FE THERMAL ANALYSIS OF 3D BALL END MILLING MODEL BASED ON ANALYTICAL STUDY OF OBLIQUE ...€¦ · · 2017-12-11BASED ON ANALYTICAL STUDY OF OBLIQUE CUTTING Abdallah Nasri, ... undeformed

Emirates Journal for Engineering Research, 16 (2), 67-78 (2011) (Regular Paper)

67

FE THERMAL ANALYSIS OF 3D BALL END MILLING MODEL

BASED ON ANALYTICAL STUDY OF OBLIQUE CUTTING

Abdallah Nasri, Mihed Ben Said, Wassila Bouzid Sai

Laboratoire de Génie de Production Mécanique et Matériaux LGPM2, ENIS, BPW 3038 Sfax, Tunisie

(Received February 2011 and Accepted November 2011)

آانت دراسة السلوك الحراري لعملية التفريز محل تحقيقات عديدة و ذلك باالستعانة بمناهج تحليلية، أو عددية أو

ثالثيةدراسة ، باستخدام نماذج تعتمد على طرق الحل العددية ذات العناصر المحددة ،قدمنالبحث في هذا .تجريبيةتمد في هذا الموضوع على نظرية قطع المعادن لتحديد الشكل نع .عملية التفريز بأداة مكورة الشكل تصور األبعاد

تم في هذا البحث .الثالثي األبعاد لألداة والمنتج والرايش و آذلك لحساب الحمل الحراري الموظف في هذا النموذجقد التطرق إلى تأثير الحرآات األساسية عند القطع مثل التغذية و سرعة القطع على نتيجة التوزيع الحراري و

تأآدت زيادة الحرارة عند زيادة السرعة والتغذية بينما تم رصد تقلص الحرارة عند إستبدال الماء بالهواء آأداة .للتبريد أو عند اإلآثار من الحدود القاطعة ألداة التفريز

.اث منشورة أستعملت فيها المقاييسهذه النتائج الحاصلة توافق نتائج معملية ألبح Thermal behaviour study in end milling has been the subject of many investigations using, either, analytical, numerical or experimental approaches. In this paper, we present a numerical analysis to predict temperature distribution in a 3D model describing a ball end milling operation. The tool, chip and workpiece geometries, and also thermal load values are issued from an analytic study based on oblique cutting theory. The effects of different cutting conditions as cutting speed, feed rate and tool rotation angle are studied. The FE model results show that temperature increases when increasing these cutting parameters. Nevertheless, the whole model temperature decreases when replacing air by water as lubricant fluid and also when decreasing tool teeth number. Finally, these results are compared to some works found in the literature and show an agreement regarding thermal evolution with cutting conditions and the coherence with measured temperatures in the same simulated conditions

1. INTRODUCTION In order to optimize cutting conditions in a metal cutting process, the common conducted investigations are concerning the evolution of the cutting forces, the strain and stress fields and the temperature distribution. These physical phenomena governing the shearing mechanisms, in a localized zone, are highly thermo-mechanical coupled. To resolve such complex problems, researchers can choose between using analytical modelling with simplified hypothesis, numerical analysis or experimental approaches. In this context, many studies are conducted to predict temperature distribution in metal cutting since it has a great effect on workpiece quality, tool life and process efficiency. It is well known that the main sources of heat generated, as a result of plastic deformation and friction occurring along the shear plane and tool-chip interface, are respectively the primary and the secondary shear zones. Also, it is reported that most of the heat flows with the chip (almost 90%) while a small portion is transferred to the tool and the workpiece material[1]. In last decades, numerical methods have become the main tool for predicting metal cutting thermal behaviour due to the advent of digital computers.

Moreover, these methods have significantly reduced the simplifying assumptions of the analytical models and the cost of inherent experimental works[2]. In general, authors use steady state or time dependent formulations to resolve the cutting heat problem where 2D or 3D models are considered. For both analyses, the needed inputs are: the model geometry, the materials’ properties, the boundary and initial conditions. Chip separation criteria, friction law and contact conditions at the secondary zone are employed only when time dependent considerations. The heat equation is numerically solved by using different techniques as finite differences, finite elements or boundary elements’ methods. In end milling, to simulate temperature distribution, researchers have, in addition, to take into account the intermittent character of the process where the undeformed chip thickness varies during cutting with the tool rotation angle. Jehnming Lin[3] has estimated the transient maximum temperature at the tool-work interface in end milling by using an inverse approach. The heat conduction equation in the system was first transformed to a 1D ellipsoidal coordinate, and then the milling temperature and heat dissipation to the workpiece were inversely solved by a finite element


68 Emirates Journal for Engineering Research, Vol. 16, No.2, 2011

method. In this study, the surface temperatures were measured by an IR pyrometer around the milled surface. However, the effects of the intermittent cut and geometry of the depth of cut on the milled surface were neglected. AbdelHamid et al.[4] developed a 3D thermo-mechanical tool-chip model to simulate the intermittent end milling cutting process and to study the role of some cutting parameters as the width of cut, the cutting speed and the tool engagement time. The authors used the finite elements technique to resolve the heat equation and imposed a uniform temperature along the shear plane and instantaneous friction heat flux on the time dependent tool-chip contact length as boundary conditions. As presented, this study was focused on the tool and didn’t consider the workpiece. Using the finite elements analysis, Ozel and Altan[5] presented an axisymmetric deformation model to predict cutting forces and tool stresses and temperatures in flat end milling process. The flow stress data of the workpiece material and friction at the chip–tool contact at high deformation rates and temperatures were implemented in the FEM model. However, the undeformed chip geometry was assumed to be circular and had an equivalent constant thickness determined from the area of undeformed chip cross section. From analytical modelling based on the predictive machining theory of orthogonal cutting, developed by Oxley[6], Shatla and Altan[7] predicted average temperature at the tool-chip interface during the ball end milling of Ti6Al4V as a function of axial distance z from the tool tip, using a 2D finite element numerical program called “OXCUT”. Lazaglu and Altintas[8] presented a 2D numerical model based on the finite difference method to predict tool and chip temperature fields in time varying milling process. The time varying chip is digitized into small elements with differential cutter rotation angles. The shear energy created in the primary zone, the friction energy produced at the rake face-chip contact zone were considered when evaluating the boundary conditions. Ming et al.[9] developed a finite element method based on Beck’s inverse heat-conduction theory through which the interface temperature distribution was back-calculated from remote measurements on the surface of the workpiece. The 3D model geometry included only the workpiece on which a boundary condition of rectangular shape of unknown heat flux in a triangular distribution moving with a constant feed rate was applied. A three-dimensional oblique model to simulate the unsteady-state process of chip formation was used by Pantalé et al.[10] in a milling application. This model, taking into account dynamic effects, thermo-mechanical coupling, constitutive damage law and contact with friction, was performed with finite

elements package Abaqus using an arbitrary Lagrangian Eulerian formulation. Recently, Brandao et al.[11] used a FEM model to estimate the energy transferred to the workpiece and the average convection coefficient in a 3D ball end milling process. In this model, considering only the workpiece geometry, a steepest decent method was employed to find the best values for both variables, based on fitting theoretical equations on the experimental temperature–time curves. As can be seen, in all the mentioned numerical studies, many assumptions are still used regarding either the intermittent character of end milling process or the real model geometry design including the tool, the chip and the workpiece, and which presents the main motivation of this investigation. The purpose of this work is to simulate the temperature distribution in 3D steady-state ball end milling model and to study the effects of some cutting parameters as cutting speed, tool rotation angle and feed rate on such process. The geometry of the 3D model, including the chip, the workpiece, the shear zones and the tool, is designed on CAD package based on experimental and analytical modelling of the ball end mill. The volumetric heat loads needed in this numerical analysis are issued from analytical formulation of cutting forces based on thermo-mechanical modelling of oblique cutting. The rest of the paper is arranged as follows. A brief review of the analytical modelling of ball end milling is presented in Section 2. Section 3 presents the techniques used to design the model geometry and thermal loads calculation. Section 4 describes some important details of the finite element simulations. Finally, Section 5 gives a summary of the main results when varying cutting conditions. 2. ANALYTICAL MODELLING OF

BALL END MILLING PROCESS The detailed geometry of ball end mill described in Fig.1 is defined in a coordinate system (O, X

uur,Yur

, Zur

) where O is the tool tip. The cutting edge lies on the tool envelope which is composed by a cylindrical surface and a spherical surface having both the same radius R0. Moreover, this cutting edge is considered helix with constant lead. In the cylindrical part, the helix angle is constant and noted i0.

For a cutting point P at height z, a local spherical coordinate system (P, re

ur, eκuur

, eψuur

) is defined. The

orthogonal projection of P on the plane (O, Xuur

,Yur

) is noted P’. The plane joining the points O, P and P’ corresponds to the reference plane Pr. The orthogonal projection of another cutting point P2 on Pr corresponds to the drawn point P1 on Fig.1(b).

FE Thermal analysis of 3D ball end milling model based on analytical study of oblique cutting

Emirates Journal for Engineering Research, Vol. 16, No.2, 2011 69

The point P, defined on the plane Pr in ir

direction by its radial distance R(z), is located on the cutting edge j by its spherical coordinates R0, κ and ψ with respect to the directions re

ur, eκuur

and eψuur

. The angular positions κ and ψ are defined respectively on the planes Pr and (O, X

uur,Yur

) (Fig.1(b)) as following:

( ) 0

0

cos R zzR

κ −= (1)

( )( )( , ) 1 2 / ( )fz j N zψ θ θ π ϕ= + − − (2)

where θ is the rotation angle of the mill, Nf is the number of teeth and ϕ (z) is the lag angle measured between the tool tip and the current point P and expressed as:

( ) ( )( )

0

0

tantan z i zz izR R z

ϕ = = (3)

Thus, the local helix angle i(z) can be written as:

( ) ( )0

0

tan tanR z

i z iR

= (4)

On the other hand, from the expression of the end mill spherical part and the equation of the cutter radius in the coordinate system (O, X

uur,Yur

, Zur

), it can be deduced that:

( )2

00

1 1zR z RR

⎛ ⎞= − −⎜ ⎟

⎝ ⎠ (5)

Figure 1. Ball end mill geometry; (a) global geometry; (b) local geometry; (c) oblique elem entary cutting edge; (d) normal plane configuration.



To predict cutting forces, for each tool rotation angle, the ball end mill is divided into infinitesimal elements along the Z-axis in a manner similar to the work of Lee and Altintas[12]. The discrete element cutting edge angle λs defined on the tangent plane Ps and formed between the cutting edge direction and the normal direction to the local cutting velocity vector Vc (plane Pr) (Fig.1(c)) is given by:

( ) ( ) ( )tan tan sins z i z zλ κ= (6)

Before cutting occurs, the material geometry in a localized cutting region has a width db(z) and a thickness tn(θ,z) as can be shown in figure 1(c).

( )( )

sindzdb z

zκ= (7)

The uncut chip thickness tn(θ,z) is function of both radial ψ(θ, z) and axial angle κ(z) and can be written as [13]: ( ) ( ) ( ), sin , sinn zt z f z zθ ψ θ κ= (8)

where fz is the feed per tooth.

The mechanical action of the cutter on the undeformed material makes the chip flowing on the rake face of the elementary cutting edge with a flow angle ηc formed between the relative direction supporter by the normal plane to the cutting edge Pn and the flow direction defined by the vector

rfeuur

(Fig.1(c)). It is commonly assumed that the flow

angle ηc is equal to the inclination of the cutting edge λs [14].

As shown in Fig.1(d), the geometry of deformed chip at the normal plane Pn is generally characterized by the primary shear angle ( )n zφ and the tool-chip interface contact length lc (z).

To calculate the normal shear angle, the proposed model of Zvorykin[15] is adopted. Hence, it can be written that:

( ) ( )( )1 2n n nA A z zφ γ β= + − (9)

A1 and A2 are constants depending on the workpiece material. β is the mean friction angle at the tool-chip interface.

The tool-chip interface contact length lc is deduced from the equilibrium, with respect to the cutting edge, of the moments of the forces applied to the chip.

( )sinsin cos cos

n n nnc

n c n

tlφ β γ

φ η β+ −

= (10)

where, ( )n zβ is the mean friction angle defined on the normal plane Pn and related to ( )zβ as following:

tan tan cosn cβ β η= (11)

The primary shear plane length is noted lp (Fig.1(d)) and is related to the tool-chip interface contact length as following:

sinn

pn

tlφ

= (12)

All the equations listed below are used for two purposes:

- calculate analytically, at each elevation z, the elementary cutting forces needed to determine heat loads for the numerical model;

- evaluate dimensional parameters characterizing chip and shear zones geometries, at each elevation z, from whom whole geometry model will be derived.

3. EXPLOITATION OF THE

ANALYTICAL STUDY: 3.1. Heat generation in primary and secondary

shear zones

The mean sources of heat generation in metal cutting are the primary (PSZ) and secondary (SSZ) deformation zones where plastic and friction works are done respectively. The primary shear zone is modelled as a thin band of constant thickness dl and length lp where dl=lp/2,6[16]. The SSZ has a rectangular shape characterized by lc and δhc where δ varies from 0.02 to 0.05 and hc is the chip thickness related to tn by the compression ratio rt (see Fig. 2).

Figure 2. Shear zones representation at an elementary cutting edge

Based on the thermo-mechanical oblique cutting model of Moufki et al.[17] assuming that the chip flow velocity on the rake face is constant, the



elementary shear force in the shear plane (dFs), the elementary frictional force between the tool rake face and the chip contact zone (dFc), the cutting velocity component along the shear plane (Vs), and the cutting velocity component along the rake face (Ve) can be written as following:

cos sinn

s hs n

tdbdF τλ φ

= − (13)

( ). cos sin

cos sin cos cosn s n

c hs n n n n c

dbtdF η βτλ φ φ β γ η

= −+ −

(14)

( )cos sin 1tancos tan

c s ns n n

s n

VV λ φ φ γη φ

⎡ ⎤= − +⎢ ⎥

⎣ ⎦ (15)

( )cos sin

cos cosc s n

ec n n

VV λ φη φ γ

=−

(16)

ηs is the angle of shearing given by the next relationship:

( )tan sin tan costan

cosc n s n n

sn

η φ λ φ γη

γ− −

= (17)

When assuming that the energy required for machining is totally converted to heat energy one can estimate, from Eq. (13-16), the elementary heat generation values dq1 and dq2 respectively in the primary and secondary shear zones as:

1 s sdq dF V= (18)

2 c edq dF V= (19)

In the numerical model, the volumetric heat loads are needed. Their values correspond to the elementary heat generation divided by the volume of each shear zone, and can be expressed as follows:

11

p

dqQl dldb

= (20)

22 . .c c

dqQh l dbδ

= (21)

3.2. Design of the model geometry

The model geometry composed by the tool, the workpiece, the chip and the shear zones, at a given rotation angle θ, is created after assessment of the cutting edge geometry. As mentioned in the works of Lazoglu [18], the cutting edge coordinates can be measured using a coordinate measurement machine (CMM). The next step consists on creating, at each point Pi with elevation z chosen on the cutter edge, various normal planes Pni. On each normal plane, one can draw the 2D chip and shear zones shapes taking into account values of the shear plane angle, the undeformed chip thickness, the tool-chip interface contact length and the geometrical characteristics of shear zones, according to analytical formulations presented in Section 2. The designed 2D shapes are, finally, joined together to obtain the 3D geometry of the chip and shear zones. Fig.3 describes this design technique. The geometry of the whole model including the tool, the chip, the shear zones and the workpiece is subsequently created.

Figure 3. Chip-shear zones design technique.



4. FE THERMAL MODEL 4.1. Heat transfer

Heat conduction is assumed to be the primary mode of heat transfer, which occurs in the chip, workpiece and tool. The governing equation of heat transfer is as following:

S

U d q n dS + Q dρ•

Ω Ω

Ω = Ω∫ ∫ ∫ (22)

where Ω is the volume of solid material with surface area S.

ρ, U•

, q and Q are respectively, the mass density, the material time rate of the internal thermal energy, and the applied surface flux density and the heat supplied externally into the body. In this work, the thermal model is steady state considered, the workpiece material is assumed to be rigid and the thermo-mechanical effects are neglected.

The boundary conditions are the following:

- in the regions exposed to the environment, convection boundaries are assumed. The heat convection coefficients h are function of the lubricant fluid parameters. Eq.(23) is used to calculate the heat convection coefficients.

1/ 3 1/ 20, 66 Pr RekhS

= (23)

Re and Pr are respectively Reynolds and Prandtl numbers concerning tool, workpiece and machining process. S is the surface area exposed to the environment, and k the thermal conductivity of the tool-workpiece. In the following simulations, convection coefficients are determined using air at room temperature.

- at the workpiece surface in contact with the machine-tool table the temperature is set to 30°C.

The used workpiece material is the 42CD4 equivalent to AISI 4142. The tool is a carbide ball end mill with radius 6 mm and a helix angle of 30 degrees. Thermal properties of these materials and used cutting conditions are presented respectively in Table 1 and Table 2.

Table 1. Thermal properties of the used materials

Material Specific heat capacity[J/Kg.°K]

Density[Kg/m3]

Thermal conductivity[W/m.°K]

AISI 4142 Carbide tool

341 +0.5 T 200

7840 – 2.7 T13000

55 – 0.0322 T 60

Table 2. Cutting Conditions

Cutting speed [m/min]

Axial depth of cut [mm]

Radial depth of cut [mm]

Feed rate [mm/rev]

200 0.5 0.5 0.2

For these cutting conditions, the considered model parameters are the following: - β=42°;A1=40°;A2=0.5 ; [19] - δ=0.05; rt=0.344; lc=0-0.38 mm; lp=0-0.30 mm; (calculated values).

4.2. Simulation results Figure 4 presents the temperature distribution in the finite element-based model.

Figure 4. Temperature distribution in the 3D ball end model



As shown in Fig. 4, the following observations can be done:

- the high temperature region occurs in the chip; - in the tool, the region affected with maximum

temperature corresponds to the tool - chip contact region;

- the temperature level and the temperature gradient in the workpiece is much lower because of the

largest surface dimension of the workpiece exposed to the ambient air.

These results are in agreement with the cutting heat balance where the majority of the heat (∼90%) is transferred to the chip. For more accuracy of the induced results, a confrontation with experimental[20] and analytical works[17] found in the literature is done (see Fig. 5).

Figure 5. Comparison between predicted and published maximum chip-tool interface temperatures in ball end milling.

As can be deduced, the numerical results are in agreement with published ones. The major difference corresponding to Vc=200 m/min is mainly due to the various machined materials used in each approach.

5. CUTTING CONDITIONS VARIATION

In the present section, the effect of cutting conditions, as cutting speed, feed rate and tool rotation angle, on the predicted temperatures will be investigated. The main parameters that will be studied are the maximum temperatures induced in the chip (MTC), in the workpiece (MTW) and in the tool (MTT).

5.1. Effect of cutting speed

The effect of the cutting speed, on the over mentioned maximum temperatures during ball end milling, is shown in Fig. 6. Four different cutting speeds are considered: 100, 150, 200 and 300 m/min at the same depth of cut and feed per tooth. The studied temperatures are found to increase linearly with the increase of the cutting speed. In fact, the MTC, for example, increases from 128°C to 1503°C when the cutting speed varies from 100 m/min to 300 m/min.

Figure 6 Effect of cutting speed on predicted maximum temperatures in the chip, in the tool and in the workpiece.



Quantitatively, it can be seen from Fig. 7 that the maximum temperature in the tool occupies a zone on the rake face corresponding to the largest tool-chip contact area.

It can be observed, also, that this zone shifts away from the cutting edge as the cutting speed decreases. Its width varies from 0.2 mm to 0.05 mm when the cutting speed increases from 100 m/min to 300 m/min.

Figure 7. Localized tool temperature distribution for various cutting speeds.

5.2 Effect of feed rate

When increasing the feed rate, the characteristic parameters of the chip geometry (undeformed chip thickness, tool-chip interface contact length, and shear plane length) increase (Eqs. (8), (10) and (12)). In the same time, the heat generation values in the shear zones decrease.

As shown in Fig. 8, the maximum induced temperatures evolution proves that the effect of chip geometry dominates the heat generation one. In fact, the MTC, the MTT and the MTW increase when the feed rate increases. A maximum temperature in the chip rise of about 545°C is observed when the feed rate varies from 0.08 mm/rev to 0.5 mm/rev.

Figure 8. Effect of feed rate on predicted maximum temperatures in the chip, in the tool and in the workpiece.

Quantitatively, the most heated zone area on the tool rake face becomes larger when the feed rate increases (see Fig. 9). This is due to the rise of the tool-chip interface contact length.

Moreover, contrarily to the effect of cutting speed, this zone shifts away from the cutting edge when increasing the feed rate.



Figure 9. Temperature distribution on the tool rake face for different feed rates.

5.3 Effect of rotation angle:

Because of the steady-state character of the finite element based model, an attempt to simulate the rotational motion of the tool is made. For that purpose, the ball end milling cutter is drawing in different angular positions θ. The results shown in Fig. 10 describe the predicted maximum temperatures in each considered angle rotation.

As can be concluded, the MTC, the MTT and the MTW reach their maximum values for θ=40°. Then, they decrease despite of the angle rotation rise. This thermal behaviour is mainly due to the rise of the tool-chip interface contact area until the rotation angle value of 40 degrees. When increasing more this parameter, the contact area decreases and the chip becomes more exposed to convection with environment. By instance, the studied maximum temperatures decrease.

Figure 10. Effect of rotation angle on predicted maximum temperatures in the chip, in the tool and in the workpiece

5.4 Effect of used lubricant fluid

The change of used lubricant fluid is traduced, in the numerical model, by the variation of convection coefficient values that can be deduced from Eq. (23). In the present work, two lubricant fluids are considered. Their thermal characteristics are presented in Table 3[21].



Table 3. Thermal characteristics of used lubricant fluids

Lubricant fluid Specific heat capacity

[J/Kg.°K]

Density[Kg/m3]

Thermal conductivity [W/m.°K]

Mean fluid speed [m/s]

Dynamic viscosity [Kg/m.s]

Air 1006 1.177 0.0262 1.000 1.850 E-5 Water 4182 1001 0.597 1.000 1.000 E-3

The results of relative simulations done with the same cutting conditions are shown in Fig. 11.

Figure 11. Effect of used lubricant fluids on predicted maximum temperatures in the chip, in the tool and in the workpiece.

5.5 Effect of tool teeth number

As can be seen, the maximum induced temperatures decrease rapidly when replacing air by water as lubricant fluid. A difference of about 200°C can be observed in all model’s components. This result is due to high convection coefficient values used when wet milling Figure 12 presents the considered geometrical models of ball end mill with different teeth number.

Figure 12. Considered geometrical models of ball end milling cutter.

When the milling cutter teeth number increases there is a rise of the material removal rate which can be modelled by the number of the chips taking into account in the whole model. In fact, in the present study, only one chip is considered when using a two or four teeth ball end milling cutter. For a six teeth tool, the chips number is doubled (see figure 13).

Figure 13. Considered geometrical models of ball end milling cutter

The results of the numerical model are summarized in figure 14.

Figure 14. Maximum temperatures in tool and workpiece

when varying tool teeth number. From figure 14, a small decrease of temperature can be observed when varying tool teeth number from two to four. However, when considering a six teeth tool, temperature increases rapidly. This is due to the



simultaneous engagement of two teeth with machined material. 6 CONCLUSION A three dimensional steady-state finite element model has been developed to analyse the temperature distribution induced in the tool, the chip and the workpiece during ball end milling process. The proposed model is based on analytical study of oblique cutting from which are deduced the volumetric load values and the geometrical model design. The found results are in agreement with heat energy balance where the maximum of temperature occurs in the chip. The variation of cutting parameters effect has leaded to the following conclusions: - the temperature increases when increasing cutting

speed and feed rate; - the most heated zone on the tool rake face shifts

away from the cutting edge when cutting speed and feed rate increases and decreases respectively;

- the temperatures in the chip, the tool and the workpiece reach their maximum values then decrease when varying the ball end mill rotation angle;

- machining with water as lubricant fluid produced the lowest temperatures since the rise of convection coefficients;

- increasing the used tool teeth number induces highest temperatures because of the simultaneous engagement of many teeth with machined material

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FE THERMAL ANALYSIS OF 3D BALL END MILLING MODEL BASED ON ANALYTICAL STUDY OF OBLIQUE ...€¦ · · 2017-12-11BASED ON ANALYTICAL STUDY OF OBLIQUE CUTTING Abdallah Nasri, ... undeformed

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