Mechanical load distribution along the main cutting edges in drilling

Journal of Materials Processing Technology 213 (2013) 245–260

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Mechanical load distribution along the main cutting edges in drilling Mihai-Bogdan Lazar a,∗ , Paul Xirouchakis b,1 a Laboratory of Computed Aided Design and Production (LICP), Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-STI-LICP, ME A1 391, Station 9, CH-1015 Lausanne, Switzerland b Laboratory of Computed Aided Design and Production (LICP), Swiss Federal Institute of Technology Lausanne (EPFL), EPFL-STI-LICP, ME A1 408, Station 9, CH-1015 Lausanne, Switzerland

a r t i c l e

i n f o

Article history: Received 25 June 2012 Received in revised form 14 September 2012 Accepted 25 September 2012 Available online 8 October 2012 Keywords: Drilling Cutting forces Mechanistic model Composite materials

a b s t r a c t Used in a very large variety of applications, drilling is one of the most complex manufacturing processes. Most of the research on drilling was done in the field of metal cutting since, in this case, high precision and quality are needed. The use of composite materials in engineering applications has increased in recent years, and in many of these applications drilling is one of the most critical stages in the manufacturing process. Delamination and extensive tool wear, affecting the quality and the costs, are among the problems which drilling of composite materials are currently facing. Understanding and predicting the cutting forces occurring during drilling of such materials would allow extending the currently used optimization methods and proposing new tool geometries and tool materials. The current paper introduces a new mechanistic model for predicting the cutting force distribution along the cutting edges of a drill. A simple, generic and effective method is proposed to relate drilling to oblique cutting using a direction cosine transformation matrix valid for any drill geometry. The oblique cutting model used considers forces on both rake and relief faces, and a simple system of empirical coefficients (their number is significantly less than other similar models). The empirical coefficients are calculated assuming the work-piece material is isotropic. The model is validated on experiments carried out on carbon-fiber and glass-fiber reinforced composites using two different drill types (tapered drill reamer and 2-facet twist drill), which are described in more detail in a previous published paper. The mathematical expression of the drill geometry is also reviewed; removing certain assumption, generalizing some definitions and introducing new drill geometry and features. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The topic of modeling the thrust and torque in drilling is, in general, approached by two methods: (i) empirical, which captures the influence of the cutting parameters and the drill diameter over the experimentally measured maximum cutting forces and (ii) mechanistic, which divides the cutting edges of a drill into small elements treated as oblique or orthogonal cutting elements and by integration along the radial direction the maximum thrust and torque values can be predicted. While empirical models cannot distinguish between drilling with 2 different drills of the same diameter, the mechanistic models have the disadvantages of their complexity and the difficulty in calibrating them on experimental data of the cutting force distribution.

∗ Corresponding author. Tel.: +41 21 693 38 68; fax: +41 21 693 35 09. E-mail addresses: bogdan.lazar@epfl.ch (M.-B. Lazar), paul.xirouchakis@epfl.ch (P. Xirouchakis). 1 Tel.: +41 21 693 29 14; fax: +41 21 693 35 53. 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.09.020

Most of the mechanistic models for drilling were developed for metals. Only Langella et al. (2005) was found to be developed and tested especially for composite laminates, although the geometry of the drill is treated in a very simplistic manner. One of the most accurate cutting force prediction models for drilling metals is presented in detail by Chandrasekharan (1996) and has also been applied to fiber reinforced composite laminates (Chandrasekharan et al., 1995). In spite of the complexity of the model and the number of empirical coefficients the geometric and dynamic effects are not fully isolated from the material properties and the model constantly underestimates the measured forces for composite materials. As the shear angle theory (introduced by Merchant, 1945a,b) has no theoretical reason to be applied to composite materials (because the cutting process is based on fracture rather than plastic deformation), the model uses a mechanistic approach in defining the cutting forces (as normal and friction forces on the rake face). Two different models (with respective empirical coefficients) are used for the cutting lips and chisel edge of the drill respectively and the contact forces on the clearance face are not considered separately (as proposed by Connolly and Rubenstein, 1968 for example). Watson

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(1985a,b) introduces another reference cutting force model which is very accurate for metal drilling. In principle, mechanistic models for cutting forces in drilling can offer information about the variation of the elementary forces along drill radius, when compared with simple empirical model for the maximum thrust and torque (such as Shaw and Oxford, 1957). The difficulty appears on calibrating the mechanistic models on relevant experimental data. The model by Chandrasekharan (1996) is also notable for introducing the idea of calibration based on the cutting forces distribution along the radius obtained experimentally, rather than the total maximum forces. Lazar and Xirouchakis (2011) have introduced an improved methodology for the analysis of the cutting forces in drilling to obtain the radial distribution of the elementary cutting forces along drill radius (or the distribution of the drilling loads among laminate plies for composite materials). Although calibration for the radial distribution of the cutting forces is much more complex, as it involves tedious post-processing of the experimental results, in theory, it should minimize the number of experiments needed for calibration and provides an accurate distribution curve along the main cutting edges of the tool. As most of the drilling tools are pointed, the distribution of the forces along the cutting edges can allow the determination of the point of application of the mechanical loads not only on the radial coordinate from the hole center, but also in depth starting from the reference point of tool tip. This allows the determination of the mechanical loads acting on each ply of a composite laminate throughout the drilling process. Current delamination models are able to determine the amount of thrust which a composite structure can withstand before a crack occurs for each ply from the bottom surface. Zhang et al. (2001) assume that thrust acts as a concentrated load on the tool tip and determines what is its maximum value (critical thrust) before delamination occurs for each of the plies starting from the bottom one. Hocheng and Tsao (2003) assume a thrust distribution acting on the last ply for several drill geometries and determine what the critical thrust is prior to delamination onset. For their FEM model for delamination, Zitoune and Collombet (2007) conduct their own experiments to determine the proportion of the thrust force developed along the chisel edge, but otherwise assume uniform distribution for each region. All of these studies model the material as isotropic, although several extension studies are approached using a quasi-isotropic material behavior. The main purpose of the current model is to provide a more accurate assessment of the thrust distribution during the drilling process to support delamination models as above. In the analysis of experimental data, Lazar and Xirouchakis (2011) are showing that the thrust distribution greatly depends on the drill geometry, but also changes with the cutting parameters, especially axial feed. It is found that most of the mechanistic cutting force models for drilling provide equations for thrust and torque, developed starting from forces defined in the oblique or orthogonal cutting model, similar to our approach. A transformation matrix to relate vectors defined in oblique cutting to drilling reference systems, assumption-free and valid for any type of drill geometry and even for the chisel edge region is employed as suggested by Altintas (2000) for milling operations. A normal/friction force model is proposed in an oblique cutting framework similar to Chandrasekharan (1996), but including the second pair of normal/friction forces on the clearance face (based on suggestions by Connolly and Rubenstein, 1968). The introduction of the later will be shown to be critical in providing accurate prediction for total forces in drilling fiber reinforced composites, characterized by lower specific values than for metals, but with similar magnitude of forces acting on the relief face. Three basic empirical coefficients (specific pressure on rake face coefficient, friction coefficient and specific pressure on relief face) are used, which can be defined as more complex functions (increasing the number of the total

empirical coefficients). In the current study configuration of only up to 4 empirical coefficients are presented which show good results. Apart from the well know strong influence of the rake angle and the minor influence of the cutting speed, the specific force coefficients for composite materials are also believed to be influenced by the fiber orientation (Hinze et al., 2011) and cutting edge rounding (Schulze et al., 2011; Faraz et al., 2009). In the current model the influence of the rake angle is accounted for; the influence of the cutting speed is evaluated from experiments on composite materials as being too small and therefore ignored, while the effects of edge rounding and fiber direction are not modeled at the current stage. The effects of the last two parameters are controlled by using only new drills which are not allowed to wear off (keeping the edge rounding almost constant) and experimenting on bi-directional woven composite laminates to lower the influence of the fiber direction. Therefore, it can be stated that the current model assumes the work-piece to be isotropic, although further developments could eliminate this assumption. The model is calibrated for experiments conducted using two different drills from a geometric point of view (tapered drill reamer and 2-facet twist drill) on two work-piece materials (woven bidirectional carbon- and glass-fiber reinforced epoxy) within a practical cutting parameter range. The experimental plan and the post-processing methodology to obtain the radial distribution of the elementary cutting forces have been detailed in a previous publication by Lazar and Xirouchakis (2011). The current paper provides only a brief introduction to the experimental plan and procedure. Two calibration strategies are tested: on maximum values of thrust and torque and on radial distribution of the axial and tangential force components. The first strategy proves very suitable for predicting the maximum thrust and torque values while the distribution obtained is not relevant. The second calibration method experiences less accurate predictions for maximum thrust and torque (while still reasonable), but captures well the distribution of the axial and tangential force components and their variation with the process parameters. The mathematical description of the drill geometry for the purpose of cutting force modeling has also been reviewed and improved. The full discussion on the improvements and the derivation of the new equations can be reviewed in the author’s doctorate thesis (Lazar, 2012). Only the resulting equations and their main features will be introduced in the current paper. Among the improvements is the consideration given to the kinematic aspect (represented by the cutting angle , as the angle between the tangential and actual velocities vectors on any point on the cutting edges) along all cutting edges (while it was previously ignored for the cutting lip region of the drill). This aspect proved critical in capturing the variation of the axial elementary cutting force distribution with the radius for varying the feed rate as will be later discussed. Additionally, the mathematical description of the drill geometry for cutting force modeling is generalized; a new type of drill (tapered drill reamer, used in previous experimental studies on composite drilling – Fernandes and Cook, 2006a,b) is introduced together with some new features of the classical twist drills.

2. Cutting force model In mechanistic models, the elementary forces defined in oblique (or orthogonal) cutting models have to be decomposed along the thrust/torque/lateral directions as defined in the drilling operation and summed for all elements of the cutting edge to obtain the total values of thrust and torque (and occasionally lateral force). The total thrust and torque values are representative for describing the cutting forces in drilling. The measured lateral component should be zero for a perfect drill (due to the symmetry along the

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particular cases of drill geometry the values of these angles and the equations used to derive them are presented in Appendix. With the help of introducing an auxiliary angle (, referred to as the 2nd Euler angle of rotation and described by Eq. (1)) the transformation matrix between the two coordinate systems can be expressed as follows:



(r) = arccos

⎡

sin(p(r)) · cos(ˇ(r)) cos(i(r))

cos()

TX  Y  Z  ,XYZ = ⎣ sin() · sin() sin() · cos()



0

− sin()

cos()

cos() · sin()

− sin()

A vector can be decomposed from tem using the following relation: {v}XYZ = TX  Y  Z  ,XYZ · {v}X  Y  Z  Fig. 1. Oblique and drilling coordinate system for a point A on the cutting lip of a twist drill.

drill axis, the lateral force on a flute is canceled by the others) and therefore is only important when one wants to study the effects of symmetry imperfections on mechanical loads. Unless the drill has a single flute, the lateral force cannot be measured effectively, and this poses a significant problem in the calibration of drilling force models based on oblique cutting as compared with milling, turning or other machining processes. The elementary forces are defined in the oblique cutting framework and then decomposed along the drilling coordinate system by employing a transformation matrix. Fig. 1 shows the two coordinate systems for a point on the cutting lip of a twist drill. XYZ is the coordinate system associated with the drilling process, with the X axis along the radial direction; the Z axis aligned with the drill axis and the Y axis corresponding to the tangential direction is perpendicular to both. The forces along the Z direction will contribute to the thrust, while the forces along the Y direction will generate the torque and along the X direction the lateral forces will be developed (which will be canceled by the action of the second flute). The X Y Z coordinate system is associated with the oblique cutting element, with the Y axis aligned according to the local velocity vector; the X axis perpendicular to the Y axis in the plane defined by the Y axis and the cutting lip and Z axis perpendicular to the X Y plane (which also includes the cutting edge). A relationship can be found between the two coordinate systems for any point A on the cutting edge (either on the cutting lip region or the chisel edge) if the following angles are known: - Point angle (p) defined as the angle between the cutting edge at point A and the tool axis (or its parallel through A–Z axis). - Web angle (ˇ) defined as the angle at point A between the radial direction (X axis) and the projection of the cutting edge in a plane perpendicular to the drill axis (XY plane). - Cutting angle () defined as the angle at point A between the local velocity vector (V/Y axis) and the tangential velocity (Vt – projection of the velocity vector on a plane perpendicular to the drill axis – XY plane). - Inclination angle (i) defined as the angle at point A between the X axis (the normal to the velocity vector (V) in the plane containing both the velocity vector and the cutting edge) and the cutting edge. All these angles are frequently used in describing drill geometry; their formulations are known and widely discussed. For the two

(1)

⎤ ⎦

(2)

cos() · cos()

X Y Z

to XYZ coordinate sys(3)

The details about how Eq. (2) is derived are presented in Lazar (2012). The equation is generic, i.e. as long as the above mentioned angles can be defined and calculated, the transformation matrix can be applied to any drill geometry. Experiments conducted by Zitoune et al. (2005) on orthogonal cutting of uni-directional long-fiber composites revealed that the cutting mechanism is fracture-based, while in the case of metals it is based on plastic deformations. Furthermore, the powder-like chip is formed by a fracture in two stages; those directions vary with the fiber orientation with respect to the cutting direction. No theoretical model was found to describe the fracture-based cutting process of long fiber reinforced composites, while for metals the shear angle theory is widely used. Mechanistic cutting force models proposed for composite materials consider the principal elemental forces as normal and tangential to the rake face and, unless ignored, on the relief face. Orthogonal cutting experiments of uni-directional glass-fiber reinforced composites (Caprino and Nele, 1996) revealed that, as in the case of metals, the depth of cut has a linear influence on both horizontal and vertical resultant force components, with the intercept above zero (attributed to forces acting on the relief face, which are believed to be independent of the change of depth of cut within practical boundaries). The same experiments report that the rake angle has a strong influence on the resultant cutting force (noted as stronger than in the case of metals) while the relief angle has little if no influence. In Schulze et al. (2011) it is noted that the inclination angle (oblique cutting) has little influence on the specific cutting forces (horizontal and vertical directions) for short glass fiber reinforced polyester, although forces in the lateral direction were not measured. For the current study, we will consider an elementary force model based on oblique cutting, considering both the forces acting on the rake and relief face as in Fig. 2. Normal (Fn1 ) and friction (Ff1 ) forces act on the rake face, while Fn2 (normal) and Ff2 (friction) are the forces acting on the relief face. As usual, the force directions are normal and tangential to the respective surfaces. Furthermore, it is assumed that the friction force on the rake face (Ff1 ) acts along the direction of chip flow, at an angle c (chip flow angle) from the normal to the cutting edge lying on the rake face, while the friction force on the relief face (Ff2 ) acts along the projection of the velocity direction (Y axis) on the relief face. In addition to the angles presented previously, determining the force vector directions in the X Y Z coordinate system requires the introduction of: - Normal rake angle (˛n ), defined as the angle between the rake face (or tangent to the rake face – if the rake face is not planar) at a point A on the cutting edge and the normal to both the cutting

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on both rake and relief faces the following relationships are obtained:

{Fn1 }X  Y  Z  = Kc · Au ·

⎧ ⎫ − cos(˛n ) · sin(i) ⎪ ⎪ ⎨ ⎬ ⎪ ⎩

− cos(˛n ) · cos(i) − sin(˛n )

(4)

⎪ ⎭

{Ff 1 }X  Y  Z  = {Kf · Fn1 }X  Y  Z  = Kf · Kc · Au ·

×

⎧ ⎫ [− sin(c ) · cos(i) − sin(˛n ) · cos(c ) · sin(i)] ⎪ ⎪ ⎨ ⎬ ⎪ ⎩

−[sin(c ) · sin(i) − sin(˛n ) · cos(c ) · cos(i)] cos(˛n ) · cos(c )

⎪ ⎭ (5)

Fig. 2. Elementary cutting forces defined in oblique cutting.

velocity vector and the cutting edge (Z axis), measured in a plane perpendicular to the cutting edge at point A. - Relief angle () is the angle at point A between the relief face and the local velocity (along the Y axis) measured in the Y Z plane (formed by the local velocity and the normal to both the velocity and the cutting edge). - Chip flow angle (c ) is a characteristic angle to oblique cutting and defines the direction of chip movement after separation from the work-piece. It is defined in the plane of the rake face (or tangent plane to the rake face at point A if the rake face is not planar) as the angle between the chip flow direction and the normal to the cutting lip. Oblique cutting experiments conducted on metals (Stabler, 1964) concluded that it is proportional to the inclination angle by a factor between 0.9 and 1.0. The same is commonly assumed in drilling cutting force models, although Watson (1985a,b) challenges this assumption by noting that the chip in drilling is usually complete across the cutting lip, and therefore there might be some restrictions in the variation of the chip flow angle in this region and recommends using a constant value rather than variable as a function of the inclination angle. As the chip is powder-like in machining fiber reinforced materials, in the current study it is assumed the material can be treated as a series of independent elements and therefore the chip flow law of Stabler (1964) is valid. For simplicity it has been assumed to be equal to the inclination angle (i). The normal rake angle (˛n ) in drilling is extensively discussed in literature. However, all the equations proposed for its derivation along the cutting lips (especially for twist drills) ignore the influence of the cutting angle (). The equation defining the normal rake angle for drilling for both the cutting lips and chisel edge has been revised to remove the assumption that the cutting angle () is zero for the cutting lip region. The details of the derivation are presented in Lazar (2012), while the final result is introduced in Appendix together with discussion about the two particular cases of drill geometry considered. It should be noted that resulting Eq. (26) is valid for drills with straight cutting edges and can be used for drills with zero helix angle but also for the chisel edge region, if the angle made by the rake face and the parallel to the drill axis can be determined. The derivation of the relief angle () in drilling is also available in the literature. It was however reworked (see Lazar, 2012) to extend its applicability to multi-staged drills (such as tapered drill reamer). The equation used in the current model is presented in Appendix. Considering both the magnitude and the direction of the elementary forces in the oblique cutting coordinate system acting

{Fn2 }X  Y  Z  = Kp · Ac ·

⎧ ⎨0 ⎩

sin() cos()

⎫ ⎬ (6)

⎭

{Ff 2 }X  Y  Z  = {Kf · Fn2 }X  Y  Z  = Kf · Kp · Ac ·

⎧ ⎨0 ⎩

− cos() sin()

⎫ ⎬ ⎭

(7)

where Kc (N/mm2 ), Kf (–) and Kp (N/mm2 ) are coefficients representing the specific cutting coefficient, friction coefficient and specific contact pressure on relief face respectively. It is assumed that the friction coefficient is identical for rake and relief faces, as the materials are identical and the relative velocities comparable. Previous studies (those not ignoring the forces on the relief face) prefer the solution of employing different friction coefficients for rake and relief faces in order to obtain an additional degree of freedom. The following generally accepted relationship is used to estimate the uncut chip area Ac (mm2 ), which can be derived from Fig. 2: Au =

f · cos() · dr N

(8)

Ac (mm2 ) represents the contact area on the relief face. The forces acting on the relief face have been modeled in a similar manner to Rubenstein (1990) and Connolly and Rubenstein (1968), which estimates the contact area (Ac ) in orthogonal cutting to be a function of the relief angle (), the cutting edge radius (Re [mm]) and a critical rake angle (˛c ) over which value plowing is believed to occur instead of cutting, according to Basuray et al. (1977). For oblique cutting the influence of the inclination angle (i) on the contact length is also considered. Therefore, Ac is modeled as: Ac = Re ·



cos(˛c ) +

[1 − sin(˛c )] tan()



·

dr cos()

(9)

It is assumed that the cutting edge radius (Re ) is constant along the cutting edges of the drill and as estimated by Basuray et al. (1977) around the value of 0.00762 mm (0.0003 in), while the critical rake angle (˛c ) is 70◦ (−70◦ if the same zero reference is considered as for the normal rake angle), according to the same Basuray et al. (1977). Using Eq. (3), the vectors can be transformed into the XYZ coordinate system. Of interest in drilling are the components in the Y and Z direction, corresponding to tangential (torque) and axial (thrust) loads. After the coordinate transformations and vector summation, the following relationships are found for the forces in the tangential (Y) and axial (Z) directions acting on an element of size dr. The

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lateral component (X) is not of interest for the current study, but can be derived in the same operation (see Lazar, 2012). Fy = Kc · Au · g1y + Kp · Ac · g2y

(10)

Fz = Kc · Au · g1z + Kp · Ac · g2z

(11)

where g1y , g2y , g1z and g2z are geometric factors as follows: g1y = sin() · sin() · [− cos(˛n ) · sin(i) + Kf · [− sin(c ) · cos(i) − sin(˛n ) · cos(c ) · sin(i)]] + · · · + cos() · [− cos(˛n ) · cos(i) − Kf · [sin(c ) · sin(i) − sin(˛n ) · cos(c ) · cos(i)]] + · · · + cos() · sin() · [− sin(˛n ) + Kf · cos(˛n ) · cos(c )]

(12)

g1z = sin() · cos() · [− cos(˛n ) · sin(i) + Kf · [− sin(c ) · cos(i) − sin(˛n ) · cos(c ) · sin(i)]] − · · · − sin() · [− cos(˛n ) · cos(i) − Kf · [sin(c ) · sin(i) − sin(˛n ) · cos(c ) · cos(i)]] + · · · + cos() · cos() · [− sin(˛n ) + Kf · cos(˛n ) · cos(c )]

249

As most of the angles vary with the radial coordinate of the point A from the drill axis (see Appendix), Fy and Fz are functions of the radius and axial feed rate (f). Several functions have been tested for the empirical coefficients (Kc , Kf and Kp ). In the current paper we will only present the results for constant coefficients (M0) and for Kc as a linear function of the sinus of the normal rake angle (M1): Kc = Kc1 · (1 − sin(˛n )) + Kc2

(16)

It is noted that the spindle speed (n [rpm]) is not a parameter in cutting forces. The experimental data analysis (extensively discussed by Lazar and Xirouchakis, 2011) shows that the nominal thrust and torque curves do not vary widely with the spindle speed. The fact that the cutting velocity plays only a small role in the magnitude of the forces for composite materials has been confirmed by previous experimental studies among which Schulze et al. (2011). To obtain the total thrust (FZ [N]) and torque (MZ [N mm]) generated during drilling functions 10 and 11 can be integrated in the following manner:



(13)

r=R

FZ =

N · Fz (r) · dr

(17)

r=0



g2y = cos() · [sin() − Kf · cos()] + cos() · sin() · [cos() + Kf · sin()]

(14)

g2z = − sin() · [sin() − Kf · cos()] + cos() · cos() · [cos() + Kf · sin()]

(15)

r=R

MZ =

N · Fy (r) · r · dr

(18)

r=0

where r is the radial coordinate, R is the outer radius of the drill and N is the number of flutes (see Appendix). The torque will further be converted to [N m] units for discussion. 3. Results and discussion

When comparing the equations obtained for the axial and tangential forces (Eqs. (10) and (11)) with previous mechanistic models for drilling we note the existence of the second term, corresponding to the contribution of the forces acting on the relief face. Additionally, most of the geometrical simplifications assumed by previous models have been removed (such as the assumption that the cutting angle  is zero at least for the cutting lips region).

The model is calibrated and compared with axial and tangential radial distribution curves obtained from experiments through a methodology introduced by Lazar and Xirouchakis (2011). The experiments were carried out using three different drills (although for modeling purposes only two are used and described geometrically in Appendix: T1 – tapered drill reamer and

Fig. 3. Sample cutting force measurement during drilling (CFRP, 2-facet twist drill (T2), n = 2750 rpm, f = 0.14 mm/rev) – (a) thrust; (b) torque)).

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Table 1 Cutting parameters. f (mm/rev)

0.02 0.08 0.14 0.20

n (rpm) 500

1625

2750

3875

5000

H01 H06 H11 H16

H02 H07 H12 H17

H03 H08 H13 H18

H04 H09 H14 H19

H05 H10 H15 H20

T2 – 2-facet twist drill) on two different work-pieces (carbon – CFRP and glass-fiber reinforced composites – GFRP). Axial feed (f [mm/rev]) and spindle speed (n [rpm]) were varied on four and five levels respectively within practical ranges (Table 1). Fig. 3 presents a thrust and torque curve obtained during drilling experiments in its raw form and smoothed for obtaining the distribution for a case where the amplitude about the nominal values (smoothed curves) is large. The curves are trimmed from entry point to full engagement of the cutting edges. It is important to note that the filtered curve is used to calibrate the model as well as for comparison. The maxima on the filtered (nominal) curves are always smaller than on the raw curves and we find that the difference increases with the spindle speed and is always larger for torque. Lazar and Xirouchakis (2011) showed that the smoothed curves (used to obtain the elementary cutting forces distribution) do not vary much with the spindle speed (n), or at least the variation is of

Fig. 5. Comparison between experimental and predicted results for axial elementary cutting force component distribution along the radius for f = 0.14 and 0.20 mm/rev (T2/CFRP).

similar magnitude as for repeating the same experimental case. As a side note, it appears that the number of fluctuations of the raw curve about the smoothed (nominal) corresponds roughly with the number of fiber layers (2 for each ply for bi-directional laminates). Such fluctuations would be of interested to an extension of the current model able to account for the influence of the fiber direction on the cutting forces. The empirical coefficients are determined for the 4 workpiece/drill combinations using two strategies: (i) fitting for the total thrust and torque values (using Eqs. (17) and (18)) and (ii) fitting on the axial and tangential force component distributing curves (using Eqs. (10) and (11)). For each combination, a number of 20 holes at

Fig. 4. Comparison between predicted (M1 model, fitted on the distribution curves) and experimental elementary cutting force components distribution along the radius for f = 0.14 mm/rev, 2-facet twist drill and both work-piece materials.

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Table 2 Obtained empirical coefficients.

M0–constant coefficients T1

T2

Fitting on axial and tangential distribution

Fitting on total thrust and torque values

CFRP

GFRP

CFRP

GFRP

Kc = 330.06 (N/mm2 ) Kf = 0.560767 Kp = 2938.45 (N/mm2 ) Kc = 279.72 (N/mm2 ) Kf = 0.278023 Kp = 4714.25 (N/mm2 )

Kc = 325.75 (N/mm2 ) Kf = 0.503423 Kp = 3758.62 (N/mm2 ) Kc = 327.33 (N/mm2 ) Kf = 0.167523 Kp = 4114.95 (N/mm2 )

Kc = 778.65 (N/mm2 ) Kf = 0.586807 Kp = 1830.35 (N/mm2 ) Kc = 163.71 (N/mm2 ) Kf = 0.62640 Kp = 3824.54 (N/mm2 )

Kc = 851.57 (N/mm2 ) Kf = 1.00824 Kp = 581.57 (N/mm2 ) Kc = 143.07 (N/mm2 ) Kf = 2.15938 Kp = 443.24 (N/mm2 )

Kc1 = −454.92 (N/mm2 ) Kc2 = 1262.58 (N/mm2 ) Kf = 0.580706 Kp = 1909.99 (N/mm2 ) Kc1 = 399.29 (N/mm2 ) Kc2 = −145.77 (N/mm2 ) Kf = 0.77615 Kp = 2269.78 (N/mm2 )

Kc1 = −1211.98 (N/mm2 ) Kc2 = 2121.28 (N/mm2 ) Kf = 1.11683 Kp = 520.48 (N/mm2 ) Kc1 = 84.26 (N/mm2 ) Kc2 = 96.58 (N/mm2 ) Kf = 1.62644 Kp = 587.26 (N/mm2 )

M1–Kc linear function of the sinus of the normal rake angle (Kc = Kc1 × (1 − sin(˛n ) + Kc2 ) Kc1 = 341.62 (N/mm2 ) T1 Kc1 = 406.43 (N/mm2 ) Kc2 = −146.34 (N/mm2 ) Kc2 = −71.91 (N/mm2 ) Kf = 0.586435 Kf = 0.520918 Kp = 2779.56 (N/mm2 ) Kp = 3562.94 (N/mm2 ) Kc1 = 499.97 (N/mm2 ) T2 Kc1 = 435.26 (N/mm2 ) Kc2 = −293.81 (N/mm2 ) Kc2 = −298.63 (N/mm2 ) Kf = 0.750408 Kf = 0.600504 Kp = 2810.35 (N/mm2 ) Kp = 1694.71 (N/mm2 )

selected values of spindle speed and axial feed rate (according to Table 1) were made. The coefficients (Table 2) are calculated in Labview (using the Levenberg–Marquardt algorithm) using all the case studies (to avoid slight variations based on different selections of the calibration cases). In theory, only one experiment is sufficient to determine a set of coefficients using the distribution curves, although at least 2 (with different feed rates) are required to obtain usable data. It is recommended to use at least 4 experiments placed at the corners of the experimental domain when fitting on

the axial and tangential elementary cutting force components distribution. When using the total values of thrust and torque, at least 2 experimental cases are needed (at the extreme values of the axial feed), but increasing the number of cases does not seem to improve the accuracy. A preliminary observation best noticed on the constant coefficients model (M0) is that the values of the coefficients vary with both the drill type and the combination of work-piece/drill material. Considering that the material of the drills is of similar

Fig. 6. Comparison between predicted and experimental (filtered) results for maximum thrust and torque for varying axial feed, T2 and both work-piece materials, M1 fitted on the experimental distributions.

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Fig. 7. Comparison between predicted and experimental (filtered) results for maximum thrust and torque for varying axial feed, T2 and both work-piece materials, M1 fitted on the total forces values.

type (solid-carbide steels for both cases although from two different suppliers – see Appendix), the desire was to obtain a set of coefficients valid for a certain work-piece/drill material combination for any drill geometry (i.e. this objective was not achieved since the values of the coefficients vary significantly with the drill type). An argument to justify the current results is that the machinability of the two work-piece materials proved to be comparable. The average maximum thrust for all tested conditions was only 3% higher for CFRP than GFRP, while the average maximum torque

was 20% higher for GFRP, and probably due to increased vibrations noted during drilling of GFRP material (further discussion are available in Lazar and Xirouchakis, 2011). Therefore, we can conclude that there are still aspects of the influence of the geometrical and cutting parameters that are not fully understand and modeled before obtaining empirical coefficients depending only on the work-piece/drill material combination. It can also be observed that there are significant differences in the solutions obtained for fitting on the distribution and total

Fig. 8. Comparison between experimental and predicted results of the current model, Chandrasekharan et al. (1995) and Isbilir and Ghassemieh (2011).

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values of thrust and torque. Fitting on the total thrust and torque values attempts to calculate the best coefficients so that the integral of the distribution curves (Eqs. (17) and (18)) matches the maxima of the thrust and torque smoothed curves (Fig. 3), while fitting on the distribution attempts to match the discreet points on the radial coordinate of the axial and tangential elementary cutting forces calculated as described in Lazar and Xirouchakis (2011) with Eqs. (10) and (11). A perfect fitting on the distribution curves would theoretically result in perfect results of the total thrust and torque values, while a perfect fitting on the total forces would not result in relevant distribution curves. When fitting on the distribution curves each point has the same importance although when integrating different points (regions) will have different impact on the total thrust and torque, depending on the size of the region (dr), number of flutes (N) and, most importantly for the tangential force, the radial coordinate (r). Therefore, errors in predicting the distribution at various points will have different impact on the total forces, i.e. an error in predicting the tangential elementary cutting force component at small radius will have almost zero impact on the total force while an error at the outer radius will have a maximum impact. Summarizing, fitting on the distribution allows obtaining relevant distribution curves of the elementary cutting forces while it is expected the total thrust and torque values to be less accurate. In the same time when fitting on the total thrust and torque values, the maximum forces can be estimated accurately while the distribution are not necessary relevant.

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Additionally, we observe that for the M1 model (specific cutting pressure Kc as a linear function of (1 − sin(˛n )) and the tapered drill reamer (T1) when fitting on the total thrust and torque values we obtain negative Kc1 values, which can be translated by the fact that the specific cutting pressure on rake face is inversely proportional to the sinus of the normal rake angle, contrary to the result for the rest of the cases. In the same time we note the large values for Kc2 (the constant), which means that the influence of the rake angle is significantly less than for 2-facet twist drill T2 (same model and fitting strategy). When analyzing Fig. 12d we note that for the most part of the radial domain the normal rake angle for tapered drill reamer (T1) remains constant, with significant changes only for the small region of the chisel edge and in its close vicinity. As the total thrust is affected more by the elementary forces along the cutting lips region (the chisel edge is rather small for this drill) and even more the torque (as the points are closer to the outer radius), the fitting algorithm probably finds that the current values offer a better solution overall. When fitting on the distribution curves, we observe that Kc1 and Kc2 align with our expectation. We start by analyzing the results obtained for the 2-facet twist drill (T1), which is extensively studied in the literature and although full distribution curves either experimental or predicted were not published there is a general knowledge about the proportion of the forces caused by the chisel edge and the cutting lips respectively from experimental drilling using pilot holes. When comparing the distributions, as expected, only the solutions found for fitting on the experimental distribution have significance, and although

Fig. 9. Comparison between predicted (M1 model, fitted on the distribution curves) and experimental elementary cutting force components distribution along the radius for f = 0.14 mm/rev, tapered-drill reamer and both work-piece materials.

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Fig. 10. Comparison between predicted and experimental (filtered) results for maximum thrust and torque for varying feed rate, T1 and both work-piece materials, M1 fitted on the total forces values.

reasonable for the model with constant coefficients (M0), the best results were obtained using the specific cutting pressure coefficient (Kc ) as a linear function of sinus of the normal rake angle (M1). Fig. 4 shows a representative case scenario of the cutting parameters (f = 0.14 mm/rev, n = 500–5000 rpm) for the 2facet twist drill (T2) for both carbon- and glass-fiber reinforced composites. Although not perfect, the predicted curves manage to capture fairly well the variation of the cutting forces along the radius for this type of drill. The predicted axial elementary force component around the chisel edge (r = 0.5 mm) of the drill seems to be always lower than the experimental results would indicate. In theory, modeling the specific cutting pressure using a power law of the normal rake angle could capture this effect better. However, from our trials, such a function will cause the model to converge at rather extreme values of the coefficients, resulting in a peak axial force around r = 0.5 mm (much higher than predicted by experiments) and negative towards the outer radius (r = 2.8 mm). Additionally, we observe that the tangential force component is in general closer to the upper bounds of the experimental values in the same region (r = 0.5 mm) and using a power law will cause the overestimation of this force component in this region as well. It should also be noted that for all distribution curves, the interval of variation of a point is rather large (as they are calculated as the derivative of the thrust and torque entry curves – see Lazar and Xirouchakis, 2011) even for identical drilling cases. In the above figure and later on, we have used 5 drilling cases with varying spindle speed to illustrate the variation interval. Furthermore, in general the variation interval is greater for the tangential force especially in the close vicinity

of the drill axis (chisel edge region) due to reasons covered by Lazar and Xirouchakis (2011). Experiments reported by Chandrasekharan et al. (1995) with pilot holes estimate that a percentage usually between 40 and 60% of the total thrust force is generated by the chisel edge region for uni-directional carbon fiber reinforced composite and various twist drills, while Zitoune and Collombet (2007) report between 40 and 50% for a twist drill and bi-directional CFRP. Although it is difficult to make an exact comparison as the drill geometry and work-piece materials vary, our predicted results for T2 drill are that roughly 45% of the total force is generated by the chisel edge for the glassfiber reinforced material while 35% for the carbon-fiber reinforced composite respectively which correspond well with our experimental data. When compared with standard twist drills, our 2-facet twist drill (T2) has the chisel edge inclined with the drill axis which should decrease slightly the loads in this region. It is also interesting to present how the model captures the changes in the distribution with increasing the feed rate, one of the most useful features of mechanistic cutting force models and a topic not covered yet by other studies. Fig. 5 presents the predicted and experimental results for the axial elementary force component distribution (T2 drill, glass-fiber reinforced composites and with coefficients determined by fitting the experimental distribution – see Table 2) for drilling cases with varying feed rate (f = 0.14 mm/rev and f = 0.20 mm/rev). It can be observed that with increasing feed rate the distribution of the axial elementary cutting force (valid for the other components as well) does not increase uniformly along the radius. The experimental results indicate an increase of the elementary forces

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Fig. 11. Drill geometries: tapered drill reamer (T1) and 2-facet twist drill (T2).

for the points bordering the point of transition between chisel edge and cutting lips, while for the rest of the regions the difference is not very obvious. Our model captures better the increase on the chisel edge area than in its vicinity on the cutting lips, an effect which can be improved by considering a power law for the specific cutting pressure on the rake face (Kc ), aspect discussed previously. From Fig. 5 we can also observe the important role of not neglecting the cutting angle along the cutting lips region. If the cutting angle was to be assumed zero along the cutting lips (as in all the previous cutting force models), a constant offset would have been observed between the two distributions curves (corresponding to the two cases with varying axial feed) along the cutting lips. As the experimental data shows clearly that increasing the axial feed will cause an increase of the elementary forces more pronounced at the side of the cutting lips bordering the chisel edge, while almost no increase for the rest of the cutting lip; we can therefore state that such a variation could only be accounted for by considering the cutting angle () in this region, and its effect on all angles involved in forces definitions and decomposition. Next, in Fig. 6 we present the results obtained for the total thrust and torque (Eqs. (17) and (18)) for the 2-facet twist drill (T2) with the coefficients (structured according to M1) obtained by fitting the distribution curves (see Table 2). The thrust (Fig. 6a and b) seems to be predicted accurately although the largest error is noted always at the maximum feed rate (f = 0.20 mm/rev). The predicted results for torque (Fig. 6c and d)

tend to underestimate the experimental values especially at higher axial feeds. The prediction capabilities of the current model for the total thrust and torque values can be greatly improved by using the coefficients obtained by fitting on the total values of the forces rather than their distributions. For the same combination of workpiece/drill as before and using the alternative set of coefficients as in Table 2, the results are presented in Fig. 7. For the twist drill family (T2) we can attempt a comparison of our model performance with previous prediction models developed or tested for composite materials. Langella et al. (2005) introduce a thrust and torque prediction model tested for drilling glass-fiber composite materials with twist drills 8 mm in diameter and of various point and helix angles. By assuming an average rake angle in their calculation, their model is not suitable for studying the cutting force distribution, but only the total thrust and torque values. Their model manages to predict the experimental results within 8% margins, while noting a variation interval of 5% for their experimental data. When fitted on the total thrust and torque values, our model predicts the total thrust and torque values within the variation interval with the exception of the thrust force for GFRP work-pieces (Fig. 7b), where the error is lower than 10%. The performance is repeated for the tapered drill reamer (T1) as will be later discussed. Another mechanistic cutting force model tested on unidirectional CFRP materials drilled by standard twist drills is

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introduced in (Chandrasekharan et al., 1995) and has an accurate geometrical description of the process and could in theory model the cutting force distribution as well. The magnitude of the forces reported in Chandrasekharan et al. (1995) is significantly greater than our values mainly due to the difference in drill diameters (5.6 mm in our case while 15.9 and 12.7 mm in the previous study). For comparison, their results are plotted together with the current ones in Fig. 8. It can be observed that the previous model greatly underestimates their experimental results for composite materials. In Chandrasekharan et al. (1995) and Chandrasekharan (1996) the same model is applied to metals with better performance. We believe that the main reason for their poor performance for composite materials while the model performs well for metals is due to

the assumption that the forces on the relief face can be ignored in both cases. If the forces on the relief face would have been ignored, the line (curve) uniting our predicted results in Fig. 8 would have been forced to pass through the origin (thrust equal zero when axial feed is zero), obtaining similar performance as Chandrasekharan et al. (1995). FEM models for composite drilling focus mostly on delamination or determination of the critical thrust force (Durao et al., 2006; Zitoune and Collombet, 2007). A recent study (Isbilir and Ghassemieh, 2011) attempt the simulation of the complete drilling process for uni-directional CFRP composites using a 8 mm twist drill. For their model validation one experimental case is compared with the simulation results showing the prediction capabilities of

Fig. 12. Variation along the radius of the most important angles affecting the cutting forces for the tapered drill reamer (T1) and 2-facet twist drill (T2).

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within 5%. The reported values are also plotted in the comparison chart (Fig. 8). The tapered-drill reamer (T2, described in Appendix) is a type of drill developed recently specifically for drilling composite materials. There is very little literature on its performance that we can relate to, except the experimental study by Fernandes and Cook (2006a,b). Although force values are reported, the study uses thin composite plates where full-engagement in not reached, therefore not easily comparable with our results. Their empirical model is reported to capture well the experimental measured force values and includes a compensation for tool wear. The experiments reported by Lazar and Xirouchakis (2011) show that the distribution of the elementary cutting forces has a totally different pattern than for the previously analyzed drill. Using the M1 model fitted on the experimental distribution of the elementary cutting forces, the following curves (Fig. 9) are predicted for the tapered-drill reamer case. It is observed that the distribution of the elementary cutting forces along the radius is not captured perfectly for this drill; especially around r = 1.8 mm (corresponding to the point of engagement of the secondary flutes – see Fig. 11a). The variations of the geometrical parameters as introduced in Appendix, do not justify the significant increase in the axial force component during these stages, such as recorded in the experiments. The following 2 hypotheses are proposed to justify this behavior, although at the current stage they cannot be accounted for in the model: (i) The point where the cutting forces start to increase unexplained is always slightly ahead of the theoretical point of engagement of the secondary pair of flutes (as in Fig. 11a). It is however probable that the secondary flutes will not engage suddenly (as assumed) but sooner, with a time interval which might vary with the feed rate (depth of cut). Prior to the theoretical point of engagement for the secondary flutes we consider only 2 flutes to be engaged. Additionally, the secondary flute seems to be (visually) poorly prepared for cutting (small relief angle) prior to the theoretical point of engagement, which could result in higher forces than usual. This hypothesis could help explain the rather large values recorded consistently for the second point of the stage 3, but fails to explain the rather large values recorded for stage 4 as well. (ii) At about r = 1.5 mm (see Fig. 11a) there is a drastic change in the point angle (see also Fig. 12a). Although this change is reflected in the variation of the 2nd Euler angle of rotation (as in Fig. 12c) which modifies the proportion between the axial and tangential elementary forces, the normal rake angle (˛n , Fig. 12d) and the inclination angle (i, Fig. 12b) which influence the efficiency of the cutting does not vary much. However, the orientation of the cutting lip changes also with respect to the fibers orientations of the composite materials. As composite materials are highly anisotropic, it might be expected that the material will behave differently and the specific cutting pressure (Kc ) might experience a sudden change. However, the current model does not take into account the relative orientation between the cutting lip and cutting direction and the fibers orientation. Although the fitting method on the distribution curves does not yield satisfying results, when fitted on the total thrust and torque values, the model becomes very accurate as outlined in Fig. 10. 4. Conclusions The current paper introduces a mechanistic model for predicting the cutting forces occurring during drilling of fiber reinforced

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composites, focused at obtaining reliable distribution along the cutting edges of the drill and assuming the material is isotropic. Unlike metals, where the structure is isotropic, composite materials have a strong anisotropy, leading to different responses under drilling loads at various points within their structure, seldom resulting in internal defects (i.e. delamination). Better understanding of this particular type of defects can be achieved by studying the distribution of the forces along the cutting edge, radius or workpiece structure. By considering 2 very different drills of roughly similar diameter the experimental results reported by Lazar and Xirouchakis (2011) show that the elementary forces are distributed differently and probably causing different extents of delamination defect. Simple empirical modeling of the cutting forces would not allow differentiating between different drill geometries, while currently available mechanistic models are not calibrated or discussed against experimental information of the forces distribution. FEM models to calculate the thrust and torque forces are reported as taking too much time (4 months for a complete drilling process run – Isbilir and Ghassemieh, 2011). Our results show that mechanistic models are able capture the distribution of the elementary cutting forces in drilling fiber reinforced composites as a function of the axial feed and the drill geometry. For relevant distributions, such models need to be calibrated on experimental distributions (a proposed solution to obtain such is presented in Lazar and Xirouchakis, 2011) rather than total values of thrust and torque. For extending the application of mechanistic models to more drill geometries, these need to be described mathematically. Providing detailed mathematical description of the active faces of the drill (at least in the close vicinity of the cutting edges) is rather difficult, because (i) drill manufacturers disclose only functional geometrical aspects of the drill geometries; (ii) there is no methodology developed for measuring these angles and (iii) it is mathematically complex to relate drill manufacturing geometrical parameters (tool-path and profiles of the helical grooves grinding process) to the geometrical parameters of interest in cutting force modeling (i.e. normal rake angle). The separate consideration given to the forces acting on the relief face proves to be the driving factor in increased accuracy of the current model in comparison with previous mechanistic models such as Chandrasekharan et al. (1995). When compared with Langella et al. (2005) the performance in predicting the total forces is similar, but the accurate modeling of the drill geometry allows the study of the cutting force distribution as well. Considering the cutting angle () along the cutting lips is believed by the authors to be critical in capturing the variation of the cutting forces distribution with the axial feed (see Fig. 5). The current model also shows that, at least for composite materials, there is no need to consider two different cutting force models for the central part of the chisel edge (as the indentation zone employed by Chandrasekharan et al. (1995) and other models developed for metals) and the remaining cutting edges. Additionally, the final number of the empirical coefficients can be maintained to about 4 (as opposed to more than 10 in Chandrasekharan et al., 1995 and 8 in Langella et al., 2005). Further increase of the number of empirical coefficients does not lead to significantly improved results and does not justify the arbitrary addition of more degrees of freedom. Hocheng and Tsao (2003), Zhang et al. (2001) or Zitoune and Collombet (2007) have proposed the concept of critical thrust force in drilling of composite laminates over which value delamination occurs. These models assume a thrust distribution along drill radius and work-piece thickness based on experimental data. The current model provides not only the magnitude of the thrust force, but also its distribution for different drill geometries and cutting parameters.

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The authors would also like to note that the cutting process for composite materials differs fundamentally from the case of metals, and as the shear angle theory was developed by Merchant (1945a,b) for metals, a similar fracture-based analytical model is needed for composite materials. Orthogonal/oblique cutting experiments with force measurements on long fiber reinforced composites could show the influence of the fiber orientation, normal rake angle and cutting velocity over the specific cutting pressure (Kc ) which could help the improvement of the current model and provide background for development of the fracture-based cutting model. The chip flow law adopted in the current model also needs to separately be confirmed (or infirmed) for drilling, while experiments using one-fluted drills would allow the calibration of mechanistic model on all 3 elementary directions (currently only 2 directions are possible). The quality of the experimental distribution of the cutting forces could be improved by considering higher diameter drills or employing rotating dynamometer for torque measurements. Acknowledgements The funding of this project was assured by Swiss Federal Institute of Technology Lausanne (EPFL), which did not impose any restrictions for study design; the collection, analysis and interpretation of data; in the writing of the report; and in the decision to submit the paper for publication. The authors would like to convey their gratitude to Dr. Ian Stroud for proof-reading the text of the current article.

functions have been used to define the point angle (p) for each tool type considered (pT1 and pT2 ), based on optical measurements:

pT1 (r) =

- T1 – tapered drill-reamer (also known as a “one-shot” drill bit) is a tool specially designed for drilling fiber reinforced composites, but not suitable for metallic materials. The tool is made of solid carbide steels with small grain size (micro-grain). It has 4 straight flutes while only 2 are engaged in cutting initially. Due to the configuration chosen of 4 flutes, the chisel edge area is very small in comparison with other drills. The tool tip has a 2-stage point angle, featuring an unusually long tip. A diameter of 5.55 mm was selected so that the tool tip was smaller than the work-piece thickness (10 mm) and full engagement was obtained, even if for a very short interval. The supplier is Starlite Industries. - T2 – 2-facet twist drill is a high-quality twist drill used extensively for drilling metallic materials. Unlike the conical twist drill, it has a straight chisel edge which makes an angle with the tool axis smaller than 90◦ . The flank faces are ground to planar surfaces, with no additional features. The material is also solid carbide steel. A 5.6 mm diameter was selected from Tusa Carbide. The drills are also employed in the experimental study reported in Lazar and Xirouchakis (2011) with the mention that only two out of the three are considered here (drill no. T3 in Lazar and Xirouchakis, 2011 is referred to as T2 in the present paper). Fig. 12 shows their photograph, measured profiles and sketches outlining their features. (a) Point angle (p) Point angle is one of the drill parameters usually supplied in drill catalogues. Its practical use is to define the angle of the revolution cone generated by the drill tip. Drills designed in stages have several point angles (as T1). In the current study the following

⎪ ⎩ 

pT2 (r) =

0 < r ≤ 0.125

59.5◦ ,

0.125 < r ≤ 1.485

8.6◦ ,

1.485 < r ≤ 2.773

80.27◦ ,

0 < r ≤ 0.5075

56.75◦ ,

0.5075 < r ≤ 2.8

⎫ ⎪ ⎬ ⎪ ⎭

(19)

(20)

These functions are plotted against the radius in Fig. 12a. (b) Web angle (ˇ) Web angle (ˇ) is defined as the angle between the radial direction at a point on the cutting edges and the projection of the cutting edge on a plane perpendicular to the drill axis. For the chisel edge area, the web angle will always be zero, while the cutting lip region is determined by the generally accepted equation: ˇ(r) = arcsin

w

(21)

r

where w is half of the web thickness. Considering the particularities of the two drills, the following equations have been used: ˇT1 (r) =

Appendix. The appendix is provided in order to introduce the geometrical aspects of the two drill types modeled in the current paper:

⎧ ◦ 90 , ⎪ ⎨

ˇT2 (r) =

⎧ ◦ ⎨0 ,

⎩ arcsin ⎧ ◦ ⎨0 , ⎩ arcsin

 0.115  r

 0.442  r

0 < r ≤ 0.125 ,

0.125 < r ≤ 2.773

(22)

0 < r ≤ 0.5075 ,

0.5075 < r ≤ 2.8

(23)

(c) Cutting angle () The cutting angle (), also known as the “feed angle” is defined as the angle at a point on the cutting edge between the local velocity vector and its tangential component. Independently of the geometry, is can be derived using the following equation: (r) = arctan



f 2··r



(24)

It can be observed that it becomes significant for values of the radius comparable with the axial feed rate (f). (d) Inclination angle (i) The inclination angle (i) is an angle characteristic to oblique cutting, defined as the angle between the cutting edge and the normal to the velocity vector in the plane containing both the velocity and the cutting edge. In drilling it is derived by the commonly accepted Eq. (25), although in many cases its simplified version is employed (with the cutting angle –  – equal to zero). i(r) = arcsin[sin(ˇ) · cos() · sin(p) + sin() · cos(p)]

(25)

Its variation along the radius for the two drills considered in the current study is outlined in Fig. 12b. It is noted that usually for the chisel edge the inclination angle is zero (as in the case of T1, for which the point angle in this region is 90◦ ). However, for T2 the chisel edge is inclined with respect to the tool axis with an angle (p) smaller than 90◦ , therefore a positive value of the inclination angle is recorded, increasing with the cutting angle (). (e) 2nd Euler angle of rotation ()

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This angle was introduced for simplification reasons as defined by Eq. (1) in the main body of the current paper. It is found interesting to present its variation along the radius (Fig. 12c) for the drills employed in the current study as it strongly affects the decomposition of the elementary cutting forces along the reference directions in drilling. (f) Normal rake angle (˛n )

 ˛n,l = arctan

tan(l ) cos(ˇ) sin(p) − cos(p) tan(l ) sin(ˇ)

sin(ˇ) cos(p) − sin(p) tan() − arctan cos(ˇ)

r R



,

 (26)

for Rc < r ≤ R

(27)

where Rc is the radius of the chisel edge area. According to the supplier’ catalogue, T2 has a helix angle () of 25◦ , while for drill T1 the helix angle is zero and therefore Eq. (26) can be simplified. Eq. (26) is valid for all points of the cutting edges as it provides a generic relationship to calculate the normal rake angle (˛n ) starting from a reference angle made by the rake face with a parallel to the drill axis measured in a plane normal to the radial coordinate –  l in the case of the cutting lip region. However, the chisel edge is ground by a different surface, and the reference angle has to be defined differently than for the cutting lips region (by Eq. (27)). Most chisel edges are ground by conical faces, and the exact determination of this angle is rather difficult as outlined by Armarego and Rotenberg (1973a,b). In cutting force modeling, the assumption is usually made that the rake face of the chisel edge (and consequently the relief face as they are symmetric) are ground by a planar face. As the inclination angle is also zero for most of the practical cases along the chisel edge; and orthogonal cutting can be assumed, the reference angle is defined as the static rake angle (˛s ). T1 (tapered drill reamer) employed in the current study has a very small chisel edge and our optical observation indicated that the surfaces forming the chisel edge are in fact planar; while 2-facet twist drill’s chisel edge is also ground by planar faces. However, the 2-facet twist drill (T2) has a chisel edge inclined with respect to the drill axis, leading to an inclination angle greater than zero and eliminating the assumption of orthogonal cutting. Therefore, the equation defining the static rake angle had to be generalized to include the second case:

 ˛s = − arctan

(29)

Summarizing, the normal rake angle for the two drills considered was calculated with the following functions, plotted in Fig. 12d: ˛n−T 1 (r) =

˛n (r),

0 < r ≤ 0.125;

˛n (r),

0.125 < r ≤ 2.773;

l = ˛s ;

w = 0.115 mm;

Rc = 0.125 mm

 = 0◦

(30)

 ˛n−T2 (r) =

˛n (r), ˛n (r),

0 < r ≤ 0.5075; l = ˛s ;

w = 0.442 mm;

0.5075 < r ≤ 2.8;

◦

Rc = 0.5075 mm

 = 25

(31)

(g) Relief angle ()

For the cutting lips region,  l (referred to as the local helix angle) can be determined from the reference helix angle (, defined at the outermost point on the cutting lip – R), provided by the drill manufacturers:



w Rc

sin( ) =





l (r) = arctan tan() ·

projection on a plane perpendicular to the drill axis and calculated based on the web thickness – w):



Normal rake angle was found to be the representative angle to describe the orientation of the rake face in oblique cutting. It is defined as the angle between the rake face and the normal to both the velocity vector and the cutting edge, measured in a plane perpendicular to the cutting edge. The following equation (Eq. (26)) is used to determine the normal rake angle in our model. Its derivation is presented in Lazar (2012), and differs from previously published equations by the removal of the assumption that the cutting angle () is zero.

259

tan(pl ) · sin( ) · sin(p) sin(p) + tan(pl ) · cos(p) · cos( )

Relief (or clearance) angle () is the angle describing the relief face, defined as the angle between this surface and the velocity vector at any point on the cutting edges of a drill, and measured in a plane formed by the velocity vector and the normal to both velocity vector and cutting edge. For the cutting lip region, it is evaluated starting from a reference relief angle ( 0 ), defined in a similar manner as the helix angle, at the outermost point of the cutting lip, between the tangent to the relief face in this point and its projection in a plane perpendicular to the drill axis. For multi-stages drill points (as in the tapered drill reamer T1) reference relief angles should be known at each stage, although they are not always provided by the drill manufacturers. Assuming the cutting lips are straight lines, for a cutting lip segment, the variation of the relief angle can be describes by the following equation, where  0 and R (the outermost radii of the segment where  0 is measured) are provided for each segment. Eq. (32) was used in the current model to calculate the relief angle for the cutting lips regions and it is similar to the ones used in previous models, although it has been slightly adapted to accommodate multi-staged drills.



l (r) = arctan

where pl is the point angle of the cutting lip segment closest to the chisel edge and is the chisel edge angle (defined as the angle between the chisel edge and the cutting lips when measured in

tan(0 ) tan(ˇ(r)) − tan(ˇ(R)) + tan(p) cos(ˇ(R))



− (r)

(32)

For calculation, we have assumed the reference relief angle  0 = 14◦ for all cutting lip segments of each drill considered hereby. For the chisel edge area, the wedge is symmetric and the static rake (˛s , calculated with Eq. (28)) and relief angles are equal, but of opposite sign. Considering also the influence of the cutting angle (), the following equations have been used to estimate the variation of the relief angle along the drill radius for the considered drills:

⎧ ◦ ⎨ 90 + ˛s − (r), T1 (r) =

⎩

0 < r ≤ 0.125;

w = 0.115 mm; ◦

l (r),

0.125 < r ≤ 1.485;

l (r),

0.1.485 < r ≤ 2.773;

0 = 14 ;

Rc = 0.125 mm R = 1.485 mm

0 = 14◦ ;

R = 2.773 mm

(33)

 (28)



cos(ˇ(r)) ·

 T2 (r) =

90◦ + ˛s − (r),

0 < r ≤ 0.5075;

l (r),

0.5075 < r ≤ 2.8;

w = 0.442 mm; 0 = 14◦ ;

Rc = 0.5075 mm

R = 2.8 mm (34)

Eqs. (33) and (34) are plotted against the radius in Fig. 12e:

260

M.-B. Lazar, P. Xirouchakis / Journal of Materials Processing Technology 213 (2013) 245–260

(h) Chip flow angle () As defined in the main body of the article, the chip flow angle was considered proportional to the inclination angle (i) by a factor of one for all drills: c (r) = i(r)

(35)

(i) Number of flutes (N) In practice, most drills have two flutes, as the case of 2-facet twist drill (T2). However, the tapered drill reamer (T1) has 4 flutes although not all of them are always engaged in cutting. It was found that:



NT1 (r) =

2;

0 < r ≤ 1.868

4;

1.868 < r ≤ 2.773

NT2 (r) = 2;

0 < r ≤ 2.8

(36) (37)

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