Research Article A Mathematical Modeling to Predict the ......In the study of macrodrilling models,...

Research ArticleA Mathematical Modeling to Predictthe Cutting Forces in Microdrilling

Haoqiang Zhang,1,2 Xibin Wang,1 and Siqin Pang1

1 Key Laboratory of Fundamental Science for Advanced Machining, Beijing Institute of Technology, Beijing 100081, China2Hebei United University, Tangshan 063009, China

Correspondence should be addressed to Haoqiang Zhang; [email protected]

Received 4 June 2014; Revised 18 July 2014; Accepted 19 July 2014; Published 6 August 2014

Academic Editor: Zhen-Lai Han

Copyright © 2014 Haoqiang Zhang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In microdrilling, because of lower feed, the microdrill cutting edge radius is comparable to the chip thickness. The cuttingedges therefore should be regarded as rounded edges, which results in a more complex cutting mechanism. Because of this, themacrodrilling thrust modeling is not suitable for microdrilling. In this paper, a mathematical modeling to predict microdrillingthrust is developed, and the geometric characteristics of microdrill were considered in force models.The thrust is modeled in threeparts: major cutting edges, secondary cutting edge, and indentation zone. Based on slip-line field theory, the major cutting edgesand secondary cutting edge are divided into elements, and the elemental forces are determined by an oblique cutting model and anorthogonalmodel, respectively.The thrustmodeling of themajor cutting edges and second cutting edge includes two different kindsof processes: shearing and ploughing. The indentation zone is modeled as a rigid wedge. The force model is verified by comparingthe predicted forces and the measured cutting forces.

1. Introduction

There has been an increasing requirement for high-accuracymicroholes in the microelectronic, automotive, computercomponents, and sensor industries. Microdrilling is experi-encing a very rapid growth in precision production indus-tries. Inmany aspects, microdrilling has fundamentally iden-tical features with conventional drilling, but the downsizingof the dimensions of the drill introduces many problems,which has a major influence on the microdrilling process,such as cutting edge radius, increased web thickness, largevibrations due to high rotation speed, and high ratio of drillbreakage. There are many factors influencing the microdrill-ing process, such as drill geometry, drill materials, drillingforces, workpiece materials, machining parameters, and vib-ration. Drilling forces are related to drill life, holes quality,and productivity.Therefore, drilling forces are one of themostimportant factors affecting the drill performance.

In general, there are four methods of modeling cuttingforces in metal machining: analytical method, experimentalmethod, mechanistic method, and numerical method [1].Many models have been developed by researchers in the

past several decades. In the study of macrodrilling models,Shaw and Oxford [2] were the pioneers. Armarego andCheng [3, 4] presented a model in which a series of obliquecutting slices was used to the drilling process with flat rakeface and conventional twist drills. Watson [5–8] produceda more detailed model of material removal in both cuttingedges and chisel edge. Stephenson and Agapiou’s model [9]simulated arbitrary drill point geometries. Chandrasekharanet al. [10, 11] developed a mechanistic model of the cuttinglips and chisel edge to predict the cutting force systemfor arbitrary drill point geometry. Strenkowski et al. [12]developed a thrust force model based on analytical finiteelement technique in drilling with twist drills. In their model,the cutting lips were regarded as a series of oblique sections,and the cutting of the chisel region was treated as orthogonalcutting. Wang and Zhang [13] presented a predictive modelfor the thrust in drilling operations usingmodified plane rakefaced twist drills. Their models were based on the mechanicsof cutting approach incorporating many tools and cuttingprocess variables. There was less literature on force modelinginmicrodrilling. Sambhav et al. [14]modeled the thrust by theprimary cutting lip of a microdrill analytically and modeled

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 543298, 11 pageshttp://dx.doi.org/10.1155/2014/543298

2 Mathematical Problems in Engineering

Indentation zoneSecondary

cutting edges Rind

Figure 1: Regions of the chisel edge.

shearing forces and ploughing forces of the major cuttingedges. Hinds and Treanor [15] analyzed the stresses occur-ring in microdrills using finite element methods in printedcircuit board drilling process, but they did not produce anymathematical model for cutting forces of microdrills.

Slip-line field theory was often used to analyze the cuttingprocess. Manymachining parameters can be predicted by theslip-line field model, such as cutting force, chip thickness,shear strain, and shear strain-rate. Merchant [16] was thefirst one who presented a mathematical model to determineshear angle by using the minimum energy principle, andhis model was the basis of all subsequent models. Lee andShaffer [17] developed a slip-line field model which wasan approximation method under certain cutting conditions.Dewhurst and Collins [18] presented a matrix technique fornumerically solving slip-line problems. Oxley [19] proposeda parallel surface shear zone model of orthogonal cuttingthat considered the change of material flow stress. Waldorfet al. [20] developed a slip-line model for ploughing by acutting tool with a definite cutting edge radius. Fang [21]presented a generalized slip-line field model for cuttingwhen edge was rounded. Fang’s model included nine effectsthat commonly occurred in machining. Then Fang [22]quantitatively analyzed orthogonal metal cutting processesbased on his slip-line model. Manjunathaiah and Endres [23]developed a new orthogonal process model that included theeffects of edge radius. Jin and Altintas [24] simplified Fang’smodel, and they considered the effects of strain, strain-rate,and temperature on the cutting process.

In microcutting applications the uncut chip thicknessis very small, typically within the range of 25 𝜇m. Sincethe cutting edge radius is typically ground with a 5–20𝜇m,the assumption of having a perfectly sharp cutting edge inmacrodrilling is not valid, so the cutting edge radius shouldnot be taken to be zero in microcutting operations. Paststudies have found that if the uncut chip thickness is belowthe minimum chip thickness 𝑡cmin, elastic deformation ora mixed elastic-plastic deformation will take place. Abovethis value, chip formation starts taking place. This is knownas the minimum chip thickness effect. However, due to the

extrusion of the material by the chisel edge region of thedrill, the drilling process can still take place, even if the chipthickness is very small.

The cutting edge is made up of the major cutting edgesand the chisel edge of the microdrill.Themajor cutting edgesare formed by the intersection of the flute surface with theflank surface of the microdrill, while the intersection of theflank surfaces forms the chisel edge. Although the length ofchisel edge is very small relative to the cutting edge of themicrodrill, the thrust created by the chisel edge is significant,and it exceeds even the thrust created by the cutting edges.In the region around the center of the chisel edge, materialremoval is by extrusion. This region is called the indentationzone, as shown in Figure 1. The portion of the chisel edgeoutside the indentation zone is termed as the secondarycutting edges. Material removal of secondary cutting edge isby orthogonal cutting with large negative rake angles.

During microdrilling, both shearing and indentingactions are happening. When the microdrill contacts theworkpiece, the drill point rubs workpiece first. Under theextrusion force of microdrill, material is squeezed around thedrill point; at the same time, the secondary cutting edges onthe chisel edge perform cutting.Then, the major cutting edgeenters intoworkpiece and begins to cut.The central portion ofthe chisel edge performs the indenting action, and the secondcutting edges on the chisel edge and the major cutting edgeson the fluted portion perform shearing.

In this paper, the thrust is modeled in three parts of amicrodrill: major cutting edges, secondary cutting edge, andthe indentation zone. The major cutting edge and secondarycutting edge force models are based on the slip-line fieldtheory, and the indentation zone is modeled as a rigid wedge.The model is, then, verified by comparing predicted thrustforce with measured data including the effects of microdrillgeometric and machining parameters.

2. Major Cutting Edge Cutting Force Models

The cutting behavior of the major cutting edge is an obliquecutting process. The cutting edge is divided into elementsand each element is approximated as a straight line, shownin Figure 2. The magnitude of the total drilling thrust (𝐹

1) is

obtained by summing the forces at all the cutting elements oneach edge and all the cutting edges on the drill.

The direction of the elemental cutting force 𝑑𝐹cut isopposite to the velocity direction; 𝑑𝐹cut is resolved into𝑑𝐹𝐶

and 𝑑𝐹𝐿. The direction of 𝑑𝐹

𝐶is along the actual

cutting direction. 𝑑𝐹𝐿is the elemental lateral force, which

is orthogonal to the cutting force and the elemental obliquecutting thrust force 𝑑𝐹

𝑇.The thrust force 𝑑𝐹

𝑇is normal to the

plane that contains the velocity vector and the cutting edge.The magnitude of those forces is given by

𝑑𝐹𝐶= 𝑑𝐹cut ⋅ cos 𝜆𝑠

𝑑𝐹𝐿= 𝑑𝐹cut ⋅ sin 𝜆

𝑠

𝑑𝐹1=cos 𝜀 sin𝜙

cos 𝜆𝑠

⋅ 𝑑𝐹𝑇− tan 𝜆

𝑠⋅ cos𝜙 ⋅ 𝑑𝐹

𝐶,

(1)

Mathematical Problems in Engineering 3

XY

Z

V

X

Y

V

Major cutting edge

r

rw

Fz

Fx

Fy

dFL

dFT

dFcut

dFC

𝜀

𝜆s

𝜙

Figure 2: Forces at an element on the major cutting edge.

where 𝜙 is the half point angle and the inclination angle 𝜆𝑠

and angle 𝜀 can be obtained by the following equations:

𝜆𝑠= sin−1 (

𝑟𝑤

𝑟sin𝜙)

𝜀 = sin−1 (𝑟𝑤

𝑟) ,

(2)

where 𝑟𝑤is half the web thickness and 𝑟 is the distance from

a point on the cutting edge to the drill axis.The normal rake angle at any point on the cutting edge is

𝛾𝑛= tan−1 (

(𝑟/𝑅) tan𝛽 cos 𝜀sin𝜙 − (𝑟

𝑤/𝑅) tan𝛽 cos𝜙

)

− tan−1 (tan 𝜀 cos𝜙)

= tan−1((𝑟/𝑅) tan𝛽√1 − (𝑟2

𝑤/𝑟2 )

sin𝜙 − (𝑟𝑤/𝑅) tan𝛽 cos𝜙

)

− tan−1(𝑟𝑤cos𝜙

√𝑟2 − 𝑟2𝑤

),

(3)

where 𝛽 is the helix angle of the drill and 𝑅 is the drill radius.The magnitude of the total drilling thrust along the axis

of the drill can be obtained by summing the forces at allthe cutting elements on each cutting edge and all the cuttingedges on the drill, so themagnitude of the total drilling thrustforce 𝐹

1is

𝐹1= 2∫𝑑𝐹

1

= 2∫(cos 𝜀 sin𝜙

cos 𝜆𝑠

⋅ 𝑑𝐹𝑇− tan 𝜆


𝐶) .

(4)

Thus, if we know the forces for each cutting element in thecutting and thrust direction in the plane perpendicular to thecutting edge, the total drilling thrust force can be calculated.

Due to the technological and material constraints inmicrodrill preparation, the major cutting edge has a definiteradius, and the uncut chip thickness is very small, so themajor cutting edge cannot be seen as completely sharp. Theslip-line field model of microcutting process for each cuttingelement of major cutting edges is shown in Figure 3.

Thematerial deformation region consisted of three zones:primary shear zone [AIBB

1I1A1A2], secondary shear zone

[𝐻𝐸𝐵𝐺𝐼𝐽], and tertiary zone [BSCD1B1]. The shape of the

slip-line field was originally proposed by Fang [21]. In Fang’smodel, the slip lines HJ and JI are defined as two basic sliplines; after their shapes are obtained, all other slip lines inthe secondary shear zone can be determined using Dewhurstand Collins’s matrix technique [18].Then, the slip-lines in theprimary and tertiary shear zones can be easily determinedfrom relevant slip-line relationships. The primary shear zoneincluded three regions: triangular region AA

1A2, convex

region AII1A1, and concave IBB

1I1. In region AA

1A2, line

AA2is a stress-free boundary; all of the slip lines in AA

1A2

intersect with AA2at a 45∘ angle. Both region AII

1A1and

region IBB1I1consist of circular arcs and straight radial lines.

Point S is the separation point for the upward and downwardmaterial bifurcating. Part of the materials flows downwardsfrom point S to point C along the rounded edge, while otherparts of the materials flow upwards from point S to point B.In order to simplify the mathematical formulas of the slip-line problem with a curved boundary, the tool edge BC isapproximately represented by two straight chords BS and SC.

BS and SC are considered to have rough surfaces; theincluded angles between them and the slip lines are 𝜉

2and

𝜉1, respectively. The intersection angle of AA

2and horizon is

𝛿. The separation angle 𝜃𝑐, the tool edge radius 𝑟

𝑛, and the

tool rake angle 𝛾𝑛determine the position of the stagnation

point 𝑆 on the rounded tool edge. Geometric analysis givesthe following set of equations:

𝛼1=

5𝜋

4− 𝜉2−

𝛾𝑛

2−

𝜃𝑐

2− 𝜃2+ 𝜃1,


Workpiece

Tool primary rake face

Rounded cutting edge

Chip

H

B

C

S

G1

I1

A1

A2

A3

A

O

I

V

45∘ 𝜃1

D1

B1

E

G

𝛿 𝛼2

𝛼1

𝜃2

rn

B

S 𝜃c

O𝛾e

𝜉2

𝜉1

C

𝜃b

𝜃s

𝜉4

𝛾n

B1

D1

J

𝜉3

Figure 3: Slip-line field model of each element cutting process of major cutting edges.

𝛿 = 𝛼1−

3𝜋

4,

𝜃𝑠= sin−1 (√2 sin 𝛿 sin 𝜉

1) ,

𝜃𝑏=

𝜋

2 + 𝛾𝑛− 𝜃𝑐

,

𝜉 =1

2cos−1 (𝜏

𝑘) ,

𝑙𝐵𝑆

= 2𝑟𝑛sin(

𝜋

4+

𝛾𝑛

2−

𝜃𝑐

2) ,

𝑙𝑆𝐶

= 2𝑟𝑛sin

𝜃𝑐

2,

(5)

where 𝜏 is the frictional shear stress and 𝑘 is the material flowstress.The tool-chip frictional shear stress along the rake facewas assumed to be constant.

Jin and Altintas evaluated the total cutting forces byintegrating the forces along the entire chip-rake face contactzone and the ploughing force caused by the round edge. Thedetailed process can be found in [24]. According to theircomputing methods, the cutting forces on the major cuttingedges of microdrill can be derived as follows.

After thematerial passes through the shear zones, the chipbegins curling freely, so the resulting force along the slip lines𝐴𝐼, 𝐼𝐽, and 𝐽𝐻 should be zero. Consider

𝐹𝑥𝐴𝐼

+ 𝐹𝑥𝐼𝐽

+ 𝐹𝑥𝐽𝐻

= 0,

𝐹𝑦𝐴𝐼

+ 𝐹𝑦𝐼𝐽

+ 𝐹𝑦𝐽𝐻

= 0.

(6)

Line 𝐴𝐴2is a stress-free boundary. The distribution of

hydrostatic pressure and shear flow stress along slip line 𝐴𝐼

can be calculated by dividing 𝐴𝐼 into several differentialelements, such as 100 differential elements, as shown in Figure4. The total force along slip line 𝐴𝐼 is obtained by summingall of the elemental forces in the𝑋 and 𝑌 directions.

H

B

C

S

A

O

I

J

Chip

E

G

X

Y

G1

I1

A1

A2

A3

𝜃1

D1B1

𝜃2

𝜉

𝜂

N1kN1

N3

N4

N2N5

pN1

Figure 4: Stress analysis in the primary shear zone.

The forces on point 𝑁1of slip line 𝐴𝐼 in the 𝑋 and 𝑌

directions are calculated as

𝑑𝐹𝑥= (𝑝𝑁1

sin 𝜂 + 𝑘𝑁1

cos 𝜂) Δ𝑙𝐴𝐼𝑤,

𝑑𝐹𝑦= (𝑝𝑁1

cos 𝜂 + 𝑘𝑁1

sin 𝜂) Δ𝑙𝐴𝐼𝑤,

(7)

where 𝑝𝑁1

is the hydrostatic pressure and 𝑘𝑁1

is the shearflow stress on the element, 𝜂 is the angular coordinate of theelement, Δ𝑙

𝐴𝐼is the length of the element, and 𝑤 is the width

of cut. Consider

Δ𝑙𝐴𝐼

=𝑙𝐴𝐼

𝑛, (8)

where 𝑛 is the number of divided differential elements.So the hydrostatic pressure 𝑝

𝐼and shear flow stress 𝑘

𝐼

of point 𝐼 can be concluded. After the total forces along slipline 𝐴𝐼 are calculated, the forces along 𝐼𝐽 and 𝐽𝐻 can bedetermined. In the second shear zone, slip line 𝐵𝐼 is dividedinto 100 angular elements in the same way; then the samenumber of slip lines is formed in the secondary shear zone.For any slip line 𝑁

2𝑁3, the shear flow stress 𝑘

𝑁2and the

hydrostatic pressure 𝑝𝑁2

at point 𝑁2are obtained from the


stress distribution in the primary shear zone. Then, the shearflow stress and hydrostatic pressure at point𝑁

3are calculated

as

𝑘𝑁3

= 𝑘𝑁2

,

𝑝𝑁3

= 𝑝𝑁2

+ 2𝑘𝑁2

⋅ (2𝜉) .

(9)

The elemental force is projected into the 𝑥 and 𝑦 direc-tions, and then the elemental forces in the 𝑥 and 𝑦 directionsat point𝑁

3are

𝑑𝐹𝑥𝐻𝐵

= [(−𝑝𝑁3

− 𝑘𝑁3

sin (2𝜉)) Δ𝑙𝐻𝐵

𝑤] cos 𝛾𝑒

− [(𝑘𝑁3

cos (2𝜉) Δ𝑙𝐻𝐵

𝑤)] sin 𝛾𝑒,

𝑑𝐹𝑦𝐻𝐵

= [− (−𝑝𝑁3

− 𝑘𝑁3

sin (2𝜉)) Δ𝑙𝐻𝐵

𝑤] sin 𝛾𝑒

− [(𝑘𝑁3

cos (2𝜉) Δ𝑙𝐻𝐵

𝑤)] cos 𝛾𝑒.

(10)

The elemental force at other points along the tool rake face𝐵𝐻 is calculated following the same procedure as point 𝑁

3,

and then the total force along 𝐵𝐻 is obtained by summing allof the elemental forces in the𝑋 and 𝑌 directions.

In tertiary shear zone, line 𝐵𝐵1is divided into 100 small

elements. The shear flow stress and hydrostatic pressure atpoint𝑁

4can be concluded from point 𝐵.Then, the shear flow

stress and hydrostatic pressure at point𝑁5are calculated as

𝑘𝑁5

= 𝑘𝑁4

𝑝𝑁5

= 𝑝𝑁4

− 2𝑘𝑁4

⋅ (2𝜉) .

(11)

The elemental forces along line 𝐵𝑆 in the 𝑥 and 𝑦 direc-tions at point𝑁 are

𝑑𝐹𝑥𝐵𝑆

= [(−𝑝𝑁5

− 𝑘𝑁5

sin (2𝜉)) Δ𝑙𝐵𝑆𝑤] cos 𝛾

𝑒

− [(−𝑘𝑁5

cos (2𝜉) Δ𝑙𝐵𝑆𝑤)] sin 𝛾

𝑒

𝑑𝐹𝑦𝐵𝑆

= [(−𝑝𝑁5

− 𝑘𝑁5

sin (2𝜉)) Δ𝑙𝐵𝑆𝑤] sin 𝛾

𝑒

+ [(−𝑘𝑁5

cos (2𝜉) Δ𝑙𝐵𝑆𝑤)] cos 𝛾

𝑒.

(12)

Similarly, along line 𝑆𝐶,

𝑑𝐹𝑥𝑆𝐶

= [(−𝑝𝑁5

− 𝑘𝑁5

sin (2𝜉)) Δ𝑙𝑆𝐶𝑤] cos 𝛾

𝑒

+ [(−𝑘𝑁5

cos (2𝜉) Δ𝑙𝑆𝐶𝑤)] sin 𝛾

𝑒,

𝑑𝐹𝑦𝑆𝐶

= [(−𝑝𝑁5

− 𝑘𝑁5

sin (2𝜉)) Δ𝑙𝑆𝐶𝑤] sin 𝛾

𝑒

− [(−𝑘𝑁5

cos (2𝜉) Δ𝑙𝑆𝐶𝑤)] cos 𝛾

𝑒.

(13)

The element forces in the plane perpendicular to themajor cutting edge along the cutting direction and the thrustdirection are obtained on the major cutting edge as

𝑑𝐹1𝑐

= (𝑑𝐹𝑥𝐻𝐵

+ 𝑑𝐹𝑥𝐵𝑆

+ 𝑑𝐹𝑥𝑆𝐶

) Δ𝐿,

𝑑𝐹1𝑡

= (𝑑𝐹𝑦𝐻𝐵

+ 𝑑𝐹𝑦𝐵𝑆

+ 𝑑𝐹𝑦𝑆𝐶

) Δ𝐿,

(14)

whereΔ𝐿 is the length of differential element ofmajor cuttingedge.

Therefore, the magnitude of the total drilling thrust alongthe axis on the major cutting edge of the drill 𝐹

1is

𝐹1= 2∫𝑑𝐹

1

= 2∫(cos 𝜀 sin𝜙

cos 𝜆𝑠

⋅ 𝑑𝐹1𝑡− tan 𝜆


1𝑐) .

(15)

3. Chisel Edge Cutting Force Model

Mauch and Lauderbaugh [25] obtained the indentation zoneradius 𝑅ind (Figure 1) for a conical drill based on the pointangle. Paul et al. [26] suggested that the dynamic clearanceangle becomes zero at the indentation zone radius. So theradius 𝑅ind of the indentation zone is given by the followingequation:

𝑅ind =𝑓

2𝜋 tan 𝛾𝑠

, (16)

where 𝛾𝑠is the static clearance angle of the chisel edge.

3.1. Secondary Cutting Edge Cutting Force Model. Since thechisel edge has a definite radius and the uncut chip thicknessis comparable in size to the edge radius, the chisel edge cannotbe seen as completely sharp but should be as a roundededge. The chip thickness at the elements on the chisel edgeis equal to half of the drill feed. The secondary cutting edgesare divided into elements and the elemental drilling thrust isdetermined; then the magnitude of the total drilling thrustalong the axis of the drill can be obtained by summing theforces at all elements for the secondary cutting edges.

Because the flank surfaces of microdrill are plane, theslip-line model of secondary cutting edge is different fromthe major cutting edge. Figure 5 shows the analytical slip-line model for machining with secondary cutting edge. Theintersection angle of 𝐴𝐴

2and horizontal line is 𝛿. Consider

𝛿 =𝜋

4− 𝜙𝑠

𝛿2= 𝜙𝑠− 𝛾𝑒− 𝜉2,

𝛿3= 𝜉1+ 𝜉2+ 𝛾𝑒− 𝜃𝑠.

(17)

The element forces in the plane perpendicular to thechisel edge along the thrust direction on the chisel edge areobtained as

𝑑𝐹2= 𝑘Δ𝐿 {(cos𝜙

𝑠− sin𝜙

𝑠) 𝑙𝐻𝐵

+ [cos 2𝜉2cos 𝛾𝑒

− (1 + sin (2𝛿2+ 2𝜉2)) sin 𝛾

𝑒] 𝑙𝐵𝑆

+ [(1 + sin (2𝛿2+ 2𝛿3+ 2𝜉1)) cos 𝜃

𝑠

− cos 2𝜉1sin 𝜃𝑠] 𝑙𝑆𝐶} ,

(18)


Chip

Workpiece

Chisel edge

Secondary cutting edge

A

A2A3

45∘

45∘

𝛿45∘

I

B2

D2

B

S

C

H

V

𝛾n

O

rn

𝛿2 𝛿3

B

S

𝜃c

O𝛾e

𝜉2

𝜉1

C

𝜃b

𝜃sA1

I1 B1

D1

VChip

𝜙s

tc

Figure 5: Slip-line field model for machining with secondary cutting edge.

where Δ𝐿 is the length of differential element, and

𝑙𝐻𝐵

=𝑡𝑐+ 𝑙𝐴𝐴2

sin 𝛿 − 𝑟𝑛(1 + sin 𝛾

𝑒)

sin𝜙𝑠

,

𝑙𝐵𝑆

= 2𝑟𝑛sin(

𝜋

4+

𝛾𝑛

2−

𝜃𝑐

2) ,

𝑙𝑆𝐶

= 2𝑟𝑛sin

𝜃𝑐

2,

(19)

where 𝑡𝑐is the uncut chip thickness, 𝑡

𝑐= 𝑓/2, and 𝑓 is feed.

Consider

𝑙𝐴𝐴2

= √2 (𝑙𝐵𝑆cos 𝜉2+ 𝑙𝑆𝐶

sin 𝜉1) . (20)

Themagnitude of the total drilling thrust can be obtainedby summing the forces at all elements for the secondarycutting edges. So the magnitude of the total drilling thrustforce 𝐹

2is

𝐹2= 2

𝐿𝑐/2

∑

𝑅ind

𝑑𝐹2

= 2𝑘

𝐿𝑐/2

∑

𝑅ind

{(cos𝜙𝑠− sin𝜙

𝑠) 𝑙𝐻𝐵

+ [cos 2𝜉2cos 𝛾𝑒

− (1 + sin (2𝛿2+ 2𝜉2)) sin 𝛾

𝑒] 𝑙𝐵𝑆

+ [(1 + sin (2𝛿2+ 2𝛿3+ 2𝜉1)) cos 𝜃

𝑠

− cos 2𝜉1sin 𝜃𝑠] 𝑙𝑆𝐶} Δ𝐿,

(21)

where 𝐿𝐶is the length of chisel edge.

3.2. The Indentation Zone Cutting Force Model. In micro-drilling processes, the ratio of web thickness to drill diameteris larger than that of macrodrilling, so the indentation zoneis quite important, and the contribution to the total drilling

Z

Xf/2

Workpiece Plastic region

Indentation zone model

𝜑

2𝛾ind

Figure 6: Indentation zone model schematic.

thrust force by the indentation zone needs to be considered.In microdrilling, although the chisel edge is circular edge,due to the major effect of the indentation zone on extrudematerial, the indentation zone can be regarded as a rigidwedge. The material is extruded on both sides of the wedge.The indentation zone model schematic is shown in Figure 6.According to the slip-line field theory, the force normal to thesurface of the wedge can be determined. Consider

𝑃𝑛1

= 2𝑘 (1 + 𝜑) , (22)

where 𝜑 is the solution of the slip line and is given by thefollowing equation:

2𝛾ind = 𝜑 + cos−1 [tan(𝜋

4−

𝜑

2)] , (23)

where 2𝛾ind is the included angle of the wedge, which is equalto twice the magnitude of the static normal rake angle at thechisel edge and is given by

𝛾ind = −tan−1 [tan𝜙 cos (𝜋 − 𝜓)] , (24)

where 𝜓 is chisel edge angle of microdrill.


(a) (b)

(c)

Figure 7: Experimental setup and microdrill. (a) Experimental setup diagram; (b) CNS7d CNC machine; (c) diameter 0.5 mmmicrodrill.

The load acted on unit length of the wedge is

𝑑𝐹3= 2𝑙𝑂𝐴

𝑃𝑛1sin 𝛾ind, (25)

where

𝑙𝑂𝐴

=𝑓

2 [cos 𝛾ind − sin (𝛾ind − 𝜑)]. (26)

So the total drilling thrust force of the indentation zonecan be expressed as

𝐹3= 2 ⋅

𝑓

2 [cos 𝛾ind − sin (𝛾ind − 𝜑)]⋅ 2𝑘 (1 + 𝜑)

⋅ sin 𝛾ind ⋅ 2𝑅ind

=4𝑘 (1 + 𝜑) 𝑓 sin 𝛾ind𝑅ind

cos 𝛾ind − sin (𝛾ind − 𝜑).

(27)

4. Experimental Validation of the ThrustForces Model of Microdrills

4.1. Experimental Work. To calibrate the thrust forces modelof microdrills, the microdrilling processes were performed

on a DMG DMU 80 monoBLOCK machining center. Theexperimental setup was shown in Figure 7(a). Workpiece isAISI 1023 carbon steel plate with a thickness of 1.5mm.Workpiece is mounted on a multicomponent dynamome-ter (Kistler, model 9257B). The material of microdrills iscemented carbide of ultrafine grain (AF K34 SF, made byGermanyAFHartmetall Group), and its performance is listedin Table 1. Microdrills were fabricated on a Makino SeikiCNS7dCNCmicrotool grindingmachine, as shown in Figure7(b). The basic parameters of microdrills are shown in Table2. The microdrill was observed under a laser microscope(KEYENE vk-x100 Series) and a stereoscopic microscope(Zeiss). Figure 7(c) shows an example of microdrills.

The material shear flow stress 𝑘 is 282.7MPa, the coeffi-cient of coulomb friction is 0.15, and the shear stress ratio 𝜏/𝑘

is 0.95. The separation angle 𝜃𝑐on the major cutting edges is

56∘ and 58.5∘ on the second cutting edges. The spindle speedis 22,000 r/min and the feed is 0.5 𝜇m/r, 1.0 𝜇m/r, 2.0𝜇m/r,3.0 𝜇m/r, and 5.0 𝜇m/r, respectively. The following equationis used to evaluate the shear angle when the second cuttingedges are cutting [14]:

𝜙𝑠= 31.48 + 0.32𝛾

𝑒. (28)


Table 1: Mechanical and physical properties of microdrills material.

Mechanical and physical properties ValueCo Content (%) 9WC including doping (%) 91Density (g/cm3) 14.3HV 30 (N/mm2) 2000HRA 94.4Transverse rupture strength (N/mm2) 4000Tungsten carbide particle size (𝜇m) 0.2

The experimental thrust force signals weremeasuredwitha dynamometer. The results are shown in Figures 8(a)–8(e).

A typical thrust profile is shown in Figure 9. In zone A,the chisel edge has contacted and extruded the workpiece; atthe same time, the second cutting edge is cutting. In zoneB,the major cutting edges are entering the hole gradually andbegin to cut. The thrust forces in zoneB consist of two partsof forces, the force generated by the chisel edge and the forcegenerated by the major cutting edges. The latter increasesgradually in zoneB, but it is always smaller than the formereven at its maximum. In zone C, the major cutting edgeshave completely entered the hole and the entire microdrill isexerting the thrust. In zone D, the chisel edge of microdrillis just out from the bottom of workpiece and the majorcutting edges are still cutting.The force in zoneD is generatedwithout the contribution of the chisel edge. Therefore, thethrust in zoneD is significantly smaller than that in zoneB.In zone E, workpiece has completely drilled through, andthere exists friction between drill and hole wall. Then, themicrodrill withdraws from the hole.

The chisel edge and cutting edges forces must be sepa-rated in order to compare them to the values predicted bythe model. The approach is to use a blind pilot hole with adiameter exactly equal to the web thickness of the microdrillused for the validation. For pilot holes, 0.15mm drills wereused, and the depth of pilot holes was kept at 0.5mm. Thetypical thrust profile for the operation is shown in Figure 10.

The experimental thrust force results are compared withthe corresponding predicted results in Figure 11.

4.2. Results and Discussion. As seen in Figure 8, the shape ofcurve in Figure 8(a) is completely different from the others,and the trend of the curve is basically the same as in Figures8(b)–8(e). At very low feed, the chip thickness is less thanthe minimum chip thickness; chips are not formed, andonly ploughing takes place. However, because the indentationzone keeps extruding the work material, the cutting processdoes take place. Figure 8(a) shows the thrust of this case; itis the main ploughing forces. As the feed increases, whenthe chip thickness exceeds theminimum chip thickness, bothshearing and ploughing take place in the cutting, so the thrustforces include shearing forces and ploughing forces, as shownin Figures 8(b)–8(e). By comparing Figures 8(a) and 8(b),we can see that the value of the minimum chip thickness isbetween 0.25 𝜇m and 0.5 𝜇m.

Table 2: Basic parameters of microdrill.

Geometric feature ValueDiameter (mm) 0.5Flute length (mm) 5.0Helix angle (∘) 25Web thickness (mm) 0.15Web taper (mm/mm) 0.03Point angle (∘) 130Primary face angle (∘) 12Chisel edge angle (∘) 42.8Major cutting edge radius (𝜇m) 2Second cutting edge radius (𝜇m) 3

As seen in Figure 11, almost all the predicted valuesare lower than the experimental ones. The data shows thatthe cutting force model of chisel edge including secondarycutting edge and indentation zone can correctly predict thethrust, and the average error is less than 5 percent. Theaccuracy of major cutting edges cutting force is relativelylower. The experimental results show that the average errorin the predicted steady state major cutting edges thrust is lessthan 10 percent.When the feed is between 0.5 and 1.0, amixedelastic-plastic deformation happens to the material; a transi-tion from the ploughing mechanism to shearing mechanismcan be seen. In general, the total drilling thrust (cutting edgesand chisel edge) is predicted with an average error of less than7 percent. Inmajor cutting edges cutting forcemodel, becausethe hydrostatic pressure and shear flow stress along tool-chip contact zone are calculated by dividing some differentialelements, the number of divided differential elements has cer-tain effect on the accuracy of themodel. On the other hand, inorder to simplify themathematical formulas, the tool circularedge is approximately represented by two straight chords,which lead to lower accuracy to some extent. Other sourcesof deviation might include the wear or local fracture of themajor cutting edges in the cutting process; these factors canlead to the increasing of the thrust during the drilling process.

The predicted and experimental results show that thethrust created by the chisel edge is quite significant. It exceedsthe thrust created by the cutting edges and represents about60–70 percent of total thrust. In this paper, the chisel edgeangle ofmicrodrill is relatively low at 42.8∘, which causes boththe length of chisel edge and the cutting force to increase.

5. Conclusions

Themathematical models to predict the microdrilling thrustare developed. The thrust is modeled in three parts: majorcutting edges, secondary cutting edge, and the indentationzone. Major cutting edge and secondary cutting edge forcemodels are based on the slip-line field theory, and theindentation zone is modeled as a rigid wedge. The majorcutting edges and secondary cutting edge are divided intoelements and the elemental forces are determined from anoblique cuttingmodel and an orthogonalmodel, respectively.Shearing and ploughing are included in the models of the


0 2 4 6 8 10−0.2

0.0

0.2

0.4

0.6

0.8Th

rust

(N)

Time (s)

(a)

0 1 2 3 4 5−0.2

0.0

0.2

0.4

0.6

0.8

Thru

st (N

)

Time (s)

(b)

0 1 2 3 4−0.2

0.0

0.2

0.4

0.6

0.8

Thru

st (N

)

Time (s)

(c)

0 1 2 3 4−0.2

0.0

0.2

0.4

0.6

0.8Th

rust

(N)

Time (s)

(d)

0 1 2 3 4−0.2

0.0

0.2

0.4

0.6

0.8

Thru

st (N

)

Time (s)

(e)

Figure 8: The thrust force profile for microdrilling. (a) Feed: 0.5 𝜇m/r; (b) feed: 1.0 𝜇m/r; (c) feed: 2.0 𝜇m/r; (d) feed: 3.0 𝜇m/r; (e) feed:5.0 𝜇m/r.


0 2 4 6 8 10−0.2

0.0

0.2

0.4

0.6

0.8

Thru

st (N

)

Time (s)

1 2 3 4 5

Figure 9: A typical profile of the thrust force formicrodrilling (feed:1.0𝜇m/r).

0 1 2 3 4 5−0.2

−0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Edge

s thr

ust

Maj

or cu

tting

Thru

st (N

)

Time (s)

Tota

l thr

ust

Figure 10: Thrust profile using pilot hole (feed: 1.0 𝜇m/r).

major cutting edges and second cutting edge. The model isapplied to a 0.5 mm ultrafine grain cemented carbide micro-drill, and the experimental and predicted values of forces arecompared.

The main conclusions from the study are as follows.

(i) Almost all the predicted values are lower than theexperimental ones.Thismight be attributed by factorssuch as drill vibrations, drill wandering, the friction ofdrill, and hole wall.

(ii) On the chisel edge, the forces of secondary cuttingedge can be modeled based on slip-line theory, andthe indentation zone can bemodeled as a rigid wedge.The model of chisel edge shows a good conformitywith the experimental results.

(iii) The accuracy of major cutting edges cutting forceis low relatively, and the average error is about 10percent. This may be due to the fact that some ofthe constants such as shear stress ratio and separationangle as well as others are calibrated for other process-ing methods and not for drilling.

Expt (a) Pred (a)Expt (b) Pred (b)Expt (c) Pred (c)

0 1 2 3 4 5 60.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Thru

st (N

)

Feed (𝜇m/r)

Figure 11: Comparison of experimental and predicted value. (a)Major cutting edges thrust; (b) chisel edge thrust; (c) total drillingthrust.

Future work should aim at two aspects: improving theaccuracy ofmajor cutting edges cutting force and consideringthe effect of the chisel edge angle and length on total drillingthrust.

Nomenclature

𝐹1: Total thrust of major cutting edges

𝐹2: Total thrust of second cutting edge

𝐹3: Total thrust of the indentation zone

𝑑𝐹cut: Elemental cutting force𝑑𝐹𝐿: Elemental lateral force

𝑑𝐹𝑇: Elemental oblique cutting thrust force

𝜏: The frictional shear stress𝑘: The material flow stress𝑝: The hydrostatic pressure𝜙: Half the drill point angle𝜆𝑠: Cutting edge inclination angle

𝛾𝑛: Normal rake angle of major cutting edge

𝛽: Helix angle𝜃𝑐: The separation angle

𝛾𝑒: Effective rake angle

𝜙𝑒: Effective shear angle

𝜓: The chisel edge angle2𝛾ind: The included angle of the wedge𝛾𝑠: The static clearance angle of the chisel edge

𝑟𝑛: The tool edge radius

𝑟𝑤: Half the web thickness

𝑟: The distance from the selected point onthe major cutting edge to drill axis

𝑅: Drill radius𝑅ind: The radius of the indentation zone𝑓: Feed


𝐿𝐶: The length of chisel edge

𝑡𝑐: The uncut chip thickness

𝑡cmin: The minimum chip thickness.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

Theauthors would like to thankTheNational Natural ScienceFoundation of China (Key Program, no. 50935001) for theirfinancial support.Without their support, this workwould nothave been possible.

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