An Integrated Mathematical Model on Thermal Phenomena

in the Cutting Process

DASCHIEVICI LUIZA, GHELASE DANIELA

Faculty of Engineering and Agronomy, Braila,

“Dunarea de Jos” University of Galati,

47, Domneasca St., Galati,

ROMANIA

Abstract: - The paper presents theoretical and experimental research to unify the dynamic and thermal

phenomena in a single comprehensive model of the cutting process identify parameters that characterize the

state of the system and provide quick information on the rate at which to produce the wear for tool edges and

how it can be influenced. Experimental and theoretical research on the temperature of the tool edge and the

medium intensity of wear established similarities between the evolutions of the two phenomena that lead to the

conclusion that by modeling the evolution of the thermal phenomenon can be determined the evolution of the

medium intensity of wear.

Key-Words: - cutting process, heat conduction, tribosystem, intensity of wear, thermal phenomena, metal

cutting, sources heat propagation modeling.

Received: August 22, 2023. Revised: February 19, 2024.. Accepted: April 16, 2024. Published: May 16, 2024.

1 Introduction

Machine parts, components of various devices,

machines, machines, commonly known as parts, are

essentially solid bodies bounded in space by several

surfaces that are characterized by geometric shape,

dimensions in different directions, degree of

smoothness, and relative position. The layers of

material whose dimensions in different directions are

given by the difference between the dimensions of

the initial surfaces and the processed surfaces and

are included between these surfaces is called the

addition of processing.

The technological process of machining by

cutting is the process of generating surfaces by

removing splinters, (Figure 1). It is the basis of the

construction of machine tools, machines that

generate surfaces by removing the addition of

processing, removal carried out by the edge of a

cutting tool that moves relative to the semi-finished

part through a well-defined movement.

It generates a new surface created by the cutting

tool edge during its relative movement to the

blank. In metal cutting, due to the action of the edge

of the tool pushed with a certain force into the

processed material, a complex state of stresses and

deformations is produced in the cutting area. A

simplified diagram of the cutting process is shown in

Figure 1 in which the machine tool through the

cutting tool exerts a force P capable of overcoming

the resistances arising in the processed material.

Fig. 1: Plastic deformation of the processed material

This diagram constitutes a basic physical model

that is completed with other specific models from the

fields of the theory of elasticity and plasticity,

thermodynamics, tribology and thus forms a

complex model, more or less detailed depending on

the requirements or claims. The process of splinter

formation is made up of a complex of physical and

mechanical phenomena, each directly influencing the

process of cutting manufacturing. The deformations

occur as a result of the complex action of the cutting

tool edge on the processed material, [1], [2], [3], [4].

The heating of the tribosystem, friction, and tool

wear are the main processes and phenomena

accompany the cutting process. As a result of the

production of plastic deformations and friction

between the elements of the tribological system, the

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Daschievici Luiza, Ghelase Daniela

E-ISSN: 2224-3410

Volume 21, 2024

energy dissipated on them is transformed into heat,

an effect that becomes decisive with the increase in

the values of the parameters of the cutting regime.

The mechanical work consumed in the cutting

process, which is completely transformed into heat,

generates the heat sources Q1, Q2, and Q3, Figure 2,

ordered according to their intensity as follows:

Q1 > Q2 > Q3, [3], [4], [5], [6].

The mathematical model elaborated in the

specialized literature corresponds in reality to a

uniform and stationary thermal state, with the same

temperature throughout the entire volume of the

splinter and the cutting tool edge and constant over

time, so quite far from reality even in a qualitative

and phenomenological analysis. The considered

hypothesis according to which, in the cutting zone,

the thermal state is uniform and stationary, can be

appreciated as a particular case which in reality is

not possible.

For these reasons, it is necessary to develop an

own model for heat sources, capable of leading to a

correct determination of the real non-stationary and

non-uniformly distributed thermal state.

As a result of the aforementioned processes that

accompany the cutting process, friction, and heating,

the wear phenomenon also appears in the

tribological system, which is very important to

know, especially concerning the cutting tool.

Through the wear of the cutting tool occurs the

modification of the geometry of its active part with

influence on the performance of the cutting process

under optimal conditions.

A basic characteristic of cutting tools is wear

resistance with direct implications on some basic

parameters of production (optimal cutting speed,

reduction of tool consumption, reduction of energy

consumption, productivity, precision, etc.). Wear

resistance is one of the main criteria for optimizing

the geometry of cutting tools. The wear of the tools

is progressive and manifests itself in several aspects

(increase in temperature, damage to the surfaces to

be processed, increase in cutting forces) which

ultimately lead to the removal from service of the

cutting tool.

In the cutting process, the different types of wear

occur rarely separately, they usually occur

simultaneously, with one or another type of wear

predominating depending on the cutting conditions,

Figure 3.

The cutting tool wear manifests under many

aspects (cutting forces increase, temperature increase

in cutting tool, processed area deterioration) finally

leading to their stop functioning and it is progressive

wear. Figure 3 presents more tool edge wear types.

This diagram results abrasive wear has the

highest influence on the total wear of cutting tool

edges.

Friction has a predominant role in heating the

elements of the tribological system workpiece tool to

be processed, therefore with a decisive influence on

the wear of the cutting tool. As the cutting speed can

vary within very wide limits and considering some

insufficiently in-depth research both quantitatively

and qualitatively regarding the influence of the

speed, I consider that it is necessary to undertake

research on the coefficient of friction for each

specific case of cutting, cutting tool edge material

and material to be processed, to highlight the fact

that also in this case the friction coefficient is non-

Colombian type.

As a general characteristic, it can be said that it is

a dry friction, on very small contact surfaces, with

high and unevenly distributed contact pressures and

at high temperatures.

The temperature of the splinter increases very

much as a result of supplying it with energy coming

exclusively from friction between the splinter and

the cutting tool and from the intercrystalline and

intercrystalline frictions that occur in the process of

splinter formation and detachment. As the

temperature increases and the splinter becomes more

plastic, certain areas of it even reach the liquid

phase, the frictions decrease in intensity leading to

the release of less energy, therefore at lower

Fig. 2: Mechanical work heat dispersing

Fig. 3: The influences of tool wear with splitting

speed or temperature

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DOI: 10.37394/232010.2024.21.6

Daschievici Luiza, Ghelase Daniela

E-ISSN: 2224-3410

Volume 21, 2024

temperatures of the splinter. These temperatures

make the splinter more solid, and more intense

friction occurs which tends to increase the

temperature again, and so on. So there is a

combination of effects with opposite tendencies, the

result being an equilibrium temperature below the

melting temperature of processed material. The

increase in speed, especially at high cutting speeds,

leads to a feed-back type chain according to Figure

For the complete tribological study of the cutting

process, very important is the heating phenomenon

of each element that participates in the process

(splinter, cutting edge, piece), their temperature

being the factor with the greatest influence on the

behavior of the tribosystem.

2 The Research and Experiment

Methodology

From the many physical models that describe the

mechanism of the appearance of cutting efforts, we

have adapted for theoretical research the orthogonal

free-cutting model developed by Merchant [4]. In

the evaluation of the external forces, of interaction

with the tool edge, we considered the friction

between the splinter and the tool edge as a non-

Colombian dry friction, so with a variable

coefficient of friction, much closer to reality, [4],

[5], [6], [7], [8].

The heat comes from non-conservative

mechanical work consumed through plastic

deformation in the area of the shear plane and from

the non-conservative mechanical friction work

on the front and back surfaces of the tool, [4], [9],

[10].

Heat propagation modeling: the heat produced by

the sources presented above propagates in a non-

homogeneous environment consisting of a splinter,

tool edge, and tool body, each of them with different

calorific coefficients (thermal conductivity, specific

heat) both as value and temperature dependence.

Modeling of heat propagation must be done to

highlight the non-stationary regime, respectively by

considering the equations of heat transfer by

conduction, convection, and radiation, which are

systems of differential equations with partial

derivatives with variable coefficients and algebraic

equations.

Assuming known thermal intensities from the

sources, respectively the volumetric density of

power in each source, Q1, Q2, Q3, having the unit of

measurement [W/m3], can be

odelled the heating

phenomenon in splinter, tool, and work piece. First

of all, interested in the temperature in the area of the

tool edge, the edge being powered thermally mainly

by the energy resulting from the mechanical work of

friction between the splinter and the front face of the

tool. Edge heating is mostly done by thermal

conduction. The thermal state at a given moment is

deduced by solving the heat transfer problem in the

tool edge.

The heat sources Q1 and Q2 depend on some

constants of material such as the coefficient of

friction, and the unit flow stress of the catted metal

layer, which in turn are dependent on temperature

and strain rate. These will create, in the physical-

mechanical model developed that feedback effect,

difficult to highlight in the models so far.

Assuming that the thermal intensities from the

sources are known, respectively the volume density

of power in each source, Q1, Q2, and Q3, the

heating phenomenon in the splinter, tool, and piece

can be modeled. First of all, the temperature in the

area of the tool edge is of interest, the edge is

thermally fed mainly by the energy resulting from

the mechanical work of friction between the splinter

and the rake face. The heating of the cutting edge is

mostly done by thermal conduction. The thermal

state at a given time is deduced by solving the heat

transfer problem in the tool edge.

The solving of heat propagation in a transient

regime and a very heterogeneous environment leads

to knowing at any time the temperature at any point

in the investigated environment.

Knowing that:

 CQ

(1)

where:

ΔQ – is the variation in the amount of heat;

C – is the caloric capacity,

cmC 

m – being the mass and c – the specific heat;

Δθ – temperature variation.

By differentiating relation (1) concerning time, the

variation of the amount of heat as a function of time

is obtained:







tm c t

  

(2)

Fig. 4: The feedback influence of temperature

increasing on the mechanical characteristics of the

processed material

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Daschievici Luiza, Ghelase Daniela

E-ISSN: 2224-3410

Volume 21, 2024

If the material is anisotropic and inhomogeneous:

 



  













tcxxx y yy z zz

   









 









 











(3)

where:

 - material density (kg/m3);

c – specific heat of the material (J/kgK);

x, y, z- thermal conductivity of the material

(W/mK).

The integration of the differential equation (3) is

analytically very difficult and the specialized

literature does not provide exact solutions for any

practical case. However, to obtain an analytical

result where  = ct; c = ct; x = ct; y = ct; z = ct; a

solution of the form is proposed:

     



t x y z T t F x y z, , , , , 

which replaced in the heat equation (3) leads to:

     











































z,y,xF

tTz,y,xF

(4)

The final solution is:

 





















































































sinB

cosA

sinB

cosA

sinB

cosAiez,y,x,t

(5)

The integration constants Axi, Bxi, Ayi, Byi, Azi, Bzi

are determined from limit conditions on the edge

surface.

Solution (5) is valid for equation (3), without a

heat source in the studied volume. If the thermal

sources mentioned above are also considered,

equation (3) becomes:

 





 

 



       













ctxxyyzzR t x y z

2, , ,

(6)

In the mathematical model, the heat exchanges

in the volume and on the surfaces of the elements of

the tribosystem are realized by knowing some limit

conditions. These conditions are extremely difficult

to describe analytically and therefore numerical

integration is preferred, the most suitable being the

finite difference method. Thus, the differential

equation is transformed into an algebraic equation

by approximating derivatives with finite differences,

time is divided into equal time increments



and

space into equal space increments



z, resulting

in a network of nodes in a space with 4 dimensions,

in which the temperature is defined.

The friction coefficient used in the calculation

program was determined experimentally. The

conclusion was that in this case, the friction is non-

coulombian. Its dependence on speed is shown in

Figure 5. The coefficient of friction used in the

calculation program was determined using a stand

made physically.

The experimental results obtained for the wear

of cutting tools were synthesized in wear diagrams,

as shown in Figure 6 for a series of cases of

processing. These wear curves, continuous over

time, allowed the study of its evolution in correlation

with the proposed mathematical modeling.

Fig. 6: VB wear

By solving the mathematical model with the

help of the developed specialized program, the

image of the thermal fields is obtained. Assessments

can be made regarding the wear and therefore the

durability of the cutting tools. With the data

Fig. 5: The variation curve of the coefficient of

friction with the relative speed

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obtained, a curve can be drawn as a function of

temperature and speed, Figure 7.

Fig. 7: The variation of temperature on cutting

speed

The similarity between the shape of the curve

0C = f(v), Figure 7, and the shape of the curves,

Figure 8 can be observed. So, on the path of

physical-mathematical modeling, data were obtained

that are comparable to the experimental data.

Fig. 8: The variation of medium intensity on cutting speed

A direct relationship: Imed=f(0C) can be

deduced from the relationships Imed=f(v) and the

relationship 0C = f(v).

Fig. 9: The variation of medium intensity on

maximum temprature

From the graphic representation of this

relationship, Figure 9, it is noticeable the good

proportionality between the average rate of wear,

Imed, and the maximum temperature in the cutting

process, a fact found experimentally and presented

in the specialized literature.

3 Conclusion

The mathematical model proposed for determining

the thermal state of the cutting tool edge depending

on the main elements of the cutting regime can

highlight the role of different parameters of the

cutting regime on the thermal state of the cutting

tool edge.

The complete and correct research of the

thermal phenomena in the cutting area is possible

only with the consideration of the feedback

relationship between the elements that physically

and phenomenological compose the tribosystem

studied and with the consideration of the movement

of the splinter over time, having as an effect, on the

one hand, a continuous feed with cold material

layers of the splinter formation zone, on the other

hand, a heat evacuation by physical transport of the

heated splinter.

Experimental research on tool edge temperature

and medium wear rate has established similarities

between the evolution of the two phenomena.

So, the evolution of the medium wear rate can

be determined by modeling the evolution of the

thermal phenomenon by applying a proportionality

constant. This constant that can be determined

experimentally for the different couple splinter-tool

edge.

References:

[1] S. Wojciechowski, T. Chwalczuk, P.

Twardowski, and G. M. Krolczyk, Modeling

of cutter displacements during ball end milling

of inclined surfaces, Archives of Civil and

Mechanical Engineering, vol. 15, no. 4, 2015,

pp. 798–805.

[2] S. Wojciechowski, R. W. Maruda, S. Barrans,

P. Nieslony, and G. M. Krolczyk,

Optimisation of machining parameters during

ball end milling of hardened steel with various

surface inclinations, Measurement, vol. 111,

2017, pp. 18–28.

[3] Khadem M, Penkov O V, Yang H K, Kim D

E. Tribology of multilayer coatings for wear

reduction: A review. Friction, 5(3), 2017,

248–262.

[4] Daschievici L, Ghelase D., Tribological

Aspects of the Dynamic and Thermal

Phenomena Modelling in the Cutting Process,

International Journal of Mechanical

Engineering, ISSN: 2367-8968, Vol. 6, pg.

45-52, 2020.

[5] V. P. Astakhov, Tribology of cutting tools,

Tribology in Manufacturing Technology,

Springer, 1-66, 2012.

WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

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Volume 21, 2024

[6] J. Rech, A. Giovenco, C. Courbon, F.

Cabanettes, U. Masciantonio, H. Karaouni, E.

D’Eramo, A. Van Robaeys, E. Jourden,

Toward a new tribological approach to predict

cutting tool wear, CIRP Annals

Manufacturing Technology, 2018.

[7] Zvonimir Dadi, Tribological principles and

measures to reduce cutting tools wear,

Mechanical Tehnology and Structural

Materials, ISSN: 1847-7917, 2013.

[8] A. Elgazzar , Sheng-Jian Zhou, Jia-Hu

Ouyang, Zhan-Guo Liu, Yu-Jin Wang and Ya-

Ming Wang, A Critical Review of High-

Temperature Tribology and Cutting

Performance of Cermet and Ceramic Tool

Materials, Lubricants 11(3):122, MDPI, 2023.

[9] Mohan Hao, Daochun Xu, Fuqiang Wei,

Qingqing Li, Quantitative analysis of

frictional behavior of cupronickel B10 at the

tool-chip interface during dry cutting,

Tribology International, Vol. 118, February

2018, Pages 163-169.

[10] J. Rech, P.J. Arrazola, C. Claudin, C. Courbo,

F. Pusavec, J. Kopac, Characterisation of

friction and heat partition coefficients at the

tool-work material interface in cutting, CIRP

Annals, Vol. 62, Issue 1, 2013, pp.79-82.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

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WSEAS TRANSACTIONS on ADVANCES in ENGINEERING EDUCATION

DOI: 10.37394/232010.2024.21.6

Daschievici Luiza, Ghelase Daniela

E-ISSN: 2224-3410

Volume 21, 2024