*Corr. Author’s Address: Technical University of Gabrovo, 4 H. Dimitar St, Gabrovo, Bulgaria, irina@tugab.bg 635
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648 Received for review: 2021-07-13
© 2021 Journal of Mechanical Engineering. All rights reserved.  Received revised form: 2021-10-23
DOI:10.5545/sv-jme.2021.7327 Original Scientific Paper Accepted for publication: 2021-11-09
Modelling and Multi-objective Optimization  
of Elastic Abrasive Cutting of C45 and 42Cr4 Steels
Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
Irina Aleksandrova1,* – Anna Stoynova2 – Anatoliy Aleksandrov1
1Technical University of Gabrovo, Bulgaria 
2Technical University of Sofia, Bulgaria
Elastic abrasive cutting is a new high-performance method to produce workpieces made of materials of different hardness, which ensures 
lower wear of cut-off wheels and higher quality machined surfaces. However, the literature referring to elastic abrasive cutting is scarce; 
additional studies are thus needed. This paper proposes a new approach for modelling and optimizing the elastic abrasive cutting process, 
reflecting the specifics of its particular implementation. A generalized utility function has been chosen as an optimization parameter. It 
appears as a complex indicator characterizing the response variables of the elastic abrasive cutting process. The proposed approach has 
been	applied	to	determine	the	optimum	conditions	of	elastic	abrasive	cutting	of	С45	and	42Cr4	steels.	To	solve	the	optimization	problem,	a	
model of the generalized utility function reflecting the complex influence of the elastic abrasive cutting conditions has been developed. It is 
based on the findings of the complex study and modelling of the response variables of the elastic abrasive cutting process (cut-off wheel wear, 
time per cut, cut piece temperature, cut off wheel temperature and workpiece temperature) depending on the conditions of its implementation 
(compression force F exerted by the cut-off wheel on the workpiece, workpiece rotational frequency nw, cut off wheel diameter ds). By applying 
a	genetic	algorithm,	the	optimal	conditions	of	elastic	abrasive	cutting	of	С45	and	42Cr4	steels:	ds = 120 mm; F = 1 daN; nw = 63.7 min–1 and 
nw = 49.9 min–1,	respectively	for	С45	and	42Cr4	steels,	have	been	determined.	They	provide	the	best	match	between	the	response	variables	
of the elastic abrasive cutting process.
Keywords: elastic abrasive cutting; multi-objective optimization; generalized utility function; С45 and 42Cr4 steels
Highlights
•	 A new approach to modelling and multipurpose optimization of elastic abrasive cutting based on a generalized utility function 
has been performed.
•	 Regression models for the elastic abrasive cutting response variables depending on the cutting conditions have been built.
•	 A model of the generalized utility function as a complex indicator characterizing the response variables of the elastic abrasive 
cutting process has been developed.
•	 By	applying	a	genetic	algorithm,	the	optimum	conditions	of	elastic	abrasive	cutting	of	С45	and	42Cr4	steels,	under	which	the	
generalized utility function has a maximum, have been determined.
•	 The proposed multipurpose optimization approach ensures the best match between the response variables of the elastic 
abrasive cutting process.
0  INTRODUCTION
Abrasive cutting is a high-performance, cost-effective 
method widely used in present-day manufacturing 
environments to produce workpieces made of ferrous 
and non-ferrous metals and alloys, and non-metallic 
materials of high hardness. This is a complex and 
varied process performed under different kinematic 
schema; the cut-off wheel has two motions: main 
rotation and radial feed [1] to [3]. The radial feed 
of the tool is ensured either by the cut-off machine 
in a kinematic way (hard abrasive cutting) or by 
maintaining a constant compression force F exerted 
by the wheel on the workpiece (elastic abrasive 
cutting) [4]. The workpiece is motionless, performs an 
oscillatory motion or rotates at a constant rotational 
frequency nw.
During elastic abrasive cutting (Fig. 1), the contact 
arc length L and the thickness h of the layer being cut 
vary, while the instantaneous cross-section area of the 
layer being cut remains constant. The introduction of 
rotary motion of the workpiece leads to a decrease 
in the working stroke of the cut-off wheel, which 
ensures shorter time per cut and a wear reduction of 
abrasive wheels. At the same time, the tool lifetime, 
process production rate, cutting forces, power rate, 
and temperature depend on the compression force 
F, workpiece rotational frequency nw, cut off wheel 
diameter ds and the type of material being machined. 
This makes the schema for elastic abrasive cutting a 
topical subject of study and optimization.
The abrasive cutting process is a sophisticated 
multi-parameter and multi-factor subject of study, 
modelling, and optimization. It is characterized by a 
number of target parameters, each having a specific 
meaning yet insufficient for its optimum control. On 
the one hand, abrasive cutting is distinguished by 
its high performance, universality and low cost. On 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
636 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
the other hand, the process is accompanied by high 
temperatures (above 1000 °С) in the cutting zone, 
which, in turn, results in intensive cut-off wheel wear, 
deterioration of the tool cutting ability, changes in the 
microstructure of the material being machined and the 
occurrence of thermal flaws [5] to [7].
Fig. 1.  Schema of elastic abrasive cutting;  
1 cut- off wheel; 2 workpiece
The response variables of the abrasive cutting 
process are determined by numerous control factors: 
the physio-mechanical properties of the materials 
being machined, cutting mode components, cut-
off wheel type and characteristics, type and way of 
supplying cooling fluids, etc. They are described in 
several publications.
Lopes еt al. [8] studied the abrasive cut-off 
operation of low and medium carbon steels, using 
aluminium oxide discs of different feed rates. They 
monitored the cutting power, disc wear, and process 
temperature and established that an increase of 130 % 
in feed rate led to a decrease of approximately 57 % in 
maximum temperature and 84 % in diametrical wheel 
wear, thus improving process efficiency, but also to an 
increase of consumed cutting power by up to 127 %. 
A simulation model of high accuracy for predicting 
temperature was proposed that could be applied to 
predict and prevent thermal damages.
Luo et al. [9] studied the characteristics and 
wear modes of diamond grits in the cut-off grinding 
BK7 glass using thin diamond wheels, as well as the 
grinding forces, the grinding ratio, the width of cut, 
the straightness of cut and chipping produced during 
the cut-off grinding. The results showed that at a lower 
transverse velocity, a freer cutting ability, a better 
grinding ratio and a better straightness of cut were 
obtained. An increase of the transverse velocity led to 
diamond wheel wear accompanied by some macro-
breakage and flattening of the tool work surface, thus 
resulting in unstable grinding forces, a low grinding 
ratio and poor straightness of cut.
Neugebauer et al. [2] studied the wear of cut-
off wheels with different abrasive grains in the 
abrasive cutting of structural and stainless steels. 
They established that cubical grits were distinguished 
by the most extended lifetime compared to splinter-
shaped grits due to their higher toughness and 
adhesion to the bond. By comparing the cutting 
performance of discs with different compositions in 
dry cutting of steel bars, Ortega et al. [10] observed 
that the abrasive type and size and disc binder were 
the most influential factors. On the basis of the results 
obtained, they concluded that discs with high grit size 
and protrusion, high grit retention by bond material, 
and closer mesh of fibreglass matrix binder were the 
optimal solution for abrasive cutting of mild steel 
parts. Riga and Scott [11] established the decisive 
influence of the type and composition of the cut-off 
wheel bond on the mechanical properties and nature 
of abrasive tool wear. The significant impact of the 
chemical composition and mechanical properties 
of the material being machined, the type and size of 
abrasive grains, the hardness of cut-off wheel and 
the cutting conditions (cut-off wheel speed and feed 
speed) on cut-off wheel wear was confirmed by the 
mathematical model developed by Yoshida et al. [12].
Ojolo et al. [13] found that the grinding/wear 
ratio and cutting time depended on the hardness and 
chemical composition of the material being machined. 
The tests were done by hard abrasive cutting of mild 
steel and stainless-steel rods. The grinding ratio in 
the elastic abrasive cutting of steel rods was also 
determined by Kaczmarek [5] and [14], using the 
results from the investigation of the cut-off wheel 
temperature. The author noted that the combination 
of a higher cut off wheel diameter and a higher feed 
force resulted in generating higher temperatures and 
obtaining lower values of G-ratio. These results were 
linked to cut off wheel wear and self-sharpening.
Levchenko and Pokintelitsa [7] analysed the 
interaction between the cut-off wheel and the cut 
pipe billet on the basis of heat-deforming analysis 
and they established the relationship between 
processing conditions and cutting conditions with 
the design parameters of the abrasive tool. To obtain 
lower temperatures and higher grinding ratios, Sahu 
and Sagar [6] proposed cut-off wheels with radial 
passages on their surfaces to supply the cutting fluid 
to the cutting zone. Upon studying four different eco-
friendly water-based fluids in cutting medium carbon 
steel, Ni et al. [15] observed that the water-based 
fluid with surfactant provided the best results, as it 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
637Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
increased the G-ratio (by 77.6 %) and decreased the 
cutting tilt (by 14.8 %) and lin span (by 34.5 %) in 
comparison with dry cutting. 
Yamasaka et al. [16] studied the application 
of abrasive cutting with diamond wheels used for 
machining the electromechanical parts of brittle 
materials, such as alumina ceramics. They found 
that the straightness of the sliced surface strongly 
correlated with the axial deflection of a thin grinding 
wheel, which was caused by the side forces acting on 
the side surfaces of the cut-off wheel. In this regard, 
they recommended decreasing the side force both 
by decreasing the depth of cut with retaining a stock 
removal rate and by improving the flatness on the side 
surfaces of the cut-off wheel to improve straightness. 
The results obtained by Braz et al. [17] in relation to 
the abrasive cutting of titanium showed that wheels 
with abrasives of silicon carbide and 30 mesh grain 
size combined with an extremely hard bond material 
led to the lowest values of depth of the affected zone.
Many studies confirm that by changing the 
abrasive cutting conditions, the thermal flows and 
the distribution of temperature in the tool, chips, 
workpiece and cut piece can be controlled, thus 
providing possibilities for enhancing tool lifetime, 
cutting process intensity and quality of machined 
surfaces.
Wang et al. [18] developed а three-dimensional 
numerical model to calculate the grinding temperature 
field distribution. The effect of the workpiece feed 
velocity, cooling coefficient and the depth of cut on 
temperature distribution were considered. Eshghy [19] 
studied the thermal flux in the chip, cut-off wheel and 
ambience and pointed out that 31 % of the thermal 
energy was released from the chips, 18 % from the 
ambience, 50 % from the wheel and only 1 % from the 
piece. Thermal optimum conditions were defined, and 
a procedure for selecting optimum down-feed rates 
was given. Hou and Komanduri [20] developed a heat 
source model and calculated heat partition in cut-off 
operations. They established that 60 % to 75 % of the 
heat was removed by the chip, 20 % to 35 % by the 
workpiece and 1 % by the cut-off wheel. Therefore, 
the authors thought that the thermal softening of the 
bond resulted from the heat released by the flying 
chips.
By using finite element (FE) analysis, Putz et al. 
[3] modelled heat distribution inside a steel bar under 
different cutting conditions. By adapting the cutting 
parameters and cooling strategy, the critical workpiece 
maximum temperature was decreased by 48 %. The 
authors proposed an alternative grinding strategy that 
combined the benefits of feed rate reduction, swing 
grinding and increased coolant velocity and ensured 
a decrease of the cutting power, wheel wear and 
piece temperature. The FE model for studying and 
analysing the temperature distribution in the machined 
bar during grinding is presented in [21], where two 
basic strategies for decreasing the temperature in the 
cutting zone are proposed: optimization of the cutting 
parameters (the feed speed) and application of the 
cubic boron nitride (CBN)-grinding technology. As 
a result, thermal damage decreases, and a cutting 
operation of high quality is ensured.
The analytical methods to calculate grinding 
temperatures and their effect on thermal damages 
were developed by Malkin and Guo [22]. According 
to them, the critical factor by means of which thermal 
damages in abrasive cutting could be controlled was 
the energy partition to the workpiece, which was 
within the range of 60 % to 85 %. This amount was 
considerably smaller in abrasive cutting with CBN 
discs with lower feed and slower workspeeds when 
using appropriate cooling fluids.
By applying the method of infrared thermography 
and planned experiment, Stoynova et al. [23] and [24] 
established that the workpiece rotational frequency 
had the greatest effect on temperature. As it increased 
(approximately 3 times), the temperatures of the 
cut piece and cut off wheel decreased (by up to 29 
% and 19 %, respectively) whereas the workpiece 
temperature increased by up to 12 %. Temperature 
models of the workpiece, chip, piece being machined, 
and cut-off wheel in abrasive cutting were also built 
[25].
The analysis of the publications related to 
abrasive cutting shows that each abrasive cutting 
process is unique and could be studied from different 
perspectives: technological, energetic, informational, 
organizational, etc. Hard abrasive cutting is a well-
studied process [3], [8] to [13] and [15] to [21] and 
the optimum conditions for its implementation are 
determined, too [26] to [28].
Elastic abrasive cutting is of great interest since 
it is a new high-performance method ensuring the 
stabilization of the dynamic and thermal phenomena 
in the cutting zone, thus reducing cut-off wheel wear 
and improving the quality of machined surfaces [14], 
[23], [29] and [30]. However, the literature referring 
to elastic abrasive cutting is scarce; thus, additional 
studies are needed. The adjustment of cutting 
operations is often done on the basis of the experience 
of the skilled staff or by using data from handbooks. 
Currently, there are no mathematical models that cover 
all aspects of elastic abrasive cutting and connect all 
its parameters with the cutting conditions. To increase 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
638 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
the efficiency and applicability of elastic abrasive 
cutting, it is necessary to study, model, and optimize 
its behaviour so as to achieve certain economic and 
technological criteria, taking into consideration the 
specific nature and conditions for its implementation. 
In this regard, the paper proposes a new approach 
to the modelling and multipurpose optimization of 
elastic abrasive cutting, which reflects the specifics 
of its particular implementation in cutting two types 
of structural steels (С45 – medium carbon steel and 
42Cr4 – alloy chrome steel), which are widely used 
in machine building for producing workpieces. The 
optimum conditions to implement the process are 
determined and verified to ensure the best match 
between the rate of cut-off wheel wear, time per cut, 
cut piece temperature, cut off wheel temperature 
and workpiece temperature. This will ensure higher 
productivity of the elastic abrasive cutting process and 
lower costs, as well as higher quality of the machined 
surfaces of the structural steels under study. 
Taking into account the high productivity and 
low cost of the elastic abrasive process, the proposed 
multipurpose optimization approach and the obtained 
theoretical-experimental results presented in this 
paper could be used in every enterprise specialized in 
machine building.
1  INVESTIGATION AND MODELLING  
OF ELASTIC ABRASIVE CUTTING RESPONSE VARIABLES
1.1  Equipment, Materials, Methods
The purpose of this study is to establish the correlation 
dependencies between the response variables of the 
elastic abrasive cutting process: cut-off wheel wear δh, 
cut off wheel temperature Ts,h, workpiece temperature 
Tw,h, cut piece temperature Td,h and time per cut tc,h 
(h is index corresponding to the type of material 
being machined; h = 1 for elastic abrasive cutting of 
С45 steel; h = 2 for elastic abrasive cutting of 42Cr4 
steel), and the conditions required to perform the 
process. Therefore, the cut-off wheel diameter ds, the 
compression force F and the workpiece rotational 
frequency nw were chosen as control factors.
To perform the elastic abrasive cutting process, a 
special attachment (Fig. 2) was designed. It is fixed 
to the main carriage of a combination lathe, supplied 
with a device for a stepless adjustment of workpiece 
rotational frequency nw. The attachment comprises an 
angle grinder (2), which ensures constant rotational 
frequency of the cut-off wheel (ns = 8500 min–1), and 
a unit for adjusting the amount of the compression 
force F of the cut-off wheel (3) exerted on the 
workpiece (1). An angle grinder (GA7020 model) is 
attached to the front end of the arm (5), which is fixed 
by a bearing (4). A counterweight (6), whose weight 
is greater than that of the angle grinder, is fixed on 
the opposite end of the arm. The compression force 
of the abrasive wheel F exerted on the workpiece is 
regulated by moving the counterweight along the arm 
(5). The cut-off wheel stroke in cutting is limited by a 
locking screw. An aspirator system (7) is included to 
arrest, collect and remove the flying chips and sparks 
during abrasive cutting. The principle description of 
the experimental setup is shown in Fig. 2.
Fig. 2.  Work stand for elastic abrasive cutting
Experimental studies have been conducted 
during counter-directional cutting with high speed 
reinforced cut-off wheels 41-180х22.2х3.0 А30RBF, 
produced by the Abrasive Tools Factory – Berkovitsa, 
Bulgaria [31]. The abrasive wheels are marked in 
compliance with EN 12413:2007: type and size of the 
cut-off wheel; abrasive material type; А = aluminium 
oxide; grit size according to ISO 8486 – 30 (Coarse); 
hardness grade, R = hard; bond, BF = fibre-reinforced 
resinoid bond.
The materials being machined are C45 (1.0503) 
and 42Cr4 (1.7045) steels (BS EN 10277-2:2015 [32]) 
(Table 1). They are in the shape of cylindrical rods of 
diameter dw = 30 mm.
The general form of the models describing the 
relationship between the studied parameters (cut-off 
wheel wear, cut off wheel temperature, workpiece 
temperature, cut piece temperature and time per cut) 
and the group of independent variables (factors ds(X1), 
F(X2) and nw(X3)) is:
    y b b X b X b X Xg h i i
i
ii i
i
ij i j
i j
,
,   
  
  0
1
3
2
1
3
 (1)
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
639Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
where g = 1 to 5, for index corresponding to the type of 
the studied elastic abrasive cutting parameter: y1,1 = δ1,
y2,1 = Ts,1, y3,1 = Tw,1, y4,1 = Td,1, y5,1 = tc,1, y1,2 = δ2, y2,2 = Ts,2, 
y3,2 = Tw,2, y4,2 = Td,2, y5,2 = tc,2.
Table 1.  Chemical composition and physio-mechanical properties 
of the steels studied
Steel 
type
Chemical composition Tensile 
strength [MPa]
Hardness 
[HB]C [%] Mn [%] Cr [%] Si [%]
C45 0.44 0.5 0.2 0.2 750 192
42Cr4 0.4 0.5 1 0.25 1000 205
The form of the model was chosen on the basis of 
the theoretical and experimental studies of the effect 
of elastic abrasive cutting conditions on cut-off wheel 
wear, temperature and cutting ability as the non-linear 
nature of experimental dependencies δh = f (ds, F, nw), 
Ts,h = f (ds, F, nw), Tw,h = f (ds, F, nw), Td,h = f (ds, F, nw), 
and Tc,h = f (ds, F, nw) were taken into account [23], [29] 
and [30].
To build the theoretical and experimental models 
in Eq. (1), multi-factor experiments were conducted 
using an orthogonal central-composite design. The 
number of trials is N = 2n + 2n +1 = 15 (n = 3 is the 
number of control factors). Three observations were 
made for each experiment. The variation limits of 
control factors are presented in Table 2. They are 
determined on the basis of the conducted experimental 
studies on the temperature in elastic abrasive cutting 
and the performance of cut-off wheels estimated by 
the parameters: wear, tool life, and cutting ability [23], 
[29] and [30].
The models were built using the measured 
values of the cut-off wheel wear, cut-off wheel 
maximum contact temperature, workpiece maximum 
instantaneous temperature, cut piece temperature at 
the end of the cut-off cycle and time per cut. Each 
measurement of wear, time per cut and temperature 
were performed three times.
Table 2.  Factor levels in the experimental design
Factors
Factor levels
–1 0 +1
X1 ds [mm] 120 150 180
X2 F [daN] 1 2 3
X3 nw [min–1] 22 91 160
The cut-off wheel wear was determined as a mean 
value of the differences between the tool diameters 
measured in two mutually perpendicular directions at 
the beginning and the end of the respective observation 
performed under specific cutting conditions. Cut-off 
wheel diameters were measured using a Mitutoyo 
Digital Calliper, ABSOLUTE, LCD of Range: 0 
mm to 200 mm, Resolution: 0.01 mm, Accuracy: 
±0.02 mm, Repeatability: 0.01 mm.
Fig. 3.  Illustration of thermography measurement process of cut-
off wheel, workpiece and cut piece temperatures, respectively in 
elastic abrasive cutting of cylindrical rods
The time per cut corresponds to the duration of the 
respective cut-off cycle. The cut-off wheel, workpiece 
and cut piece, and respective maximum contact 
temperatures during the elastic abrasive cutting of 
cylindrical rods, were measured using a thermal 
imaging camera. Non-destructive thermography 
measurement leads to a reduction of measurement 
temperature errors compared to thermocouple 
measurement, as described in [23] and [33]. The whole 
cutting process was registered by infrared cameras 
ThermaCam FLIR SC640, with an image resolution 
of 640 × 480 pixels, temperature measurement 
range from –40 ºC to +2000 ºC, reading accuracy 
of ±2 % and IP-link using FireWire. Thermovision 
SDK software was used to manage, change settings, 
and control the calibration of the cameras. The 
ResearchIR MAX software package was also used for 
real-time monitoring and temperature measurements 
of the cutting area, selected as the region of interest 
(ROI). Surface temperature measurements were 
recorded simultaneously in two ROI positions: 
cutoff wheel profile and face view of side (Fig. 3a). 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
640 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
Fig. 3b illustrates real-time monitoring and surface 
temperature measurement of the cutting area using 
ResearchIR MAX. 
Further thermograms processing was performed 
in MATHLAB, for noise reduction, contouring, 
additional analysis, recognition and classification of 
the areas with the highest and lowest temperature or 
other ROI interesting areas.
1.2  Experimental Results and Modelling
The designs of the experiments, including the values 
of the studied response variables of the elastic abrasive 
cutting process, are presented in Table 3. 
After statistical analysis of the experimental 
results and by applying the regression analysis 
method and QstatLab software [34], theoretical and 
experimental models of the cut-off wheel wear, cutoff 
wheel temperature, workpiece temperature, cut piece 
temperature and time per cut were made (Table 4). 
The regression models include only the significant 
regression coefficients determined in accordance with 
the condition t > tg,h(α/2,ν), where t  and tg,h(α/2,v) are, 
respectively, the calculated values of Student’s 
t-criterion for each coefficient b0, bi, bij, bii of the 
regression equations, the general form of which is 
presented by Eq. (1) and the tabular value of Student’s 
t-criterion (α = 0.05 is the significance level; ν = N – k 
is the number of degrees of freedom; k is the number 
of coefficients in the model), presented for each model 
in Table 4. The constructed models are relevant since 
the condition F g h , > Fg,h(α,ν1,ν2) has been met with a 
confidence level of 95 %. The calculated F g h ,  and 
tabular Fg,h(α,ν1,ν2) values of the Fisher criterion 
(α = 0.05; ν1 = k – 1 and ν2 = N – k are degrees of 
freedom) for each regression model are presented in 
Table 4. The theoretical and experimental models 
extremely accurately describe the dependencies 
between the studied response variables and the control 
factors. The values of the determination coefficients 
are Rg h ,
2
= 0.880 to 0.999 (Table 4).
The effect of cutting conditions on the cut-off 
wheel wear, cutoff wheel temperature, workpiece 
temperature, cut piece temperature and time per cut 
according to the created theoretical and experimental 
models (Table 4) is graphically presented in Fig. 4. 
To determine the impact rate of control factors on 
the studied response variables of the elastic abrasive 
cutting process, an analysis of variance (ANOVA) 
was conducted.
1.3  Analysis of Experimental Results
The investigation of the response variables of the 
elastic abrasive cutting process shows that they 
depend, to a large extent, on the conditions for 
implementing the process. However, the effect of 
the cut-off wheel diameter, compression force, and 
workpiece rotational frequency on the cut-off wheel 
wear, time per cut, cut piece temperature, workpiece 
temperature and cut off wheel temperature is different, 
and it is related to the theoretically established 
influence of ds, F and nw on the length of the contact 
arc between the cut-off wheel and the workpiece, the 
Table 3.  Design of the experiments and response variables of the elastic abrasive cutting process of C45 and 42Cr4 steels
Control factors
Response variables
Cut-off wheel wear Cut-off wheel temperature Workpiece temperature Cut piece temperature Time per cut
ds [mm] F [daN] nw [min-1] δ1 [mm] δ1 [mm] Ts,1 [°C] Ts,2 [°C] Tw,1 [°C] Tw,2 [°C] Td,1 [°C] Td,2 [°C] tc,1 [s] tc,2 [s]
120 1 22 0.977 1.051 160 164 970 989 201 215 10.90 12.08
180 1 22 0.551 0.583 195 200 830 850 220 235 9.91 10.80
120 3 22 1.882 1.981 175 179 989 1008 208 223 8.90 10.06
180 3 22 1.422 1.513 205 210 863 880 235 251 7.90 8.86
120 1 160 3.141 3.272 135 144 1093 1149 145 154 12.9 14.88
180 1 160 2.294 2.390 150 160 950 1000 170 180 11.87 13.64
120 3 160 4.476 4.865 160 172 1125 1184 152 161 10.94 12.69
180 3 160 4.307 4.631 180 193 965 1015 184 195 9.87 11.45
120 2 91 2.528 2.689 149 156 907 936 173 184 10.96 12.49
180 2 91 2.109 2.221 183 190 838 864 207 222 9.89 11.37
150 1 91 1.652 1.721 157 165 865 889 193 205 11.40 12.80
150 3 91 2.948 3.189 173 182 879 905 206 219 9.40 10.52
150 2 22 1.218 1.282 185 196 805 838 215 230 9.45 10.58
150 2 160 3.374 3.628 156 164 933 956 180 192 11.41 13.12
150 2 91 2.308 2.455 165 175 872 900 202 215 10.43 10.43
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
641Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
depth of cut and the thickness of the layer of material 
being cut by one abrasive grain [29] and [35].
The analysis of the theoretical and experimental 
models (Table 4) and the graphics plotted on the 
basis of them (Fig. 4), as well as the interpretation 
of ANOVA results, allow us to draw the following 
conclusions:
(1) The workpiece rotational frequency has the 
highest effect on the elastic abrasive cutting process 
response variables. The increase of nw  within the range 
being studied results in increasing cut-off wheel wear 
(from 2.4 to 4.2 times depending on the compressive 
force and cut off wheel diameter), time per cut (by 
up to 25 %), and workpiece temperature (by up to 18 
%), as well as in decreasing cut piece temperature (by 
up to 29 %) and cut off wheel temperature (by up to 
17.6 %). The intensified cut-off wheel wear and the 
increased time per cut are connected with the increased 
thickness of the layer being cut off by one abrasive 
grain, thus resulting in increasing the load of abrasive 
grains and establishing preconditions for faster filling 
of the cut off wheel pores with chips [29] and [35], as 
well as with the change of the ratio between the normal 
and main cutting force, which results in deteriorating 
the possibilities for abrasive grain cutting [30]. The 
decrease of cut piece temperature is caused by the 
enhanced heat removal resulting from the thicker 
layer being cut and the cross section of the chip being 
cut off by one abrasive grain [29] and [35], as well as 
the time per cut. The increased workpiece temperature 
and the decreased cut off wheel temperature are linked 
to an increase of the contact area between the cut-off 
wheel and the workpiece, as well as to the fact that 
in the course of cutting the workpiece appears a tool 
coolant absorbing part of the released heat, which 
consequently is transferred to the chip. 
(2) As the cut-off wheel diameter ds decreases 
within the range under study, both the tool wear size 
and time per cut increase, from 11 % to 94 % and from 
9 % to 14 %, respectively. That tendency could be 
explained with the decrease in the number of abrasive 
grains involved in the removal of the layer being cut 
per revolution of the cut-off wheel, the increase of the 
thickness of the chip being cut by one abrasive grain, 
respectively, the load on the abrasive grains, as well 
as with the inevitable decrease of the cut-off wheel 
speed since the rotational frequency ns is constant [29] 
and [35]. All this deteriorates the cutting ability of the 
abrasive grains and logically results in intensifying 
tool wear and increasing time per cut. The influence 
of cut-off wheel diameter on tool wear and time per 
cut depends on the compression force, workpiece 
rotational frequency, and the effect of ds on the wear 
decreases as the compression force and workpiece 
Table 4.  Theoretical and experimental models of cut-off wheel wear, temperature and time per cut during elastic abrasive cutting and 
statistical characteristics of the models
Steel, 
type
Response  
variables
Models
Student’s 
criterion
Fisher criterion Determination 
coefficientCalculated Tabular
C45
Cut-off wheel wear 1 1 218 0 008 0 382 0 011 0 003    . . . . .d F n Fns w w 2.228 294.382 3.478 0.988
Temperature of the 
cut-off wheel
T d F ns s w, . . .1 103 865 0 447 8 0 201    2.201 48.371 3.587 0.910
Temperature of the 
workpiece
T d F d
F n
w s s, . . . .
. .
1
2603 317 19 479 194 928 0 0578
51 557 0 005
2
2
    
  w
2
2.262 38.234 3.482 0.930
Temperature of the 
cut piece
T d F n dd s w s, . . . . .1 88 426 3 69 4 667 0 359 0 011
2      2.228 98.168 3.478 0.965
Time per cut
t d F n F
d n
c s w
s w
,
. . . . .
. .
1
13 486 0 017 0 881 0 015 0 029
6 6 10
2
6
     
 
2.262 19098,664 3.481 0.999
42Cr4
Cut-off wheel wear 2 1 344 0 008 0 391 0 011 0 004    . . , . .d F n Fns w w 2.228 370,829 3.478 0.991
Temperature of the 
cut-off wheel
T d F ns s w, . . . .2 105 799 0 46 8 583 0 168    2.201 35.092 3.587 0.880
Temperature of the 
workpiece
T d F d
F n
w s s
w
,
. . . .
. .
2
2695 024 21 143 140 028 0 063
37 403 0 006
2
2
    
  2
2.262 38.494 3.482 0.931
Temperature of the 
cut piece
T d F n dd s w s, . . . .2 89 928 3 887 5 0 394 0 011
2      2.228 90.495 3.478 0.962
Time per cut t d F nc s w, . . . .2 15 212 0 02 1 062 0 019    2.201 863.138 3.587 0.994
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
642 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
% to 11 %. The minimum effect of the compression 
force is related to the fact that when F increases, 
elastic abrasive time per cut decreases and, in contrast, 
the length of contact arc and the depth of cut increase 
[29] and [35].
(6) The nature and level of influence of workpiece 
rotational frequency, cut off wheel diameter and 
compression force on cut-off wheel wear, time per 
cut and elastic abrasive cutting temperature are 
equal for the two materials being machined (C45 and 
42Cr4 steels). Nevertheless, the temperatures of cut-
off wheel, workpiece, and cut piece are higher when 
machining 42Cr4 steel (by 1.9 % to 7.5 %), which 
is related to the higher hardness and strength of this 
material. The values of cut-off wheel wear and time 
per cut are also higher, respectively by 4.2 % to 8.7 % 
and by 9 % to 16 %.
2  OPTIMIZATION OF ELASTIC  
ABRASIVE CUTTING CONDITIONS
2.1  Optimization Method
The undertaken investigation, modelling, and analysis 
of the response variables of the elastic abrasive 
cutting process show complicated and different 
nature of change of cut-off wheel wear, cut-off 
rotational frequency increase. As the workpiece 
rotational frequency increases and the compression 
force decreases, the effect of ds on the time per cut 
increases.
(3) As ds decreases, the temperatures of the cut 
piece and cut-off wheel also decrease, whereas the 
workpiece temperature increases within a range of 
11 % to 17 %. That is linked to an increase of the 
thickness of the layer of material being cut and the 
cross-section of the chip being cut by one abrasive 
grain [29] and [35], which enhances heat removal and 
to an increase of time per abrasive cut and a decrease 
of cutting speed.
(4) The increase of the compression force F 
results in decreasing elastic abrasive time per cut 
(by 14 % to 20 %), respectively, increasing process 
performance in increasing cut-off wheel wear (from 
1.6 to 2.7 times). That is linked to the increase of the 
length of contact arc and the depth of cut [29] and 
[35], as a result of which cut off wheel pores are filled 
with chips and abrasive particles faster. The effect 
of the compression force increases when the cut-off 
wheel diameter increases and the workpiece rotational 
frequency decreases.
(5) The compression force has little effect on 
temperature. As it increases, the temperatures of the 
cut piece, cut-off wheel and workpiece increase by 5 
a)      b)      c) 
d)      e) 
Fig. 4.  Effect of the elastic abrasive cutting conditions on the cut-off wheel wear δs ; a), cut off wheel temperature Ts , b), workpiece 
temperature Tw , c), cut piece temperature Td , d) and time per cut tc , e) (for steel 42Cr4, ds = 150 mm)
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
643Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
wheel temperature, workpiece temperature, cut piece 
temperature and time per cut depending on the cut-
off wheel diameter, the compression force exerted by 
the cut-off wheel on the workpiece and the workpiece 
rotational frequency. This implies that the optimum 
values of the investigated response variables will be 
obtained at different combinations of values of control 
factors (ds, F and nw). Hence, the optimization of the 
elastic abrasive cutting process by one parameter 
is irrelevant. Multi-objective optimization will 
provide much more information so as to make a 
justified decision on the selection of optimum cutting 
conditions. 
The common multi-objective optimization 
methods can be classified into three main groups [36] 
to [38]. The first group includes methods that use one 
of the response variables as an objective function and 
the rest of the response variables are considered to be 
limits. The major disadvantage of those methods is 
that they do not apply the main idea of multi-objective 
optimization: specifically, all response variables to be 
considered simultaneously. The proposed procedures 
from this category would result in unreal solutions, 
especially when there are conflicting objectives. In 
addition, in many cases, it is difficult to choose one 
of the response variables as an objective function. 
When applying the methods from the second group, 
a domain where different response variables meet 
specific requirements occurs. This approach is 
effective in the case of a small number of control 
factors (2 or 3) and response variables (up to three). 
The third group comprises methods that combine a 
set of response variables in a generalized objective 
function defined as a utility function, desirability 
function, loss function, or proportion of conformance. 
In those cases, the optimization problem is solved as a 
single-objective one. 
To define the optimum elastic abrasive cutting 
conditions, the generalized utility function method 
was chosen [36], [37] and [39]. It is based on the idea 
that the quality of a product or process with a set 
of response variables is unacceptable if one of the 
response variables is outside the utility limits. This 
method defines the result as a combination of response 
variables and chooses a set of factors where the result 
is a maximum value. The generalized utility function 
has many advantages compared to other combining 
methods, mainly due to its flexibility since it makes 
it possible to maximize some output quantities and 
minimize others simultaneously. 
The generalized utility function could be defined 
as a geometric-mean value ΦG,h or an arithmetic-mean 
value ΦA,h of the partial utility functions ηg,h obtained 
by transforming the studied and modelled response 
variables of the elastic abrasive cutting process 
yg,h  (Table 4) into dimensionless quantities [37] and 
[39]. To solve the specific optimization problem, the 
generalized geometric-mean utility function was 
chosen as an optimization parameter since if one of 
the response variables of the elastic abrasive cutting 
process does not meet the requirements of utility 
limits, then ΦG,h = 0. In this case, the arithmetic 
generalized utility function is ΦA,h ≠ 0 and could have 
a maximum value but the elastic abrasive cutting 
conditions under which this value of ΦA,h has been 
obtained are not optimum. 
The generalized geometric-mean utility function 
is a complex indicator, which is determined in 
conformity with the dependency [37] and [39]:
 
G h g hg
g g h g u
gg
k y y
y, ,
,
, 
 

 

 
1
5
1
5
5  (2)
where kg is utility coefficient (kg = +1), when the 
increase of the studied response variable yg,h is useful; 
(kg = –1), when the decrease of yg,h is useful); yg,u is 
the most useless result of the response variable yg,h, 
obtained within the limits of the permissible space; 
Δyg = yg max – yg min; yg max and yg min is utility limits 
(maximum and minimum value of yg,h).
The solution of the optimization problem is 
reduced to defining the combination of the control 
factors of the elastic abrasive cutting process; cut-off 
wheel diameter ds, compression force F and workpiece 
rotational frequency nw, where the generalized 
geometric-mean utility function has a maximum. The 
problem is solved for each material being machined 
(С45 and 42Cr4 steels).
2.2  Modelling of Generalized Utility Function
To solve the optimization problem, mathematical 
models for defining the generalized geometric-mean 
utility function depending on the control factors of 
the elastic abrasive cutting process were built. The 
general form of the models, assumed on the basis 
of the performed analysis of the influence of elastic 
abrasive cutting conditions on its response variables: 
cut off wheel wear δh , cut-off wheel temperature Ts,h , 
workpiece temperature Tw,h , cut piece temperature 
Td,h , and time per cut tc,h , is:
 
G h i i
i
ii i
i
ij i j
i j
A A X A X
A X X A X X X
,
.
   
 
 

 

0
1
3
2
1
3
123 1 2 3  (3)
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
644 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
The models in Eq. (3) under the conditions of 
elastic abrasive cutting of С45 and 42Cr4 were built 
on the basis of the results from the experiments 
carried out according to an optimum design involving 
the following number of trials N = 2n + 2n +1 = 15 (n = 3 
is the number of control factors) in Table 5. With each 
trial, the generalized geometric-mean utility function 
was determined in compliance with dependency in 
Eq. (2), as the values of the most useless result and 
the utility limits of the response variables of the 
elastic abrasive cutting process were determined in 
compliance with the following equations:
δ(u) = (δh)max; Ts(u) = (Ts,h)max; Tw(u) = (Tw,h)max; 
Td(u) = (Td,h)max; tc(u) = (Tc,h)min;
Δδ = (δh)max – (δh)min; ΔTs = (Ts,h)max – (Ts,h)min; 
ΔTw = (Tw,h)max – (Tw,h)min; ΔTd = (Td,h)max – (Td,h)min;
Δtc = (tc,h)max – (tc,h)min; 
where (δh)min, (δh)max, (Ts,h)min, (Ts,h)max, (Tw,h)min, 
(Tw,h)max, (Td,h)min, (Td,h)max, (Tc,h)min, (Tc,h)max are, 
respectively, the minimum and maximum values of cut 
off wheel wear, cut-off wheel temperature, workpiece 
temperature, cut piece temperature and time per cut, 
calculated by using the regression models in Eq. (1) 
(Table 4).
The models in Eq. (3), Table 6, were built on 
the basis of the defined values of the generalized 
geometric-mean utility function (Table 5) by applying 
the regression analysis method and QstatLab software 
[34]. They include only the significant regression 
coefficients defined according to the condition 
t > th(α/2,ν), where t  and th(α/2,ν) are respectively 
the calculated values of Student’s t-criterion for 
each coefficient of the regression equations and the 
tabular value of Student’s t-criterion (α = 0.05) is the 
significance level; ν = N – k is the number of degrees of 
freedom; k is the number of coefficients in the model). 
The tabular values of Student’s t-criterion for each 
model are: t1(0.025, 6) = 2.447 and t2(0.025, 9) = 2.262. The 
constructed models are relevant since the condition 
Fh > Fh(α, ν1, ν2) has been met with a confidence level 
of 95 %. They extremely accurately describe the 
dependencies between the studied response variables 
and control factors. The calculated Fh  and tabular 
Fh(α, ν1, ν2) values of Fisher criterion ( Fh
  = 0.05; 
ν1 = k – 1 and ν2 = N – k are degrees of freedom), as 
well as the values of the determination coefficient 
Rh
2
 for each regression model in Eq. (3) are shown 
in Table 6.
The effect of cutting conditions on the generalized 
utility function according to the created theoretical 
and experimental models (Table 6) is graphically 
presented in Figs. 5 and 6. 
Table 5.  Design of the experiment and generalized utility functions 
during elastic abrasive cutting of C45 and 42Cr4 steels
Control factors Generalized utility function
ds [mm] F [daN] nw [min–1] ΦG,1 ΦG,2
120 1 22 0.604 0.497
180 1 22 0.437 0.000
120 3 22 0.486 0.365
180 3 22 0.474 0.000
120 1 160 0.484 0.000
180 1 160 0.563 0.400
120 3 160 0.335 0.000
180 3 160 0.400 0.215
120 2 91 0.593 0.473
180 2 91 0.498 0.285
150 1 91 0.608 0.478
150 3 91 0.475 0.323
150 2 22 0.480 0.311
150 2 160 0.541 0.395
150 2 91 0.574 0.447
To determine the impact rate of control factors 
on the generalized utility function, ANOVA was 
a)      b)      c) 
Fig. 5.  Generalized utility function in elastic abrasive cutting of C45 steel; a) nw = 63.7 min–1 ,  b) F = 1 daN; and c) ds = 120 mm 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
645Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
conducted. QstatLab software [34] was used, and the 
interaction between the factors was considered. 
It has been established that the elastic abrasive 
cutting conditions have different effects, in terms of 
nature and level, on the generalized geometric-mean 
utility function, depending on the type of material 
being machined. In the elastic abrasive cutting of C45 
steels, the highest effect is exerted by the compression 
force. As F decreases, the generalized utility function 
value ΦG,1 increases. The effect of the compression 
force increases as the workpiece rotational frequency 
nw increases and the cut-off wheel diameter ds 
decreases. In the elastic abrasive cutting of 42Cr4 
steels, the highest effect on the generalized utility 
function ΦG,1 is exerted by the workpiece rotational 
frequency and the relationship ΦG,2 = f (nw) has a 
maximum whose value depends on the cut-off wheel 
diameter and compression force. The effect of nw on 
ΦG,2 increases as the cut-off wheel diameter decreases 
and the compression force increases. It is highest when 
cutting with abrasive wheels of diameter ds = 120 mm 
and compression force F = 3 daN.
2.3 Definition of Optimum Elastic Abrasive Cutting 
Conditions
The optimization problem was solved upon elastic 
abrasive cutting of С45 and 42Cr4 steels by using a 
genetic algorithm and QStatLab software [34]. The 
defined optimum conditions (cut-off wheel diameter 
ds, compression force F and workpiece rotational 
frequency nw), where the generalized utility function 
ΦG,h has a maximum, are presented in Table 7 and 
Figs. 5 and 6. The elastic abrasive cutting under these 
conditions ensures the best possible match between 
the rate of cut off wheel wear, time per cut, cut piece 
temperature, cut-off wheel temperature and workpiece 
temperature, respectively, in the elastic abrasive 
cutting of С45 steels: d1 = 1.532 mm, Ts,1 = 1.527 °C, 
Tw,1 = 975 °C, Td,1=177.8 °C and tc,1 = 8.21 s; in the 
elastic abrasive cutting of 42Cr4 steels: d2 = 1.524 mm, 
Ts,2 = 161.2 °C, Tw,2 = 977.4 °C, Td,2 = 212.6 °C, and 
tc,2 = 12.7 s.
Under the predicted optimum elastic abrasive 
cutting conditions (cut-off wheel diameter 
ds = 120 mm, compression force F = 1 daN; 
workpiece rotational frequency nw = 63.7 min–1 and 
nw = 49.9 min–1, respectively for С45 steels and 42Cr4 
steels), confirmation run experiments were performed, 
where cut off wheel wear, cut-off wheel temperature, 
workpiece temperature, cut piece temperature and time 
per cut were determined. The experimental values of 
the elastic abrasive cutting process response variables 
are presented in Table 7. They are determined as an 
arithmetical mean of the four observations done for 
each of them. A comparison between the experimental 
a)      b)      c) 
Fig. 6.  Generalized utility function in elastic abrasive cutting of 42Cr4 steel; a) nw =  49.9 min–1, b) F = 1 daN, c) ds = 120 mm
Table 6.  Theoretical and experimental models of the generalized utility function and statistical characteristics of the models
Steel type Models
Fisher criterion Determination  
coefficientCalculated Tabular
C45
 
G w s wF n d n, . . . . . . .
.
1
5 2 5 2
0 899 0 25 0 003 1 791 10 1 318 10
0 00
     

 
1 0 002 4 08 10 1 058 10
5 5Fn d F d n d Fnw s s w s w  
 
. . . . .
5.425 4.147 0.717
42Cr4
 
G w s w
s
n d n
d
,
. . . . . .
. .
2
5 2 5 2
5
0 917 0 007 3 06 10 3 61 10
9 107 10
    

 
 n d Fw s

3 057 10
4
. .
12.553 3.482 0.804
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
646 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
and the predicted, according to the models, Eq. 
(1), Table 4, values of the elastic abrasive cutting 
process response variables (Table 7) shows that error 
percentage is: 2.37 % to 2.92 % for cut off wheel 
wear; 1.48 % to 3.47 % for cut off wheel temperature; 
1.28 % to 1.56 % for workpiece temperature; 2.84 
% to 4.43 % for cut piece temperature; 1.44 % to 
1.93 % for time per cut. These results prove that the 
recommended elastic abrasive cutting conditions are 
optimum and correct.
3  CONCLUSIONS
This paper proposes a new approach for optimizing 
the elastic abrasive cutting process, which takes 
into consideration the specifics of its particular 
implementation. The generalized geometric-mean 
utility function has been chosen as an optimization 
factor. It appears to be a complex indicator 
characterizing the technological and economic 
parameters of elastic abrasive cutting.
The proposed optimization approach has been 
applied to determine the optimum conditions of 
elastic abrasive cutting of С45 and 42Cr4 steels. The 
following results have been achieved:
(1) To determine the generalized utility function, 
theoretical and experimental models reflecting 
the complex influence of the control factors of the 
process (cut-off wheel diameter ds, compression 
force F and workpiece rotational frequency nw) 
have been developed. The models are based on 
the findings of complex studies and modelling 
of the parameters of the elastic abrasive cutting 
process (cut off wheel wear, time per cut, cut 
piece temperature, cut-off wheel temperature, 
and workpiece temperature) depending on the 
conditions of its implementation. 
(2) By applying the generalized utility function and 
genetic algorithm, the optimum conditions for 
implementing the process have been determined 
as follows: cut-off wheel diameter ds =120 
mm; compression force F =1 daN; workpiece 
rotational frequency nw = 63.7 min–1 and nw = 49.9 
min–1, respectively for С45 and 42Cr4 steels. 
The authenticity of the determined optimum 
conditions has been proven by an experimental 
study of the response variables of the elastic 
abrasive cutting process. It has been established 
that they ensure the best possible match between 
the rate of cut off wheel wear (δ ≤ 1.58 mm), 
the cut off wheel temperature (Ts ≤ 167 °C), the 
workpiece temperature (Tw ≤ 965 °C), the cut 
piece temperature (Td ≤ 212 °C) and the time per 
cut (tc ≤ 12.95 s).
The results obtained in applying the new 
optimization approach provide possibilities for 
enhancing efficiency and controlling the elastic 
abrasive cutting process by choosing optimum 
conditions in accordance with the specifics and 
implementation of the particular abrasive cutting 
process. They also confirm the key role of the 
workpiece rotational frequency, compression force 
and cut-off wheel diameter for the cut off wheel 
lifetime, the productivity of the elastic abrasive cutting 
process and the temperature distribution within the 
tool, workpiece and cut piece. Taking into account the 
high productivity and low cost of the elastic abrasive 
process, the proposed multipurpose optimization 
approach and the obtained theoretical-experimental 
results, presented in this paper, could be used in every 
enterprise specialized in machine building.
4  ACKNOWLEDGEMENTS
This work was supported by the European Regional 
Development Fund within the OP “Science and 
Education for Smart Growth 2014 - 2020”, Project CoC 
“Smart Mechatronic, Eco- and Energy-saving Systems 
and Technologies“, No BG05M2OP001-1.002-0023 
and the Research fund of the Technical University of 
Gabrovo (Project no 2008 M).
Table 7.  Optimum elastic abrasive cutting conditions
Steel 
type
Optimal conditions of elastic 
abrasive cutting
Response variables
Generalized 
utility 
function, 
ΦG,h
Cut off  
wheel wear,  
δh [mm]
Temperature of 
the cut off wheel, 
Ts,h [°C]
Temperature of 
the workpiece, 
Tw,h [°C]
Temperature of 
the cut piece, 
Td,h [°C]
Time per cut, 
tc,h [s]
ds [mm] F [daN] nw [min–1] PV EV PV EV PV EV PV EV PV EV
C45 120 1 63.7 1.532 1.578 152.7 155 975 960 177.8 183 8.21 8.33 0.634
42Cr4 120 1 49.9 1.524 1.561 161.2 167 977.4 965 202.6 212 12.7 12.95 0.528
EV = experimental value; PV = predicted value
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
647Modelling and Multi-objective Optimization of Elastic Abrasive Cutting of C45 and 42Cr4 Steels 
5  REFERENCES
[1] Ganev, G. (2013). Elastic Abrasive Cutting of Rotational 
Workpieces in Mechanical Engineering. PhD Thesis, Technical 
University of Gabrovo, Gabrovo. (in Bulgarian)
[2] Neugebauer, R., Hess, K.-U., Gleich, S., Pop, s. (2005). 
Reducing tool wear in abrasive cutting. International Journal 
of Machine Tools and Manufacture, vol. 45, no. 10, p. 1120-
1123, DOI:10.1016/j.ijmachtools.2005.01 .002.
[3] Putz, M., Cardone, M., Dix, M. (2017). Cutoff grinding of 
hardened steel wires - modelling of heat distribution. Procedia 
CIRP, vol. 58, p. 67-72, DOI:10.1016/j.procir.2017.03.198.
[4] Nenkov, N., Аleksandrova, I., Ganev, G. (1999). Methods of 
abrasive cutting of workpieces. Маshinostroene, no. 5-6, p. 
38-40. (in Bulgarian)
[5] Kaczmarek, J. (2011). Using a thermovision method for 
measuring temperatures of a workpiece during abrasive cutoff 
operation. Advance in Manufacturing Science and Technology, 
vol. 35, no. 4, p. 85-95.
[6] Sahu, P., Sagar, R. (2006). Development of abrasive cutoff 
wheel having side grooves. The International Journal of 
Advanced Manufacturing Technology, vol. 31, p. 37-40, 
DOI:10.1007/s00170-005-0138-2.
[7] Levchenko, E., Pokintelitsa, N. (2017). Investigation 
of thermal processes in abrasive pipe samplin. 
МATEC	 Web	 of	 Conferences, vol. 129, DOI:10.1051/
matecconf/201712901082.
[8] Lopes, J.C., Ribeiro, F.S.F., Javaroni, R.L., Garcia, M.V., Ventura, 
C.E.H., Scalon, V.L., de Angelo Sanchez, L.E., de Mello, H.J., 
Aguiar, P.R., Bianchi, E.C. (2020). Mechanical and thermal 
effects of abrasive cutoff applied in low and medium carbon 
steels using aluminum oxide cutting disc. The International 
Journal of Advanced Manufacturing Technology, vol. 109, p. 
1319-1331, DOI:10.1007/s00170-020-05753-5.
[9] Luo, S.Y., Tsai, Y.Y., Chen, C.H. (2006). Studies on cutoff 
grinding of BK7 optical glass using thin diamond wheels. 
Journal of Materials Processing Technology, vol. 173, no. 3, p. 
321-329, DOI:10.1016/j.jmatprotec. 2005.11.036.
[10] Ortega, N., Martynenko, V., Perez, D., Krahmer, D.M., López 
de Lacalle, L.N., Ukar, E. (2020). Abrasive disc performance in 
dry-cutting of medium-carbon steel. Metals, vol. 10, no. 4, art. 
ID 538, DOI:10.3390/met10040538.
[11] Riga, A., Scott, C. (2001). Failure analysis of an abrasive cut-
off wheel. Engineering Failure Analysis, vol. 8, no. 3, p. 237-
243, DOI:10.1016/S1350-6307(00)00014-5.
[12] Yoshida, T., Nagasaka, K., Kita, Y., Hashimoto, F. (1986). 
Identification of a grinding wheel wear equation of the 
abrasive cutoff by the modified GMDH. International Journal 
of Machine Tool Design and Research, vol. 26, no. 3, p. 283-
292, DOI:10.1016/0020-7357(86)90006-5.
[13] Ojolo, S., Orisaleye, J., Adelaja, A. (2010). Development of a 
high speed abrasive cutting machine. Journal of Engineering 
Research, vol. 15, p. 1-8.
[14] Kaczmarek, J. (2008) The effect of abrasive cutting on the 
temperature of grinding wheel and its relative efficiency. 
Archives of Civil and Mechanical Engineering, vol. 8, no. 2, p. 
81-91, DOI:10.1016/S1644-9665(12)60195-2.
[15] Ni, J., Yang, Y., Wu, C. (2019). Assessment of water-based 
fluids with additives in grinding disc cutting process. Journal 
of Cleaner Production, vol. 212, p. 593-601, DOI:10.1016/j.
jclepro.2018. 12.066.
[16] Yamasaka, M., Fujisawa, M., Oku, T., Masuda, M., Okamura, 
K. (1990). improvement of straightness in precision cut-off 
grinding using thin diamond wheels. CIRP Annals, vol. 39, no. 
1, p. 333-336, DOI:10.1016/S0007-8506(07)61066-X.
[17] Braz, C.O., Ventura, C.E.H., De Oliveira, M.C.B.,·Antonialli, 
A.Í.S., Ishikawa, T. (2019). Investigating the application of 
customized abrasive cut-off wheels with respect to tool wear 
and subsurface integrity in metallographic cutting of pure 
titanium. Metallography, Microstructure, and Analysis, vol. 8, 
p. 826-832, DOI:10.1007/s13632-019-00587-4.
[18] Wang, Z., Li, Y., Yu, T., Zhao, J., Wen, P.H. (2018). Prediction 
of 3D grinding temperature field based on meshless method 
considering infinite element. The International Journal of 
Advanced Manufacturing Technology, vol. 100, p. 3067-3084, 
DOI:10.1007/s00170-018-2801-4.
[19] Eshghy, S. (1968). Thermal Aspects of the Abrasive Cut off 
Operation. Part 2 - Partition Functions and Optimum Cut off. 
Journal of Engineering for Industry, vol. 90, no. 2, p. 360-364, 
DOI:10.1115/1.3604641.
[20] Hou, H., Komanduri, R. (2004). On the mechanics of the 
grinding process, Part III - thermal analysis of the abrasive 
cutoff operation. International Journal of Machine Tools and 
Manufacture, vol. 44, no. 2-3, p. 271-289, DOI:10.1016/j. 
ijmachtools.2003.09.009.
[21] Putz, M., Cardone, M., Dix, M., Wertheim, R. (2019). Analysis 
of workpiece thermal behaviour in cutoff grinding of high-
strength steel bars to control quality and efficiency. CIRP 
Annals, vol. 68, no. 1, p. 325-328, DOI:10.1016/j.cirp.2019. 
04.023.
[22] Malkin, S., Guo, C. (2007). Thermal analysis of grinding. 
CIRP Annals, vol. 56, no. 2, p. 760-782, DOI:10.1016/j.
cirp.2007.10.005.
[23] Stoynova, A., Aleksandrova, I., Aleksandrov, A., Ganev, G. 
(2018). Infrared thermography for elastic abrasive cutting 
process monitoring. MATEC Web of Conferences, vol. 210, 
DOI:10.1051/matecconf/201821002018.
[24] Stoynova, A., Aleksandrova, I., Aleksandrov, A., Goranov, 
G. (2019). Non-contact measurement and monitoring of 
cutoff wheel wear. 28th International Scientific Conference 
Electronics, DOI:10.1109/ET.2019.8878321.
[25] Ortega, N., Bravo, H., Pombo, I., Sánchez, J., Vidal, G. (2015). 
Thermal analysis of creep feed grinding. Procedia Engineering, 
vol. 132, p. 1061-1068, DOI:10.1016/j.proeng.2015.12.596.
[26] Nagasaka, K., Yoshida, T., Kita, Y., Hashimoto, F. (1987). 
Optimum combination of operating parameters in abrasive 
cutoff. International Journal of Machine Tools and 
Manufacture, vol. 27, no. 2, p. 167-179, DOI:10.1016/S0890-
6955(87)80048-2.
[27] Shaw, M. (1975). The rating of abrasive cut-off wheels. 
Journal of Engineering for Industry, vol. 97, no. 1, p. 138-146, 
DOI:10.1115/1.3438526.
[28] Vologin, K. (2002). Selection of rational conditions for cutting 
workpieces of difficult to machine material with abrasive 
Strojniški vestnik - Journal of Mechanical Engineering 67(2021)12, 635-648
648 Aleksandrova, I. – Stoynova, A. – Aleksandrov, A.
wheels. PhD Thesis, Моscow State University of Technology 
Stankin, Моscow. (in Russian)
[29] Aleksandrova, I., Ganev, G., Hristov, H. (2019). Performance 
of cutoff wheels during elastic abrasive cutting of rotating 
workpieces. Tehnički	vjesnik	-	Technical	Gazette, vol. 26, no. 3, 
p. 577-583, DOI:10.17559/TV-20151223184850.
[30] Aleksandrova, I., Ganev, G., Hristov, H. (2016). Cutting 
forces and power during elastic abrasive cutting of rotating 
workpieces. Tehnički	vjesnik	-	Technical	Gazette, vol. 23, no. 2, 
p. 343-348, DOI:10.17559/TV-20131108115506.
[31] Catalogue of the Abrasive Tools Factory - Berkovitsa (2020). 
from https://zai-bg.com/wp-content/uploads/2020/01/ZAI-
2020.pdf, accessed on 2021-10-16.
[32] Steel and Cast Iron Standards EN 10277 (2008). Steel 
grades/numbers, from http://www.steelnumber.com/en/
standard_steel_eu.php?gost_number=10277, accessed on 
2021-10-16.
[33] Jamuła, B. (2021). Temperature measurement analysis in the 
cutting zone during surface grinding. Journal of Measurements 
in Engineering, vol. 9, no. 2, p. 106-116, DOI:10.21595/
jme.2021.21894.
[34] Vuchkov, I.N., Vuchkov, I.I. (2009). Statistical methods of 
quality control, robust engineering, planning, modeling and 
optimization, QStatLab, v 5.3. (in Bulgarian)
[35] Aleksandrova, I., Hristov, H., Ganev, G. (2011). Dynamic and 
technological characteristics of the process elastic abrasive 
cutting of rotating workpieces. Journal of the Technical 
University Sofia, Branch Plovdiv: Fundamental Sciences and 
Applications, vol. 16, p. 123-128.
[36] Mukherjee, I., Ray, P. (2006). A review of optimization 
techniques in metal cutting processes. Computers & Industrial 
Engineering, vol. 50, no. 1-2, p. 15-34, DOI:10.1016/j.
cie.2005.10.001.
[37] Stoyanov, S. (1993). Optimization of Manufacturing Processes. 
Tehnika, Sofia.
[38] Yusup, N., Zain, A., Hashim, S. (2012). Evolutionary 
techniques in optimizing machining parameters: Review 
and recent applications (2007-2011). Expert Systems with 
Applications, vol. 39, no. 10, p. 9909-9927, DOI:10.1016/j.
eswa.2012.02.109.
[39] Аleksandrov, A., Аleksandrova, I. (2012). Theory of Experiment. 
Еx-Press. Gabrovo. (in Bulgarian)