na mestu vpetja
Figure 1: Damage to the 3TW on the TG1
Slika 1: Po{kodba 3TW na TG 1
Table 1: Investigated samples
Tabela 1: Vzorci za preiskave
Blade No. Low-pressurepart State Material
152 3NT Destroyed
modified 12 % Cr
martensitic steel AK1
TD.9
156 3NT Laboratorybroken
modified 12 % Cr
martensitic steel AK1
TD.9
106 3NT Additionallyidentified
modified 12 % Cr
martensitic steel AK1
TD.9
New
material
1.4939
Identified 1.4939 + QT
344 NT 4a OK X2CrNiMo13-4 steel(WEV 400)
2.2 Investigation methods
2.2.1 Evaluation of the chemical and microchemical
composition of the turbine-blade material
Three different devices were used for the chemical
analysis: a ARL 34600 OE quantimeter, a Belec Vario
Lab spectrometer, and an Electron Microanalysis Came-
bax MICRO CAMECA with a KAVEX energy-disper-
sion analyzer. These various devices were used because
of the great variability in the chemical composition
within the different melts.
2.2.2 Verification of the mechanical properties of the
turbine blades
The mechanical properties were determined with
tensile, Charpy impact and Brinell hardness tests. Two
different types of samples with respect to the turbine-
blade orientation were extracted. The first were extracted
from the connection part perpendicular (PN) to the blade
axis, the second samples extracted from the blade base in
a parallel orientation (PO) to the blade axis.
Tensile tests were carried out on an INSTRON
100kN Series IX Automated Materials Testing System
according to the testing conditions of the standard ^SN
EN 10002-1. A Padostroj Charpy 300 J was used for the
Charpy impact tests, according to the standard (with a
test temperature of 20 °C and specimen dimensions of
(10 × 10 × 55) mm with the V-notch) ^SN EN 10045-1.
The Brinell hardness was determined with an EMCO-
TEST M4C universal hardness tester according to the
standard ^SN EN 6506-1.
2.2.3 Light-microscopy examination of metallographic
samples
Qualitative and quantitative microstructural analyses
of the base material and of the fracture areas were
carried out for the perpendicular (PN) and parallel (PO)
samples cut out from the blades. The samples were taken
from the fracture zone and from the middle parts of the
blade bodies. For the examination a Zeiss-Neophot 32
light microscope with suitable magnifications was used.
For the X2CrNiMo13-4 steel, a solution of 5 mL HCl, 1
mL TNP (2,4,6-trinitrofenol) and 95ml of ethyl alcohol
was used as an etching agent. The 12% Cr martensitic
steel AK1 TD. 9 was etched in a water solution with 2.5
mL of HNO3.
2.2.4 Electron-microscopy examination of
metallographic samples
The samples for examination with transmission
electron microscopy were prepared with spark erosion
and cut parallel to the crack-propagation plane. From the
thin plates, discs of 2.7 mm in diameter were prepared
with spark-erosion cutting and mechanically thinned to
70–80 µm. Thin foils for the electron-microscopy
observation were prepared with electrolytic polishing in
TENUPOL equipment, applying a 6 % solution of
HClO4 in methanol, a temperature of 35 °C and a current
of 150 mA. The samples were examined in a JEOL
JEM2000FX transmission electron microscope and the
energy-dispersion microanalyzer of an RTG emission
LINK AN10000 was used for the chemical analyses.
3 RESULTS AND DISCUSSION
3.1 Chemical composition and mechanical properties
The results of the chemical analysis and the measure-
ments of the mechanical properties (yield strength,
ultimate strength, ductility, reduction of cross-section
area, Charpy impact strength, hardness) are summarized
in Table 2, Table 3 and Table 4. It is evident that the
requirements of the standards AK1 TD.9 and WEV 400
are fulfilled almost completely. Only the lower Charpy
impact strength was determined for the material of the
blades 152, 156, 106. This can be due to the operation of
the components for a long time or the non-standard
operating conditions.
3.2 Optical microscopy
The microstructure of the base material was typical
for modified 12 % Cr martensitic steel. The microstruc-
ture of the heat-treated (quenched and annealed) steel is
shown in Figure 4. The large amount of -ferrite present
in the microstructure was aligned in rows, oriented
parallel with the initial crack growth and may have
P. HÁJKOVÁ ET AL.: INFLUENCE OF -FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
228 Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232
Low-pressure part Turbine Wheel number Blade number
3LP 3. TW 152, 156, 106
1LP 4. TW 344
3LP 3. TW (suggestion) 1.4939
Figure 3: Scheme of broken blades within the whole turbine vessel
Slika 3: Shema prelomljenih lopatic v turbinskem ohi{ju
greatly influenced the propagation of the fatigue cracks
(see Figure 5).
The microstructure of the base steel of
X2CrNiMo13-4 consisted of sorbite, and was typical for
the heat-treated (quenched and annealed) steel. No
-ferrite inserts were found within the structure, even at
a magnification of 400 (see Figure 6). The quantitative
analysis of the amount of -ferrite is shown in Figure 7.
The size of the original austenitic grains was evaluated
according to the standards ^SN ISO 643 and ASTM E
112. A combination of Nital 3 % and Vilella-Bain
etching agents was used to highlight the original
austenitic grains. The obtained values showed that its
size varied between 4.5 and 5.5 for both steel types
(modified 12 % Cr martensitic steel and X2CrNiMo13-4
steel).
3.3 TEM
Figure 9 shows a grain of -ferrite in the modified 12
% Cr martensitic steel (Figure 8). The TEM revealed
that the microstructure of blade No. 106 was similar to
the microstructures of blades No. 152 and 156. Accord-
P. HÁJKOVÁ ET AL.: INFLUENCE OF -FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232 229
Table 2: Chemical composition (w/%)
Tabela 2: Kemi~na sestava jekel (w/%)
Sample C Si Mn P S Cr Ni Mo V W
Specification
AK1 TD.9 0.10–0.16 < 0.60 0.60 < 0.025 < 0.015 10.5–12 1.5–1.8 0.35–0.5 0.18–0.30 1.6–2.0
152 0.142±0.004
0.371
±0.002
0.46
±0.004
0.022
±0.002
0.01
±0.001
11.71
±0.14
1.853
±0.022
0.448
±0.004 0.3 ±0.006
1.77
±0.142
156 0.143±0.002
0.374
±0.002
0.462
±0.004
0.021
±0.002
0.011
±0.002
11.62
±0.022
1.843
±0.012
0.451
±0.006
0.302 ±
0.006
1.768
±0.044
106 0.16±0.004
0.31
±0.006
0.41
±0.006
0.024
±0.004
0.010
±0.002
11.44
±0.014
1.90
±0.042
0.44
±0.004
0.29
±0.008
1.59
±0.022
Specification
1.4939 0.15 < 0.35 0.9 0.020 0.015 12.50 3.00 2.00 0.40 –
New mat.
1.4939+QT
0.314
±0.004
0.102
±0.004
0.829
±0.018
0.011
±0.002
0.007
±0.001
11.57
±0.20
2.498
±0.076
1.568
±0.014
0.345
±0.008
0.222
±0.036
Specification
WEV 400 max. 0.05 0.15–0.35 0.20–0.80 max. 0.020max. 0.025 12.0–14.0 3.5–4.5 0.30–0.50 – –
344 0.013±0.002
0.29
±0.002
0.76
±0.004
0.015
±0.008
0.002
±0.001
12.88
±0.012
4.14
±0.022
0.46
±0.003
0.02
±0.002
0.017
±0.014
Table 3: Mechanical properties
Tabela 3: Mehanske lastnosti
Sample (blade No.)
Yield Strength
Rp 0.2/MPa
Ultimate strength
Rm/MPa
Ductility
A/%
Contraction
Z/%
152/1 PN 845 957 20.0 55.4
152/2 PN 860 969 20.0 55.4
156/1 PO 830 948 18.3 55.4
156/2 PO 835 966 16.7 55.4
156/3 PN 820 962 16.7 55.4
106/1 PO 860 965 16.7 50.9
106/2 PO 849 957 17.3 55.4
106/3 PN 865 976 16.7 50.9
344/1 PO 951 989 18.3 81.2
344/2 PO 950 984 18.0 79.7
344/3 PN 952 987 18.3 79.7
Specification AK1 TD.9 min. 700 850–1000 min. 14 min. 40
Specification WEV 400 min. 950 1000–1100 min. 12 min. 45
Table 4: Charpy impact test and hardness [HBW]
Tabela 4: Charpyjeva `ilavost in trdota
Temperature, T/°C + 20 °C Average valueHBW 2.5/187.5
Blade No. Sample KV/J KCV/(J/cm3) Required R/(J/cm3)
152 1PO-2PN-3PN 27 28 28 35
50
255
156 1PO-2PO-3PN 41 28 27 40 255
106 1PO-2PO-3PN 36 31 32 41 264
344 1PO-2PO-3PN 206 201 196 251 295
P. HÁJKOVÁ ET AL.: INFLUENCE OF -FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
230 Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232
Figure 10: Blade No. 344, martensitic microstructure
Slika 10: Lopatica {t. 344, martenzitna mikrostruktura
Figure 7: Content of δ-ferrite for different materials determined with
different methods
Slika 7: Vsebnost -ferita v razli~nih jeklih, dolo~ena z razli~nimi
metodami
Figure 6: Blade No. 344 (PO), 200x, microstructure of base steel
Slika 6: Lopatica {t. 344 (PO), mikrostruktura osnovnega jekla
(200-kratna pove~ava)
Figure 5: Blade No. 106, 200x, longitudinal crack and aligned inserts
of -ferrite
Slika 5: Lopatica {t. 106, podol`na razpoka in usmerjeni vlo`ki
-ferita (200-kratna pove~ava)
Figure 4: Blade No. 152 (PN), 200x, microstructure of base steel
Slika 4: Lopatica {t. 152 (PN), mikrostruktura osnovnega jekla
(200-kratna pove~ava)
Figure 8: Blade No. 152, martensitic microstructure of base steel
Slika 8: Lopatica {t. 152, martenzitna mikrostruktura osnovnega jekla
Figure 9: Blade No. 152, detail of -ferrite particle
Slika 9: Lopatica {t. 152, detajl delca -ferita
ing to the electron diffraction results, the carbide
particles were of the M23C6 type and the microstructure
was practically without -ferrite.
Blade No. 344 of the steel X2CrNiMo13-4 (Figure
10) contained very fine precipitates of the intermetallic
phase Fe2Mo (Figure 11). Similar fine particles were
also found in the newly developed steel 1.4939, which
was free of coarser particles typical for the other
analysed steel. Nevertheless, the local occurrence of a
very limited quantity of these carbide particles cannot be
excluded.
3.4 Fractographical analysis of damaged blades
(modified 12 % Cr martensitic steel)
The fracture surfaces were covered with a thick layer
of corrosion products that complicated the investigation
of the micromorphology in the region of the fatigue
cracks. These layers were removed with ultrasonic
cleaning. The fractographical analysis showed that the
cause of the blade damage was high-cycle fatigue-crack
growth in the vicinity of the connection part of the
blades and the gas-turbine shaft.
The cracks started on the blade surfaces in the upper
groove of the connection area (see Figure 12) and then
propagated with transcrystalline decohesion. It can be
assumed that the -ferrite particles acted as the initiation
points of the fracture. Many progressing lines were
detected on the fracture areas of the damaged blades.
The presence of these lines indicates that the loading was
not constant and this is typical for fatigue fracture. From
the number of progressing lines, the number of starting
regimes of the gas turbine could be estimated.
From the width of certain growth rings it can be
concluded that during stable operating conditions the
crack-propagation stopped. From the microfractography
it can be further concluded that the fatigue failure
propagated with the striation mechanism (see Figure
13a).
In addition, the presence of the progressing line
determined the location and the shape of the fatigue
crack tip at the moment of the overloading cycle. The
dependence between these lines and the operating
conditions made it possible to reconstruct the whole
history of the failure process (see Figure 13b).
Estimation of the time-dependent failure propagation
(modified 12 % Cr martensitic steel)
On the blades from the 3.TW (modified 12 % Cr
martensitic steel AK1 TD. 9) on LP3 extensive maps
were prepared (at various magnifications up to 300-
times). This made it possible to obtain detailed informa-
tion about the history of the fatigue-crack propagation.
P. HÁJKOVÁ ET AL.: INFLUENCE OF -FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232 231
Figure 13: Characteristic microfractographical signs of fatigue
failure. (a) Blade No. 152, field of submicron striations, (b) Blade No.
152, propagation lines
Slika 13: Karakteristi~ni mikrofraktografski znaki utrujenostnega
preloma. (a) Lopatica {t. 152, podro~je submikrometrskih brazd, (b)
lopatica {t. 152, brazde propagacije
Figure 11: Blade No. 344, fine precipitates displayed in dark field
Slika 11: Lopatica {t. 344, izlo~ki, prikazani v temnem polju
Figure 12: Fracture area of blade No. 152 (the white arrow shows the
front line beneath the striking edge)
Slika 12: Prelomna povr{ina lopatice {t. 152 (svetla pu{~ica ka`e na
~elno ~rto pod udarnim robom)
The maps were closely analyzed with respect to the
operation history of the gas turbine.
Figure 14 shows the propagation lines identified by
these observations. These lines correspond to the data on
the left-hand side of the micrograph. As mentioned
above, the lines are related to the table operation of the
turbine, and from the known data of the specific factory
test the propagation regime on the fracture can be
derived. The main results of the fatigue fraction history
of the blades from 3.TW on LP3 in the fracture area can
be seen in Figure 15.
4 CONCLUSION
The mechanical properties confirm that the require-
ments for the mechanical properties of most of the
blades from the modified 12% Cr martensitic steel are
fulfilled. However, the following exceptions were identi-
fied:
1. The blade No. 344 (X2CrNiMo13-4 steel) does not
fulfil the requirements for ultimate strength. The
lower value of 1.5 % is negligible, but it also positi-
vely affects the increase in the ductility parameters
A5 and Z.
2. The Charpy impact tests of blades No. 152, 156 and
106 revealed that measured values do not fulfil the
requirement for impact toughness.
3. In contrast to the impact toughness of blade No. 344
(X2CrNiMo13-4 steel) was 2.5 times higher than
required.
4. The hardness values fell within the required interval
that allows higher hardness values within the blade
No 344 (X2CrNiMo13-4 steel) in comparison with
the modified 12 % Cr martensitic steel.
The low resistance to fatigue failures (mainly the
resistance to fatigue-crack propagation) of the used
materials in the working conditions can be stated as the
main reason for the fatigue cracks’ initiation.
The initiation of the fatigue cracks was a result of the
presence of -ferrite (modified 12 % Cr martensitic
steel). The fractographical reconstruction revealed that
the initiation of the fatigue cracks in the blades started
quite early after the initialization of the working process.
From the microstructural analysis it was concluded
that the presence of -ferrite (the shape and the particles)
negatively influences the resistance of the modified 12 %
Cr martensitic steel to crack initiation. The subsequent
propagation of fatigue cracks is controlled by the main
operating load. The new material, 1.4939, without
tungsten, which eliminates the presence of -ferrite to a
large extent, is proposed for the manufacture of 3.TW
blades.
5 REFERENCES
1 J. Janovec, J. Siegl, CTU in Prague, 12/2008, Research report No.
10-08 Pøí~iny poru{ení lopatek 3. obì`ných kol NT3 a NT1 TG1
ETE na 1. HVB (~ást 1) – (Causes of failure of 3th low pressure
turbine wheel in LP3 and LP1 of TG1 in 1. HVB (part 1))
2 J. Siegl, CTU in Prague, FJFI V-KMAT-746/08 – Research report
Fraktografická analýza poru{ených lopatek NT dílù TG1 ETE (~ást
2) – (Fractographical analysis of failure blades LP parts of TG1 ETE
(part 2))
3 J. Siegl, CTU in Prague, FJFI V-KMAT/2009 Presentation Frakto-
grafická analýza poru{ených loptek NT2 dílu TG2 ETE (Fracto-
graphical analysis of failure blades in LP2 parts of TG2 ETE))
4 J. Janovec, J. Siegl, CTU in Prague, 12/2008 Presentation Pøí~iny
poru{ení lopatek 3. obì`ných kol NT3 a NT1 TG1 ETE na 1. HVB
(Causes of failure 3rd in LP3 and LP1 of TG1 ETE )
5 J. Zenkl, O. Novák, ^EZ, a.s. ETE, ^EZ, a.s. EDU, Analýza pøí~in
po{kození lopatek na NT dílech TG 1000 MW (Analysis of the
causes of damage to the blades on LP parts TG 1000 MW)
P. HÁJKOVÁ ET AL.: INFLUENCE OF -FERRITE ON THE FATIGUE RESISTANCE OF BLADE MATERIALS
232 Materiali in tehnologije / Materials and technology 44 (2010) 4, 227–232
Figure 15: Blade No. 156. Dates, when the crack front reached the
line of final blade failure
Slika 15: Lopatica {t. 156. Na makrografiji je ozna~en datum, ko je
~elo razpoke doseglo mejo kon~nega preloma lopatice
Figure 14: Blade No. 156. Propagation lines on the fracture area and
dates of turbine operation
Slika 14: Lopatica {t. 156. Linije propagacije razpoke in datumi rasti
razpoke