Analytical modeling of thrust force and torque in drilling

Abstract

The prediction of thrust force and torque in drilling remains a key issue. There are three main ways to determine these forces based on experimental, numerical and finally analytical approaches. The major drawback with numerical and analytical methods concerns their reliability compared to phenomenological models. As a consequence, several studies use a resetting method in order to correct parameters of their analytical or numerical models so that they correspond to experimental results. The goal of this article is to introduce a new analytical model in drilling based on the discretization of the cutting edge. Local forces are estimated with a semiorthogonal analytical model based on a modified Merchant’s model. Parameters have been identified by a basic semiorthogonal cutting test for a large range of cutting speed and feed rates, by friction tests for a range of sliding velocities and by a variable shear angle model. The macroscopic feed force and torque are estimated by the sum of each local force along the cutting edge. Two drills applied in a large range of cutting conditions are investigated to validate this approach.

Keywords

Cutting drilling thrust force analytical modeling resetting procedure

Introduction

Context

In the context of global competition, the industry requires rapid methods in order to predict forces in drilling without making experimental tests on real costly prototype tools. In order to optimize the machining operations, it is important to predict the forces produced by cutting processes. Three main ways are commonly used.

The first approach consists in identifying a phenomenological model based on experimental measurements. This model provides a feed force and torque depending on the cutting speed and feed rate for a specific cutting tool and work material. For instance, the standard tool material pair (TMP)¹ is a way to optimize the cutting configuration by chip fragmentation control. Stephenson and Agapiou,² Altintas³ or Audy⁴ exposed the advantages and limits of phenomenological models, which give accurate values, but poor extensibility. Additionally, any modification of the tool design or manufacturing process requires at least 2 months before machining any experimental validation. As a consequence, the industry is keen to develop virtual models to predict rapidly the forces for a large range of tool designs. Two approaches can be used: numerical and analytical models. Numerical cutting models become even more popular, thanks to the improvement of computer and computational mechanics. Most of the studies were concerned by orthogonal cutting.^5,6 Concerning numerical drilling models, few studies have been published^7,8 even if a lot of advertisement can be found on the websites of software dealers. The major limitation of these drilling models is related to the geometrical complexity of the cutting operation, which increases the computational duration (several days) to simulate not more than two revolutions of the tool. Additionally, these models use constitutive mechanical models identified for a very limited number of work materials. The identification of each new one is a tough and costly job.⁹ Finally, these models do not consider any variation of the friction coefficient over the cutting configuration (velocity and pressure), even if the authors have shown a strategic influence of this parameter^10–13 on chip formations and forces. As a consequence, the application of numerical models is limited to some research laboratories to predict qualitative results.

The third approach consists in using analytical models. Most articles use the discretization of cutting edges,^14–22 as illustrated in Figure 1. The macroscopic feed force and torque are calculated by the sum of each local force along the cutting edge. Local forces depend on the rake angle $γ_{n}$ , inclination angle $λ_{s}$ , cutting speed $V_{c}$ , uncut chip thickness $t_{1}$ , friction coefficient $μ_{t}$ and work material behavior.

Figure 1.

Plane orientation for measuring angle on a drill.²³

Various methods have been used to predict all the local forces. Some authors use drilling tests with one specific investigated tool and one specific work material.^14–22 In this case, the local forces are estimated after tests with specific geometric and decomposition models. The main disadvantage of this method is that these specific drills have to be ground before any experimental identification of the model parameters (as mentioned earlier, almost 2 months are required for manufacturing and control). So, this limits its interest compared to phenomenological models that would be more accurate.

The second way consists in using generic cutting tests in order to identify the analytical flow stress model of the work material. For instance, Elhachimi et al.^18,24 and Dargnat et al.²⁰ use the Oxley model, whereas Jrad et al.¹⁹ have chosen a Johnson–Cook model. Elhachimi and Jrad use model parameters from the literature, which are limited to few work materials. Based on this status, Dargnat has developed a strategy to identify the parameters from orthogonal cutting tests (mainly with a high-speed camera). This approach is efficient, but its replication for a large-scale application is limited.

Finally, all models consider a constant friction coefficient along the cutting edge but it has not been quantified by any tribological tests. In particular, Zemzemi et al.²⁵ have highlighted the fact that the friction coefficient can vary from 0.2 to 0.6 when the cutting speed decreases from 200 to 10 m min⁻¹, which is consistent with the variation of the cutting speed along the edge of a drill for steel.

As a consequence, it became necessary to develop a new methodology capable of defining a local force model that can be easily identified through basic cutting and tribological tests. Thereby, this model will not be dependent on specific tool geometry and can therefore be used for complex tools. This force model will be validated for orthogonal and oblique cutting in the context of an AISI 4142 steel (300 HB) machined with TiN-coated solid carbide tool. Next, it will be applied to complex tools, such as drills. Two modern carbide drills with web thinning, instead of conventional high-speed steel (HSS) drills with chisel edge commonly used in the literature, will be used for validation. The objective of this article is to present this new methodology.

Description of the proposed methodology

As has been shown earlier, three main factors have to be closely studied: tool geometry, orthogonal cutting experiments and friction law. The following Figure 2 exposes this methodology. From a defined tool and work material, two major consequences can be estimated by experience or previous knowledge: the a priori accurate tool geometry (defined by the tool manufacturer) and conventional cutting conditions. The latter give the framework for cutting and friction tests, mainly the cutting speed and feed rate. From tool grinding (or directly from a virtual tool, as in this article), the local cutting configuration can be determined. Coupled with a cutting force model identified from cutting and tribological tests, the local forces can be calculated. Finally, by a sum of each local element, the global forces can be computed. In the next part, each factor (geometry, cutting and friction) will be studied in order to feed the model.

Figure 2.

Synoptic survey of the proposed method.

Analytical force model

Drill geometry

In this study, two geometries of TiN-coated solid carbide drills (K40 grade, with 10% of cobalt) are used. The main characteristics of these tools are listed in Table 1. Tool 1 exposed a wavy main cutting edge, a corner edge chamfer, two relief faces, a thinning face and a special flute section in order to promote smooth chip fragmentation and evacuation. Tool 2 has a straight main cutting edge and a wide flute opening.

Table 1.

Tool characteristics.

	Tool 1	Tool 2
Diameter (mm)	6	9
Helix angle (°)	30
Point angle (°)	145
Edge preparation	Rounded 30 µm	Rounded 30 µm
First clearance angle (°)	10	12
Illustration

The first step is to identify the local geometry at each point of the drill edge. The cutting edge of the drill is splitted into a finite number of elementary edges. For each of them, it is possible to define the local cutting configuration of the point considered. The French standard²³ has been chosen in order to measure the cutting edge geometry and its evolution with the radial position.

It is therefore possible to apply this method to each elementary cutting edge. The evolution of angles is shown in Figure 3 for both tools, expressed as a radius function. The Zone 1 corresponds to the main cutting edge, while Zone 2 stands for the chisel edge. The oblique configuration is defined by a nonzero $λ_{s}$ . In this case, the center of the drill is an orthogonal cut. In contrast, the main edge is always oblique. Finally, the cutting speed is $V_{c} = 90 m {min}^{- 1}$ at the external diameter of the tool.

Figure 3.

Angle evolution for both Sirius and Polaris geometry types, expressed as a function of the radius.

Cutting configuration

According to Figure 3, it is possible to determine the cutting configuration on each elementary edge of the tool. This leads to Figure 4, which represents the cutting conditions retained for our orthogonal and oblique cutting test. In fact, we choose a complete range of cutting speeds and feed rates for the orthogonal cutting (1) because of the real cutting configuration on the drill and (2) in order to be in line with Merchant’s model hypothesis. For the rake angle, $γ_{n}$ , the highest value is reached in the outer diameter, where the cutting speed is the highest. For the inclination angle, the nonzero zone (oblique cutting) is in the middle of the cutting edge, where the cutting speed ranges from $15$ to $60 m {min}^{- 1}$ . It is to be noted that in the center part of the drill, it is well known that the indentation phenomenon prevails over cutting. So the cutting configuration of the chisel edge is not taken into account in this study.

Figure 4.

Cutting configurations for the elementary cutting test.

Friction coefficient

While Merchant retains a constant friction coefficient and does not include the cutting speed, this approach uses a friction coefficient that depends on the sliding velocity $V_{g}$ at the tool/chip interface. This method has been developed by Zemzemi et al.²⁵ and applied to the machining of AISI 4142 at a hardness of 300 HB with carbide TiN-coated tools and tool–piece speed friction ranging from $40$ to $300 m {min}^{- 1}$ . It uses a tribometer especially designed to simulate the friction conditions at the tool/chip interface.

Furthermore, linking the overall cutting speed $V_{c}$ to the tool/chip scrolling speed $V_{g}$ , the compression ratio $t_{1} / t_{2}$ is used (where $t_{1}$ is the uncut chip thickness, that is, the feed rate, and $t_{2}$ is the chip thickness). To maintain the consistency of units, this law uses a ratio between the cutting speed and a reference speed, chosen among the range of experimentation here, $V_{ref} = 150 m / {min}^{- 1}$ . The model used in this study has been developed for this tool/work material/coating. It is based on the studies by Zemzemi et al.,²⁵ Claudin et al.²⁶ and Naisson et al.²⁷ The coefficient of friction $μ_{t}$ at the tool/chip interface or the angle of friction $λ_{t}$ is connected by relation (1)

μ_{t} = 0.26 \times {(\frac{t_{1}}{t_{2}} \times \frac{V_{g}}{V_{ref}})}^{- 0.43}

(1)

Cutting force models based on Merchant’s approach

The cutting model from Merchant^28,29 is chosen. It is a model in which three physical phenomena are taken into account. The first one concerns the behavior law for shearing the work material. Merchant uses a Bridgman-type model³⁰ with two parameters $S_{0}$ (shear stress under zero normal stress) and $k$ (slope of shear stress vs normal stress) depending on the work material. This material law links the shear stress $S_{s}$ to the normal stress $σ_{n}$ in the case of plasticity, as expressed in equation (2). Merchant uses this model to calculate the stress in the shear band. This model does not take into account the temperature and strain rate. These two parameters can be selected from the literature, but conventional methods can shift to better agreement with experimental data

S_{s} = S_{0} + k σ_{n}

(2)

The model gives access to cutting forces and advances in orthogonal cutting. The cutting conditions (speed and angle) are identified in the plane $P_{n}$ . Notation configurations are repeated in Figure 5. The main hypothesis formulated by Merchant is a steady-state cutting process and plane strain, with no variation in the chip section and a cutting speed fast enough to have no built-in edges. The pressure in the shear band is assumed constant.

Figure 5.

Orthogonal and oblique cutting configuration and notation.

The principle for determining the primary shear angle $φ$ is based on minimizing the total power dissipated during the cutting operation. The latter assumption leads to the following equation

φ = \frac{π}{4} + \frac{γ_{n} - λ_{t}}{2}

(3)

where $λ_{t}$ is defined as $λ_{t} = \tan^{- 1} (μ_{t})$ .

However, Moufki et al.¹⁰ and Molinari and Moufki³¹ have shown that Merchant’s hypotheses are not consistent with experimental measurements of the shear angle. They have introduced the following formulation

φ = A_{1} + A_{2} \times (γ_{n} - λ_{t})

(4)

where $A_{1}$ and $A_{2}$ are material-dependent coefficients.

A geometric relationship (5) gives the chip thickness $t_{2}$ from the uncut chip thickness (or feed rate in this case) $t_{1}$ and output from the shear zone, the shear angle $φ$ and the cutting angle $γ_{n}$

\frac{t_{1}}{t_{2}} = \frac{\sin φ}{\cos φ - γ_{n}}

(5)

The perfect plastic behavior requires the knowledge of two parameters related to the material, $k$ and $S_{0}$ , to determine two intermediate variables $S'_{s}$ and $C$ (equations (6) and (7)), necessary to calculate the cutting forces (equations (8) and (9)). C corresponds to the condition of plasticity, while $S'_{s}$ is the average shear stress

S'_{s} = \frac{S_{0}}{1 - k \tan (φ + λ_{t} - γ_{n})}

(6)

C = 2 φ + λ_{t} - γ_{n}

(7)

Finally, with these parameters, it is possible to determine the cutting and feed forces $F_{c}$ and $F_{f}$ as follows

F_{c} = w \times t_{1} \times S'_{s} \times (\tan (C - φ) + \frac{1}{\tan φ})

(8)

F_{f} = w \times t_{1} \times S'_{s} \times (\frac{\tan (C - φ)}{\tan φ} - 1)

(9)

where $w$ is the width of cut.

Oxley³² has shown that it is possible to obtain coherent radial cutting forces in the case of the oblique cut (with a small error) by taking an orthogonal cutting model for the cutting and penetration forces, while the radial stress in relation (10) is obtained from these efforts and the flow angle of the chip $η_{c}$ . In contrast, the cutting conditions (speed and angle) must be identified in the plane $P_{n}$ (which is consistent with the previous choices for local cutting edge orientation systems)

F_{r} = \frac{F_{c} (\sin λ_{s} - \cos λ_{s} \sin γ_{n} \tan η_{c}) + F_{f} (\cos γ_{n} \tan η_{c})}{\sin λ_{s} \sin γ_{n} \tan η_{c} + \cos λ_{s}}

(10)

The chip flow angle $η_{c}$ is difficult to measure. Jrad et al.¹⁹ show that the simple hypothesis $η_{c} = λ_{s}$ is not suitable in the case of drilling, mainly because of chip continuity. The Oxley formulation of $η_{c}$ is chosen here

η_{c} = \frac{λ_{s}}{V_{c}^{j}}

(11)

In the present case, the parameters to identify are $k$ , $S_{0}$ , $A_{1}$ , $A_{2}$ and $j$ . To do so, an inverse method will be used. Test cutting can give access to macroscopic quantities necessary to obtain the model parameters. These tests are presented in the following paragraph.

Experimental resetting procedure

Cutting test

The effort pattern is readjusted over a wide range of cutting conditions, in terms of both speed and configuration angle (rake and inclination angle). These cutting configurations are chosen according to the ones observed on the selected drills presented earlier (Figure 4). To make the test feed the model, it is necessary to design devices for placing the cutting edge in the correct position. The real cutting edge of the drill exposes various cutting configurations. The developed mounting has to reproduce these configurations; 3-mm-thick and 80-mm-diameter disks of AISI 4142 (300 HB) were used. Efforts in the three directions were recorded by a Kistler dynamometer with a sampling frequency up to 2 kHz. The efforts of the stabilized machining operation are selected thereafter. It is possible to perform tests with configurations of rake angle $γ_{n}$ and inclination angle $λ_{s}$ at 0° , 15° and 30° (Figure 6).

Figure 6.

Cutting test setup.

The tests were conducted on a computer numerical controlled (CNC) lathe. Solid carbide (10% Co) TPKN 1603 PPR inserts without chip breaker coated with TiN are used to machine the disk. These inserts have undergone a special sharpening operation not only to improve and stabilize the edge preparation $R_{β}$ at 30 μm but also for a clearance angle up to 35° (instead of 11° normally) to avoid tailgating when the angle of cutting $γ_{n}$ is 30°.

The cutting tests provide a great amount of data. First, forces in all directions are posttreated (divided by the disk width). Moreover, as shown earlier, it is necessary to know the thickness of the chips to solve the equations of Merchant (among others to find the shear angle $φ$ ). The chips are all without scalloping (nonsegmented Merchant type 2 chips). The chips are coated and then measured under a binocular microscope. For each configuration used, two tests are performed. The average repeatability error is 3% on the cutting force and 6% on the penetration effort. Regarding the measurement of the chip thickness, the error is around 6%. Tables 2 and 3 hereafter contain these measurements, according to the cutting configuration.

Table 2.

Orthogonal cutting configurations, AISI 4142 steel at 300 HB.

Configuration	Vc ( $m / mi n^{- 1}$ )	f ( $mm {rev}^{- 1}$ )	$F_{c}$ (N)	$F_{f}$ (N)	$F_{r}$ (N)	$t_{2}$ ( $μ m$ )
1	40	0.05	231	213	−3	206
2	75	0.05	195	186	−3	140
3	100	0.05	183	183	−2	132
4	25	0.1	324	207	−4	266
5	40	0.1	338	274	−4	286
6	75	0.1	300	242	−2	219
7	100	0.1	277	220	−1	186
8	175	0.1	263	212	−4	171
9	20	0.18	586	423	−6	537
10	40	0.18	531	409	−5	448
11	75	0.18	473	344	−5	335
12	175	0.18	421	294	−5	281
13	25	0.25	698	508	−9	624
14	40	0.25	663	485	−9	586
15	75	0.25	588	389	−7	439
16	100	0.25	581	377	−6	371
17	140	0.25	550	339	−7	356
18	175	0.25	544	337	−6	346
19	210	0.25	533	339	−6	332
20	250	0.25	520	361	−6	325
21	175	0.32	646	354	−8	422
22	175	0.4	724	513	−1	501

Table 3.

Oblique cutting configurations, AISI 4142 steel at 300 HB.

Configuration	Vc ( $m / {min}^{- 1}$ )	f ( $mm re v^{- 1}$ )	$γ_{n}$ ( $\circ$ )	$λ_{s}$ ( $\circ$ )	$F_{c}$ (N)	$F_{f}$ (N)	$F_{r}$ (N)	$t_{2}$ (µm)
23	75	0.05	0	0	164	132	2	168
24	100	0.05	0	0	164	149	1	181
25	25	0.1	0	0	331	230	2	417
26	40	0.1	0	0	294	211	4	305
27	10	0.18	0	0	567	324	3	770
28	20	0.05	0	15	158	93	12	225
29	75	0.05	0	15	163	132	17	165
30	100	0.05	0	15	165	151	20	170
31	20	0.1	0	15	330	214	27	459
32	75	0.1	0	15	285	218	31	264
33	20	0.25	0	15	713	527	78	845
34	75	0.25	0	15	562	321	85	469
35	40	0.05	0	30	188	141	33	195
36	100	0.05	0	30	164	152	40	167
37	40	0.1	0	30	312	227	69	346
38	100	0.1	0	30	276	217	80	246
39	40	0.25	0	30	635	406	169	616
40	100	0.25	0	30	582	339	200	468
41	20	0.05	15	0	149	77	−2	154
42	75	0.05	15	0	130	66	−1	158
43	20	0.1	15	0	263	129	−4	302
44	75	0.1	15	0	238	126	−4	246
45	20	0.25	15	0	556	213	−6	703
46	75	0.25	15	0	492	177	−5	498
47	40	0.05	30	0	126	57	−1	175
48	100	0.05	30	0	114	46	−1	134
49	40	0.1	30	0	219	82	−4	243
50	100	0.1	30	0	194	62	−3	196
51	40	0.25	30	0	395	49	−5	453
52	100	0.25	30	0	401	59	−6	415
53	75	0.1	30	30	237	108	43	229

Resetting

These tests provide the cutting forces and chip thickness, while the friction coefficient is a variable depending on the cutting speed. For each test, $γ_{n}$ is set by the mounting, and the friction coefficient is calculated using the cutting speed and equation (1), knowing the feed rate (equal to the uncut chip thickness) and the chip thickness. On the whole dataset, the model is calculated and optimized using a simplex method,³³ which is a direct search method that does not require heavy computed resources to find an optimal value. The five parameters ( $S_{0}$ , $k$ , $A_{1}$ , $A_{2}$ and $j$ ) are computed in order to minimize root mean square (RMS) error between the model and experimental data, which are $t_{2}$ , cutting, feed and radial forces.

The validity limits of our model are based on Merchant’s model. We have therefore chosen experimental conditions to readjust the model of material plastic behavior over a wide range. The objective is to estimate the error of the cutting force with operating configurations that are far from optimal conditions but are found all along the cutting edge in many machining operations such as end milling or drilling.

Comparison between model and experimental results

In order to validate the model, the experimental values are compared with the ones calculated in the corresponding cutting conditions (Table 4).Figure 7 shows the evolution of the predicted forces versus the experimental ones. Moreover, an identification of Merchant’s original model is performed (with a fixed friction coefficient of 0.45 and shear angle defined as relation (3)) on the same set of data, in order to measure the benefits of the proposed optimization (friction coefficient function of cutting speed, variable shear angle and optimized chip flow angle). It appears that the present model gave better results than the old one, especially on the feed force. Moreover, it appears that the friction law makes it possible to reproduce the decrease of forces with the increase of the cutting speed. Nevertheless, the cutting force is better predicted than the feed force in every cutting model. In this case, with a simple Merchant’s model, the accuracy is consistent with his hypothesis. The feed force is mainly driven by the thermal aspect in the material behavior law,^6,34 which is not taken into account in this simple model. Edge preparation is not included in the model and generates an additional feed force due to the plowing effect.

Table 4.

Coefficients of analytical model of orthogonal and oblique cutting for an AISI 4142 steel machined with a TiN-coated insert.

Model parameters	$S_{0}$	$k$	$A_{1}$	$A_{2}$	$j$
Value	1248	−0.21	0.889	0.499	0.019

Figure 7.

Comparison between experiments, the original Merchant model and the modified one.

Application to drilling

Drilling tests have been performed in order to compare force and torque with the simulated ones. As was shown in the previous part, the cutting model can fit with experiments under a relative wide range of cutting conditions. This model is applied to predict the drilling thrust force and torque in Zone 1, the main cutting edge. The drills used in this study have been presented earlier. As mentioned previously, the present model is purely about cutting, so it cannot be applied to model the indentation phenomenon occurring in the chisel edge. That is why predrilled holes of Ø $1.5$ for Tool 1 and Ø $2.5$ for Tool 2, which correspond to the web diameter, are performed. In fact, the cutting model is applied to the main cutting edge and the corner edge. Figure 8 shows the evolution of thrust force and torque versus time in the case of the predrilled holes. As one can see, while the torque is mainly driven by the main cutting edge (about 80%), the thrust force depends on the center of the drill, where the cutting speed is too low to speak about a cutting phenomenon. So the model cannot be used to predict force and torque in this zone. The depth of the hole is $10 mm$ , in order to have a stabilized measurement. The forces have been measured with a two-component Kistler rotating dynamometer 9125A. The sampling frequency is fixed to 2 kHz.

Figure 8.

Thrust force and torque for Tool 2 with a Ø $2.5$ predrilled hole in AISI 4142 at 300 HB.

With the previous model, it is possible to calculate the forces for any local cutting configuration along the cutting edge, which has been defined previously for the two types of drill. Figure 9 shows how the cutting, penetration and radial forces are oriented in the drill reference coordinate system. In order to have the total thrust force $F_{p}$ and torque $C_{z}$ on the tool, we have to project local forces, knowing mainly the edge direction angle $κ_{r}$ . Finally, $F_{p}$ and $C_{z}$ are calculated as the sum of each local force. The discretization step is arbitrarily set up to 50 subedges (about $0.075 mm$ in the case of Tool 1 with predrilled hole of Ø $1.5$ ). The discretization step has no influence over the final results (thrust force or torque) if the edge is splitted into more than 30 subedges (or $0.12 mm$ subedge size).

Figure 9.

Forces on a local edge of the tool, according to the reference plane.

Figure 10 compares modeled and experimental forces for Tool 1. The torque is well represented by the present approach. Thrust force and torque increase when the feed rate increases. The thrust force is mainly dependent on the penetration force and radial force, which expose a more significant error than the cutting force. Since the model underestimates the feed force, the global thrust force is lower than the experimental one.

Figure 10.

Thrust force and torque for main cutting edge with Ø $1.5$ predrilled hole, solid carbide drill Tool 1 Ø $6$ TiN coated in AISI 4142 at 300 HB.

For Tool 2 geometry, the thrust force is always underestimated, as can be seen in Figure 11. In the $0.1 mm {rev}^{- 1}$ feed rate case, the penetration force is calculated with about 60% error, which decreases as the feed rate increases. This is mainly due to the poor results of the cutting model in the particular case of the low feed rate for feed force. Nevertheless, in the case of this tool geometry, Zone 1 represents less than 25% of the thrust force, while it is 40% for Tool 1 geometry. For the torque, the new model gives good results.

Figure 11.

Thrust force and torque for main cutting edge with Ø $2.5$ predrilled hole, solid carbide drill Tool 2 Ø $9$ TiN coated in AISI 4142 at 300 HB.

So, as one can see, the model presented, applied to the modeling of thrust force and torque in the case of two drill geometries, allows to predict the working of the tool with good accuracy. The feed rate is the main parameter in this case, while we also find a slight diminution of forces with the increase of the cutting speed. The torque is better modeled than the thrust force and for feed rates higher than $0.2 mm {rev}^{- 1}$ . This is directly linked with the resetting of our model, which exposes better results for the cutting force (which completely drives the torque).

Two limits have been exposed in our approach. First, by discretization, the chip formation is assumed to be independent in each subedge, which is false in the case of drilling. The chip size evolution is much more complex and has to be more closely studied. Another limit is related to the partial simulation of thrust force and torque. In fact, it does not take into account the extrusion phenomenon occurring in the center of the drill. This zone of the tool drives about 60%–80% of the total thrust force and 10%–25% of the torque. The next step will be to add an analytical model, based on indentation theories, for example, and linked with a material behavior law.

Conclusion

The main interest of the approach presented in this article consists in a general cutting force model applied to a particular cutting edge. A really simple analytical force model has been used, coupled with three optimizations: variable friction coefficient, variable shear angle and a more realistic value for the chip flow angle. These ploys allow to find better results in the case of a pure cutting test, both in orthogonal and in oblique cutting configurations. The parameters of Merchant’s model are fitted on various cutting conditions (rake and inclination angles, cutting speed and feed rate) for a simple cutting edge. In this context, it is necessary to readjust the parameters of the plastic behavior of the material. To do so, a strategy has been defined, and an experimental device has been designed for varying operating conditions over a wide area. The results obtained show that the model predicts efforts with a reasonable error on a wide range of cutting conditions. This simple model has been applied to two complex tool geometries. The main cutting edge of the tool has been discretized, in order to analyze the evolution of the cutting configuration along the edge. Then, the local forces have been computed for each subedge. The model proposed in this article applied to drilling gives good results. Predicted forces compared to experimental results show good accuracy. Moreover, trends are respected, and the thrust force is underestimated by about 15%, while torque prediction has less than 10% error. Considering a simple Merchant’s model, and restrictive hypothesis, this result is acceptable. Moreover, this approach allows to predict forces for a virtual tool, which does not have to be ground already. Drills have been used here, but the analytical model is not restrictive, and it can be applied to other complex tool such as reamers or mills.

Footnotes

Acknowledgements

The authors wish to thank all the partners of this project, especially the Tivoly Group for drill supplying.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This study has been carried out in the framework of Forage à grande vitesse vibratoire—FGVV project, which is supported by Viameca and Arve Industries group.

References

AFNOR. NF E66-520-8: domaine de fonctionnement des outils coupants—couple outil-matière—partie 8: mode d’obtention du couple outil-matière en perçage dans le plein. 1st ed. Saint Denis, France: AFNOR, 2000.

Stephenson

Agapiou

. Metal cutting theory and practice: manufacturing engineering and materials processing. 2nd ed. New York: Taylor & Francis, CRC Press, 2005.

Altintas

. Manufacturing automation: metal cutting mechanics, machine tool vibrations and CNC design. New York: Cambridge University Press, 2006.

Audy

. A study of computer-assisted analysis of effects of drill geometry and surface coating on forces and power in drilling. J Mater Process Tech 2008; 204(1–3): 130–138.

Outeiro

Umbrello

M’Saoubi

. Experimental and numerical modelling of the residual stresses induced in orthogonal cutting of AISI 316l steel. Int J Mach Tool Manu 2006; 46(14): 1786–1794.

Llanos

Villar

Urresti

. Finite element modeling of oblique machining using an arbitrary Lagrangian–Eulerian formulation. Mach Sci Technol 2009; 13(3): 385–406.

Marusich

Usui

. Finite element modeling of drilling processes with solid and indexable tooling in metals and stack-ups. In: 10th CIRP international workshop on modeling of machining operations, Reggio Calabria, Italy, 27–28 August 2007, pp.1–6.

Isbilir

Ghassemieh

. Finite element analysis of drilling of titanium alloy. Procedia Eng 2011; 10: 1877–1882.

Burns

Mates

Rhorer

. Dynamic properties for modeling and simulation of machining: an update of results from the NIST pulse-heated Kolsky bar. In: 12th CIRP conference on modelling of machining operations, San Sebastián, Spain, 7–8 May 2009, pp.255–262.

10.

Moufki

Dudzinski

Molinari

. Thermoviscoplastic modelling of oblique cutting: forces and chip flow predictions. Int J Mech Sci 2000; 42(6): 1205–1232.

11.

Özel

. The influence of friction models on finite element simulations of machining. Int J Mach Tool Manu 2006; 46(5): 518–530.

12.

Bonnet

Valiorgue

Rech

. Identification of a friction model—application to the context of dry cutting of an AISI 316l austenitic stainless steel with a tin coated carbide tool. Int J Mach Tool Manu 2008; 48(11): 1211–1223.

13.

Arrazola

Özel

. Investigations on the effects of friction modeling in finite element simulation of machining. Int J Mech Sci 2010; 52(1): 31–42.

14.

Shaw

Oxford

. On the drilling of metals 2—the torque and thrust in drilling. Trans ASME 1957; 79: 139–148.

15.

Armarego

EJA

Wright

. Predictive models for drilling thrust and torque—a comparison of three flank configurations. CIRP Ann: Manuf Techn 1984; 33(1): 5–10.

16.

Watson

. Drilling model for cutting lip and chisel edge and comparison of experimental and predicted results. I: initial cutting lip model. Int J Mach Tool D R 1985; 25(4): 347–365.

17.

Chandrasekharan

Kapoor

Devor

. A mechanistic model to predict the cutting force system for arbitrary drill point geometry. J Manuf Sci E: T ASME 1998; 120(3): 563–570.

18.

Elhachimi

Torbaty

Joyot

. Mechanical modelling of high speed drilling. 2: predicted and experimental results. Int J Mach Tool Manu 1999; 39(4): 569–581.

19.

Jrad

Devillez

Dudzinski

. Geometrical and mechanical approach of drilling. In: Fifth international conference on high speed machining, 14–16 March 2006, Metz, France, LBMM, p.1.

20.

Dargnat

Darnis

Cahuc

. Energetical approach for semi-analytical drilling modelling. Mach Sci Technol 2008; 12(3): 295–324.

21.

Claudin

Poulachon

Lambertin

. Correlation between drill geometry and mechanical forces in MQL conditions. Mach Sci Technol 2008; 12(1): 133–144.

22.

Guibert

Paris

Claudin

. Identification of thrust force models for vibratory drilling. Int J Mach Tool Manu 2009; 49(6): 730–738.

23.

AFNOR. NF E66-502: géométrie de la partie active des outils coupants: notions générales système de référence, angles de l’outil et angle en travail. 1st ed. Saint Denis: France: AFNOR, 1978.

24.

Elhachimi

Torbaty

Joyot

. Mechanical modelling of high speed drilling, 1: predicting torque and thrust. Int J Mach Tool Manu 1999; 39(4): 553–568.

25.

Zemzemi

Rech

Ben Salem

. Identification of a friction model at tool/chip/workpiece interfaces in dry machining of AISI4142 treated steels. J Mater Process Tech 2008; 209(8): 3978–3990.

26.

Claudin

Mondelin

Rech

. Effects of a straight oil on friction at the tool–work material interface in machining. Int J Mach Tool Manu 2010; 50(8): 681–688.

27.

Naisson

Rech

Paris

. Characterization of friction properties during machining of various stainless steels. In: Eighth international conference on high speed machining, 9–11 December 2010, Metz, France, 2010, p.1.

28.

Merchant

. Mechanics of the metal cutting process. I: orthogonal cutting and a type 2 chip. J Appl Phys 1945; 16(5): 267–275.

29.

Merchant

. Mechanics of the metal cutting process. II: plasticity conditions in orthogonal cutting. J Appl Phys 1945; 16(6): 318–324.

30.

Bridgman

. Studies in large plastic flow and fracture with special emphasis on the effects of hydrostatic pressure. McGraw-Hill, Cambridge USA: Harvard University Press, 1952.

31.

Molinari

Moufki

. The Merchant’s model of orthogonal cutting revisited: a new insight into the modeling of chip formation. Int J Mech Sci 2008; 50(2): 124–131.

32.

Oxley

. The mechanics of machining: an analytical approach to assessing machinability. Prentice Hall USA, 1989.

33.

Lagarias

Reeds

Wright

. Convergence properties of the Nelder–Mead simplex method in low dimensions. SIAM J Optimiz 1998; 9(1): 112–147.

34.

Courbon

Mabrouki

Rech

. On a new local contact definition in dry cutting simulations of AISI 1045 steel. In: Eighth international conference on high speed machining, 9–11 December 2010, Metz, France, p.1.