Mathematical Problems in Engineering

Volume 2014 (2014), Article ID 543298, 11 pages

http://dx.doi.org/10.1155/2014/543298

## A Mathematical Modeling to Predict the Cutting Forces in Microdrilling

^{1}Key Laboratory of Fundamental Science for Advanced Machining, Beijing Institute of Technology, Beijing 100081, China^{2}Hebei United University, Tangshan 063009, China

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

Academic Editor: Zhen-Lai Han

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

#### Abstract

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

#### 1. Introduction

There has been an increasing requirement for high-accuracy microholes in the microelectronic, automotive, computer components, and sensor industries. Microdrilling is experiencing a very rapid growth in precision production industries. In many aspects, microdrilling has fundamentally identical features with conventional drilling, but the downsizing of the dimensions of the drill introduces many problems, which has a major influence on the microdrilling process, such as cutting edge radius, increased web thickness, large vibrations due to high rotation speed, and high ratio of drill breakage. There are many factors influencing the microdrilling process, such as drill geometry, drill materials, drilling forces, workpiece materials, machining parameters, and vibration. Drilling forces are related to drill life, holes quality, and productivity. Therefore, drilling forces are one of the most important factors affecting the drill performance.

In general, there are four methods of modeling cutting forces in metal machining: analytical method, experimental method, mechanistic method, and numerical method [1]. Many models have been developed by researchers in the past several decades. In the study of macrodrilling models, Shaw and Oxford [2] were the pioneers. Armarego and Cheng [3, 4] presented a model in which a series of oblique cutting slices was used to the drilling process with flat rake face and conventional twist drills. Watson [5–8] produced a more detailed model of material removal in both cutting edges and chisel edge. Stephenson and Agapiou’s model [9] simulated arbitrary drill point geometries. Chandrasekharan et al. [10, 11] developed a mechanistic model of the cutting lips and chisel edge to predict the cutting force system for arbitrary drill point geometry. Strenkowski et al. [12] developed a thrust force model based on analytical finite element technique in drilling with twist drills. In their model, the cutting lips were regarded as a series of oblique sections, and the cutting of the chisel region was treated as orthogonal cutting. Wang and Zhang [13] presented a predictive model for the thrust in drilling operations using modified plane rake faced twist drills. Their models were based on the mechanics of cutting approach incorporating many tools and cutting process variables. There was less literature on force modeling in microdrilling. Sambhav et al. [14] modeled the thrust by the primary cutting lip of a microdrill analytically and modeled shearing forces and ploughing forces of the major cutting edges. Hinds and Treanor [15] analyzed the stresses occurring in microdrills using finite element methods in printed circuit board drilling process, but they did not produce any mathematical model for cutting forces of microdrills.

Slip-line field theory was often used to analyze the cutting process. Many machining parameters can be predicted by the slip-line field model, such as cutting force, chip thickness, shear strain, and shear strain-rate. Merchant [16] was the first one who presented a mathematical model to determine shear angle by using the minimum energy principle, and his model was the basis of all subsequent models. Lee and Shaffer [17] developed a slip-line field model which was an approximation method under certain cutting conditions. Dewhurst and Collins [18] presented a matrix technique for numerically solving slip-line problems. Oxley [19] proposed a parallel surface shear zone model of orthogonal cutting that considered the change of material flow stress. Waldorf et al. [20] developed a slip-line model for ploughing by a cutting tool with a definite cutting edge radius. Fang [21] presented a generalized slip-line field model for cutting when edge was rounded. Fang’s model included nine effects that commonly occurred in machining. Then Fang [22] quantitatively analyzed orthogonal metal cutting processes based on his slip-line model. Manjunathaiah and Endres [23] developed a new orthogonal process model that included the effects of edge radius. Jin and Altintas [24] simplified Fang’s model, and they considered the effects of strain, strain-rate, and temperature on the cutting process.

In microcutting applications the uncut chip thickness is very small, typically within the range of 25 *μ*m. Since the cutting edge radius is typically ground with a 5–20 *μ*m, the assumption of having a perfectly sharp cutting edge in macrodrilling is not valid, so the cutting edge radius should not be taken to be zero in microcutting operations. Past studies have found that if the uncut chip thickness is below the minimum chip thickness , elastic deformation or a mixed elastic-plastic deformation will take place. Above this value, chip formation starts taking place. This is known as the minimum chip thickness effect. However, due to the extrusion of the material by the chisel edge region of the drill, the drilling process can still take place, even if the chip thickness is very small.

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

During microdrilling, both shearing and indenting actions are happening. When the microdrill contacts the workpiece, the drill point rubs workpiece first. Under the extrusion force of microdrill, material is squeezed around the drill point; at the same time, the secondary cutting edges on the chisel edge perform cutting. Then, the major cutting edge enters into workpiece and begins to cut. The central portion of the chisel edge performs the indenting action, and the second cutting edges on the chisel edge and the major cutting edges on the fluted portion perform shearing.

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

#### 2. Major Cutting Edge Cutting Force Models

The cutting behavior of the major cutting edge is an oblique cutting process. The cutting edge is divided into elements and each element is approximated as a straight line, shown in Figure 2. The magnitude of the total drilling thrust () is obtained by summing the forces at all the cutting elements on each edge and all the cutting edges on the drill.

The direction of the elemental cutting force is opposite to the velocity direction; is resolved into and . The direction of is along the actual cutting direction. is the elemental lateral force, which is orthogonal to the cutting force and the elemental oblique cutting thrust force . The thrust force is normal to the plane that contains the velocity vector and the cutting edge. The magnitude of those forces is given by where is the half point angle and the inclination angle and angle can be obtained by the following equations: where is half the web thickness and is the distance from a point on the cutting edge to the drill axis.

The normal rake angle at any point on the cutting edge is where is the helix angle of the drill and is the drill radius.

The magnitude of the total drilling thrust along the axis of the drill can be obtained by summing the forces at all the cutting elements on each cutting edge and all the cutting edges on the drill, so the magnitude of the total drilling thrust force is

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

Due to the technological and material constraints in microdrill preparation, the major cutting edge has a definite radius, and the uncut chip thickness is very small, so the major cutting edge cannot be seen as completely sharp. The slip-line field model of microcutting process for each cutting element of major cutting edges is shown in Figure 3.

The material deformation region consisted of three zones: primary shear zone [*AIBB*_{1}*I*_{1}*A*_{1}*A*_{2}], secondary shear zone , and tertiary zone [*BSCD*_{1}*B*_{1}]. The shape of the slip-line field was originally proposed by Fang [21]. In Fang’s model, the slip lines HJ and JI are defined as two basic slip lines; after their shapes are obtained, all other slip lines in the secondary shear zone can be determined using Dewhurst and Collins’s matrix technique [18]. Then, the slip-lines in the primary and tertiary shear zones can be easily determined from relevant slip-line relationships. The primary shear zone included three regions: triangular region* AA*_{1}*A*_{2}, convex region* AII*_{1}*A*_{1}, and concave* IBB*_{1}*I*_{1}. In region* AA*_{1}*A*_{2}, line* AA*_{2} is a stress-free boundary; all of the slip lines in* AA*_{1}*A*_{2} intersect with* AA*_{2} at a 45° angle. Both region* AII*_{1}*A*_{1} and region* IBB*_{1}*I*_{1} consist of circular arcs and straight radial lines. Point* S* is the separation point for the upward and downward material bifurcating. Part of the materials flows downwards from point* S* to point* C* along the rounded edge, while other parts of the materials flow upwards from point* S* to point* B*. In order to simplify the mathematical formulas of the slip-line problem with a curved boundary, the tool edge* BC* is approximately represented by two straight chords* BS* and* SC*.

*BS* and* SC* are considered to have rough surfaces; the included angles between them and the slip lines are and , respectively. The intersection angle of* AA*_{2} and horizon is . The separation angle , the tool edge radius , and the tool rake angle determine the position of the stagnation point on the rounded tool edge. Geometric analysis gives the following set of equations:
where is the frictional shear stress and is the material flow stress. The tool-chip frictional shear stress along the rake face was assumed to be constant.

Jin and Altintas evaluated the total cutting forces by integrating the forces along the entire chip-rake face contact zone and the ploughing force caused by the round edge. The detailed process can be found in [24]. According to their computing methods, the cutting forces on the major cutting edges of microdrill can be derived as follows.

After the material passes through the shear zones, the chip begins curling freely, so the resulting force along the slip lines , , and should be zero. Consider

Line is a stress-free boundary. The distribution of hydrostatic pressure and shear flow stress along slip line can be calculated by dividing into several differential elements, such as 100 differential elements, as shown in Figure 4. The total force along slip line is obtained by summing all of the elemental forces in the and directions.

The forces on point of slip line in the and directions are calculated as where is the hydrostatic pressure and is the shear flow stress on the element, is the angular coordinate of the element, is the length of the element, and is the width of cut. Consider where is the number of divided differential elements.

So the hydrostatic pressure and shear flow stress of point can be concluded. After the total forces along slip line are calculated, the forces along and can be determined. In the second shear zone, slip line is divided into 100 angular elements in the same way; then the same number of slip lines is formed in the secondary shear zone. For any slip line , the shear flow stress and the hydrostatic pressure at point are obtained from the stress distribution in the primary shear zone. Then, the shear flow stress and hydrostatic pressure at point are calculated as

The elemental force is projected into the and directions, and then the elemental forces in the and directions at point are

The elemental force at other points along the tool rake face is calculated following the same procedure as point , and then the total force along is obtained by summing all of the elemental forces in the and directions.

In tertiary shear zone, line is divided into 100 small elements. The shear flow stress and hydrostatic pressure at point can be concluded from point . Then, the shear flow stress and hydrostatic pressure at point are calculated as

The elemental forces along line in the and directions at point are

Similarly, along line ,

The element forces in the plane perpendicular to the major cutting edge along the cutting direction and the thrust direction are obtained on the major cutting edge as where is the length of differential element of major cutting edge.

Therefore, the magnitude of the total drilling thrust along the axis on the major cutting edge of the drill is

#### 3. Chisel Edge Cutting Force Model

Mauch and Lauderbaugh [25] obtained the indentation zone radius (Figure 1) for a conical drill based on the point angle. Paul et al. [26] suggested that the dynamic clearance angle becomes zero at the indentation zone radius. So the radius of the indentation zone is given by the following equation: where is the static clearance angle of the chisel edge.

##### 3.1. Secondary Cutting Edge Cutting Force Model

Since the chisel edge has a definite radius and the uncut chip thickness is comparable in size to the edge radius, the chisel edge cannot be seen as completely sharp but should be as a rounded edge. The chip thickness at the elements on the chisel edge is equal to half of the drill feed. The secondary cutting edges are divided into elements and the elemental drilling thrust is determined; then the magnitude of the total drilling thrust along the axis of the drill can be obtained by summing the forces at all elements for the secondary cutting edges.

Because the flank surfaces of microdrill are plane, the slip-line model of secondary cutting edge is different from the major cutting edge. Figure 5 shows the analytical slip-line model for machining with secondary cutting edge. The intersection angle of and horizontal line is . Consider

The element forces in the plane perpendicular to the chisel edge along the thrust direction on the chisel edge are obtained as where is the length of differential element, and where is the uncut chip thickness, , and is feed. Consider

The magnitude of the total drilling thrust can be obtained by summing the forces at all elements for the secondary cutting edges. So the magnitude of the total drilling thrust force is where is the length of chisel edge.

##### 3.2. The Indentation Zone Cutting Force Model

In microdrilling processes, the ratio of web thickness to drill diameter is larger than that of macrodrilling, so the indentation zone is quite important, and the contribution to the total drilling thrust force by the indentation zone needs to be considered. In microdrilling, although the chisel edge is circular edge, due to the major effect of the indentation zone on extrude material, the indentation zone can be regarded as a rigid wedge. The material is extruded on both sides of the wedge. The indentation zone model schematic is shown in Figure 6. According to the slip-line field theory, the force normal to the surface of the wedge can be determined. Consider where is the solution of the slip line and is given by the following equation: where is the included angle of the wedge, which is equal to twice the magnitude of the static normal rake angle at the chisel edge and is given by where is chisel edge angle of microdrill.

The load acted on unit length of the wedge is where

So the total drilling thrust force of the indentation zone can be expressed as

#### 4. Experimental Validation of the Thrust Forces Model of Microdrills

##### 4.1. Experimental Work

To calibrate the thrust forces model of microdrills, the microdrilling processes were performed on a DMG DMU 80 monoBLOCK machining center. The experimental setup was shown in Figure 7(a). Workpiece is AISI 1023 carbon steel plate with a thickness of 1.5 mm. Workpiece is mounted on a multicomponent dynamometer (Kistler, model 9257B). The material of microdrills is cemented carbide of ultrafine grain (AF K34 SF, made by Germany AF Hartmetall Group), and its performance is listed in Table 1. Microdrills were fabricated on a Makino Seiki CNS7d CNC microtool grinding machine, as shown in Figure 7(b). The basic parameters of microdrills are shown in Table 2. The microdrill was observed under a laser microscope (KEYENE vk-x100 Series) and a stereoscopic microscope (Zeiss). Figure 7(c) shows an example of microdrills.

The material shear flow stress is 282.7 MPa, the coefficient of coulomb friction is 0.15, and the shear stress ratio is 0.95. The separation angle on the major cutting edges is 56° and 58.5° on the second cutting edges. The spindle speed is 22,000 r/min and the feed is 0.5 *μ*m/r, 1.0 *μ*m/r, 2.0 *μ*m/r, 3.0 *μ*m/r, and 5.0 *μ*m/r, respectively. The following equation is used to evaluate the shear angle when the second cutting edges are cutting [14]:

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

A typical thrust profile is shown in Figure 9. In zone ①, the chisel edge has contacted and extruded the workpiece; at the same time, the second cutting edge is cutting. In zone ②, the major cutting edges are entering the hole gradually and begin to cut. The thrust forces in zone ② consist of two parts of forces, the force generated by the chisel edge and the force generated by the major cutting edges. The latter increases gradually in zone ②, but it is always smaller than the former even at its maximum. In zone ③, the major cutting edges have completely entered the hole and the entire microdrill is exerting the thrust. In zone ④, the chisel edge of microdrill is just out from the bottom of workpiece and the major cutting edges are still cutting. The force in zone ④ is generated without the contribution of the chisel edge. Therefore, the thrust in zone ④ is significantly smaller than that in zone ②. In zone ⑤, workpiece has completely drilled through, and there exists friction between drill and hole wall. Then, the microdrill withdraws from the hole.

The chisel edge and cutting edges forces must be separated in order to compare them to the values predicted by the model. The approach is to use a blind pilot hole with a diameter exactly equal to the web thickness of the microdrill used for the validation. For pilot holes, 0.15 mm drills were used, and the depth of pilot holes was kept at 0.5 mm. The typical thrust profile for the operation is shown in Figure 10.

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

##### 4.2. Results and Discussion

As seen in Figure 8, the shape of curve in Figure 8(a) is completely different from the others, and the trend of the curve is basically the same as in Figures 8(b)–8(e). At very low feed, the chip thickness is less than the minimum chip thickness; chips are not formed, and only ploughing takes place. However, because the indentation zone keeps extruding the work material, the cutting process does take place. Figure 8(a) shows the thrust of this case; it is the main ploughing forces. As the feed increases, when the chip thickness exceeds the minimum chip thickness, both shearing and ploughing take place in the cutting, so the thrust forces include shearing forces and ploughing forces, as shown in Figures 8(b)–8(e). By comparing Figures 8(a) and 8(b), we can see that the value of the minimum chip thickness is between 0.25 *μ*m and 0.5 *μ*m.

As seen in Figure 11, almost all the predicted values are lower than the experimental ones. The data shows that the cutting force model of chisel edge including secondary cutting edge and indentation zone can correctly predict the thrust, and the average error is less than 5 percent. The accuracy of major cutting edges cutting force is relatively lower. The experimental results show that the average error in the predicted steady state major cutting edges thrust is less than 10 percent. When the feed is between 0.5 and 1.0, a mixed elastic-plastic deformation happens to the material; a transition from the ploughing mechanism to shearing mechanism can be seen. In general, the total drilling thrust (cutting edges and chisel edge) is predicted with an average error of less than 7 percent. In major cutting edges cutting force model, because the hydrostatic pressure and shear flow stress along tool-chip contact zone are calculated by dividing some differential elements, the number of divided differential elements has certain effect on the accuracy of the model. On the other hand, in order to simplify the mathematical formulas, the tool circular edge is approximately represented by two straight chords, which lead to lower accuracy to some extent. Other sources of deviation might include the wear or local fracture of the major cutting edges in the cutting process; these factors can lead to the increasing of the thrust during the drilling process.

The predicted and experimental results show that the thrust created by the chisel edge is quite significant. It exceeds the thrust created by the cutting edges and represents about 60–70 percent of total thrust. In this paper, the chisel edge angle of microdrill is relatively low at 42.8°, which causes both the length of chisel edge and the cutting force to increase.

#### 5. Conclusions

The mathematical models to predict the microdrilling thrust are developed. The thrust is modeled in three parts: major cutting edges, secondary cutting edge, and the indentation zone. Major cutting edge and secondary cutting edge force models are based on the slip-line field theory, and the indentation zone is modeled as a rigid wedge. The major cutting edges and secondary cutting edge are divided into elements and the elemental forces are determined from an oblique cutting model and an orthogonal model, respectively. Shearing and ploughing are included in the models of the major cutting edges and second cutting edge. The model is applied to a 0.5 mm ultrafine grain cemented carbide microdrill, and the experimental and predicted values of forces are compared.

The main conclusions from the study are as follows.(i)Almost all the predicted values are lower than the experimental ones. This might be attributed by factors such as drill vibrations, drill wandering, the friction of drill, and hole wall.(ii)On the chisel edge, the forces of secondary cutting edge can be modeled based on slip-line theory, and the indentation zone can be modeled as a rigid wedge. The model of chisel edge shows a good conformity with the experimental results.(iii)The accuracy of major cutting edges cutting force is low relatively, and the average error is about 10 percent. This may be due to the fact that some of the constants such as shear stress ratio and separation angle as well as others are calibrated for other processing methods and not for drilling.

Future work should aim at two aspects: improving the accuracy of major cutting edges cutting force and considering the effect of the chisel edge angle and length on total drilling thrust.

#### Nomenclature

: | Total thrust of major cutting edges |

: | Total thrust of second cutting edge |

: | Total thrust of the indentation zone |

: | Elemental cutting force |

: | Elemental lateral force |

: | Elemental oblique cutting thrust force |

: | The frictional shear stress |

: | The material flow stress |

: | The hydrostatic pressure |

: | Half the drill point angle |

: | Cutting edge inclination angle |

: | Normal rake angle of major cutting edge |

: | Helix angle |

: | The separation angle |

: | Effective rake angle |

: | Effective shear angle |

: | The chisel edge angle |

: | The included angle of the wedge |

: | The static clearance angle of the chisel edge |

: | The tool edge radius |

: | Half the web thickness |

: | The distance from the selected point on the major cutting edge to drill axis |

: | Drill radius |

: | The radius of the indentation zone |

: | Feed |

: | The length of chisel edge |

: | The uncut chip thickness |

: | The minimum chip thickness. |

#### Conflict of Interests

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

#### Acknowledgment

The authors would like to thank The National Natural Science Foundation of China (Key Program, no. 50935001) for their financial support. Without their support, this work would not have been possible.

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