UNIT 1 METAL CUTTING AND CHIP FORMATION

UNIT 1 METAL CUTTING AND CHIP FORMATION Metal Cutting and Chip Formation Structure 1.1 Introduction Objectives 1.2 Material Removal Processes 1.3 Chip Formation 1.3.1 Deformation in Metal Machining 1.3.2 Chip Types 1.3.3 Types of Cutting 1.3.4 Mechanics of Chip Formation 1.3.5 Geometry of Chip Formation (Orthogonal Cutting) 1.4 Force Analysis 1.5 Velocity Relationships 1.6 Shear Strain and Shear Strain Rate 1.7 Shear Angle Relationships 1.8 Summary 1.9 Key Words 1.10 Answers to SAQs 1.1 INTRODUCTION Manufacturing processes can be broadly divided into four categories, viz., primary (casting, forging, moulding, etc), secondary (machining, finishing, etc.), tertiary (fabricating processes like welding, brazing, riveting, etc.), and fourth level processes (painting, electroplating, etc.). Secondary manufacturing processes are as important as any other level processes. These processes involve removal of material in the form of chips or otherwise, to give the desired shape, size, surface roughness, and tolerance on the workpiece obtained from the primary manufacturing processes. The machined components can be used as it is, or one can be assembled (sometimes using fabricating processes) and if required, given an aesthetic look by electroplating, painting, etc. This block/unit will discuss the fundamentals of traditional material removal processes (nontraditional material removal processes are discussed in Block 4). This unit will discuss basic principles of metal cutting including mechanics of chip formation, velocity and force analysis, and some of the models proposed to evaluate the shear angle relationships. Objectives After studying this unit, you should be able to understand classification scheme for various types of material removal processes, identify various types of metal cutting processes, types of chip formed, mechanism of chip formation and geometry of chips, analyse forces and velocities in cutting process, and know various schools of thought regarding shear angle relationships. 5

Theory of Metal Cutting 1.2 MATERIAL REMOVAL PROCESSES Material removal processes can broadly be divided into two categories : traditional and advanced (non-traditional). Each of these categories can be further sub-divided into bulk removal processes (or cutting) and finishing processes. The classification of various types of material removal processes is shown in Figure 1.1(a). This unit will discuss only the basics of traditional cutting processes. Traditional Cutting Traditional cutting processes can be classified as those which produce parts having surfaces of revolution and those which produce prismatic shapes. Another scheme for the classification of metal cutting processes is provided in Figure 1.1(a). This classification is based on the type of motion imparted to the work and tool. Cutting tools used for material removal are classified in two categories : single point cutting tool and multi-point cutting tool (cutting tools having more than one cutting edge). The following section discusses how material removal takes place by using a single point cutting tool. Similar principles are applicable to the multiple point cutting tools as well. 6 Figure 1.1(a) : Classification of Material Removal Processes The process of metal cutting is effected by providing relative motion between the workpiece and the hard edge of cutting tool. Such relative motion is produced by a combination of rotary and translating movements either of the workpiece or of the cutting tool or both. Depending on the nature of the relative motion, metal cutting process is called either turning or planning or boring, etc. For different types of operations, one needs to have different types of machine tools. For example, lathe for turning, planer for planning, grinder for grinding, etc. Some of these machines (say, lathe, boring m/c, and drill) generate surfaces of revolution whereas others (planer, milling m/c, and shaper) make prismatic (or flat surfaces) parts. With the help of different types of tools, a lathe can perform various kinds of operations (Figure 1.1(b)). Conventionally, the translatory displacement of the cutting edge of the tool along the work surface during a given period of time is called feed ( f ), while the relative rate of traverse of work surface past the cutting edge is designated as the cutting velocity or simply speed (V c ). In case of single point turning, V c is the peripheral velocity of the rotating workpiece in meters per minute. In case of slab milling, it is the peripheral velocity of the milling cutter in meters/minute.

Metal Cutting and Chip Formation Figure 1.1(b) : Various Operations that can be Performed on a Lathe [Kalpakjian, 1989] Table 1.1 Operation Motion of Job Motion of Cutting Tool Figure of Operation Turning on a lathe Rotary motion of the work Axial movement of the tool Boring on a lathe Work rotation Axial tool movement Drilling on a drill machine Fixed Rotations as well as translatory feed Planning Translatory Intermittent Translation Milling Translatory Rotation Grinding Rotary/Translatory Rotary 7

Theory of Metal Cutting 1.3 CHIP FORMATION 1.3.1 Deformation in Metal Machining Figure 1.2 shows a schematic diagram of material deformation during cutting, and subsequently removal of the deformed material from the workpiece by a single point cutting tool. Because of the relative motion between the tool and the workpiece, material ahead of the tool face (rake face) is compressed (elastically and then plastically). Further, movement of the tool into the workpiece deforms the work material plastically and finally separates the deformed material from the workpiece. This separated material flows on the rake face of the tool called as chip. The chip near the end of the rake face is lifted away from the tool, and the resultant curvature of the chip is called chip curl. Figure 1.2 : Schematic Diagram of Chip Deformation The study of the mechanism of chip formation involves deformation process of the chip ahead of the cutting tool. Theoretical study of the material deformation in metal cutting is difficult and therefore experimental techniques have been resorted to for analyzing the process of deformation in chips. The methods commonly employed for this purpose are : 8 (i) (ii) (iii) Use of movie camera for taking pictures of chip. Observing grid deformation during cutting. Examination of frozen chip samples obtained by the use of quick-stop device. Experimental study of chip deformation process has revealed that : (i) (ii) (iii) 1.3.2 Chip Types During machining of ductile materials, a plastic deformation zone is formed in front of the cutting edge (Figure 1.2). The distinctive zone of separation between the chip and workpiece where deformation gradually increases towards the cutting edge is called the primary deformation/shear zone. In shear zone extensive deformation occurs. The width of shear zone is very small. The plastic deformation involved in the formation of chips affects the hardness of material (strain hardening). Strain hardening increases when a layer undergoes deformation in the shear zone. The type of chip obtained from a machining process is characterized by a number of parameters e.g., the type of tool-work engagement, work material properties and the cutting conditions. Ernst has classified the chips obtained in machining processes into three categories : Type 1 : Discontinuous chip,

Type 2 : Continuous chip, and Type 3 : Continuous with built-up edge. When ductile materials at high cutting speed are cut by a single point cutting tool, ribbon like continuous chip (Figure 1.3(a) and 1.3 (b)) is obtained. The conditions that promote formation of continuous chips in metal cutting are sharp cutting edge, low feed rate (or small chip thickness), large rake angle, ductile work material, high cutting speed, and low friction at chip-tool interface. As shown in Figure 1.2, major deformation takes place in primary shear deformation zone (PSDZ) resulting in the formation of chip. Due to ductile nature of work material and reasonably high temperature in the PSDZ, the deforming material flows on the rake face of the tool as continuous mass rather than the one fractured/ruptured at small distances at the underneath of the chip as in discontinuous chip. Continuous chip results in good surface finish, high tool-life, and low power consumption. But disposal of large coiled chips is a serious problem, for many industries where tons of chips are produced every week. To get rid of this problem various types of chip breakers are used which are in the form of step or groove on the rake face of the tool (Figure 1.4). The chip strikes with this step/groove and gets broken in the form of small segments. Disposal of such small chips is not a problem. If the friction between tool and chip while machining ductile materials is high, some part of the chip gets welded to the rake face of the tool near its cutting edge. The welded material is extremely hard and its size keeps on increasing with time. Because of the hardness of the adhered materials onto the cutting edge, it participates in cutting to a certain extent. That is why it is named as built up edge (Figure 1.5). As the size of the BUE grows larger, it becomes unstable and it breaks. Some part from the broken BUE is carried away by the chip as well as on the machined surface (Figure 1.3). Metal Cutting and Chip Formation Figure 1.3 : Different Kinds of Chips : (a) Continuous; (b) Photograph of Continuous Chip; (c) Continuous Chip with Built Up Edge; and (d) Discontinuous Chip The chip with the adhered parts of the BUE is known as continuous chip with BUE. The adhered parts of the BUE on the machined surface make the machined surface rough, but the BUE protects the actual cutting edge of the tool from wear. Thus, cutting with BUE enhances the tool life (or tool cuts longer before regrind). 9

Theory of Metal Cutting Figure 1.4 : Different Types of Chip Breakers Figure 1.5 : Development of Built Up Edge [Rao, 2000] Discontinuous or segmented chips are produced while machining brittle materials or ductile materials at low speeds and high friction conditions. The basic difference between the mechanism of formation of discontinuous chip and continuous chip is that, instead of continuous shearing of the material ahead of the cutting tool, rupture occurs intermittently producing segments of chip (Figures 1.3 and 1.6). These chips are smaller in length hence easy to dispose off, and give good surface finish on the workpiece. Discontinuous chips are formed when cutting brittle materials, or cutting ductile materials at low speed, or cutting with tools of small rake angle. 10 (a) Tear Type 1.3.3 Types of Cutting (b) Shear Type Figure 1.6 : Hypothesized Discontinuous Chip Formation [Rao, 2000] Principally, there are two types of cutting : (i) (ii) Orthogonal Cutting Orthogonal cutting, and Oblique cutting. Orthogonal cutting operation is the simplest type of cutting operation, in which the cutting edge is straight, parallel to the original plane surface of the workpiece and perpendicular to the direction of cutting, and in which the length of the cutting edge is greater than the width of the chip removed (Figures 1.7(a) and (b)). This orthogonal cutting is also known as Two Dimensional (2-D) Cutting. A few of the cutting tools perform orthogonally, such as lathe cut-off tools (Figure 1.7(a)), straight (not helical) milling cutters, broaches, etc. In actual machining, majority of the cutting operations (turning, milling, etc.) are three dimensional (3-D) in nature and are called as oblique cutting. In oblique cutting, the cutting edge of the tool is inclined to the line normal to the cutting direction, and this angle is known as angle of obliquity. This is also called the inclination angle, i (Figure 1.7(c)). Oblique cutting can be defined as the cutting

operation in which the cutting edge is straight and parallel to the original surface of the workpiece, but is not perpendicular to the cutting direction, being inclined to it. An angle of interest in this case is the chip flow angle, η c which is defined as the angle measured in the plane of cutting face between the chip flow direction and the normal to the cutting edge (Figure 1.7(c)). Both i and η c are zero in case of orthogonal cutting. The certain practical limitations to orthogonal cutting are mitigated by three dimensional tooling. Metal Cutting and Chip Formation Figure 1.7 : (a); (b) Orthogonal Cutting System; and (c) Oblique Cutting System Generally for the mathematical analysis of the mechanics of metal cutting, orthogonal cutting is considered because it is simpler than the oblique cutting. The results so obtained can be used for oblique cutting operations. 1.3.4 Mechanics of Chip Formation Plastic deformation is the main factor that governs formation of chips. Initially, researchers (Merchant and others) proposed that deformation of the material takes place along a plane (called shear plane) just ahead of the cutting tool and runs up to free surface of the workpiece (Figure 1.8(a)). Once the deforming material crosses the shear plane, it slides along the rake face of the tool due to the velocity of cutting tool (relative motion between the tool and workpiece). This hypothesis of a shear plane is useful from the analysis of metal cutting point of view but has theoretical drawbacks. Here, the transition from the un-deformed to the deformed material takes place along a shear plane by changing cutting velocity from V c (velocity of tool with respect to workpiece) to V f (chip velocity relative to the tool). For this change to take place, the acceleration across the plane (plane thickness equal to zero) has to be infinite. This also applies to the stress-gradient across the shear plane. Due to the above anomaly, researchers (Oxley and others) experimentally studied the deformation zone by freezing the cutting process with the help of a quick stop device. When they studied the deformed zone under a microscope, they found that the deformation takes place within a finite zone (thin or thick depending upon various governing parameters). This is called as primary shear deformation zone (PSDZ) (Figures 1.8 (b) and (c)). They also found that under certain machining conditions, deformation also takes place at the tool-chip interface 11

Theory of Metal Cutting (Figures 1.8 (b) and (c)). This deformation is known as secondary shear deformation zone (SSDZ). Figure 1.8 : (a) Shear Plane; (b) Primary and Secondary Shear Deformation Zone in Chip Formation; and (c) Frozen Chip Obtained by a Quick Stop Device 1.3.5 Geometry of Chip Formation (Orthogonal Cutting) Figure 1.9 shows a simple geometry of chip formation in case of continuous chip (type 2). The uncut chip thickness t u (equal to feed in turning) is deformed to give chip thickness t c which experiences two velocities V f (chip sliding velocity) and V s (shear velocity) along the tool face and shear plane, respectively. From this geometry, it is possible to calculate the shear angle (φ) in terms of measurable or known quantities t u, t c and α. Figure 1.9 : Geometry of Continuous Chip Formation From right angle triangles, ABC and ABD (BD is perpendicular to AD drawn from B), AB t u / Also, AB t c / sin (90 (φ α)) t c / cos (φ α) t t u c cos ( φ α) 12

tu tc as is called chip thickness ratio or chip thickness coefficient (r c ) which can be written Metal Cutting and Chip Formation 1 r c cosφcosα + sin α or, or, 1 r c cot φcosα + r c sin α r c cosα (1 rc sin α) tan φ r cos tan c α φ... (1.1) 1 r c sin α To determine the shear plane angle (φ) for a given cutting condition the chip thickness t ratio u r c, should be known. But to determine t c with a micrometer is somewhat tc inaccurate. Hence, an indirect approach to this problem is to assume that the density of metal during the cutting process does not change. Hence, the volume of uncut chip is equal to the volume of metal removed (or deformed chip). Since the width of chip (b) is equal to the width of metal being cut (in orthogonal cutting), therefore : Lctc Lutu blc tc Lutub (volume constancy condition) or, tu tc rc L c Lu tu L t c c L... (1.2) u where, L c is length of chip, and L u is corresponding length of material removed from the workpiece (or uncut chip length). L c can be easily measured, and it (L c / L u ) will give more accurate results than (t u / t c ) because of the difficulties and inaccuracies involved in the measurement of thickness of the deformed chip (t c ). 1.4 FORCE ANALYSIS Let us analyse the forces acting on the chip in orthogonal cutting. These are shown in Figure 1.10 (a) and are as follows : Force, F s, is the resistance to shear of the metal in forming the chip. F s acts along the shear plane. Force, F n, is normal to the shear plane and is a backup force on the chip provided by the workpiece. Force N acting on the chip is normal to the cutting face of the tool and is provided by the tool. Force F is frictional resistance offered by the tool to the chip flow. The latter force acts downwards against the motion of the chip as it slides upwards along the tool face. Figure 1.10 (b) shows the free body diagram of the forces acting on the chip. Forces F s and F n are represented by the resultant R, and F and N are replaced by the resultant R'. This means that only two combined forces are acting on the chip, i.e., R and R'. There are external couples on the chip which curl it, and they may be negated in this approximate analysis. If equilibrium is to exist when a body is acted upon by two forces, they must be equal in magnitude, and be collinear. Hence, R and R' are equal in magnitude, opposite in direction and collinear (Figure 1.10). 13

Theory of Metal Cutting Figure 1.10 : (a) Force Components Acting on a Chip; and (b) Free Body Diagram of a Chip Figure 1.11 shows a composite diagram in which the two force triangles of Figure 1.10, have been superimposed by placing the two equal forces R and R ' together. Since the angle between F s and F n is a right angle, the intersection of these forces lies on the circle with diameter R as shown. Also, F and N may be replaced by R to form the circle diagram (Figure 1.11). 14 Figure 1.11 : Force Circle Diagram The horizontal cutting force F c and vertical force F t can be measured in a machining operation by the use of a force dynamometer. The electric strain gauge type of transducer is used in the dynamometer. After F c and F t are determined, they can be laid off as in Figure 1.11 and their resultant is the diameter of the circle. The rake angle α can be laid off, and the forces F and N can then be determined. The shear plane angle φ can be measured approximately from a photomicrograph or by measuring t c and t u, or length of chip and corresponding length of unmachined chip (discussed elsewhere). From Figure 1.11, the following vector Eqs. can be written ρ ρ ρ R' F + N ρ ρ ρ ρ ρ ρ R Fs + Fn Fc + Ft R' Merchant represented various forces in a force circle diagram in which tool and reaction forces have been assumed to be acting as concentrated at the tool point instead of their actual points of application along the tool face and the shear plane. The circle has the diameter equal to R (or R') passing through tool point. After F c, F t, α and φ are known, all the component forces on the chip may be determined from the geometry. For instance, the average stress on the shear plane can be determined by using force F s and the area of the shear plane. Another useful quantity is the coefficient of friction (µ) between the tool and chip. Using force circle diagram, it can be shown that

and, F F t cosα + Fc sin α... (1.3) N F c cosα Ft sin α... (1.4) Metal Cutting and Chip Formation Then, the coefficient of friction (µ) is calculated as µ tanβ F N F α + α t cos Fc sin Fc cosα Ft sin α... (1.5) where, β is the friction angle. or, We can also write : F + α µ t Fc tan Fc Ft tan α... (1.6) β tan 1 ( µ ) + α β tan tan 1 F t F... (1.7) c Fc Ft tan α From Figure 1.11, we get : Fs Fc cosφ Ft... (1.8) F n Ft cosφ + Fc F n F s tan( φ + β α)... (1.9) Also, from Figure 1.11, F F F c s c s R cos(β α) F R cos( φ+ β α) cos(β α) cos( φ + β α) or, cos(β α) F c F s... (1.9(a)) cos( φ + β α) Shear plane area is equal to : t b A u s... (1.10) If τ be the shear strength of the work material, then, t b F u s τ... (1.11) Substituting in Eq.(1.9 (a)), we get t b F u τ cos( β α) c cos( φ + β α)... (1.12) hence, t u bτ R... (1.12(a)) cos( φ + β α) 15

Theory of Metal Cutting From Figure 1.11, Ft Rsin( β α) t τ ub sin ( β α) cos ( φ + β α) (From Eq. 1.12(a))... (1.13) From Eqs. (1.12) and (1.13), we can write : Ft tan ( β α)... (1.14) Fc From the above analysis, unknown forces in the force circle diagram and the value of coefficient of friction can be calculated provided F c, F t, α, t u and t c are known measured. During machining operations, chips are formed as a result of plastic deformation. Hence, chips experience stresses and strains. At shear plane, two normal forces simultaneously act, i.e., F s and F n. Shear stress (τ) can be found as Mean shear stress ( τ) Fs As ( F c cosφ Ft ) btu... (1.15) Mean Normal stress (σ) Fn As ( Ft cosφ+ Fc )... (1.16) btu where, A s Shear plane area b /. t u 1.5 VELOCITY RELATIONSHIPS Since the chip is thicker than the uncut chip, the velocity of the chip as it moves along the tool face must be less than the cutting speed (assuming volume constancy during cutting, and width of cut before machining and after machining remains same). Different velocities during cutting can be estimated as follows : Assume that the cutting velocity of the tool relative to the workpiece is V c which is known before hand. The chip slides along the cutting (rake) face of the tool with a velocity relative to the tool equal to V f (chip flow velocity). The newly cut chip elements move relative to the workpiece along the shear plane with a velocity equal to V s (shear velocity). From the principle of kinematics that the relative velocity of two bodies (here tool and chip) is equal to the vector difference between their velocities relative to the reference body (the workpiece). Employing this principle, Figure 1.12 has been drawn. Using sine rule, from ABC, we get : V V c f Vs... (1.17) sin (90 ( φ α)) sin (90 α) or, V V c f Vs cos( φ α) cosα The chip flow velocity along the tool rake face is given by Vc V f V c. r c... (1.18) cos( φ α) whereas the shear velocity V s is obtained as : 16

V cosα V c s... (1.19) cos( φ α) Metal Cutting and Chip Formation Figure 1.12 : Velocity Relationship 1.6 SHEAR STRAIN AND SHEAR STRAIN RATE Since most practical cutting processes are geometrically complex, let us first study the orthogonal machining and extend the theory of orthogonal cutting to more complicated cutting process involving oblique cutting. Due to simplicity and fairly wide applications, the continuous chip without BUE has been most extensively studied. There are conflicting evidences about the nature of the deformation zone in metal cutting. This has led to two basic schools of thought in the approach to the analysis. Many workers, such as Piispanen, Merchant, Kobayashi and Thomson have favoured the thin plane model while Palmer and Oxley, and Okushima and Hitomi have based their analysis on thick deformation region (Figure1.8). Experimental evidences indicate that the thick zone model may describe the cutting process at low speeds, but at high speeds most evidences indicate that a thin shear plane is approached. Thin zone model is more useful in practical cutting and its analysis is simpler hence it has received more attention. Thin Zone Model Merchant developed an analysis based on the thin shear plane model. He made the following assumptions : The tool tip is sharp and no rubbing or ploughing occurs between the tool and the workpiece. The deformation is two dimensional, i.e., no side spread. The stress on the shear plane is uniformly distributed. The resultant force R on the chip applied at the shear plane is equal, opposite and collinear to the force R' applied to the chip at the tool-chip interface. Strain and strain rate are determined as follows : To derive an expression for shear strain, the deformation can be idealized as a process of block slip (or preferred slip planes), as shown in (Figure 1.13). Shear strain (γ) is defined as the deformation per unit length. s γ y AB CD AD CD + DB CD tan ( φ α) + cot φ... (1.20) sin ( φ α) cosφ + cos( φ α) sin ( φ α) + cosφ cos( φ α) cos( φ α) cos( φ+α φ) cosα cos( φ α) cos( φ α) 17

Theory of Metal Cutting or, cosα γ... (1.21) cos( φ α) Figure 1.13 : Strain and Strain Rate in Orthogonal Cutting Strain may also be expressed in terms of the shear velocity (V s ) and the chip velocity (V f ) γ V c Vs (from Eq.(1.19)) Therefore, shear strain rate (γ&) in cutting is given by s dγ y s 1 γ& dt t t y V V γ& s c cosα y cos( φ α) y y is mean thickness of PSDZ.... (1.22) 1.7 SHEAR ANGLE RELATIONSHIPS 18 A number of attempts have been made to study the mechanics of cutting process. In designing a metal cutting operation, it would be helpful to predict the position of the shear plane (angle φ). Attempts have been made to derive a fundamental relationship of the shear plane angle φ in terms of rake angle (α) and friction angle (β). Several theories have been proposed to establish a relationship between φ, α and β. Some of the theories have been discussed below. Ernst-Merchant derived a relationship using the minimum energy criterion, that is, the shear plane is located where the least energy is required for shear. The derivation of Ernst-Merchant equation is based on the following assumptions : (i) (ii) (iii) (iv) cutting is orthogonal, the shear strength of the metal along the shear plane is independent of the magnitude of compressive (normal) stress acting on that plane, the chip is continuous type with no built up edge, and the energy of separation of chip elements is neglected and the minimum energy criterion establishes the plane on which shearing deformation occurs. As the cutting progresses in the beginning, the cutting force (F c ) increases gradually, the shear stress on various planes ahead of the tool also increases. However, the shear stress will not be same on all the planes ahead of the tool because the shearing components of the forces on the planes are not the same, nor is the extent of areas the same. On one of

the planes, however, the shear stress will be greater than on any other plane, and as F c is further increased, the shear stress will reach the yield strength in shear of the material being cut and plastic deformation will occur along that plane, thus forming the chip. The cutting force required to cause shear deformation along that plane will then be the lowest cutting force. Once the shear deformation begins along one plane, the cutting force cannot exceed that minimum value. Metal Cutting and Chip Formation To determine shear-plane angle φ, express the cutting force F c in terms of φ, differentiate it with respect to φ, equate the derivative to zero, and solve it for the angle φ as follows : cos( β α) Fc Fs cos τ tub cos( β α) { φ + ( β α) } cos( φ + β α) (Eq.1.12) Here, except φ all other parameters can be taken as constant during machining (assuming that no strain hardening takes place). It would give the condition for the minimum energy if the derivative of F c with respect to φ is equated to zero. d Fc τt d φ u cosφcos( φ+β α) sin φ( φ+β α) b cos( β α) 0 2 2 cos ( φ+β α) Therefore, cosφcos ( φ +β α) sin ( φ+β α ) 0 or, cos (2 φ +β α ) 0 or, (2 φ+β α ) π 2 π 1 Hence, φ ( β α)... (1.23) 4 2 where, φ, β and α are shear angle, friction angle and rake angle, respectively. Eq. (1.23) indicates that the shear angle φ is a unique function of the tool rake angle and the angle of friction in metal cutting. Merchant further introduced a modification to this theory and assumed that the shear strength of a polycrystalline metal is affected by temperature, rate of shear, shear strain (plastic) and the stress acting normal to the shear plane. While it is known that the normal compression stress on a plane does not affect the shear strength of a single crystal however, the shear strength of polycrystalline material is affected. The modified Eq. is 1 φ C ( β α)... (1.24) 2 2 where, C depends on the slope of the shear strength vs. compressive stress curve for the given material. 'C' is also known as machining constant. In 1949, another approach to the analytical solution of the shear plane angle was made by Lee and Shaffer. They assumed that the material being cut behaves as an ideal plastic which does not strain harden. It was assumed that the shear plane coincides with the direction of the maximum shear stress (Figure 1.14). Based on these assumptions, they applied slip line field theory and derived the relationship given by Eq. (1.25). π φ + ( α β)... (1.25) 4 As a modification, later on Lee and Shaffer considered the effect of a small built up edge or nose, and its effect on the stress field referred to above and arrived at an expression for the shear angle (φ) which included an additional angle θ, which depends on the size of the built up edge, 19

Theory of Metal Cutting π φ + ( α β) + θ 4 In 1952, Shaw, Cook and Finnie extended the Lee and Schaffer theory by further analytical and experimental investigations, and arrived at the following relationship : π φ + ( α β ) + η 4 While deriving the above relation, they assumed that the shear plane is not a plane of maximum shear. Here, η is established by the analytical method and it is not constant. η is the angle between the shear plane and the direction of the maximum shear stress. To determine the value and sign of the η, it is necessary to draw the Mohr s circle diagram. Figure 1.14 : Shear Plane Model of Lee and Shaffer Based on the experimental study of the mechanics of chip formation and the flow of grains in the material during cutting, Palmer and Oxley observed that the deformation does not take place along a plane, rather it takes place in a narrow wedge shaped zone. But for analytical simplicity, it was considered as a parallel sided shear zone (Figure 1.15). 20 Figure 1.15 : Shear Zone Model by Oxley A further contribution towards the solution of this problem was made by R Hill in 1954, who analyzed the state of stress at the shear zone, using a new principle On the limits set by plastic yielding to the intensity of singularities of stress. But in 1959, Eggleston, Herzog and Thomsen tried to show by their test results that none of the three Eqs. (by Ernst and Merchant, Lee and Shaffer, and Hill) was correct which implies that metal in the shear zone under the existing conditions of stress, high rates of strain and elevated temperature does not behave as ideal plastic solid. Since no single criterion is applicable to the shear angle relationship in metal cutting, and since a satisfactory theory has not been advanced at present to explain the experimental observations adequately, the challenge exists for a closer solution to the problem of angle relationship. This problem is so tedious because the complexity is created by the simultaneous presence of so many variables at a time, for example : (i) (ii) (iii) plastic deformation, work hardening, external and internal friction,

(iv) (v) (vi) temperature effect, diffusion, oxidation, and (vii) local heating, etc. Example 1.1 Solution Show that in case of ideal orthogonal cutting operation the shear strain undergone by the chip during its removal from the workpiece would be minimum if the chip thickness ratio is 1. In Figure 1.13 the shear strain in general and shear strain in cutting are shown. Here, s is in the direction of force, y is in the direction to the force. Shear strain in another term of interest is associated with the cutting process. The s shear strain is defined as and hence in cutting (Figure 1.13), y Metal Cutting and Chip Formation s AB AD DB γ + tan ( φ α ) + cot φ y CD CD CD We want the condition when γ should be minimum. Hence, differentiate γ with respect to φ and equate the derivative equal to zero. dγ d dφ dφ sec 2 sec { tan ( φ α) + cot φ} 0 ( φ α) + ( cosec 2 ( φ α) cosec or, sin φ cos ( φ α). Take the under root to both sides, ± 2 ± cos( φ α) 2 2 2 φ) 0 φ or, cos( φ α)... (A) cos φ cosα + sin α 1 cosαcot φ + sin α... (B) Question is that at the condition (A) whether the chip thickness ratio is 1 or not. We know that chip thickness ratio is given by t γ u c tc If, γ 1, then 1 cos( φ α) cos( φ α) cos ( φ α)... (C) By comparing Eqs. (A) and (C), we find that both are the same. Hence, it is proved that shear strain will be minimum only when the chip thickness ratio is unity. 21

Theory of Metal Cutting Example 1.2 Solution In orthogonal turning operation with +10 back rake angle tool, the following observations were made: cutting speed 160 m/min, width of cut 2.5 mm, F c 180 kgf, F t 50 kgf, deformed chip thickness 0.27 mm, tool chip contact length 0.63 mm and feed rate 0.20 mm/rev. Determine the following : chip thickness ratio, shear angle, friction angle, resultant force, shear force and shear strain. (i) Chip thickness ratio, r c tu tc 0. 20 0.74 0. 27 r c 0.74 (ii) Shear angle, φ tan -1 rc cosα 1 rc sin α 0.74 cos 10 tan -1 1 0.74sin 10 φ 39.94 o (iii) Friction angle, β tan -1 µ tan - F 1 N F N F + α t Fc tan Fc Ft tan α 50 + 180 tan10 180 50 tan10 0.477 β tan - 1 (0.477) β 25.52 o (iv) R F c cos( β α) 180 cos(25.52 10) R 186.81 kg (v) F S R cos (φ + β α) 186.8 cos (39.9 + 25.5 10) F S 106.07 kg (vi) Shear strain tan (φ α) + cot φ tan 29.9 + cot 39.9 0.575 +1.196 22

Example 1.3 γ 1.771 A cylindrical bar has a blind hole of 15 mm diameter. Its face is being turned (facing operation) from inner diameter to the outer periphery (Figure given below) at a speed of 600 RPM, feed 0.20 mm/rev., and depth of cut 1.0 mm. Calculate the cutting speed (m/s) and total volume removed at the end of 15s. Metal Cutting and Chip Formation Solution Arrow in the figure shows the tool movement. To find, (i) Revolution/second (N s ) 600/60 10 V 15 cutting speed at the end of 15 seconds of facing operation. (ii) V 15 volume of material removed at the end of 15 seconds. (i) V t πdν s t 1000 where, N s t 15 10150 rev.(# of revolutions made by the the work at the end of 15s) V t cutting speed at time 't' D d + 2f N s t (Figure above) D Diameter of the workpiece at which the tool tip will be after the time of machining 15s. In one revolution of the workpiece, the diameter at which the tool will be cutting, will increase by 2f. (or in one revolution the diameter to which the tool tip reaches is increased by 2f). where, f feed rate D 15 15 + (2 0.20 10 15) V15 75 mm π 75 10 1000 V 15 2.36 m/s (ii) Volume of the material removed in 15s. 23

Theory of Metal Cutting V 15 total area machined depth of cut 4 π (D 2 d 2 ) 1 mm 3 Example 1.4 Solution 4 π (75 2 15 2 ) 1 V 15 4241 mm 3 During orthogonal turning of a pipe of 100 mm diameter, the rake angle of the tool was 20 o. The ratio of the cutting force to feed force was 3.0.The feed rate, depth of cut and chip thickness ratio were 0.275, 0.687 and 0.4 respectively. With the help of a dynamometer, feed force was measured as 460 N. Workpiece was rotating at 450 revolution per minute. Determine chip velocity, shear strain, shear strain rate and mean width of PSDZ. We know from Eq.(1.12) that Vc V V s f sin(90 + α φ) sin(90 α) V f V c sin (90 + α φ) sin (90 α) V s V c sin (90 + α φ)... (A)... (B) But, we do not know the values of φ and V c. They can be evaluated as follows : r tan φ c cosα 1 rc sin α 0.4cos 20 1 0.4sin 20 0.436 φ tan -1 0.436 φ 23.54 o... (C) πdn π 100 450 V c 1000 1000 V c 141.37 m/min Substitute the values of φ and V c in Eqs.(A) and (B). 141.37 sin 23.54 V f sin (90 + 20 23.54)... (D) 24 We also know, V f 56.56 m/min 141.37 sin(90 20) V s sin (90 + 20 23.53) V s 133.11 m/min

γ tan (φ α) + cot φ tan (23.54 20) + cot (23.54) γ 2.357 Metal Cutting and Chip Formation cosα γ c... (E) dsv.cot ( φ α) Here, we do not know the value of ds. Using Lee and Shaffer s theory, ds can be derived as [Jain and Pandey, 1980] 1 f sin (90 + α φ) ds 2 2 sin (45 α + φ) Therefore, from Eq. (E) 2 1 ds 0.324 mm 0.275. 2 sin (25.54) sin (90 + 20 25.54) sin (45 20 + 25.54) γ 141.37 1000 0.324 γ 6830 s - 1 cos20 cos(23.53 20) Note that the shear strain rate in metal cutting is very high as compared to the one obtained in classical deformation test. Example 1.5 Solution Prove that the specific cutting pressure in an ideal orthogonal cutting is given by τ cot φ, provided 2 φ + β α π/2 holds good (τ shear stress). F c u Specific cutting pressure bt From Eq. (1.12), t τ F c u b cos( β α). cos( φ + β α) It is given that,... (A)... (B) (β α) + 2 φ π/2... (C) Substitute the value of (β α) from (C) in (B), t F c u bτ cos( π/ 2 2φ). cos( φ + π/ 2 2φ) t τ u b sin 2φ. t τ Sp. cutting press u b. sin 2φ. 1 bt u 2cosφ τ sin φ Sp. cutting press 2 τ cot φ proved Example 1.6 25

Theory of Metal Cutting Following data were recorded during orthogonal machining : Bar diameter 40 mm, depth of cut 0.125 mm, length of chip obtained 62.5 mm/rev, horizontal cutting force 220 kgf, vertical cutting force 85 kgf, α 7 ο, spindle speed 500 RPM. Find out friction angle, chip thickness ratio, shear angle, chip velocity and shear velocity. Solution t Chip thickness ratio u l c 62. 50... (A) tc lu lu We know, undeformed chip length l u π D N r π 40 1 120.66 mm From (A), r c 62.5/120.66 0.479 r c 0.479 From Eq. (1.1), Substitute the values, φ tan 1 rc cosα 1 rc sin α φ tan 1 0.494 0. 526 0.939 β φ 27.7 ο Ft + Fc tan α Fc Ft tan α Substitute the values in the above equation πdn Cutting velocity, V c 1000 (Eq. 1.7) 85 + 220tan 7 β 0. 53 ο 220 85tan 7 β 28.12 ο π 40 500 1000 V c 62.83 m/min V f V c cos( φ α) 62.83 0.465 0.935 V f 31.22 m/min. o 26

V s V c sin (90 α) sin (90 + α φ) Metal Cutting and Chip Formation sin (90 7) 62.83 sin (90 + 7 27.7) V s 66.66 m/min. SAQ 1 Write the most appropriate option from the given ones (i) In actual practice, chip thickness ratio is (a) > 1, (b) < 1, (c) 1. (ii) In oblique cutting, the number of forces that act on the tool are (a) one, (b) two, (c) three, (d) none of these. (iii) Which of the following is the chip removal process? (a) rolling, (b) extruding, (c) die casting, (d) broaching, (e) none of these. (iv) Time taken to drill a hole through a 2.5 cm thick plate at 3000 RPM at a feed rate 0.025 mm/rev. will be (a) 20 s, (b) 10 s, (c) 40 s, (d) 50 s. (v) Shear plane angle is the angle between (a) shear plane and the cutting velocity vector, (b) shear plane and tool face, (c) shear plane and horizontal plane, (d) rake face and vertical plane. (vi) In orthogonal cutting, the cutting edge should be (a) straight, (b) parallel to the original plane surface of the workpiece, (c) normal to the direction of cutting, (d) all of these, (e) none of these. (vii) Continuous chip with BUE (a) yields good surface finish, (b) yields poor surface finish, (c) has no effect on surface roughness. (viii) The ratio of cutting velocity to chip velocity is usually (a) >1, (b) <1, (c) 1. 1.8 SUMMARY Various types of metal cutting processes can be classified in two types: orthogonal and oblique cutting. During cutting, depending upon the type of workpiece material and machining conditions, one of the three types of the chips will be obtained (continuous, continuous with BUE, or discontinuous). Chip formation takes place due to the shearing action. In the process of chip formation, various types of forces act simultaneously. Magnitude of force decides the power requirement. Chip velocity can be theoretically evaluated using the analysis presented in this unit. One of the important parameters is shear angle, which can be determined theoretically or experimentally. Two schools of thought prevail regarding PSDZ thin zone model and thick zone model other than shear plane. 27

Theory of Metal Cutting 1.9 KEY WORDS Chip Primary Shear Deformation Zone (PSDZ) Secondary Shear Deformation Zone (SSDZ) Orthogonal Cutting Oblique Cutting : It is the material which is separated from the workpiece when the tool moves into the workpiece. : Finite zone (thin or thick depending upon various governing parameters) within which deformation takes place. : Deformation which takes place at tool-chip interface. : Two dimensional (2-D) cutting in which cutting edge is straight, parallel to the original plane surface of the workpiece and perpendicular to the direction of cutting. : Cutting operations are 3-D in nature. In this type cutting edge at the tool is inclined to the line normal to the cutting direction. 1.10 ANSWERS TO SAQs SAQ 1 (i) (b) (ii) (c) (iii) (d) (iv) (a) (v) (a) (vi) (d) (vii) (a) (viii) (b) EXERCISES 28 Q 1. (i) (ii) (iii) (iv) Derive a relationship to calculate shear angle in terms of measurable/known parameters. Draw force circle diagram proposed by Merchant for orthogonal cutting conditions showing different forces acting on tool, chip, and work system. From the diagram, derive the expression for (a) (b) shearing force on the shear plane, friction force on the tool face in terms of cutting force, thrust force, rake angle, and shear angle. Define orthogonal cutting. Draw Merchant's force circle diagram for the orthogonal cutting. Using the Figure in Q.1 (ii) (a), derive the expression for friction force. What are the factors which affect the formation of different types of chip obtained in cutting.

(v) (vi) Determine the condition when φ tan 1 (r c ) where, φ is the shear angle and r c is the chip thickness ratio. (Hint : in the original Eq., Substitute tan φ r c ) (Ans. α 0) Determine the condition for which chip flow velocity is equal to the cutting velocity, assuming α 0. (Ans. φ 45 o ) (vii) Find the ratio of F c / F t for an imaginary case of machining if α β π/4. Q 2. Mild steel rod is being turned at the speed of 27.3 m/min. Feed rate used is 0.25 mm/rev, and deformed chip thickness is equal to 0.30 mm. Rake angle and shear angle of the tool are 20 o and 30 o, respectively. Calculate the shear flow velocity. Q 3. For orthogonal cutting of a M.S. rod, the following data are obtained : width of cut 0.125'', feed 0.007'' per rev., α 15 o, β 30 o, and machining constant, C 70 o.the dynamic shear strength of the work material 80000 lb/in 2. Calculate Fc and Ft. Q 4. During orthogonal cutting of a tube at 100 m/min, the tangential force (in the direction of cutting velocity) measured by the 3-D dynamometer is 200 kgf, and the axial force is 100 kgf. Assume the rake angle as 10 o. Calculate the work in shearing the metal if the shear angle 30 o. Also, derive the velocity relationship used. Metal Cutting and Chip Formation 29

Theory of Metal Cutting BIBLIOGRAPHY Armarego, E. J. A. and Brown, R. H. (1969), The Machining of Metals, Prentice Hall, Englewood Cliffs, NJ. Jain, V. K. and Pandey, P. C. (1980), An Analytical Approach to the Determination of Mean Width of Primary Shear Deformation Zone (PSDZ) in Orthogonal Machining, Proc. 4 th International Conference on Production Engineering, Tokyo, pp 434-438. Kalpakjian, S. (1989), Manufacturing Engineering and Technology, Addison-Wesley Publishing Co., New York. Pandey, P. C. and Sing, C. K. (1998), Production Engineering Science, Standard Publishers Distributors, Delhi. Rao, P. N. (2000), Manufacturing Technology : Metal Cutting and Machine Tools, Tata McGraw-Hill Publishing Co. Ltd., New Delhi. Shaw, M. C. (1984), Metal Cutting Principles, Oxford, Clarendon Press. 30