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Stress analysis of metal cutting tools Jemal, Girma 1992

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Stress Analysis of Metal Cutting ToolsbyGirma JemalB. Tech (Honours), Mechanical EngineeringUniversity of Calicut, IndiaA THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESMECHANICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© Girma Jemal, 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of Mechanical EngineeringThe University of British ColumbiaVancouver, CanadaDate DE-6 (2/88)AbstractMetal cutting tools experience cutting forces distributed over a small chip-toolcontact area. When the magnitude of the stresses induced by the cutting forcesexceeds the tool material fatigue strength, failure of the cutting tool results. In thisthesis, stress analysis in cutting tools is presented in order to predict the location andmodes of tool failures.The stress analysis of cutting tools is presented using both analytical and numer-ical (Finite Element) based methods. First, various cutting force distributions onthe rake face of the tool and analytical cutting tool stress solutions available in theliterature are surveyed. It is then shown that the previous analytical solutions are in-correct because they directly applied the infinite wedge solution to determine stressesin the loaded region of the cutting tool. In this thesis, the tool and the boundarystresses are considered both in the loaded and free region. For a polynomial boundarystresses on the rake face and zero boundary stresses on the flank face, the stressesin a two-dimensional cutting tool are determined using the infinite wedge solution.The analytical cutting tool stress distributions obtained agrees well with finite ele-ment solutions and published photoelastic experimental stress distributions. Fromthe stress distribution obtained, it is shown that the critical maximum tensile stressoccurs at the end of chip-tool contact and it results in initiation of cracks and finalfracture of the whole loaded region. The critical maximum compressive stress occurson the flank face close to the cutting edge which results on cutting edge permanentdeformation. The critical maximum shear stress occurs at the cutting edge and itresults in cutting edge chipping.The possible extension of the two-dimensional solution to determine stresses iniiend mill flutes is considered. A comparison of a finite element solution of an end millflute and the two-dimensional solution obtained above (for the same wedge angle andboundary load distribution) shows agreement at the cutting edge while at the end ofchip-tool contact the two-dimensional solution gives an upper bound estimate. Thusthe conclusions reached for tool failure in the loaded region from the two-dimensionalsolution is also applicable in end mill flutes. At the end mill shank, stress predictionsusing a cantilevered beam solution agrees with a finite element solution. The stressdistribution shows shank fracture either at the fixed end of the end mill where it isattached to the chuck or at the flute section closest to the circular portion of the endmill.In this study for both orthogonal cutting tools and end mills, good correlation isobtained between predicted and observed in-service cutting tool failures. Therefore,the proposed cutting tool stress analysis approach may be recommended for cuttingtool design and selection of optimum machining conditions.iiiContentsAbstract	 iiList of Figures	 viNomenclature	 xAcknowledgement	 xii1 Introduction	 12 Literature Review	 52.1 Introduction  	 52.2 Metal cutting  	 72.3 Boundary load distributions  	 102.3.1 Split-tool method 	 102.3.2 Photoelasticity  	 132.3.3 Slip-line theory  	 232.4 Analytical stress analysis of cutting tools  	 272.4.1 Analytical stress analysis of cutting tools for a concentrated load ap-proximation.  	 272.4.2 Analytical stress analysis of cutting tools for distributed boundaryloads.  	 292.5 Previous analytical cutting tool stress solutions  	 322.5.1 Cutting edge stresses  	 322.5.2 Stress distribution in the loaded region for parabolic load  	 352.6 Objectives of the present work and methods of investigation  	 39iv3 Stress Calculation in Orthogonal Cutting Tools 413.1 Introduction 	 413.2 Analytical cutting tool stress calculation 	 423.3 Analytical cutting tool stresses for parabolic load distributions 	 483.4 Finite element cutting tool stress analysis for parabolic load distributions 553.5 Discussion and conclusions 	 594 Stresses for Higher Speed Photoelastic Boundaries 654.1 Introduction 	 654.2 Analytical solution for high speed photoelastic boundary load distributions 664.3 FEM solution for higher cutting speed photoelastic boundary load distributions 744.4 Point-load analytical solution for higher cutting speed photoelastic boundaryload distributions 	 764.5 Failure in cutting tools 	 794.6 Discussions and conclusions 	 875 Stress Analysis of an End Mill 935.1 Introduction 	 935.2 Shank stresses in an end mill 	 955.2.1	 Force distribution along end mill cutting edge 	 955.2.2	 Analytical stress analysis in an end mill shank 	 995.2.3	 Finite element stress analysis in an end mill shank 	 1045.2.4	 Discussions and conclusions 	 1105.3 End mill flute stresses 	 1125.4 End mill cutting edge stresses 	 1165.4.1	 End mill cutting edge stresses using elasticity 	 1175.4.2	 End mill cutting edge stresses using FEM 	 1195.5 Conclusions 	 1226 Concluding Remarks 1256.1 Summary 	 1256.2 Conclusions 	 1256.3 Recommendations 	 128VList of Figures2.1 Schematics of an orthogonal cutting process  	 72.2 Definition of terms used in cutting  	 82.3 Photomicrograph of a partially formed chip, Trent [13]  	 92.4 Principles of split-tool dynamometer  	 102.5 Cutting boundary load distribution result from split-tool dynamometer, Kato[15], Barrow [16], and Childs [17]  	 122.6 Orthogonal cutting of a lead workpiece with a photoelastic tool, Amini [20] 	 142.7 Numerical integration of the equation of equilibrium to determine normalstresses  	 162.8 Cutting boundary load distribution result from photoelasticity at low cuttingspeeds, Betaneli [7], Usui [18], and Chandrasekaren [19]  	 182.9 Cutting boundary load distribution result from photoelasticity at higher cut-ting speeds, Amini [20] and Ahmad [21]  	 182.10 Stress components on shear planes on an infinitesimal element  	 232.11 Derivation of co-ordinate stresses from the shear plane stresses  	 242.12 Boundary stresses at the cutting edge using slip line theory  	 262.13 Concentrated cutting load at the cutting edge  	 282.14 Distributed load on the rake face  	 302.15 Stress components in polar co-ordinate on an infinitesimal element of a cut-ting tool  	322.16 Linear cutting load distribution assumption, Archibald [6]  	 332.17 Radial cutting edge stresses as a function of wedge angle and friction coeffi-cient, Archibald [6]. Here 0 is the wedge angle  	 362.18 Principal stress distributions in the loaded region for a parabolic load, Be-taneli [7]. Here a is the depth of cut and /3 is the wedge angle  	 38vi3.1 Polynomial boundary load distribution determined by photoelasticity at lowcutting speeds, Betaneli [7]  	 533.2 Analytical principal stress distributions in the loaded region of a cutting tooldone to verify Betaneli's result  	 543.3 Analytical solution of rake face stresses for the parabolic boundary load dis-tribution made to verify Betaneli's result  	 543.4 Elastic deformation of the loaded region of a cutting tool  	 553.5 Finite element model of the cutting tool  	 563.6 Finite element model of the loaded region of the cutting tool, contact length=/,  	 573.7 FEM solution for the maximum principal stress distributions in the loadedpart of the tool for the boundary shown in Fig. 3.1  	 583.8 FEM solution for the rake face stresses for the parabolic boundary load dis-tribution of Fig. 3.1, contact length lc = 0.5 mm  	 583.9 Comparison of FEM and analytical maximum principal stresses al along therake face for the boundary shown in Fig. 3.1, contact length /c = 0.5 mm .	 593.10 The FEM solution satisfies the boundary load conditions both in the loadedand free region of the rake face, contact length lc = .5 mm  	 603.11 The Analytical solution does not satisfy the boundary load conditions in thefree region of the rake face, contact length /, = .5mm 	 613.12 Comparison of FEM and analytical solutions for the boundary load distribu-tion shown in Fig. 3.11  	 623.13 Tool deformation and maximum principal stress distribution for the bound-ary shown in Fig. 3.11  	 634.1 Ahmads's [21] polynomial functions for his photoelastic data do not satisfythe free boundary condition after chip-tool contact (l, = 4.5 mm)  	 674.2 A polynomial function which approximately satisfy the free loading conditionafter chip-tool contact for a metal cutting tool (lc = 1 mm)  	 674.3 The FEM boundary completely satisfy the free loading condition after chip-tool contact (lc = lmm)  	 704.4 Analytical solution for the maximum principal stress distribution for theboundary shown in Fig. 4.2  	 734.5 Analytical solution for rake face stresses for the boundary shown in Fig. 4.2 	 734.6 FEM solution for the maximum principal stress distribution for the boundaryshown in Fig. 4.3  	 74vii4.7 FEM solution for the minimum principal stress distribution for the boundaryshown in Fig. 4.3  	 754.8 FEM solutions for rake face stresses for boundaries shown in Fig. 4.3 . . .  	 764.9 Comparison of analytical and FEM principal and maximum shear stress dis-tributions for the boundary shown in Fig. 4.3  	 774.10 A comparison of rake face principal stress al by different methods for a 620wedge angle and boundary load distribution shown in Fig. 4.3, chip-toolcontact Ic = 1 mm 	  784.11 Fracture locus of a brittle material  	 804.12 Brittle failure of a cutting tool near the end of chip-tool contact, Tlusty [27] 834.13 Cutting edge deformation due to maximum compressive stress, Wright [36]	 844.14 Equivalent stress contour lines for the boundary shown in Fig. 4.3  	 854.15 cutting edge chipping, Tlusty [27]  	864.16 Bending stress calculation at the fixed end (point A) of a cutting tool . .  	 874.17 Maximum shear stress distribution from photoelasticity , Amini [20] 	 894.18 Twice the maximum shear stress distribution obtained for the photoelasticcutting boundary of Fig. 4.3 	  904.19 Effect of change in wedge angle on the critical principal stresses of a cuttingtool  	 915.1 Schematic diagram of an end mill 	  955.2 Schematics for the cutting operations of an end mill  	 965.3 Cutting force distribution along the cutting edges of an end mill, Kline [28] 	 975.4 Definition of terms used in elemental force calculation  	 985.5 Developed surface of a four flute end mill having helix angle 0 	  995.6 Determination of the equivalent diameter for four flute cutters 	  1025.7 Representation of an end mill by an equivalent solid stepped bar 	  1035.8 Cross-sectional geometry of one flute of a four flute cutter at z = 0, Fig. 5.1 1055.9 Circular cross-section of a flute at the beginning of the circular section atz=c, Fig. 5.1 	  1065.10 Finite element model of an end mill with elemental forces applied 	  1075.11 Finite element solution for end mill bending stress distribution 	  1095.12 Common types of end mill shank breakages, Bouse [33] 	  1115.13 End mill FEM flute model and its minimum principal stress a3 distributionfor the boundary load distribution shown in Fig. 4.2 	  114viii5.14 Comparison of maximum principal stress distributions o-i for a two dimen-sional wedge and an end mill flute for the boundary load distribution shownin Fig. 4.2 	 1155.15 Boundary stress distribution considered for cutting edge stress analysis . . 	  1175.16 Finite element solution for the minimum principal stress 0-3 at the cutting edge1205.17 Schematic diagram showing the portion of the ball end mill considered forcutting edge FEM model 	  1215.18 Finite element solution for the maximum shear stress at the cutting edge 	 1235.19 Comparison of maximum principal stress distributions cri for a two-dimensionalwedge and a ball end mill 	  124ixNomenclaturea	 distane from the fixed end of an end mill to force F, mma„b,,ci, d,	 stress distribution constantswidth of cut, mmend mill flute axial length, mmcutter diameter, mmde	 equivalent cutter diameter, mmFt	 tangential elemental forces, NSF,.	 radial elemental forces, Nz	 axial element length, mmfeed per tooth, mm/toothfringe constantresultant cutting force in an end mill, NF,	 shear force on the rake face, kgnormal force on the rake face, kgF.S	 factor of safetytool thickness, mmyield shear strengthKt	 tangential force constant, M PaKr	 radial force constant, (-)ic	 contact length, mmbending moment at the shank, kN — mmexponent of the parabolic boundary load distributionN,	 polynomial normal stress distribution coefficientsNr,	 rake face polynomial normal stress distribution coefficientshydrostatic pressureresultant cutting force at the cutting edge(r, 0)	 polar coordinate of any point in the cutting wedgeR,	 polynomial radial stress distribution coefficientsSri	 rake face polynomial shear stress distribution coefficientsS,	 polynomial shear stress distribution coefficientsS/	 difference of maximum and minimum principal stressesSy	 tensile yield strengthSut	 ultimate tensile strength, M PaSuc	 ultimate compressive strength, M Padepth of cut, mmcutting torque on end mill, kN — mmaxial depth of cut, mmrake distance from cutting edge, mmGreek Symbolswedge anglefriction angle7	rake anglee	angle between finished surface and axial cutting edge elementA	 angle between wedge axis of symmetry and force Pfriction coefficientnormal stress on the rake face, kg Imm2cri	 maximum principal stress, kg Imm2•3	 minimum principal stress, kg Imm2o	 normal boundary stress at the cutting edgear	 radial normal stress in polar coordinateax	 normal stress in the x-directionay	 normal stress in the y-directionCry	 tangential normal stress in polar coordinateaav	 average normal stress on the rake faceGreg	 equivalent stressshear stress on the rake face, kg Imm2To	 shear boundary stress at the cutting edgeTav	 average shear stress on the rake faceTr!)	 shear stress in polar coordinateTxy	 shear stress in rectangular coordinateangle between al and x-axis; swept angleangle between a-line and x-axisçb	 stress function; helix anglexiAcknowledgementI like to express my deep gratitude to Professor Douglas P. Romilly and ProfessorYusuf Altintas for their guidance and advice during all stages of this project. I alsowant to thank Professor I. Yellowley for the helpful discussions.My thanks are also extended to Mr. Gerry Rohling for his assistance with thecomputers.I want also to thank my family and friends for their moral and emotional support.Finally, I want to thank CIDA for its financial support during my stay at UBC.)diChapter 1IntroductionDesign and failure analysis of cutting tools and selection of optimum cutting con-ditions require knowledge of the stress distribution in the cutting tools. The stressanalysis approach used to determine the stresses depend on where the stress predic-tions are required. When the stresses far from the cutting edge (i.e. at the shank) arerequired, beam equations with a concentrated load at the cutting edge can be used.When the stresses outside the loaded region within the wedge shape are needed, anelasticity solution with a concentrated load at the cutting edge could be used. How-ever, when the stresses within the loaded region must be predicted then the actualload distribution along tool-chip contact should be used in the elasticity solution.The boundary load distributions on cutting tools are determined using one or moreof the following techniques:• Split-tool method,• Photoelasticity,• Slip-line solution.1CHAPTER 1. INTRODUCTION	 2Once the boundary load distribution is determined using one of the methods listedabove, the stresses within the cutting tool may be determined either:• Analytically, or• Numerically (Finite Element Method or MeIlin Transform).The analytical solution for a concentrated load at the cutting edge is based onthe elasticity solution given by Frocht [1]. The analytical solution for the distributedload along tool-chip contact is based on Michell's [4] general stress function for two-dimensional problems which was subsequently applied to an infinite wedge prob-lem by Timoshenko [5]. Archibald [6], Betaneli [7], and others applied the infinitewedge solution to determine the stress distribution in the finite cutting tool, i.e.they assumed the infinite wedge solution to be applicable to determine stresses intwo-dimensional cutting tools.For a given cutting tool geometry and boundary load distribution, numerical meth-ods could be used to determine the stress distribution in a cutting tool. The two nu-merical methods used in the stress analysis of cutting tools are the MeIlin Transformand the Finite Element Method (FEM). The Mellin Transform method is described byTranter [9], and cutting tool stress solutions using this method are given by Thomason[10].From the discussion above, the analytical approach available to determine thestresses in the loaded region of a cutting tool is based on the assumption that theinfinite wedge solution can be applied to determine the stresses in the finite cuttingtool. It is important to verify this assumption because the solutions obtained by usingthis approach may sometimes lead to incorrect conclusions. To critically investigateCHAPTER 1. INTRODUCTION 	 3this assumption and to conclude whether and under what conditions this approachcould be applied is the purpose of the first phase of this work.In the second phase of this work, the critical regions of cutting tools where breakageis likely to occur will be predicted. For this an orthogonal cutting tool and an end millwill be considered. Analytical methods are first used to predict the critical stressesthen these solutions will be verified with numerical methods. This verification isuseful to indicate whether analytical solutions can be applied in cutting tool failureanalysis.A brief description of the contents of the chapters which follow are given below.The principles of the experimental methods used to determine the boundary loaddistribution in cutting, and some results by previous workers of these distributionsare discussed in Section 2.3. In Section 2.4 and 2.5 the analytical stress equationsand the solutions obtained by previous workers are presented.In Section 3.2 the method for calculationg stresses in a cutting tool are calculatedfor a general boundary load distribution is shown. Section 3.4 gives the finite elementsolution for the low speed photoelastic boundary load distribution. Finally in thischapter previous analytical solutions and finite element solutions will be compared,and the types of load distributions that the analytical solutions may be used will beidentified.Analytical and finite element solutions will be compared in Chapter 4 for the typesof load distributions where the analytical solution may be applied. These results willalso be compared to previous experimental results for similar boundary conditions.In Section 4.5 the critical regions in two-dimensional cutting tools and their modesof failure will be determined.CHAPTER 1. INTRODUCTION	 4In Chapter 5 the stress analysis of an end mill is presented. The critical regionsfor breakage of an end mill is shown. The possibility of using a simpler analyticalsolution to determine the stresses in these critical region is discussed.Finally in Chapter 6 the results are summarized and conclusions drawn. Therecommendations for future work for a better understanding of the stress distributionin cutting tools are given.Chapter 2Literature Review2.1 IntroductionStress analysis of a metal cutting tool deals with the determination of the stressdistribution in the cutting tool produced by cutting forces. This stress distributionprovides the critical regions of the cutting tool where breakage is likely to occur, andit also helps in selecting the maximum cutting force that the cutter can withstandwithout breakage. Experience in metal cutting indicates that tools usually breaknear the cutting edge where the loads are applied, and therefore to predict thesetool failures stresses near the applied loads have to be determined. The method ofstrength of materials which is only applicable further away from applied loads andfixed boundaries can not be used and therefore in this case the more general theoryof elasticity method is used.Stress-strain relations and the basic equations of theory of elasticity is described byTimoshenko [5]. In elasticity the solution of a two dimensional problem is the stressfunction which satisfy the biharmonic and the boundary conditions of the problem.Once the stress function for the given problem is known then the stress components5CHAPTER 2. LITERATURE REVIEW 	 6can be determined by differentiation of the stress function and satisfying the boundaryconditions of the problem.In cutting tool stress analysis, the boundary conditions are determined experi-mentally using one of the following methods:• Split-tool• Photoelasticity• Slip-line field.The principle behind these methods and some boundary load distribution results byprevious workers are reviewed in Section 2.3.Once the boundary load distributions are known, the stresses within the tool maybe determined using one of the following:1. Analytical methods with:• concentrated load approximation.• distributed boundary loads.2. Numerical methods:• Mellin transform• Finite Element Method (FEM).The analytical methods are reviewed in Section 2.4. The MeIlin Transform methodis described by Tranter [9], and cutting tool stress solutions using this method aregiven by Thomason [10]. The finite element method is described by Cook et al [8].CHAPTER 2. LITERATURE REVIEW	 7VFigure 2.1: Schematics of an orthogonal cutting processThe determination of boundary conditions on cutting tools require the under-standing of the principles and terminologies used in metal cutting. Therefore, thischapter is started by reviewing relevant metal cutting principles.2.2 Metal cuttingMetal cutting principles are discussed by Shaw [12], Trent [13], Boothroyd [14] andothers. Here metal cutting principles and terminologies required in this study will bebriefly summarized.All metal-cutting operations are likened to the fundamental process illustrated inFig. 2.1 in which a wedge-shaped tool with a straight cutting edge is constrainedto move relative to the workpiece in such a way that a layer of metal is removed inthe form of a chip. The thickness of material removed from the workpiece is knowncuttingedgemachinedsurfaceCHAPTER 2. LITERATURE REVIEW	8rakeanglerakefaceclearanceor flankfaceFigure 2.2: Definition of terms used in cuttingas the depth of cut t. The width of the material removed is the width of cut b. Ifthe cutting edge is at right angles to the direction of the relative work-tool motion,Fig. 2.1, the cutting mechanism is said to be orthogonal, otherwise the term obliquecutting is used. Two-dimensional orthogonal cutting is widely used in research worksbecause the principles developed is also generally applicable to the three-dimensionaloblique cutting. The surface along which the chip flows (Fig. 2.2) is known as therake face of the tool, and it intersects the tool flank face to form the cutting edge.The angle the rake face makes with the vertical to the machined surface is called therake angle -y. The clearance angle is the angle between the machined surface andthe flank face. This angle is necessary to prevent rubbing between the tool and themachined surface.The mechanism of chip formation could be explained using a photomicrographCHAPTER 2. LITERATURE REVIEW	9-•%A;Figure 2.3: Photomicrograph of a partially formed chip, Trent [13]of a partially formed chip shown in Fig. 2.3, Trent [13]. Such a photomicrographis obtained by suddenly stopping the cutting operation and taking a photographthrough a microscope of a polished and etched section of a partially formed chip. Thedevice used to suddenly stop the cutting operation is called a 'quick stop' mechanism.Examination of Fig. 2.3 shows there is a plane AB in which some change in the metalstructure is taking place. This plane is called the shear plane. If the work materialis ductile so that it does not fracture first, then there will be plastic flow along thatplane and the chip will be created and will glide along the rake face of the tool. Thesecond point to be observed from Fig. 2.3 is that close to the cutting edge, where thenormal loads are very high, the chip is in intimate contact or is under seizure withthe rake face of the tool. Under this condition relative movement cannot occur at theinterface between tool and chip and instead the movement involves shearing withinfront	reartool	 toolCHAPTER 2. LITERATURE REVIEW	 10Figure 2.4: Principles of split-tool dynamometerthe chip close to the interface. Thus, the shear stress applied on the cutting tool atthe cutting edge is close to the yield shear strength of the workpiece material.2.3 Boundary load distributions2.3.1 Split-tool methodStress analysis of a cutting tool requires prior knowledge of the applied normal andshear cutting force distributions or boundary stresses along chip-tool contact. Bound-ary stresses in cutting tools have been determined using split-tool dynamometers byKato et al [15], Barrow et al [16] and Childs et al [17].The working principle of the split-tool dynamometer can be described using theschematic diagram of the dynamometer shown in Fig. 2.4. The dynamometer consistsof two parts separated by an air gap (about 0.1 mm) and supported independently.CHAPTER 2. LITERATURE REVIEW	11The air gap on the rake face is parallel to the cutting edge. The two tool parts areidentified as the front-tool and the rear-tool which indicate their relative positionin the composite tool. Usually, only forces transmitted through the rear-tool aremeasured.For a given machining condition, the distance of the rear-tool from the cuttingedge xi, the shear and normal forces transmitted through the rear-tool (F.), and(F„), respectively, are measured (Fig. 2.4). Then the front-tool of the compositetool is replaced by a front-tool having a different land width resulting in a new valuex2+1 for the distance of the rear-tool from the cutting edge. For the same machiningconditions, the change in x, results in a change in the chip and rear-tool contactlength, and therefore change in the forces on the rear-tool. For the new value of xa+1,the shear and normal forces (F8)1+1 and (F)2+1 transmitted through the rear-tool aremeasured.From the above measurements, the boundary stresses at a distance of (xi -Fxi+i)/2from the cutting edge can be calculated using(F„)2+1 — (F„),Cis =  	 (2.1)b(Xj+i — xi)(F3)2+1 — (Fs). Ti = b(Xj+i — xi)where,= width of cutx,	 = distance of the rear-tool from the cutting edge for the ith front-toolCrt	 = normal stress at a distance of (xi + xi+i )/ 2 from the cutting edgeTi	 = shear stress at a distance of (x. xi+i)/ 2 from the cutting edge(Fn)i = normal force on the rear-tool for the ith front-tool(2.2)rakeic distanceCHAPTER 2. LITERATURE REVIEW	 12stressFigure 2.5: Cutting boundary load distribution result from split-tool dynamometer, Kato[15], Barrow [16], and Childs [17]= shear force on the rear - tool for the ith front - toolBarrow [16] has shown that a better way to obtain a smoothed boundary stresses isto first plot the normal and shear force versus rake distance curves and to use theinstantaneous slope of the curves to determine the stresses instead of the averageslopes indicated by Eqs. (2.1) and (2.2).Previous boundary stress distribution results using split-tool dynamometerAs mentioned above, various workers have used the split-tool technique to determinethe boundary load distribution on a cutting tool.Kato et al [15] determined the boundary load distribution using high-speed steelsplit-tool dynamometer while machining aluminium, copper, zinc and lead workpieces.The general boundary load distribution obtained is shown in Fig. 2.5. They foundCHAPTER 2. LITERATURE REVIEW	 13that the region of constant shear stress (also called sticking region) is equal to thechip thickness and the contact length lc is about twice the chip thickness. They alsofound that the ratio of the contact length to the depth of cut depend on the workpiecematerial, being maximum for aluminium and minimum for zinc.Barrow et al [16] used a carbide split-tool dynamometer to determine the boundaryload distribution while machining a nickel-chromium-steel workpiece. The resultsobtained are also as shown in Fig. 2.5. They found that the sticking region andcontact length increase with increasing depth of cut, but decrease with increasingvelocity.Childs et al [17] modified the conventional split-tool by changing the inclinationof the air gap with the cutting edge from 00 to 450• They used brass, aluminium,and mild steel workpieces. Their results are again similar to those shown in Fig. 2.5.However, in this case the ratio of sticking region to contact length for steel is muchhigher (about 0.7). In their conclusions, Childs et al. have indicated that more workis required to assess the reliability of their method.Due to design limitations in these experiments (i.e. minimum front-tool strengthand air-gap requirement) it was not possible to determine the boundary stresses nearthe cutting edge (within 0.2 mm) It was also observed (Barrow [16]) that at a lowdepth of cut, the air-gap interferes in the chip formation processes and therefore altersthe forces. This implies that the boundary load distribution near the cutting edgefrom this method may not be reliable.2.3.2 PhotoelasticityAnother technique used to determine the boundary load distribution in a cutting toolis the photoelastic method. The photoelastic method of stress analysis is describedCHAPTER 2. LITERATURE REVIEW	14Figure 2.6: Orthogonal cutting of a lead workpiece with a photoelastic tool, Amini [20]in detail by Frocht [1]. Here a brief review of this technique applied to cutting willbe made.Certain materials, notably plastics such as epoxy resin, celluloid and bakelitetransmit polarized light along the principal stress axes at different velocities whenstressed. This difference in velocity of the transmitted light produce interferencefringe patterns called isochromatics which are used in the measurement of stresses inthe photoelastic method. The apparatus used to determine stresses in the photoelasticmethod is called a polariscope.In the photoelastic method of cutting tool stress analysis, a photoelastic tool isused to machine a low strength workpiece at low cutting speeds (Fig. 2.6). Leadis typically selected as the workpiece because it is soft enough to be cut withoutbreaking the photoelastic cutting tool. The cutting tests are conducted at low speedsCHAPTER 2. LITERATURE REVIEW	 15to avoid the rapid drop in strength with increase in temperature of the plastic tool.A milling machine is generally used in these orthogonal machining tests.The cutting loads on the photoelastic tool produce fringe patterns, like thoseshown in Fig. 2.6, which are related to the principal stress difference by— cr3 = f0.1n11h	 (2.3)where,N = fringe orderh = tool thickness= the algebraically larger principal stress0-3 = the algebraically smaller principal stressf, = fringe constantIn addition to the fringe patterns, the polariscope also gives the angle 0 betweencri and the x—axis. The locus of points having the same angle of inclination 0 iscalled an isoclinic. Once the isochromatic N and isoclinic 0 are determined, theshear stresses everywhere in the body are calculated using Eq. (2.3) and the shearstress equationT =yx— 0-3 .2	 sin 20 (2.4)Then one of the normal stresses is determined by numerical integration of the equationof equilibriumaa aTxii-EY=0aY	 ox (2.5)CHAPTER 2. LITERATURE REVIEW	16Figure 2.7: Numerical integration of the equation of equilibrium to determine normalstressesfor zero body forces (Y = 0), as (see Fig. 2.7)y (97-xv 	dycr =- (70fy0 (IXn ,c,cryo	 cOY.;41(2.6)In the above equation ayo is determined from the principal stresses at the clearanceface (Fig. 2.7) using the normal stress equationay = o sin2 qS + (73 cos2	 (2.7)and Eq. (2.3). The clearance face is a free boundary; as a result al = 0. Thereforealong the clearance face where a+ q5 = 7r/2, cry° = —L,Nsin2 a lh and is directlydetermined from the fringe patterns and tool geometry.CHAPTER 2. LITERATURE REVIEW	 17Once the stress in the y-direction everywhere in the tool is determined using Eq.(2.6), then the normal stress in the x-direction is determined using the normal stressrelation given byo-. = o•1, + .V(71 — c73)2 — 47-2.1i (2.8)Thus, the photoelastic method can be used to determine the stress distributionthroughout the photoelastic tool including the boundary load distribution at tool-chip interface.Many workers have used the photoelastic method to determine the stress distribu-tion in a cutting tool. The boundary load distribution results obtained by this methodcan be roughly divided into two categories. The first category is the distribution forlow cutting speeds of about 1 in/min as obtained by Usui [18], Chandrasekaran [19]and Betaneli [7] and is shown in Fig. 2.8. The second category is for the distributionat relatively higher cutting speeds of about 100 in/min as obtained by Amini [20] andAhmad [21] and is shown in Fig. 2.9. The fringe patterns which are proportional tothe maximum shear stress contour lines, from the photoelastic cutting tests of Amini[20], are shown in Fig. 2.6.Boundary load distributions at lower cutting speeds.Photoelastic boundary load distribution results at lower cutting speeds will be dis-cussed here. Usui [18] obtained a value of 0.7 for the ratio of the sticking region tothe contact length for a tool having a seven degree rake angle. Chandrasekaran [19]showed that the load distribution depends on the rake angle used: the sticking regionand the contact length for a positive rake tool increase with decrease in rake angle.However, the ratio between sticking region to contact length was found to be almostrakeic	 distancestressrakeic distanceCHAPTER 2. LITERATURE REVIEW	18stressFigure 2.8: Cutting boundary load distribution result from photoelasticity at low cuttingspeeds, Betaneli [7], Usui [18], and Chandrasekaren [19]Figure 2.9: Cutting boundary load distribution result from photoelasticity at higher cuttingspeeds, Amini [20] and Ahmad [21]CHAPTER 2. LITERATURE REVIEW	 19independent of the rake angle and has a value close to 0.4. Betaneli [7], for a toolhaving a 200 rake angle, obtained a value of 0.4 for the ratio of the sticking region tothe contact length.Betaneli [7] has fitted a power law function for his lower cutting speed boundaryload distribution. This function isIa = Cfc1(1 — (rificin (2.9)where,o-o' =- maximum normal stress at the cutting edge of the photoelastic tool= contact length in photoelastic cuttingr' = rake distance from cutting edge of the photoelastic tooln = exponent of the parabolaBy cutting lead with a photoelastic tool at 25.4 mm/min, depth of cut of .75 mm,rake angle of 20° and wedge angle of 62°, Betaneli [7] found that the exponent of thepower law function n = 3.3.The photoelastic normal distribution given by Eq. (2.9) can be used to determinethe boundary load distribution on metal cutting tools. If the shapes of the boundaryload distribution obtained from photoelasticity are assumed to be similar to thosein metal cutting, then the linear relationship between the two distributions can bewritten as(2.10)(2.11)where,CHAPTER 2. LITERATURE REVIEW	 20a-, cr'	 = normal stresses distribution in metal cutting and photoelastic cuttingr, r'	 = rake distance in metal cutting and photoelastic cutting respectivelyao, a: = cutting edge applied normal stresses in metal cutting and photoelastic cuttingSubstituting Eqs. (2.10) and (2.11) into Eq. (2.9), the boundary normal load distri-bution on the metal cutting tool is obtained asa = cr„,(1 — (r/1)")	 (2.12)where,cro = maximum normal stress at the cutting edge of a metal cutting tooldc = contact length in metal cuttingr = rake distance from cutting edge of a metal cutting tooln = exponent of the parabolaTo determine the boundary load distribution from Eq. (2.12) for a metal cuttingtool, the contact length 4 and the maximum normal stress cro should be determinedfrom metal cutting tests and the exponent n is obtained from photoelastic cuttingtests.The maximum normal stress in Eq. (2.12) could be determined from the averagenormal stress on the cutting tool. This average stress is given byF„a =av	 bl,where,crot, = average normal stress= cutting force normal to rake face(2.13)CHAPTER 2. LITERATURE REVIEW	 21b = width of cut/c = contact lengthThe average normal stress is also related to the normal boundary load distributionby0.. =1 fic udr	 (2.14)/c oSubstituting the parabolic distribution given by Eq. (2.12) into Eq. (2.14) andsimplifying, the expression obtained for the maximum normal boundary stress isn + 1a 0 -= 	 (Invn(2.15)The normal boundary load distribution of Eq. (2.12) is thus completely determinedfrom the photoelastic results which gives the exponent n, and from actual metalcutting tests which provide the contact length, normal cutting force, width of cutand the average normal stress. The boundary shear stress is determined by assumingthe shear stress to be related to the normal stress through the coefficient of friction.Therefore, the boundary shear stress is given byT = ACT	 (2.16)In the above equation the friction coefficient is calculated using the relationF,14 — Fnwhere,F,., = boundary force normal to rake faceF. = boundary force parallel to rake face(2.17)CHAPTER 2. LITERATURE REVIEW 	 22Boundary load distribution for higher cutting speedsPhotoelastic boundary load distribution result at higher cutting speeds will be dis-cussed here. Ahmad [21] has fitted a polynomial function for his higher speed bound-ary load distributions and they are given by= 2.91 — 1.53r' + .214r'2 — .0033r'3 (2.18)Ti = 1.63 — 1.24r' + .315r'2 — .0266r'3 (2.19)where,a' = normal boundary stress on the photoelastic tool, kg/mm2Ti = shear boundary stress on the photoelastic tool, kg/mm2r' = rake face distance from the cutting edge of the photoelastic tool, mm.These equations were obtained for a photoelastic tool model having a rake an-gle of 6°, depth of cut of .203 mm, width of cut of 5.1 mm, cutting speed of 2.93m/min and contact length of 4.5 mm. As described in the previous section, the aboveequations can also be used to determine the stress distribution in metal cutting toolsby assuming the shape of the boundary stress on both photoelastic tool and metalcutting tool to be similar.In the photoelastic boundary load distribution for higher cutting speeds shown inFig. 2.9, there is little sticking region (constant shear stress near the cutting edge)and this fact has been confirmed by Barrow [16] using split-tool dynamometer wherehe shows a decrease in the sticking region with increase in cutting speed. Therefore,at higher cutting speeds where the sticking region is small, the shape of the boundaryload distribution from split-tool and photoelasticity are approximately similar.CHAPTER 2. LITERATURE REVIEW	23Figure 2.10: Stress components on shear planes on an infinitesimal element2.3.3 Slip-line theoryIn the previous sections the experimental methods used to estimate the boundary loaddistribution in metal cutting tools were discussed. In this section a purely theoreticalmethod from plasticity theory can be used to approximate the applied normal andshear stresses at the cutting edge is discussed.This plasticity solution, as proposed by Loladze [37], makes use of the propertiesof slip-lines along the shear plane to determine the normal and shear stresses at thecutting edge. The state of stress throughout a rigid, perfectly plastic solid materialunder deformation can be represented by a constant yield shear stress k in plane strain,and a hydrostatic stress p which in general varies from point to point throughout thematerial. The two maximum shear stress planes at each point of the material underdeformation are perpendicular to each other and form two orthogonal families ofCHAPTER 2. LITERATURE REVIEW	 24Figure 2.11: Derivation of co-ordinate stresses from the shear plane stressescurves known as slip-lines. These slip-lines are labelled as a and 0 as shown in Fig.2.10. The directions of these slip-lines are not arbitrary and are determined fromthe direction of the shear stresses. The shear stress directions and the correspondingdirection of the slip-lines are as shown in Fig. 2.10. Another parameter which isimportant in slip-line theory is the angle 0 measured in the counter-clockwise directionfrom the x-axis to the a slip-line, Fig. 2.10. The stresses in the x and y-directionsin terms of the hydrostatic pressure and yield shear stress can be derived from theMohr's circle shown in Fig. 2.11, and are given byuz -.= —p — k sin 20	 (2.20)Cry = —p + k sin 20	 (2.21)rii, = k cos 20	 (2.22)CHAPTER 2. LITERATURE REVIEW	 25In the above equations p is considered positive when in compression. Substitutingthe above equations into the equations of equilibrium for the case of no body forces,and taking the a-line to coincide with the x-axis at the origin (Johnson [23]) yieldsalong an a-lineOp = —21c60	 (2.23)and along f3-lineOp = 21c60	(2.24)where,Op = change in hydrostatic pressurek = shear yield stressScb = change in the angle between x-axis and a slip-line (radian).These equations are known as the Hencky equations and together with the com-ponent stresses given by Eqs. (2.20) to (2.22) are all that are required to determinethe normal and shear boundary stresses at the cutting edge.The use of the above equations to estimate the applied stresses at the cuttingedge B of Fig. 2.12 is discussed next. Since the shear plane passing through A andB of Fig. 2.12 is a maximum shear stress line it is a slip-line. From the knowledgeof the direction of the shear stress along the shear plane, the orientation of the sliplines are as shown in Fig. 2.12 and the shear plane is a /3-line. At point A the planeperpendicular to the x-direction is a free boundary (u. = 0 and -r.i, = 0), thereforeit is a principal plane. The angle between the normal to the principal plane and themaximum shear plane is 7r/4, and therefore at point A the angle between the x-axisand the a-line is OA = 7r/4 + 7r/2 = 37r/4. Applying Eq. (2.20) to point A, theCHAPTER 2. LITERATURE REVIEW	26Figure 2.12: Boundary stresses at the cutting edge using slip line theoryhydrostatic compressive stress is found to be PA = k. It is known that along the rakeface, at the cutting edge, the work material is in shear yielding and therefore thisface is an a-slip line. From Fig. 2.12 it can be seen that at point B, OB =- 7r — 7 andtherefore the change in angle 0 from A to B is 80 = 7r/4--y. For a # slip-line, from Eq.(2.24) the hydrostatic stress at the cutting edge therefore is pB = 2k(1/2 -1-7r/4 —Finally, from Fig. 2.12 at point B it can be seen that the applied normal stress isequal to the hydrostatic pressure and the applied shear stress is equal to the yieldshear strength of the workpiece, and these yieldac, = 2k(1/2 ir/4	 (2.25)To = k	 (2.26)where,ao = normal boundary stress at the cutting edgeCHAPTER 2. LITERATURE REVIEW	 27To = shear boundary stress at the cutting edgek = yield shear strength of the work material7 = rake angleThe above equations provide the applied normal and shear stresses at the cuttingedge and they can be used to estimate the cutting edge stresses. They do not takeinto account the strain-hardening and strain-rate effects of real materials, and there-fore their correlation with experimental values are approximate. However, as shownby Chandrasekaren [19] for non-strain hardening material at lower strain-rates goodcorrelation is obtained.2.4 Analytical stress analysis of cutting tools2.4.1 Analytical stress analysis of cutting tools for a concentrated loadapproximation.When the stresses far from the cutting edge within the wedge shape are required, thesolution with single load approximation by Frocht [1] may be used. This solution isderived from the stress functionik ,-- CrO sin 0	 (2.27)where,C = a constant,r and 0 are as defined in Fig. 2.13.The stress components are derived from the stress function using the relations320cr8	 37-2	 (2.28)1 alk	 1 a20 (2.29)(Tr r Or + r2 ae2CHAPTER 2. LITERATURE REVIEW	 28Figure 2.13: Concentrated cutting load at the cutting edgea 1 aoTro =	 ( 	 )ar r ao(2.30 )where V) is the stress function.By substituting Eq. (2.27) into Eqs. (2.28) to (2.30), and applying force equi-librium for the cutting tool in Fig. 2.13, the stress distributions within the tool areobtained as2P (cos A cos 0 sin A sin 0)cir = 	 +  	 (2.31)	rb a + sin a	 a — sin a )	0-8 = 0.	 (2.32)	7-re = 0.	 (2.33)where,P = concentrated load on the tool, P = ‘ / Fn2 + F32F, = shear force on rake faceCHAPTER 2. LITERATURE REVIEW	 29Fn = normal force on rake faceb = width of cuta = wedge angler = radial distance from cutting edgep = average friction coefficient, A = F,IF„A = angle between tool axis of symmetry and force P, A = tan'(1/p) — a/20 = angle measured from tool axis of symmetry, positive when as shown in Fig. 2.13The component stresses are shown in Fig. 2.15.The above equations were used by Kaldor [2] to determine the optimal tool ge-ometry and by Chandrasekaran [3] to calculate fracture stresses in milling cutters.Examination of Eq. (2.31) shows that at the cutting edge where r = 0, the radialstresses is undefined and therefore this solution can not be used to estimate stressesclose to the cutting edge. However, far from the cutting edge within the wedge shapethis solution may be applicable. This will be investigated in the current study. Animportant conclusion that can be made from Eq. (2.31) is that for a given cuttingtool and cutting conditions, further from the cutting edge (i.e. as r increases) theradial stress decreases. This decrease in stress further from the cutting edge is dueto the increase in resisting area or section modulus of the wedge.2.4.2 Analytical stress analysis of cutting tools for distributed boundaryloads.When the stresses near the cutting edge where the cutting loads are applied are re-quired, the actual load distribution along tool-chip contact should be used. Elasticitysolution applicable for a distributed load on an infinite wedge may be used to de-termine the stress distribution in the loaded region of the cutting tool. The infiniteCHAPTER 2. LITERATURE REVIEW	 30Figure 2.14: Distributed load on the rake facewedge solution for distributed loads is discussed in this section.The generalized stress function for two dimensional problems in polar coordinateswas derived by Michell [4]. This stress function was then used by Timoshenko [5]to determine the stress distribution in an infinite wedge for polynomial boundaryload distributions. The infinite wedge solution was letter used by Archibald [6] andBetaneli [7] and others to determine the stress distribution in metal cutting tools.The generalized stress function derived by Michell [4] is given asV) = ao log r + bor 2 + c0r2 log r + d0r20 + 40+ —a1re sin 6' -I- (bir3 + a:1r + blir log r) cos 192ci— —r0 cos 0 + (d1r3 + c; r_1 + ce r log r) sin 02	 iCHAPTER 2. LITERATURE REVIEW	 3100+ E(anrn + bnrn+2 + anr-n + bn r -n-F2 ) COS nOn=200+ E(enrn + dnrn+2 + cni r + dn r- n-I-2 ) sin nOn=2(2.34)where,r and 0 are as defined in Fig. 2.14 and a0, b0, c0, c/0, ... are constants.The stress components are determined by substituting Eq. (2.34) into the compo-nent stress equations (2.28) to (2.30) and taking only terms containing rn with n > 0.The results of this substitution as given by Timoshenko [5] and Archibald [6] for thetangential (a9), radial (cr,) and shear (r,.9) stress components as shown in Fig. 2.15are:ce = 2b0 + 240 + 2a2 cos 20 + 2c2 sin 2000+ (n + 1)(n + 2)rn[bn cos nO + dn sin nOn=1+ an+2 cos (n + 2)9 + cn+2 sin (n + 2)0], 	 (2.35)a-, = 2b0 + 2c/00 — 2a2cos29 — 2c2 sin 21900— E (n + 1)rn[(n — 2)(bn cos nO + 4 sin n8)n=1+ (n + 2){an+2 cos (n + 2)0 + en-0 sin (n + 2)01],	 (2.36)Tr° = —d0 + 2a2 sin 29 — 2c2 cos 29CHAPTER 2. LITERATURE REVIEW	32Figure 2.15: Stress components in polar co-ordinate on an infinitesimal element of a cuttingtool00+ E rn(n + 1)[n(by, sin ne — d, cos ri9)n=1+ (n + 2).(an+2 sin (n + 2)8 — c„+2 cos (n + 2)01] 	 (2.37)How the above equations may be used to determine the stress distribution withinthe loaded region of the tool will be discussed in the following section and in Chapter3.2.5 Previous analytical cutting tool stress solutions2.5.1 Cutting edge stressesArchibald [6] used the stress component equations (2.35) to (2.37) to determine thestresses at the cutting edge (r = 0 and 8 = 0). He assumed a linear boundaryCHAPTER 2. LITERATURE REVIEW	 33rakedistanceFigure 2.16: Linear cutting load distribution assumption, Archibald [6]load distribution to determine the boundary stresses from cutting forces. This linearnormal load distribution is given bya = ao(1— r/lc)	 (2.38)where,a = normal boundary load distributionr = rake distance from cutting edgeic = contact length.The normal stress at the cutting edge determined by Archibald [6] from his linearload distribution assumption and power measurements is given bycosi3a°, 3.36(88.24 -y)cos(i3 —	 ,y)	 (2.39)where,CHAPTER 2. LITERATURE REVIEW	 34-y = rake angle (deg)/3 = friction angle (deg)a° = maximum boundary normal stress at the cutting edge (kpsi)In his analysis, Archibald used a chip-tool contact length of twice the depth of cut(dc = 2t). He indicated this length to be determined experimentally. The boundaryshear stress is determined from the coefficient of friction and the normal boundaryload distribution as T = lair . The clearance angle used was 50 and the wedge angle ais related to the rake angle 'y by a = 85 — 'y.For the linear boundary load distribution, from the stress component equations(2.35) and (2.37), n = 0 and 1 and the eight constants bo, do, az, cz, b1, d1, az, and czcan be determined by substituting the linear boundary load distributions on the rakeface, and the zero load distributions on the flank face, into Eqs. (2.35) and (2.37)and solving the equations obtained simultaneously for the constants. Then the stresscomponents at the cutting edge are determined by substituting the determined con-stants back into equations (2.35) to (2.37). The solution obtained for the rake faceradial stress at the cutting edge by Archibald [6] using this procedure is given byUr =a	 tan a(a tan a + 1) — a	  CFI 01 	 ITO Itan a — a	 (tan a — a) tan a (2.40)where,Cr r	 = radial stress parallel to the rake face at the cutting edgea	 = wedge angle, radiansI o-01 = magnitude of normal stress at the cutting edge on the rake faceiTol = magnitude of shear stress at the cutting edge on the rake faceCHAPTER 2. LITERATURE REVIEW	 35The radial stresses from Archibald's [6] solution for various coefficients of friction(it = ro/cro) as a function of the wedge angle a, as determined from Eq. (2.40) andEq. (2.39), are shown in Fig. 2.17. In the current study the applicability of Eq.(2.40) in cutting edge stress analysis will be verified numerically.2.5.2 Stress distribution in the loaded region for parabolic loadAs discussed in the previous section, Archibald [6] used the stress component equa-tions for distributed loads, Eqs. (2.35) to (2.37), to determine the stresses at thecutting edge of a tool. Betaneli [7] then went further and used these equations todetermine the stress distribution in the whole loaded region.The method used by Betaneli [7] was similar to that used by Archibald [6] excepthe replaced the linear load assumption by an experimental one. The experimentalboundary load distribution result by Betaneli [7] was described in Section 2.3.2. InSection 2.3.2 it was noted that in addition to the results from photoelasticity, forceand contact length measurements are required to completely determine the boundarystresses in metal cutting tools. Betaneli [7] took data from Zorev's [24] extensivemachining test results in order to determine the boundary load distribution on ametal cutting tool. Zorev's results which were used by Betaneli [7] as discussed inSection 2.3.2 are given in Table (2.1)....11cce-	 Po-peers•45 	•Grr	 At3	r.5- 0 -311111•1111111	111NM11111111111111111EMENIKII"WiZZELIINENIMME-44MTINLIN=WIM11113MLISLIMMIIE1111111111111111111111151MMIll11111111111111111 MIMESG	 60	 7072.0, 1:ect001-0 AuecLe # DCGRE(17JO	 .90111111111111	100CHAPTER 2. LITERATURE REVIEW	 36Figure 2.17: Radial cutting edge stresses as a function of wedge angle and friction coefficient,Archibald [6]. Here /3 is the wedge angleCHAPTER 2. LITERATURE REVIEW	 37Table 2.1. Machining parameterswhen cutting steel in water, Zorev [24].-y = 200, v =.7m/mintMinlcminIt aavkg/nun.'0.2 0.5 0.36 60.0where,= rake angle= depth of cutlc = contact length= velocity of tool relative to the workpiece= coefficient of friction at chip-tool contact= average normal stress along chip-tool contactUsing the relations given in Section 2.3.2 and the data from Table (2.1), theboundary load distributions on a metal cutting tool can be determined completely.For this load distribution, the stress distribution in the loaded region is determinedfrom the stress component equations (2.35) to (2.37). The principal stresses fromBetaneli's [7] solution are shown in Fig. (2.18).The results discussed in the previous sections are the available analytical ap-proaches in analytical cutting tool stress analysis. The limitation of these approachesand the objectives of the current study are discussed in the next section.CHAPTER 2. LITERATURE REVIEW	 38Principal stresses at and as plotted against r and0 (with y . 2; p. 62" a = 0.2,1a and n . 3.33):I - 6 . y . 20'; 2 - 0 - 3CP; 3 - 6 . 40'; 4 - 0 . 50';5 -0.  60 3; 6 - 0 . 7l; 7 - 0 . 80 '; 8 - 0 . /3 + y . Pr.Figure 2.18: Principal stress distributions in the loaded region for a parabolic load, Betaneli[7]. Here a is the depth of cut and # is the wedge angleCHAPTER 2. LITERATURE REVIEW	 392.6 Objectives of the present work and methods of investi-gationFrom the discussion in the previous sections, stress on cutting tools have been esti-mated analytically both at the cutting edge and in the whole loaded region. However,these solutions were obtained on the assumption that the infinite wedge solution canbe applied to determine the stresses in the finite cutting tool. It is important toverify this assumption because the solutions obtained by using this approach maysometimes lead to incorrect conclusions. To critically investigate this assumption andcome up with a conclusion as to whether this approach could be applied or not, andif it could be applied then to identify the conditions is the purpose of the first phaseof this work.In the second phase of this work, the critical regions of cutting tools where breakageis likely to occur will be predicted. For this an orthogonal cutting tool and an endmill will be considered. Analytical methods will first be used to predict the criticalstresses then these solutions will be verified numerically. This comparison will indicatewhether analytical solutions could be applied in the failure analysis of cutting tools.The steps that will be taken in this study are listed below:• Compare the analytical solution by previous researchers with a finite elementsolution for the same geometry and 'same' boundary load distribution.• If there is difference between the two solutions then determine the cause for thisdifference.• From the understanding of the reason for the difference between the two solu-tions identify the conditions (boundary load distribution) where the analyticalCHAPTER 2. LITERATURE REVIEW	40solution may be used and verify this using an example.• Compare the solution reached in the last step above with previous experimentalresult for similar geometry and boundary load distribution.• From the solution obtained above identify the critical regions of the two-dimensionalcutting tool and its modes of failure.• Determine the stresses at critical regions of an end mill for shank breakage usinganalytical methods.• Verify the analytical solution for shank stresses obtained above numerically.This result will indicate whether analytical methods are applicable in end millshank stress analysis.• Compare the analytical solution obtained for the orthogonal cutting tool abovewith numerical solution for an end mill flute. This result will indicate whetheranalytical methods are applicable or not in end mill flute stress analysis.• Finally give a conclusion of this work and suggest what future work should bedone to get an improved understanding of stress distribution in cutting tools.Chapter 3Stress Calculation in OrthogonalCutting Tools3.1 IntroductionThe analytical stress equations (2.35), (2.36), and (2.37) of the previous chapter wereused by Archibald [6], Betaneli [7] and others to determine stresses in cutting tools.This analytical approach is based on the assumption that the infinite wedge solutioncould be used to determine the stress distribution in the finite cutting tool. One ofthe motivations of the present research is to critically investigate this assumption.We begin this investigation by first verifying previous analytical solutions and thencomparing these results with numerical solutions.The boundary load distribution used by Archibald [6] is linear and that of Betaneli[7] is parabolic. These boundary load distributions are limited since they can notinclude the different possible boundary load distributions on a cutting tool. In Section3.2 an analytical solution for a polynomial load distribution which provides a moregeneral case is developed. The previous analytical solutions are compared with currentanalytical solutions in Section 3.3. This is done to verify the previous analytical41CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 42solutions and also to test the computer program for our analytical solutions. InSection 3.4 these analytical solutions are compared with FEM results for 'identical'boundaries and geometry. In Section 3.5 the results of this chapter are discussed andconclusions regarding these analysis is drawn.3.2 Analytical cutting tool stress calculationIn this section, the analytical method used to determine the stress distribution in acutting tool for a general polynomial boundary load distribution is discussed.The stress equations (2.35), (2.36), and (2.37), can be rewritten in the followingforma-0 = 2a0 + 21)09 + 2c0 cos 20 + 2d0 sin 20E(i + 1)(i + 2)r1[ai cos i8 bi sin it)ci cos (i + 2)9 + di sin (i 2)9],	 (3.1)(Tr = 2a0 2b09 — 2c0cos28 — 243 sin 2900- E(i + 1)71(i — 2)(ai cos i8 bi sin i9 )(i	 2){ci cos (i	 2)0 + di sin (i	 2)01],	 (3.2)CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 43TT° = —bo 2c0 sin 29 — 2d0 cos 20+ > r(i 1)[i(ai sin i9 — bi cos it9(i 2)(c2 sin (i 2)0 — di cos (i 2)8)]	 (3.3)where• = tangential stress• = radial stressTr() = shear stress ai, bi, ci, and d, for i > 0 are constants3 and 9 , and the positive direction of the component stresses are as shown in Fig. 2.15.The above equations can be written in a simplified form in terms of normal (N,),radial (Ri) and shear (S,) polynomial coefficients as given below:000-0 = E Niricoo-r = E Rri=o00Tro = E SiTii=0where the coefficients of the polynomial in the above three equations, determinedfrom equations (3.1) to (3.3), are given by(3.4)(3.5)(3.6)CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 44for i = 0,and for i > 1,N0= [2 20 2 cos 20 2 sin 20] {Ro = [2So = [020— 1— 2 cos 202 sin 202 sin 20]— 2 cos 20]aobocodoabo°codo[cos i0 sin i9 cos(i + 2)0 sin(i + 2)0]i= m [cos i0z + 2 sin i0 cos(i + 2)0 sin(i + 2)0 I+ 2	 i — 2	 i — 2 ](3.7)(3.8)(3.9)aici(3. 1 0)aici(3.11 )CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 45Si ai	= j L s.ini0 	cos i9 sin(i + 2)0	 cos(i + 2)0 J bi2+2 i+2 (3.12)where,j = i(i + 1)(i + 2)m	 (i — 2)(i + 1)(i + 2)= Normal polynomial distribution coefficientsR, = Radial polynomial distribution coefficientsSi = Shear polynomial distribution coefficientsai, b, ci, and di are constants to be determined from the boundary load distributions.The stress components in Eqs. (3.1) to (3.3) at any point (r, 0) can be determinedif the constants ai, bi, ci, and d, for i > 0 are known. These constants are determinedfrom the boundary load distributions of ae and .74 at the rake and flank faces. FromEqs. (3.4) and (3.6), the boundary loads can be written in polynomial form so that atthe boundaries Ni and S, take the values of the coefficients of the polynomial normaland shear boundary load distributions respectively. From these boundary conditions,as is shown below, the arbitrary constants and therefore the stresses within the cuttingtool can be determined. The coordinate axes and tool geometry used are shown inFig. 2.15.If the normal (as) and shear (Trio) boundary load distributions on the rake andflank faces are assumed to be represented by (n + 1) finite terms of a polynomial,then they can be written as(00)9=0,.=E Nfiri	 (3.13)i=oCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 46where,(040.=-9/ = E Nfirii=0(Tr8)0=9,. = E srirsi=0(rr0)0=01	 E Sfirin=0(3.14)(3.15)(3.16)or = -yOf = 7 + ar = radial distance from cutting edgen = degree of the polynomial boundary load distribution(u9)9=0, and N. are the rake face normal load distribution and its coefficients(u8)19=91 and Arfi are the flank face normal load distribution and its coefficients(Tre)9=9,. and Sr, are the rake face shear load distribution and its coefficients(r7.9)9=91 and Sfi are the flank face shear load distribution and its coefficientsSubstituting the polynomial coefficients for the normal and shear load distribu-tions at the rake and flank faces into the equations for N, (Eqs. (3.7) and (3.10)) andS, (Eqs. (3.9) and (3.12)), for 0 < < n yields (n+1) sets of matrix equations:for i = 0- 2 2OT 2 cos 20,. 2 sin 20r ao{Nr0A 10} 2 219f 2 cos 29f 2 sin 2Of b0S r0 0 —1 2 sin 2OT —2 cos 2Or cof 0 0 —1 2 sin 2Of —2 cos 2Of do(3.17)CHAPTER 3. STRESSfor 1 < i < nCALCULATION IN ORTHOGONAL CUTTINGcos Or	 sin iOr	 cos(i + 2)0,.	 sin(i + 2)0,.TOOLS	 47Nri CO5 i91 sin Of cos(i + 2) 0 f sin(i + 2) 0 fNfi _ . aibSriSfi- sin iB,. cos Or sin(i + 2)8,. COS(i	 2)0,. Cli + 2 i + 2sin Of cos Of sin(i + 2) 0 f cos(i + 2) 9ji + 2 i + 2(3. 1 8)where,j	 = i(i + 1)(i + 2)Or = -yf = -y + a= rake anglea = wedge anglei	 = powers of the polynomialNri = rake face polynomial normal load distribution coefficientsN1 = flank face polynomial normal load distribution coefficientsSri = rake face polynomial shear load distribution coefficientsSfi = flank face polynomial shear load distribution coefficientsai, b„ c, and d, for i > 0 are constants.Eqs. (3.17) and (3.18), with the knowledge of the boundary normal and shearload distribution and the geometry of the tool, are sufficient to determine the con-stants that appear in the stress equations (3.1), (3.2) and (3.3). From Eqs. (3.17)CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 48and (3.18), for each term i of the boundary load distribution there are four unknownconstants and four equations, and therefore each of the 4x4 matrix equation are suf-ficient to determine the unknowns ai, bi, ci and d,. The number of arbitrary constantsto be determined depends on the number of terms of the polynomial boundary loaddistribution. For a polynomial expression of degree n, the number of terms are n 1and therefore 4(n + 1) constants must be determined which can then be substitutedinto equations (3.1) to (3.3) to obtain the stress components.The equations given here is used in the next section to determine stresses in cuttingtools for a parabolic load distribution, and in Chapter 5 to determine stresses for ageneral polynomial load distribution.3.3 Analytical cutting tool stresses for parabolic load dis-tributionsThe solution developed to determine stress distribution for the general polynomialload in the previous section is used here to solve the particular case of a parabolicload distribution. This is done to verify the stress distribution result obtained byBetaneli [7] and also to test the computer program results of the current study.The experimentally determined normal boundary load distribution by Betaneli [7](Section 2.3.2) has a parabolic distribution given by= cro	 cro pcn ) r n= Nro N,rn	 (3.19)and the shear boundary stress distribution is determined from the normal boundaryCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 49stress as	= 	 _ (two! icn)rn	=	 Sro Srnrn	 (3.20)From the above equations for a and T, the coefficients of the boundary stresses onthe rake face areNro	Nrn	 { —00:00111:	Sro	 "Lao	Srn	 —110-0M(3.21)The flank face is assumed to be free from loads and as a result N10, Nfn, Sic) and Sinare zero. Substituting these coefficients into Eqs. (3.17) and (3.18), the equationsthat are sufficient to obtain the constants and therefore the stress distribution incutting tools for the parabolic boundary load distribution assumption arefor i	 0cro - 2 20T 2 cos 2Or 2 sin 29r0 2 20f 2 cos 20f 2 sin 2Oftwo 0 —1 2 sin 20,. —2 cos 20,.0 0 —1 2 sin 219f —2 cos 20(3.22 )I al):codoCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 50for i ncos nO,	 Sin ner	 cos(n + 2)9,cos nO f	 sin nO f cos (n + 2) Ofsin nO,	 cos nO, sin(n + 2)0,.n + 2	 n + 2sin nOf	 cos nOf sin(n + 2) Ofn+2	 n+2sin(n + 2)0,. -nsin(n + 2)Ofcos(n + 2)8,.cos(n + 2)8f  anbncndn(3.23)where,n(n + 1)(n +2)Or = 7Of = 'Y7 = rake anglea	 wedge angle= coefficient of friction= exponent of the parabolic boundary load distribution= applied normal stress at the cutting edgedc = contact length al), 1)0, co, do, an, bn, cn, dn are the constants to be determined.In the above equations a consistent sign convention for stresses must be used.Normal tensile stresses are positive. Positive shear stresses relative to the coordinateaxes chosen are as shown in Fig. 2.15.For given values of cro, contact length lc, parabolic exponent n, average frictioncoefficient p, rake angle -y and wedge angle a, the eight constants can be determinedCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 51by solving Eqs (3.22) and (3.23) simultaneously. The constants obtained can then besubstituted into Eqs. (3.1) to (3.3) to determine the stress distribution in the loadedregion of the cutting tool.For expedience, these equations were programmed using Fortran and this programis given in Jemal [40] (Appendix C.1). The outputs from this computer program arethe polynomial coefficients Ni, R and S1; the component stresses ory, 07. and 7-r9; andthe principal stressesa1,3 = (a, + 08)12 + V(a, — 09)214 + 7-7.20	 (3.24)which can then be used to determine the region of maximum stresses and thus predictfailure in the cutting tool.For a given boundary load distribution and tool geometry, the stress distributionis independent of the choice of the coordinate axes and therefore the rake face canbe taken as the x-axis by substituting -y = 0 (see Fig. 2.14). Thus, at the rake face= ay, ar cr. and TrO = Try •The boundary load distribution used by Betaneli [7] (Section 2.3.2) is shown inFig. 3.1. This boundary load distribution, as discussed in Section 2.5.2, is for acutting condition having a contact length lc=.5 mm, parabolic exponent n=3.3, fric-tion coefficient A=0.36 and average normal stress o-,,,,=60 kg/mm2 For the parabolicboundary load distribution, the applied maximum stress at the cutting edge is re-lated to the average normal stress by ac, = (n+1)cravIn (Eq. 2.15), therefore cro = 78kg/mm2. The wedge angle of the tool considered is 62°, so Of = 627r/180 radians.Substituting these values into Eqs. (3.22) and (3.23) yields-78.0 2 0.000 2.000 0.0000.0 2 2.164 -1.118 1.658-28.1=}0 -1.000 0.000 -2.0000.0 0 -1.000 1.658 1.1181CodoCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 52for i = 0for i = n = 3.3768.2 22.79 0.000 22.79 0.000	 -0.0 _ -20.72 -9.49 19.45 -11.87276.6{	1- 0.00 -14.19 0.000 -22.790.0 -5.91 12.90 -11.87 -19.45 _Solving the above equations simultaneously, the results obtained for the non zeroconstants are:1 ao }bocodo = 1 -14.7- 15.8- 24.3 } '21.9 1 anbnCr,4 19.8 11.8} = { 13.9-13.3These constants are then substituted into Eqs. (3.1) to (3.3) to determine thestress components at any point (r, 0) of the cutting tool. From these stress componentsthe principal stresses are determined using Eq. (3.24) and the maximum shear stressfromTniaz = (cri - a-3)/2 (3.25)The analytical solution for the principal stress distribution in the loaded region isshown in Fig. 3.2. This figure shows the maximum principal stress distribution fordifferent values of angle 8 (see Fig. 2.14) as a function of distance from the cuttingedge.The rake face stress distributions from the analytical solution, together with theboundary loads, are shown in Fig. 3.3. In this figure ay and r 	 the normal andAnal tical load boundarCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 53-75CNI-50uj - 25CU6-1ch,0.0	 0.1	 0.2	 0.3	 0.4	 05Rake distance (mm)Figure 3.1: Polynomial boundary load distribution determined by photoelasticity at lowcutting speeds, Betaneli [7]shear boundary load distributions respectively and a. is the normal stress parallel tothe rake face. This result predicts the maximum principal stress al at the cutting edgewhile giving zero stress at the end of chip-tool contact. This conclusion is identicalwith that given by Betaneli's [7] which is shown by the dashed curve labelled '1' inFig. 2.18.This result of a zero maximum principal stress at the end of chip-tool contact doesnot explain breakage of cutting tools that are observed in practice in this region. Thisresult, as shown below, also does not agree with conclusions reached of stresses fromthe elastic deformation of the loaded region of the cutting tool. During cutting theloaded region of the cutting tool elastically deforms as shown in Fig. 3.4. As shownin this figure, the rake face locally bends about point A near the end of chip-toolcontact, producing a bending stress (non zero tensile stress) at this point.tical solution.Anal050.2	 0.3	 0.40.01111 1 11 1 11 1 11 1 11 1 11 10.0-5CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 54A  Principal stress distribution. 3- I20	0.0	 0.1	 0.2	 0.3	 0.4	 05Radial distance (mm)Figure 3.2: Analytical principal stress distributions in the loaded region of a cutting tooldone to verify Betaneli's resultRake distance (mm)Figure 3.3: Analytical solution of rake face stresses for the parabolic boundary load distri-bution made to verify Betaneli's resultCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 55Figure 3.4: Elastic deformation of the loaded region of a cutting toolThus, the validity of Betaneli's [7] analytical result needs be investigated. Toverify this result a finite element model as described in the next section is used.3.4 Finite element cutting tool stress analysis for parabolicload distributionsIn the previous section it was mentioned that the analytical solution obtained for therake face principal stress distribution by Betaneli [7] does not explain cutting toolfailure near the end of chip-tool contact and therefore this solution need be verified.To verify this analytical result the finite element method will be used.One of the more powerful methods of numerical analysis is the Finite ElementMethod (FEM). By means of FEM, solutions can be obtained for a wide range ofpractical problems. The finite element method is described by Cook et al [8] andothers. To assist in development of the finite element model, ANSYS [25] finiteNO.S.11t.445" R"'......*111111:04tatallatanie.4144riotattiN:71,4, MOM*.4 Nritt 411110 NNW OS,f7	 44,4140 aft‘'.***4-•t V41+.Fixed boundaryCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 56Figure 3.5: Finite element model of the cutting toolelement software was employed.The finite element model of the tool developed in this study is shown in Fig.3.5. The zero displacement boundary were applied at a radial distance of 1 cm fromthe cutting edge. This distance was chosen because its further increase does nothave any significant effect on the accuracy of the FEM solution. A combinationof two-dimensional quadrilateral and triangular isoparametric elements with mid-side nodes were used. These types of element were selected to obtain a. reasonablerepresentation of the steep stress gradient in the loaded region. These elements weregenerated automatically after the element sizes around the tool boundary were chosen.A magnified view of the loaded region of the tool is shown in Fig. 3.6. The finiteelement boundary load distribution in the loaded region was determined from theparabolic equations (2.12) and (2.16) at the mid-point of each element (Fig. 3.6)along the rake face. The normal stresses were applied directly to the element sides asCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 57Figure 3.6: Finite element model of the loaded region of the cutting tool, contact length=lcpressures. The shear stresses were first converted to forces by multiplying them bythe element length in the loaded region (0.02 mm in this case) and then were appliedat the nodes. Very small elements were used to accurately represent the steep stressgradient that exists in the loaded region. The thickness of the tool was taken as unity.The ANSYS finite element input data and other programs used in this analysisare given in Jemal [40] (Appendix C.2). The results of the FEM analysis for theprincipal stress distribution for the parabolic load distribution of Fig. 3.1 is shownin Fig. 3.7. This figure shows that the principal stress al at the rake face reachesits maximum at the end of chip-tool contact, and within the wedge it increases fromflank face to the rake face.The rake face stress distributions from the FEM solution together with the bound-ary load distributions (ay and Try) are shown in Fig. 3.8. In this figure a. is the normalCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 58Figure 3.7: FEM solution for the maximum principal stress distributions in the loaded partof the tool for the boundary shown in Fig. 3.1FEM solution.11111111111I11111111111111111111111111111111111IIIII50al_100	0.1	 0.2	 0.3	 0.4	 0.5	 0.6	 07Rake distance (mm)Figure 3.8: FEM solution for the rake face stresses for the parabolic boundary load distri-bution of Fig. 3.1, contact length lc = 0 5 mmCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 59Stress comparisonIII	 I	 Jill	 TI TI TI7550= Finite element.- - - - = Analytical.0.0;11111111111t11141111111111111111111114111111111111111 0.25	 0.50	 0.75	 1.00	 1.25	 1.)0Rake distance, mmFigure 3.9: Comparison of FEM and analytical maximum principal stresses al along therake face for the boundary shown in Fig. 3.1, contact length I = 0.5 mmstress parallel to the rake face. The above FEM results and the analytical result of theprevious section are compared in Fig. 3.9, and will be discussed in the next section.3.5 Discussion and conclusionsFrom the analytical and FEM solutions obtained in the previous sections, the maxi-mum principal stresses cri on the rake face from the two solutions are compared in Fig.3.9. From this figure, it is clear that the principal stress results from the analyticaland FEM solution for the parabolic load distribution of Fig. 3.1 do not agree. Theanalytical solution shows the maximum principal stress at the cutting edge, while thenumerical solution shows the maximum principal stress just after chip-tool contactlength lc. In addition, the maximum principal stress in the tool given by the numeri-cal solution is more than two times that given by the analytical solution. The reason100 -ill"0.1 0.2	 0.3	 0.4 0.5 0.650CHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 60FEM boundar load.Rake distance (mm)Figure 3.10: The FEM solution satisfies the boundary load conditions both in the loadedand free region of the rake face, contact length lc = .5 mmfor this disagreement between the two solutions is explained below.Since identical geometry were used in both solutions, clearly the geometry is notthe cause of the difference in the results shown in Fig. 3.9. The next step should beto check to see if the applied boundary load distribution are the same everywhere inthe tool for the two solutions.Upon comparison in the loaded region, both the FEM and analytical boundaryload distributions are the same. Outside of the loaded region, the finite elementsolution satisfies the zero boundary load conditions as shown in Fig. 3.10. To de-termine the boundary load distribution value taken by the analytical solution outsidethe region of chip-tool contact it is necessary to extrapolate the parabolic normalboundary curve of equation (2.12) beyond the chip-tool contact length lc = 0.5 mm.This extrapolation is shown in Fig. 3.11 and this clearly does not satisfy the freeCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 61Parabolic load boundar/•-n 50CNI<EE--a, 0.0.-NdN.."CAtd(1)I.,cis -500.25 0.50	 0.75Rake distance (mm)Figure 3.11: The Analytical solution does not satisfy the boundary load conditions in thefree region of the rake face, contact length /c = .5mmboundary condition after chip-tool contact. It is this difference in applied boundaryloading which is responsible for the difference between the analytical and FEM solu-tions. Thus, the comparison made was for two different problems where in the first(analytical) there is load after chip-tool contact while in the second (FEM) there isno load after chip-tool contact. To check this hypothesis, the extrapolated analyti-cal boundary load distribution was applied on the FEM model. The finite elementinputs for this problem are given in Jemal [40] (Appendix C.3) and the results areshown in Fig. 3.12. From this figure it can be seen that the two solutions are now inagreement. This result shows the analytical solutions can be used to determine stressdistribution in a wedge geometry whose applied loading is continuous. However, whenanalytical solutions are used to determine the stress distribution in cutting tools, itshould be made sure that the analytical expression chosen to represent the boundaryCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 62Comparison of results.-100	It111111111iIIIIIIIIIIi1111111111 11111110il0.1	 0.2	 0.3	 0.4	 05Rake distance (mm)Figure 3.12: Comparison of FEM and analytical solutions for the boundary load distributionshown in Fig. 3.11load distribution satisfy the zero loading condition after chip-tool contact, otherwisethe solution obtained might be for an entirely unrealistic boundary load distributionwhere there are loads on the cutting tool after chip-tool contact.The result of applying the boundary load distribution which does not satisfy thezero loading condition after the end of chip tool-contact is shown in Fig. 3.13. In thisfigure the position of the cutting edge before the load is applied was at the origin ofthe coordinate system. Thus, it can be seen that for this condition the cutting tooldeforms upwards, i.e. opposite to the direction of the cutting force. Since Betaneli's[7] results were for a tool which behaved as shown in Fig. 3.13, his conclusions areincorrect. Further examination of Fig. 3.12 shows a negative maximum principalstress u1 near the end of chip-tool contact. This is because the analytical solution isbased on the two-dimensional case and therefore does not include the zero principalCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 63Figure 3.13: Tool deformation and maximum principal stress distribution for the boundaryshown in Fig. 3.11stress perpendicular to the plane of the tool when the value for the algebraicallygreatest stress al is calculated. Thus, whenever both the principal stresses in the planeof the tool, from the analytical solution, are negative then the maximum principalstress al should be zero as shown by the FEM solution.Other conclusions that can be drawn from the results in this chapter are:• Analytical and FEM solutions for the maximum principal stress at the cuttingedge do agree (Figs. 3.9 and 3.12). This result supports Archibald's [6] analyticalsolution for cutting edge stresses.• The maximum principal stress increases from flank face towards the rake face(Fig. 3.7).In the boundary load distribution types shown in Fig. 3.10 there is a discontinuityCHAPTER 3. STRESS CALCULATION IN ORTHOGONAL CUTTING TOOLS 64of slope at the end of chip tool contact, lc = .5 mm. For these type of boundariesit is difficult to get a polynomial expression that can reasonably represent the curveshown in Fig. 3.10 both in the loaded and free regions of the rake face. However, forthe boundary load distribution types of Amini [20] and Ahmad [21] as determinedfrom photoelastic studies at higher cutting speeds (see Fig. 2.9), it is possible to geta polynomial expression which approximately satisfies the boundary condition bothin the loaded and free region of the rake face. This is because in these particularboundary load distributions there is little discontinuity of slope at the end of chip-tool contact and therefore it can be approximated by a polynomial function. Thenit can be proposed that for these types of boundary load distributions, the solutionsfrom the analytical and FEM solutions will be close and as a result the analyticalmethod could be used to determine the stress distribution in the loaded region ofcutting tools. This is verified in the next chapter.The split-tool dynamometer result by Barrow [16] shows reduced constant shearstress region (sticking region) with increase in cutting speed. Therefore for this casethe shape of the boundary load distribution from the high speed photoelastic cuttingand split-tool dynamometer are approximately similar (see Figs. 2.9 and 2.5). Thus,the high speed photoelastic boundary load distribution can be assumed to reasonablyrepresent the boundary load distribution in metal cutting at higher cutting speeds.Chapter 4Stresses for Higher SpeedPhotoelastic Boundaries4.1 IntroductionIn Chapter 3 it was shown that when an analytical solution is used to determinecutting tool stresses, the analytical function chosen to represent the boundary loaddistribution should satisfy the zero loading condition after chip-tool contact. Thephotoelastic boundary load distribution at higher cutting speeds shown in Fig. 2.9are almost tangent to the rake face at the end of chip-tool contact. For such acontinuous distribution it is possible to fit a polynomial which can approximatelysatisfy the boundary condition both in the loaded and free regions of the rake face.In this chapter, the stress distribution in a cutting tool for such polynomial bound-ary load distribution is determined analytically, and then these results are verifiedwith numerical solutions. The result is compared with a photoelastic experimentalresult given by Amini [20] for a similar boundary load distribution and tool geometry.Finally, from the stress distributions obtained, the critical regions of the orthogonalcutting tool and its modes of failure will be discussed.65CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 664.2 Analytical solution for high speed photoelastic bound-ary load distributionsThe boundary load distribution determined from photoelasticity at higher cuttingspeeds by Ahmad [21], as discussed in Section 2.3.2, is given byo-' = 2.91 — 1.53r' + .214ra — .0033ra 	 (4.1)T1 = 1.63 — 1.24r' + .315ra — .0266r'3	 (4.2)where,o-' = normal boundary stress on the photoelastic tool, kg/mm2Ti = shear boundary stress on the photoelastic tool, kg/mm2r' = rake face distance from the cutting edge of the photoelastic tool, mm.In order to determine whether this polynomial boundary load distribution satisfiesthe free loading condition after chip-tool contact, these equations are plotted beyondthe contact length of 4.5 mm in Fig. 4.1. From this figure, it can be seen that thefree boundary conditions on the rake face are not satisfied. Therefore, Eqs. (4.1)and (4.2) as they are cannot be used as boundary conditions to determine the stressdistribution in cutting tools. The second reason why these equations cannot be useddirectly in metal cutting tool stress analysis is because these boundary stresses weredetermined from photoelastic cutting test where the magnitude of the stresses arevery low and the contact length is very high for metal cutting tools.A boundary load distribution which approximately satisfies the boundary condi-tions in the free region of the rake face and having stress magnitudes experiencedby metal cutting tools is shown in Fig. 4.2. The least squares method was used toCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 67Rake distance (mm)Figure 4.1: Ahmads's [21] polynomial functions for his photoelastic data do not satisfy thefree boundary condition after chip-tool contact (lc = 4.5 mm)Tangential tyr boundaries.n•••n —100-50Tvy0.0n0.0I	 I	 I	 1	 I	 I	 I	0.5 	 1.01	 1	 1	 1 1.5	 20Rake distance (mmlFigure 4.2: A polynomial function which approximately satisfy the free loading conditionafter chip-tool contact for a metal cutting tool (/, -= 1 mm)CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 68determine the polynomial function which provides a satisfactory approximation of thefree loading condition after chip-tool contact. In the determination of this boundaryload distribution function, Zorev's [24] cutting data which gives a friction coefficientof 0.36 when cutting steel in water with a cutting tool having a rake angle 200 wastaken. These data were used by Betaneli [7] in his analytical cutting tool stressanalysis. This particular metal cutting condition was selected in this current studybecause it is similar to the cutting conditions used in photoelasticity and thereforethe analytical results to be obtained could be compared with previous photoelasticresults. The contact length and the width of cut were assumed to be 1 mm. Atthe cutting edge, the shear stress applied on the cutting tool was assumed to reachthe yield shear strength of the workpiece material being machined, Fig. 2.3. ForN.E. 9445 steel, the shear yield strength quoted by Merchant [26] is 40.5 kg/mm3.This value was assumed as the applied shear stress value at the cutting edge. Theshear stress and the normal stresses were assumed to obey the relation 7- = fLa. Theequations for the boundary load distributions shown in Fig. 4.2 are given bya- = —112.5 + 242.1r — 170.5r2 38.4r3 3.1r4 — 1.5r5T = —40.5 + 87.2r — 61.4r2 13.8r3 1.1r4 — .5r5or they could be represented in simplified form as:5	= E Nrie	(4.5)5T =	 sriri	(4.6)where,—112.5 Sr 0242.1 S„—170.5 S238.4	 1' S33.1 Sr 4—1.5 Sr 5{N70N71N7 2N73N,4Nr5—40.587.2—61.413.81.1—0.5CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 69where,a- = rake face normal stress distribution, kg/mm2T = rake face shear stress distribution, kg/mm2Nri = rake face normal load distribution coefficientsSri = rake face shear load distribution coefficients= rake face distance from cutting edge, mm.From Fig. 4.2, it can be seen that there is a good approximation of the freeboundary after chip-tool contact (1 <r < 2) mm and for this distribution analyticaland FEM solutions will be compared. The FEM boundary can be made to satisfythe free loading conditions completely as shown in Fig. 4.3.As described in Section 3.2, the arbitrary constants that appear in the stressequations (3.1) to (3.3) can be determined from the boundary conditions and thegeometry of the tool using Eqs. (3.17) and (3.18). For quick reference Eqs. (3.17)and (3.18) are also given by Eqs. (4.7) and (4.8) below.for i = 0N70 2 29, 2 cos 29,. 2 sin 29r aoN10 2 2Of 2 cos 29f 2 sin 29f boSr0 0 —1 2 sin 28,. —2 cos 29r COS 10 0 —1 2 sin 20f —2 cos 2Of dO(4.7)0.5	 1.0	 1.50.050eze100(4.8)I at);CsCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 70Rake distance (mm)Figure 4.3: The FEM boundary completely satisfy the free loading condition after chip-toolcontact (I, = lmm)for 1 < i < n- COS 9r sin iOr COS(i	 2)Or sin(i + 2)19,.N. cos ie f sin i0 cos(2, + 2)	 f sin(i + 2) OfS . sin iOr cos iOr sin(i + 2)8r COS(i	 2)OrSf i-1-2sin Of cos Of sin(i + 2) 0 f cos(i + 2) O fi-1-2 i+2where,j	 = i(i + 1)(i + 2)=-CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 71Of =-yd-a= rake anglea = wedge anglei	 = powers of the polynomial= rake face polynomial normal load distribution coefficientsNfi = flank face polynomial normal load distribution coefficientsS,., = rake face polynomial shear load distribution coefficientsSfi = flank face polynomial shear load distribution coefficientsai, bi, ci and di for i> 0 are constants.The calculation of the constants and then the stress distribution in the cuttingtool is discussed below. The boundary conditions are already given by Eqs. (4.5)and (4.6), where the degree of the polynomial n = 5. The x-axis is taken along therake face and therefore Or -= 0 (Fig. 2.14). The wedge angle is 62°, so Of = 627r/180radians. The flank face is assumed to be free from loads, therefore, Nfi 0 andSfs= 0. Substituting these values into Eqs. (4.7) and (4.8) yieldsfor i 0—112.5 2 0.000 2.000 0.0000.0 2 2.164 —1.118 1.658 bo—40.5 0 —1.000 0.000 —2.000 co0.0 0 —1.000 1.658 1.118 dofor i 1242.1 - 6.000 0.000 6.000 0.0000.0 2.817 5.298 —5.967 —0.627 1)187.2{0.000 —2.000 0.000 —6.0000.0 1.766 —0.939 —0.627 5.967for i = 51 —1.50.0—0.50.0-	 42.000 0.000 42.000 0.000_ 26.997 —32.174 11.577 40.373— 0.000 —30.000 0.000 —42.000—22.981 —19.284 40.373 —11.5771 a5 1b5C5d5}={ —21.2—35.1 1'—22.731.6133.3—11.0= 7.1 , • • • ,1	 —10.8 11 —0.02 1—0.00—0.010.021 aoboCodo 1 albiCldi. { a5b5C5d5 } =.CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 72Solving the above equations simultaneously, the results obtained for the non zeroconstants are:These determined constants are then substituted into Eqs. (3.1) to (3.3) to deter-mine the stress components at any point (r, 0) of the cutting tool. From these stresscomponents, the principal normal stresses can be determined from Eq. (3.24, and themaximum shear stress from Eq. (3.25). For convenience, the above calculations wereperformed using a Fortran program which is given in Jemal [40] (Appendix D.1).The analytical result for the principal stress distribution in the loaded regionwithin the cutting tool is shown in Fig. 4.4. This result shows increase in principalstress from the flank face towards the rake face. The stress distributions along the rakeface are shown in Fig. 4.5. In this figure ay is the applied normal load distribution,Ty is the applied shear stress distribution and az is the normal stress parallel to therake face. In the next section this analytic solution is compared to an FEM solutionfor the boundary load distribution shown in Fig. 4.3.0.5	 1.0 1.5500.0-50CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 73Anal tical solution.Rake distance (mm)Figure 4.4: Analytical solution for the maximum principal stress distribution for the bound-ary shown in Fig. 4.275N<	 50EEba9 25......CtLI)tl)	 0.0rIDI-44..)-25O.(Max. principal stress dist.,-2 — " III "Mr HIM" i--150.---30°_	I IIIIII I I1/11 n 11111 1 11111 1 1111111111E-45°0.25	 050	 0.75	 1.00Radial distance, mmFigure 4.5: Analytical solution for rake face stresses for the boundary shown in Fig. 4.2CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 74Figure 4.6: FEM solution for the maximum principal stress distribution for the boundaryshown in Fig. 4.34.3 FEM solution for higher cutting speed photoelastic bound-ary load distributionsThe FEM procedure described in Section 3.4 was again used to determine the stressdistribution in a cutting tool for the boundary load distributions shown in Fig. 4.3.The finite element model used is shown in Figs. 3.5 and 3.6. The element lengthalong the rake face in the loaded region was 0.04 mm. The input data files and otherprograms used in this analysis are given in Jemal [40] (Appendix D.2).The FEM solution for the maximum principal stress in the loaded region, for theboundary load distribution of Fig. 4.3, is shown in Fig. 4.6. This result shows anincrease in maximum principal stress magnitude from flank face towards the rakeface. On the rake face, the principal stress is a maximum at the end of chip-toolCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 75Figure 4.7: FEM solution for the minimum principal stress distribution for the boundaryshown in Fig. 4.3contact (point A). These conclusions agree with those given in Section 3.4. Thesetwo solutions were for two different types of boundary load distributions, but thelocation of the maximum principal stress from both solutions was found to be at theend of chip-tool contact.The minimum principal stress distribution is shown in Fig. 4.7. This figure showsthat the magnitude of the compressive stress is a maximum in the flank face close tothe cutting edge (point A).The FEM solutions of the rake face stress distributions are shown in Fig. 4.8. Inthis figure, au is the applied normal load distribution and Tx is the applied shear stressdistribution. The maximum principal stress reaches its maximum value at the end ofchip-tool contact (lc--=-1 mm) and then drops off gradually. A comparison between theanalytical and FEM solution is shown in Fig. 4.9. This figure shows a satisfactoryFEM solution.1111111111111111111IIIIIIIIIIIIIIFIIIIIIIIIIIIIIII-100	 !,11,1111,11,11,1,1,,,,11,1,11,1111.1.1111. 0.25	 0.50	 0.75	 1.00	 1.25CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 76Rake distance (mm)Figure 4.8: FEM solutions for rake face stresses for boundaries shown in Fig. 4.3agreement between the two results and therefore indicates that analytical methodscan be used to determine stresses in cutting tools for certain types of boundary loaddistributions with reasonable accuracy. The results obtained in the previous sectionswill be discussed in Section 4.6.4.4 Point-load analytical solution for higher cutting speedphotoelastic boundary load distributionsAs discussed in Section 2.4.1, the point load analytical solution was used by Kaldor [2]to determine optimal tool geometry and by Chandrasekaran [3] to determine fracturestresses in milling cutters. The point load solution (Eq. 2.31) cannot be used toestimate stresses close to the cutting edge. In this section, the way in which the pointload solution and the current analytical solution may together be used to estimatethe stress distribution close to and far from the cutting edge is examined.CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 77lull111111111111111111111111111Comparison of max. stresses. C-1---—..--V1-- l 0()-_1111111111111111111111111111111111 0.25	 0.50	 0.75	 1.30Rake distance (mm)Figure 4.9: Comparison of analytical and FEM principal and maximum shear stress distri-butions for the boundary shown in Fig. 4.3The maximum principal stress distribution crj at the rake face from the point-loadapproximation (Eq. 2.31) is compared with the analytical and finite element solutionsobtained in the previous sections in Fig. 4.10. The magnitude of the concentratedforce P was calculated from the areas of the distributed shear and normal boundarystresses (Fig. 4.3 in this case). From Fig. 4.10, it can be seen that in the loadedregion (rake distance < 1 mm) the point-load approximation provides a significantlyless accurate result. However, after the chip-tool contact the result from the point-loadapproximation improves indicating that an estimate of the maximum principal stressin the cutting tool could be made by using the value of the point load solution where itintersects the current analytical distributed load solution (Fig. 4.10). The advantagesof this method are that it is very simple to use, and the maximum principal stress islinearly related to the resultant cutting force by Eq. (2.31). Since the cutting forceStress comparison1CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 78Labels:1 = Analytical concentratedload solution.2 = Analytical distributedload solution.3 = Finite element distributedload solution.Rake distance, mmFigure 4.10: A comparison of rake face principal stress ai by different methods for a 620wedge angle and boundary load distribution shown in Fig. 4.3, chip-tool contact lc = 1 mmCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 79is also considered to be linearly related to the uncut chip thickness, the point loadsolution gives a linear relation between the uncut chip thickness and the maximumprincipal stress. This relationship can be used to determine the maximum uncut chipthickness (i.e. feed) corresponding to the allowable tensile stress of the tool material.Thus the current distributed analytical solution can be used to estimate the stressdistribution in the loaded region while the point load distribution can be used toestimate the principal stress distribution outside the loaded region. This procedureprovides a complete analytical solution for the prediction of the critical stresses thatcauses failure in the cutting tool.It is noted from Fig. 4.10 that the maximum normal stresses given by the point-load solution are higher than the solutions given by the other methods. This featurecan be explained by the fact that the resultant of the normal distributed boundarystresses which act at the distribution's centroid is instead applied at the cutting edgein the point-load solution. This results in a larger bending moment and consequentlyhigher bending stresses. Thus the solution obtained from the above procedure for thecritical maximum principal stress and its location on the rake face, as shown in Fig.4.10, are higher than the numerical results.In order to use the stress analysis results obtained in the previous sections topredict the critical regions of cutting tools and their types of failures, cutting toolfailures will be considered next.4.5 Failure in cutting toolsThe two modes of failure of a loaded body are ductile and brittle failure. If theamount of permanent deformation occurring in the loaded body at rupture is large,CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 80Figure 4.11: Fracture locus of a brittle materialthe failure is ductile, if the deformation is small it is brittle.Brittle failureThe Coulumb-Mohr theory is often used to predict fracture of brittle materials.This theory indicates that for plane stress conditions, a2 = 0, the loaded body issafe as long as (cri, a3) falls within the area shown in Fig. 4.11. In this figure,the ultimate compressive stress IS2,1 is typically greater than the ultimate tensilestrength Std. This is because flows such as microcracks or cavities which are presentin materials weaken it in tension, while it does not appreciably affect its compressivestrength.When the principal stresses al and 03 at a critical point in the tool are bothgreater or equal to zero then the factor of safety F.S = Sutlai. If both the principalstresses are less than or equal to zero, the factor of safety F.S = 1,9.1/1(731. WhenCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 81al > 0 and 03 < 0, the equation to be used is0-1	 0-3	 1c + ---0 = D 0...Jut	 iluc	 X' • 43 .(4.9)In this equation both Suc and 03 are negative quantities.From the above discussion, the stresses to be considered for brittle failure of acutting tool are the maximum principal stress al and the minimum principal stressescr3. Since in Fig. 4.9, the results obtained in the loaded region from the analytical andnumerical solutions are similar, either solution can be used to determine the criticalregions of the cutting tool. The principal stress distributions are shown in Figs. 4.6and 4.7, where the critical points are shown by point A in both figures. These resultscan be explained from the elastic deformation of the loaded region. During cutting,the loaded region of the cutting tool deforms as shown in Fig. 3.4 and results inthe shortening of the flank face close to the cutting edge and bending of the rakeface at a point near the end of chip tool contact. The shortening of the flank faceresults in the minimum principal stress to be in this face close to the cutting edge asshown in Fig. 4.7. The rake face bending about the point near the end of chip-toolcontact results in the maximum principal stress to be at this point as shown in Fig.4.6. Inside the cutting tool wedge, the stresses are within these two extreme valuesincreasing algebraically from the flank face towards the rake face. From Figs. 4.6 and4.7, since the maximum stresses occur on free surfaces, the other principal stresses atthese critical points are zero.From the principal stress distributions shown in Figs. 4.6 and 4.7 the ratio of thecritical minimum principal stress to the critical maximum principal stress is 2.7. Thisratio, when compared with the ratio of the compressive strength to tensile strengthof the tool material used, provides an indication of whether the tool failure is due toCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 82compressive stress on the flank face close to the cutting edge, or due to maximumtensile stress near the end of chip-tool contact. If the ratio of the critical compressivestress to the critical tensile stress is assumed not to vary with change in cuttingforces, then when the magnitude of the critical maximum principal stress al = Sut(say) then the magnitude of the critical compressive stress 53 = 2.7Sut. Also, if thetensile strength of the tool material is S. and its compressive strength S. = N Sut,then the factor of safety for tool failure due to maximum principal stress is F.S,, =Sut/cri = 1, and the factor of safety for tool failure due to minimum principal stressis F.Sus = Suck-3 = N/2.7. When F.Sui > F.S,8, the tool fails due to compressivestresses, while when the inequality is reversed it fails due to tensile stresses. Therefore,when N = Suc/ Sut for the tool material is lower than 2.7, then the tool fails due tocritical compressive stress at the cutting edge and if higher it fails due to criticaltensile stress at the end of chip-tool contact. For example, the typical ratio of thecompressive strength to the tensile strength of high-speed steel is 2, of carbide is 3and that of ceramics is 5 (Loladze [37]). From these values, for the boundary loaddistribution and tool geometry considered in this study (53/ai = 2.7), the high-speedsteel fails due to maximum compressive stress at the cutting edge while both thecarbide and ceramic tools fail due to the maximum tensile stress near the end of chip-tool contact. This result explains why tool materials that have low tensile strength(carbides, ceramics and diamond) are used only with robust tool geometry (largewedge angles or low rake angles) where the maximum tensile stresses are low. Theabove analysis is an example of the use of cutting tool stress distributions in selectionof tool material properties.With increase in depth of cut in the carbide and ceramic tools, cracks initiateCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 83Figure 4.12: Brittle failure of a cutting tool near the end of chip-tool contact, Tlusty [27]near the end of chip tool contact, point A in Fig. 4.6, due to maximum principalstresses and results in breakage of the whole loaded region of the cutting tool. Aphotograph by Tlusty [27] which shows this failure of the whole loaded region of acarbide cutting tool due to maximum tensile stresses is shown in Fig. 4.12. Failuredue to compressive stresses result in permanent deformation of the cutting edge asshown in Fig. 4.13, Wright [36]. Thus the analytical results obtained for tool failurescorrelate well with observed cutting tool failures and this verifies the approach usedin this study.Experimental investigations by Trent [34] show, cutting tool materials loose theirstrength with increase in temperature. Therefore, the effect of temperature on theproperties of the tool material should be taken into account when failure analysis ofthe cutting tool is performed.CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 84Figure 4.13: Cutting edge deformation due to maximum compressive stress, Wright [36]Ductile failure The theory of failure commonly used to predict ductile failure is the von Mises-Hencky theory. According to this theory, also known as the distortion energy theory,a loaded body is safe as long as the maximum value of the elastic distortion energyper unit volume in that material remains smaller than the distortion energy per unitvolume required to cause yield in a tensile-test specimen of the same material. For theplane stress case this theory indicates that a loaded body is safe as long as aey < Sy,where2	 2Greg = N/0.1 — al U3 + U3 (4.10)where,a1, a3 are the algebraically largest and smallest stressesa-ey = an equivalent stress1.433 mmMNCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 85F"--Figure 4.14: Equivalent stress contour lines for the boundary shown in Fig. 4.3Sy = yield tensile strength of the material.To determine the ductile cutting tool failure regions, the equivalent stress contourlines in the loaded region were plotted and are shown in Fig. 4.14. The results in thisfigure can be explained from the maximum and minimum principal stress distributionsshown in Figs. 4.6 and 4.7. At a free surface the equivalent stress is proportional tothe non zero principal stress. Since the magnitude of the minimum principal stressclose to the cutting edge in Fig. 4.7 is about three times the maximum principalstress at the end of chip tool contact in Fig. 4.6, the equivalent stress therefore willbe maximum near the cutting edge as shown in Fig. 4.14.From Fig. 4.14, it can be seen that the region of maximum distortion energy isclose to the cutting edge. Near the cutting edge the temperature is very high and thisreduces the strength of the tool and increases its ductility and results in the shearingof the cutting edge which is known as edge chipping. A photograph by Tlusty [27]CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 86Figure 4.15: cutting edge chipping, Tlusty [27]which shows chipping of the cutting edge is shown in Fig. 4.15. It is interesting to seethe good correlation between the equivalent stress lines in Fig. 4.14 and the profileof the chipped surface.To complete the failure analysis of the cutting tool its stress at the shank whereit is attached to the tool post needs to be considered. This was done by consideringthe bending stresses at the fixed end for a width of cut of 5 mm. This analysis forthe tool geometry shown in Fig. 4.16 shows the maximum tensile stresses near thecutting edge is four times the bending stress at the fixed end (point A). This resultshows shank design is not only based on cutting tool breakage at the fixed end butalso is based on the rigidity of the tool.CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 875F 50Fr, =38 kgFs =14 kg5 F5	2° /////15x15units in mmFigure 4.16: Bending stress calculation at the fixed end (point A) of a cutting tool4.6 Discussions and conclusionsIn the above sections, both analytical and FEM cutting tool stress distributions forhigh speed photoelastic boundary load distribution were determined. The results forthe principal stresses and maximum shear stress distribution from both solutions inthe loaded region are in good agreement. The results are identical at the cutting edge(verifying Archibald's [6] result), however, further from the cutting edge the differencebetween the two solution increases. The value for this difference at half the chip-toolcontact length is 8% for the maximum principal and shear stresses. The results forthe minimum principal stress are almost identical everywhere in the loaded region.If the boundary conditions after the tool-chip contact were completely satisfied thenthe above differences would vanish.From Fig. 4.9, the analytical solution for the distributed boundary load gives aCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 88similar result to the finite element solution in the loaded region where the stressesare most critical. Therefore, either solution can be used to predict the failure regionsand their mode of failure of the cutting tool. From these results it was found thatthe maximum principal stress in the cutting tool occurs on the rake face (Fig. 4.6)and the minimum principal stress on the flank face (Fig. 4.7). These results can beexplained from the local deflection of the loaded region shown in Fig. 4.18. Due tocutting forces the loaded region bends downwards about point A creating a maximumbending stress (tensile) at this point. On the other hand, the flank face, as shown inthe figure, shortens and is therefore under compression. Thus cutting tool failures areeither at the cutting edge or near the end of chip-tool contact. At the cutting edgethe failures are either due to maximum shear stress (Fig. 4.14) which causes cuttingedge chipping as shown in Fig. 4.15, or due to maximum compressive stress (Fig. 4.7)which causes permanent deformation of the cutting edge as shown in Fig. 4.13. Atthe end of chip tool contact the failure is due to the maximum tensile stresses (Fig.4.6) which causes initiation of cracks at this point and results in the fracture of thewhole loaded region of the cutting tool as shown in Fig. 4.12.The stress distribution at the rake face from the analytical and finite elementsolutions were compared with point-load elasticity solutions. This comparison isshown in Fig. 4.10. From this figure the two analytical solutions together can beused to determine the stress distribution in a cutting tool, where the distributed loadsolution is used in the loaded region while the point load solution is used after theloaded region. It is interesting to note that after twice the contact-length the resultfrom the point load solution and finite element solution in Fig. 4.10 are in agreementand therefore in this region the point load solution is very satisfactory.CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 89Figure 4.17: Maximum shear stress distribution from photoelasticity , Amini [20]The stress distribution obtained can also be compared with a photoelastic resultby Amini [20]. The photoelastic result is shown in Fig. 4.17. In this figure theisochromatics or fringe patterns (see Section 2.3.2) are proportional to maximumshear stress lines. Amini used a tool having a wedge angle of 760, but the analysismade in this study was for a 620 wedge angle. Therefore, in order to get goodcomparison of our solution with the experimental result the analysis was repeatedfor a wedge angle of 760. The contour lines which are equal to twice the maximumshear stress, obtained for the boundary load distribution of Fig. 4.3 and wedge angleof 76°, are shown in Fig. 4.18. Comparison of this figure with Fig. 4.17 shows theexperimental and maximum shear stress contour lines of this study agree verifyingthe proposed solution.To see the effect of the wedge angle on stresses in a cutting tool, minimum principalCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 90Figure 4.18: Twice the maximum shear stress distribution obtained for the photoelasticcutting boundary of Fig. 4.3stresses at the flank face and maximum principal stresses at the rake face for twowedge angles (62° and 76°) were compared. This comparison is shown in Fig. 4.19.This figure shows the effect the wedge angle has on the principal stresses distribution.At the end of chip-tool contact length, 1, = 1 mm, the maximum principal stress hasdecreased by 60% for the same boundary load distribution with the increase in thewedge angle. This result explains why tool materials having low transverse rupturestrength (like carbide, ceramics and diamonds) are made with higher wedge anglesor have low rake angles. At the cutting edge, with the increase in the wedge anglefor the same boundary load distribution the magnitude of the minimum principalstress has decreased by 45%. These are the reasons why rough cutting operations areperformed using tools that have high wedge angles.Close to the cutting edge the temperatures and the compressive stresses are very-200—biI I I	 I	 I	 n 1 11 1	 II]	 I	 t	 I	 I	 I	 11111..ma0.0 --------0'3n•n•CHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 91Labels:=- Rake face max. principal stress.cr3 = Flank face min. principal stress.	 = For a wedge angle of 620- - - - = For a wedge angle of 7600.5	 1.0	1.5	 2.0	 2.5Radial distance, mmFigure 4.19: Effect of change in wedge angle on the critical principal stresses of a cuttingtoolCHAPTER 4. STRESSES FOR HIGHER SPEED PHOTOELASTIC BOUNDARIES 92high, and therefore tool materials that maintain their compressive strength at thishigh temperature are superior. This is the main reason why carbide tools are betterthan high speed steel tools in their cutting performance.This chapter is summarized by listing the important results obtained:• An analytical stress distribution in a cutting tool which agrees with numericaland previous experimental results for the high speed photoelastic boundary loaddistribution was obtained.• From the solutions obtained the failure regions of the cutting tool and theirmodes of failure were identified. These failure predictions were found to correlatewell with observed cutting tool in-service failures.Chapter 5Stress Analysis of an End Mill5.1 IntroductionMaximum production in metal cutting requires a large chip load. However, at veryhigh chip loads breakage of the cutting tool may occur. Therefore, to select theoptimum cutting condition (maximum cutting force) for maximum production, with.-out the danger of tool breakage, knowledge of the relationship between cutting toolforces and tool stresses is required. These stress distributions also help to predict thelocation and types of failures in the cutting tool.The stress distribution in an end mill can be determined using numerical methods,however, these methods require much time and effort. Analytical solutions may nothave these shortcomings, and therefore the possible use of analytical solutions in endmill stress analysis is investigated.An end mill could fail in two ways due to stresses developed in the tool duringcutting. It could fail at the shank section due to excessive bending stresses or it couldfail within the flutes where the cutting forces are applied. Thus, both of these criticalregions must be considered for potential failure of the end mill.The cutting force distribution along the flutes of the end mill must be known to93CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 94determine the stress distribution in an end mill. When the stress distribution faraway from the cutting edges (shank stresses) are required, the cutting forces can beassumed to be distributed as concentrated forces along the cutting edge. However,when stresses close to the cutting edge are required, then the actual cutting forcedistributions across tool-chip contact along the cutting edge (like those determinedfrom photoelastic cutting) must be used.Once the cutting force distribution and the geometry of the end mill are selected,the stress distribution in the tool (at the shank or the flute) may be determinedanalytically or numerically. The analytical solution to be obtained in this study willbe verified with numerical solutions. In the literature there is little work done onend mill stress analysis, while there is sufficient work done on end mill deflectionand cutting force analysis. In the end mill shank stress analysis, the cutting forcedistribution models established in the literature will be used. The critical regions forshank breakage obtained in this study will be compared with shank failure regionsobserved in practice. Shank stress analysis will be considered in Section 5.2.In end mill flute stress analysis, the stresses required are close to the applica-tion of the cutting forces. In this case, the actual force distribution across tool-chipcontact must be used. Photoelastic cutting force distributions will be used in theseflute stress analysis. From the two-dimensional stress analysis done in the previouschapters, it was seen that the critical stresses occur close to the loaded region, be-cause further from the loaded region the load resisting area or section modulus ofthe wedge increases and this reduces the magnitude of the stresses. Thus, in the endmill flute stress analysis only a portion of the end mill close to the loaded regionwill be considered. Finite element solutions of flute stresses are obtained, and theywill be compared with the two-dimensional solutions. End mill flute stress analysisCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 95 No of flutes, T = 4Helix angle = 30 deg.Units in mm-I = 76.23C = 54.45h = 40d = 25.4Figure 5.1: Schematic diagram of an end millis considered in Section 5.4.5.2 Shank stresses in an end mill5.2.1 Force distribution along end mill cutting edgeFig. 5.1 shows a schematic diagram of an end mill and Fig. 5.2 shows its cuttingoperation.When stress predicitions in the shank of an end mill are required, the section isfurther from the application of the cutting forces and therefore these stresses can bedetermined from the resultant cutting force on the tool. The resultant cutting force Fon the tool acts at a distance a from the fixed end of the cutter as shown in Fig. 5.2.This resultant force can be calculated using the force distribution model of Kline etal [28], an example of which is shown in Fig. 5.3. In this figure, the forces shown arethe radial and tangential forces which act on each axial elements of the end mill andCHAPTER 5. STRESS ANALYSIS OF AN END MILL	96View A—AI = .127 mm/toothN = 190 rpmw = 25.4 mmHellix = 30 degFigure 5.2: Schematics for the cutting operations of an end millwhich vary in magnitude and direction along the cutting edge. From these elementalforces the resultant cutting force F and its point of application a can be determined.In the end mill cutting edge force distribution model (also called mechanisticmodel), the three-dimensional milling operation is assumed to be equivalent to anaggregation of orthogonal cuts by each axial elements of the cutter. On each axialelement, as shown in Fig. 5.4, the tangential elemental cutting force is proportionalto the radial depth of cut Sr and the axial length of the element 8z. From this figurethe radial depth of cut is related to the feed rate per tooth f by Sr = f sin 0, where0 is the position of the cutting edge of the axial element measured from the normalto the finished surface. Therefore, the elemental tangential force is given by.5Ft = Kt f 8z sin 9	 (5.1)where,CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 97Figure 5.3: Cutting force distribution along the cutting edges of an end mill, Kline [28]Kt = tangential force constant, MPaThe elemental radial force is proportional to the elemental tangential force andthus can be written asSF, = Kr8Ft	 (5.2)where,Kr = radial force constant.Kline [28], has shown that the tangential and radial force constants depend on thecutting conditions, which are mainly the feed rate, the radial depth of cut and theaxial depth of cut. For the cutting conditions and the standard end mill geometryshown in Figs. 5.1 and 5.2, the values of the force constants for steel workpiece fromKline [28] are Kt = 2660 MPa and Kr = .553. These values are used in the shankCHAPTER 5. STRESS ANALYSIS OF AN END MILL	98Figure 5.4: Definition of terms used in elemental force calculationstress analysis presented here.In Eq. 5.1, to determine the elemental tangential force, the angle 0 shown in Fig.5.4 for each axial element along the cutting edge should be determined. This anglecan be determined from the equation for the swept angle 0 of the cutter tooth shownin Fig. 5.2. The swept angle is measured about the cutter axis and is defined as theangle swept from the beginning of a cut to the end of the cut by one tooth of the endmill in one revolution. The swept angle 0, from the geometry of Fig. 5.5, is given by0 = 7rdwhere,w = axial depth of cut, mm0 = helix angled = cutter diameter, mm360w tan 0 (5.3)CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 99i 	 7r d Figure 5.5: Developed surface of a four flute end mill having helix angle 00 = swept angle, degreeFrom Eq. (5.3), angle 8 which is required in the elemental force calculation alongthe cutting can be determined just by replacing w with the corresponding axial dis-tance z (< w) of the element under consideration (see Fig. 5.5).The above equations are sufficient to determine the elemental cutting forces, theresultant cutting force and its point of application on the end mill. In the followingsections, these equations will be used to determine the forces to be applied on theend mill in the analytical and numerical shank stress analysis.5.2.2 Analytical stress analysis in an end mill shankIn the literature, there is little information on the stress distribution in end mills. Inthis section an analytical solution for the determination of shank stresses in an endmill will be presented.CHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 100At the shank, which is far away from the application of the cutting forces, thestresses can be determined from the resultant cutting force F acting at a distancea shown in Fig. 5.2. The critical regions of the end mill for shank breakage couldeither be at the circular section close to the fixed end denoted by A in Fig. 5.2, orit could be across the flutes section close to the circular section denoted by B in thesame figure.Shank stress at the fixed endAt point A, the section is circular and the maximum bending stress can be de-termined from cantilever beam equations. If the resultant bending moment at thissection is denoted by Ma, then the maximum bending stress for the circular sectionhaving diameter d is given bycra 	32Ma	7r 3	 (5.4)The maximum shear stress for the circular section of the end mill due to cuttingtorque T is given byT = 	16T	ird3	 (5.5)One of the important properties of cutting tool materials is their high hardness.With an increase in hardness, the rupture strength of materials reduces and theybecome brittle. Therefore the critical stress to be consider for shank failure of an endmill is the maximum (tensile) principal stress. This maximum principal stress can bedetermined from the normal and shear stresses using the relation= cra / 2 + cfa / 2 )2 + T  (5.6)The above equations, together with the mechanistic force distribution model givenin Section 5.2.1, can be used to determine the stresses at the fixed end of the endmill.CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 101The above equations will be used to determine the shank stresses of the end mill forthe cutting conditions and tool geometry given in Figs. 5.1 and 5.2. The calculationat the fixed end for the resultant bending moment and torque, using the mechanisticmodel, is given in Jemal [40] (Appendix E.1). From these calculations the resultantbending moment at the fixed end is Mo = 337 kN-mm, and the resultant cuttingtorque is T = 56.2 kN-mm.From the resultant moment and torque at the fixed end and from the stress equa-tions given above, the critical stresses at point A of Fig. 5.2 can be calculated. Theresults of these calculations are summarized in Table 5.1.Table 5.1: Stresses at the fixed end, point A,of the end mill shown in Fig. 5.2, (MPa)Ta Cra 6117.5 209 210Table 5.1 shows the difference between the bending stress cra and the principalstress cri is only 1% and this means that the contribution of the shear stress towardsthe principal stress is negligible and therefore the bending stress is sufficient to predictshank breakage. These analytical results will be verified with numerical solutions inSection 5.2.3.Shank stress at the end of the flute sectionIt is not easy to determine analytically the maximum bending stress at the endof the flute section closest to the circular section (point B of Fig. 5.2) because of thecomplicated shape of the flute. However, an approximate solution can be obtainedby using the concept of equivalent diameter. The equivalent diameter is defined asthe diameter of a circular bar which gives the same deflection as the end mill underthe same cutting forces. Kops et al. [29] have used finite element end mill deflectionCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 102iTO = end mill dia, dFigure 5.6: Determination of the equivalent diameter for four flute cutterssolutions and beam deflection equations to determine the equivalent diameter. Aneasier method of calculating the equivalent diameter for four flute cutters from Kopset al. [29] is to take the moment of inertia of the equivalent circular bar, 7r4/64, tobe equal to the moment of inertia of a square section ABCD shown in Fig. 5.6 aboutx-axis, d4/48. Then this yields for the equivalent diameter the relation de = .8d.When applying the equivalent diameter concept, it is imagined that the fluteportion of the end mill is replaced by a circular section of the same length but smallerdiameter, de = .8d, as shown in Fig. 5.7.Once the flute section at point B is approximated by a circular section as shown inFig. 5.7, the stresses can be determined in exactly similar manner as described abovefor shank stresses at the fixed end. The resultant moment and torque at section Bare also given in Jemal [40] (Appendix E.1), and the value for the bending momentis Mb = 231 kN-mm, while the resultant cutting torque is still T = 56.2 kN-mm.CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 103Units in mmo=69.2b=47.4d=25.4de=20.3F=4867 NMo=aFMb=bFd	 d eFigure 5.7: Representation of an end mill by an equivalent solid stepped barFrom the resultant moment and torque at section B and from the stress equations(5.4) to (5.6), the critical stresses at point B of Fig. 5.7 can be calculated. Theresults of this calculation are summarized in Table 5.2.Table 5.2: Stresses at point B of theequivalent end mill shown in Fig. 5.7, (MPa)Tb ab cri34 281 286Table 5.2 shows the difference between the bending stress al, and the principalstress ui is less than 2% and this means that the contribution of the shear stresstowards the principal stress is negligible and therefore the bending stress is sufficientto predict shank breakage. The reason for the negligible effect of the shear stresson the shank principal stress is due to the large cutter axial length to cutter radiusCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 104ratio found in end mills. Comparison of the results in Tables 5.1 and 5.2 shows thatthe bending stress at the critical flute section is higher than that at the fixed end.This is because the effective section modulus of the flute section is much lower thanthe section modulus of the circular section. This lower section modulus at the flutesection offsets the smaller bending moment at this section and makes it the mostcritical point for shank breakage of the end mill for the cutting condition and toolgeometry considered in this analysis.The analysis above can be extended to more complex end mills such as taperedend mills. However, in this case, or when the number of flutes of the cutter aredifferent from four, the equivalent diameter has to be determined using finite elementdeflection solutions and beam deflection equations as described by Kops et al. [29].In the next section numerical solutions will be used to verify the analytical solu-tions presented above for shank critical stresses.5.2.3 Finite element stress analysis in an end mill shankIn the previous section analytical methods were used to determine critical shankstresses for the end mill geometry and cutting conditions shown in Figs. 5.1 and5.2. In this section these analytical solutions will be verified using the finite elementmethod (FEM) for the same cutting condition and tool geometry. The ANSYS [25]finite element software will be used in this analysis.The finite element model of the end mill shown schematically in Fig. 5.1 will bedeveloped from the geometry of the cross-section of one flute at the free end of theend mill shown in Fig. 5.8. In this figure the co-ordinate positions of nodal pointsthat define the geometry of the cross-section are also given. From this cross-sectionthe flutes of the end mill can be developed. The flute has a helix angle, therefore as0CHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 105NODE X(mm) Y(mm)1-8.8900000 0.000000002-8.8900000 3.81000003-8.2550000 6.98500004-6.9850000 9.20750005-5.0800000 10.7950006-2.5400000 12.0650007 0.00000000 12.7000008-0.25400000 11.4300009-0.63500000 9.525000010-2.2225000 6.667500011-4.4450000 4.445000012-6.6675000 2.222500013 0.00000000 8.890000014 0.00000000 4.445000015-2.2225000 2.222500016-4.4450000 0.0000000017 0.00000000 0.00000000Figure 5.8: Cross-sectional geometry of one flute of a four flute cutter at z 0, Fig. 5.1we move along the axis of the end mill (z-axis) to generate new nodes, the flute cross-section shown in Fig. 5.8 has to be rotated. The amount of this rotation in degrees peraxial distance can be calculated from Eq. 5.3 and is given by 0/z = 360 tan 7,1) 1 (r d),where is the helix angle and d is the diameter of the end mill. From this angleand the nodes of the flute cross-section shown in Fig. 5.8, the nodes for one flutemodel having any axial length c can be made. Fig. 5.1 shows, after the end of theflute section z = c, the end mill has a circular cross-section. This circular section canbe made by adding nodes to the flute cross-section at z = c (this cross-section hasrotated by some angle relative to the free end's cross-section shown in Fig. 5.8) asshown in Fig. 5.9. From these nodes, the nodes for the circular portion of the endmill are generated. This results in nodes for one quarter of the four flute end mill,and to obtain its finite element model these nodes are filled with six-node triangularprism elements. These elements are selected because they take less computer time,CHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 106X(mm)	 Y(mm)6.99	-5.484.64	-8.482.18	-10.59-.18	-11.55-2.66	-11.62-5.36	-11.51-7.83	-9.99-6.85	-9.15-5.37	-7.88-2.36	-6.610.755	-6.243.87	-5.86-5.48	-6.99-2.74	-3.490.377	-3.123.49	-2.740.000	 0.000-2.85	-12.37-0.221	-12.692.64	-12.426.15	-11.109.99	-7.839.15	-6.857.88	-5.37NODE3013023033043053063073083093103113123133145 315316317451452453454455456457Figure 5.9: Circular cross-section of a flute at the beginning of the circular section at z=c,Fig. 5.1CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 107Figure 5.10: Finite element model of an end mill with elemental forces appliedand because the stress gradients in the shank are low so they can be modeled withsufficient accuracy using these linear elements. Finally, to obtain the complete finiteelement model of the four flute end mill, new nodes are generated from the nodesalready defined by rotating them by 90° about the end mill axis four times and fillingthem with elements. This results in the finite element model of the complete end millshown in Fig. 5.10. The finite element ANSYS input used to generate this end millmodel is given in Jemal [401 (Appendix E.2). In this model the nodes and elementswere generated manually and the numbering kept systematic in order to simplify thegeneration of elements and to enable identification of their position in the model whenthe boundary conditions are to be applied.The concentrated forces to be applied along the cutting edge can be calculatedfrom the mechanistic model described in Section 5.2.1. For the cutting condition andend mill geometry given in Figs. 5.1 and 5.2, An example of cutting force calculationCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 108is given in Jemal [40] (Appendix E.1). The calculated tangential and radial cuttingedge forces are then applied on the model as indicated in Fig. 5.10 to obtain theshank stresses in the end mill using the finite element approach.ResultsThe finite element results obtained for shank stresses are discussed here. In Section5.2.2 it was shown that for end mills, due to its large axial length to radius ratio, thecontribution of the shear stress towards the maximum principal stress is negligible,and as a result the bending stress distribution is similar to the maximum bendingstress distribution. The finite element solution for the bending stress distribution,az, of the end mill is shown in Fig. 5.11. From this figure, the bending stressesare maximum at the fixed end, point A, and at the flute end close to the circularsection, point B. Thus, this finite element solution predicts shank failure in thesetwo regions.Comparison of the stress results at points A and B of Fig. 5.11 shows thebending stress at the critical flute section is higher than that at the fixed end. Thisis because the effective section modulus of the flute section is much lower than thesection modulus of the circular section at the fixed end. This lower section modulusat the flute section offsets the smaller bending moment at this section and makes itthe most critical point for shank breakage. Another contribution to a higher stressat point B could be stress concentration. In our FEM end mill model the transitionfrom the flute section to the circular section is not continuous and this results inhigher stresses. The relative magnitude of the critical stresses at points A and Bdepends on the effective section modulus of the flute, the section modulus of thecircular section, and the axial length of the circular section. Therefore, in general,it is not possible to say the stress in this region is always higher than in the otherSTRESS RNHLY5JS IN AN ENDMILL.CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 109ANSVS 4.45DEC 21 199123,41,28PLOT NO.	 1POST1 STRESSSTEP=1ITER=1SZ	 (RVGI5 GLOBALDMX =0.083633CNN =-254.94SMN6=-306.178SMX =329.269SMX8=48I .564XV =1CV =0.5ZV =16I31=39.524XF =0.967E-03ZE- =38.115PRECISE HIDDEN-254.94• -190.028-125.116-60.2044.70869.621134.5331111 199.445264.357329.269Figure 5.11: Finite element solution for end mill bending stress distributionCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 110region. So in determining shank stresses both the critical regions shown in Fig. 5.11must be checked.5.2.4 Discussions and conclusionsIn the previous sections analytical critical shank stresses for an end mill were deter-mined, and for the same cutting condition and tool geometry finite element bendingstress distribution were obtained. The analytical results are summarized in Tables 5.1and 5.2, and the finite element solution is shown in Fig. 5.11. Both of the analyticalresults lie within the range of values corresponding to the red coloured regions of thefinite element solution and therefore they are in good agreement. Calculations showthat the difference in the critical stresses between the two solutions are less than9%, and this verifies the applicability of the analytical approach presented in Section5.2.2. Improved solution could be obtained by reducing the element size used in thefinite element analysis. However, this lead to the need for larger computer memoryand time so a compromise element size was taken.The analysis above gives the location for shank breakage of end mills. This resultcould be compared with common types of end mill shank breakage observed duringthe manufacture of turbine components by Bouse [33] of General Electric Companywhich is shown in Fig. 5.12. Comparison of this figure with Fig. 5.11 shows thesolutions obtained in this study correctly predict the location of end mill shank in-service breakages.From the results obtained, the allowable cutting force Fail corresponding to theallowable tensile stress craii of the tool material could be obtained from the linearforce-stress relationship. This relation gives Fat = (F / cr)aaii, where F and a are theresultant cutting force and critical tensile stress relations obtained in this study forCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 111Figure 5.12: Common types of end mill shank breakages, Bouse [33]CHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 112the cutting condition and tool geometry considered. This allowable force gives theoptimum force for maximum production without the danger of tool shank breakage.5.3 End mill flute stressesThe general objective of this study to investigate how far the analytical solutions cango in determining the critical end mill stresses. In the previous sections, an analyticalapproach was presented which was used to determine shank stresses, and it was foundthat the solutions obtained are in reasonable agreement with numerical results. Thesecond critical region for an end mill failure is its flute. In this section end mill flutestresses will be considered.Comparison will be made between the two-dimensional cutting tool stress solutionsalready obtained in Section 4.3 and finite element solution of a. straight end mill flutefor the same wedge angle of 62° and boundary load distribution shown in Fig. 4.2. Thesimple straight flute is selected because it is the most likely end mill flute geometrywhere its stress distribution could be approximated by the two-dimensional solution.If the two solutions are similar then helical end mills will be considered, but if thesolutions are different then it can be concluded that end mill flute stresses can not bedetermined from two dimensional solutions and their solutions have to be obtainedusing three-dimensional numerical methods.The same procedure as described in Section 5.2.3 was used to develop the finiteelement model of the flute shown in Fig. 5.13. Since in flute stress analysis the concernis for local stresses, only two flutes of a four flute end mill near the loaded region wasconsidered. The ANSYS input used in generating the model and determining thestress distribution is given in Jemal [40] (Appendix F.1). The boundary loads wereapplied on face ABC D where AB is the active cutting edge (also equal to the axialCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 113depth of cut), while AD and BC are equal to the chip-tool contact length. Theapplied distributed loads along any plane perpendicular to the cutting edge AB areidentical and are as shown in Fig. 4.2. The chip-tool contact length for this boundaryload distribution is 1 mm. This length denoted by BC and AD in Fig 5.13 is dividedinto 20 elements of length 0.05 mm. The length of the elements along the cuttingedge is 0.5 mm. The boundary normal stresses were calculated at each of the elementsalong the contact length. The normal stresses were applied directly as pressures onthe element faces. The shear stresses at each element was first converted to forces bymultiplying them by the element area (0.025 mm2 in this case) and they were appliedat the nodes. Very small elements were used to accurately represent the distributedshear and normal boundary load distribution shown in Fig. 4.2.Fig. 5.13 also shows the minimum principal stress distribution in the loaded re-gion. This figure clearly shows the increase in the magnitude of the minimum principalstress on the flank and rake faces when moving towards the cutting edge. The min-imum compressive stress value given in this figure agrees with the two-dimensionalsolution given in Fig. 4.7. Therefore, the critical cutting edge stress for straight endmill flutes can be determined from two-dimensional solutions. It is also interestingto see how the stress distribution is uniform along the cutting edge, and how thestress variations are localized to the loaded region alone. This observation verifies thevalidity of considering only a portion of the end mill in flute stress analysis.The rake face maximum principal stress distribution of the flute model is comparedwith the two-dimensional result in Fig. 5.14. From this figure the critical maximumprincipal stress in the end mill flute is less (by 35%) than that in the two-dimensionalcutter. Therefore, two-dimensional solutions cannot be used to accurately determinethe maximum principal stress in the loaded region of an end mill flute. However, theyCHAPTER 5. STRESS ANALYSIS OF AN END MILL	 114qNSYS 4.49MHR 3 199216,55.27PLOT ND.	 1POST! STRESSSTEP=1ITER=18103 (AVG.OMX =0.832E-03SMN =-149.308SMN8=-174.818SAX =6.266SW03=27.538XV =ITV =1ZV =I*DEST=3.087*XF =-2.756*YF =12.916*ZF =1.189FACEHIDDEN-149.308-132.022-114.736-97.45-80.164-62.878-45.592-28.306-11.026.266Figure 5.13: End mill FEM flute model and its minimum principal stress cr3 distributionfor the boundary load distribution shown in Fig. 4.27550250.0Two-dimensional wedgeEnd mill fluteI I	 11111111111111 CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 1150.5	 1.0	 1.5	 20Rake distance, mmFigure 5.14: Comparison of maximum principal stress distributions 01 for a two dimensionalwedge and an end mill flute for the boundary load distribution shown in Fig. 4.2can give an upper bound value as shown in Fig. 5.14. Fig. 5.14 also indicates thatin both two-dimensional wedge and end mill flute, the maximum principal stressesreach their peak at the end of chip-tool contact (in this case at lc = 1 mm).The reason for the difference between the end mill flute and the two-dimensionalcutting tool solutions observed in Fig. 5.14 may be explained as follows: In two-dimensional cutting tools the planes perpendicular to the cutting edge (side faces ofthe tool) are free from resisting stresses. On the other hand, in the end mill flute,resisting stresses act on the sides faces and provides support for the cutting edge.This effect results in lower deflection of' the loaded region and lower tensile stress atthe end of chip-tool contact as shown in Fig. 5.14.A finite element solution was also obtained to see the effect of change in axial depthof cut AB on the flute stress distribution. Results of this flute stress distribution forCHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 116the same boundary load distribution, but different axial depth of cut show similarstresses. From this result, flute failures and shank failures could be compared for theend mill geometry given in Fig. 5.1. For the axial depth considered in Fig. 5.13,the maximum tensile stress in the flute is found to be four times the stress at thefixed end. Therefore, with further increase in radial depth of cut, the flute fails firstcompared to the shank. If, however, the axial depth of cut is increased to morethan four times, then the shank stress becomes higher, and in this case with furtherincrease in radial depth of cut the shank fails first. At moderate axial depth of cut,failures at either the shank or flute is possible.The results from the two methods for the maximum principal stresses as shownabove are different even for the simple straight end mill, and therefore when thesestresses are required in the complex helical end mill a three-dimensional numericalsolution has to be used.5.4 End mill cutting edge stressesIn the previous section stresses for straight flute end mill were considered. Theseresults show two dimensional solutions can be used to determine cutting edge stressesin straight end mill flutes. To complete the investigation for the possible use ofanalytical solutions in end mill stress analysis, the effect of the shape of the cuttingedge on the stress distribution is considered in this section. For this analytical two-dimensional cutting edge stresses will be compared with numerical stresses for acurved cutting edge.III 111111111_100Evs^ —50 —CUV)(4- 25 —i411111111111111 0.0	 0.25	 0.50 0.75 1.00CHAPTER 5. STRESS ANALYSIS OF AN END MILL	117Rake distance, mmFigure 5.15: Boundary stress distribution considered for cutting edge stress analysis5.4.1 End mill cutting edge stresses using elasticityIn this section the analytical equations for cutting edge stresses will be discussed.Then as an example, this equations will be used to determine the cutting edge stressesfor the boundary load distribution shown in Fig. 5.15 and a wedge angle of a standard1 inch end mill. The analytical solution to be found will then be compared with finiteelement solutions for the same boundary load distribution and wedge angle in orderto verify the applicability of the analytical solutions in cutting edge stress analysis.Analytical solution for the cutting edge stresses are discussed in Section 2.5.1.The solution for the radial rake face stress is given by Eq. (2.40) and isa	 tan cx(cg tan a + 1) —— 	  aoi 	1„01tan a — a i	 (5.7)(tan a — a) tan awhere,= radial stress parallel to the rake faceCHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 118a = wedge angle, radianslo-01 = magnitude of normal stress at the cutting edge on the rake faceNI = magnitude of the shear stress at the cutting edge on the rake faceThe wedge angle of the end mill to be analyzed is a = 76° and it is measured froma standard 1 inch diameter end mill. The boundary stress distribution to be appliedon the tool is the high speed photoelastic boundary (Section 2.3.2) shown in Fig.5.15. For the wedge angle considered, from Eq. (5.7), the radial stress at the cuttingedge is given by ar = .51501 — 2.217-01. Once this stress component is determined, theprincipal stresses can be determined using Eq. (3.24). These stress results at thecutting edge are summarized in Table 5.3.Table 5.3: Cutting edge stresses for the boundaryof Fig. 5.15 and wedge angle of 76°, (kg/mm2)Ur U2 U3 SI = 0-1 - 0-3-12 -4 -113 113In Table 5.3 since both the principal stresses in the cutting tool plane are negative,the maximum principal stress for the plane stress problem ui = 0 was taken.Examination of Eq. (5.7) shows that the cutting edge stresses are functions ofthe applied normal and shear stress at the cutting edge and the wedge angle only.The magnitude of the shear stress at the cutting edge on the rake face is close to theyield shear strength of the workpiece material being machined and is independentof the cutting conditions. The normal stress at the cutting edge on the rake faceis approximately related to the shear stress at the cutting edge. This relation byLoladze [37] is given by Eq. (2.25) in Section (2.3.3) which iscro , 2k(1/2 + 7/4 — 7)	 (5.8)CHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 119where k is the yield shear strength of the workpiece material and -y is the rake an-gle of the tool. Since the rake angle is directly related to the wedge angle for agiven clearance angle, the applied normal stress is a function of only the yield shearstrength of the workpiece material and the wedge angle. This means that for thesame work material and wedge angle, a change in cutting conditions or change intool geometry (meaning change in cutting forces) do not result in a change in cuttingedge stresses. For the same work material and tool wedge angle, a change in cuttingconditions results in a change in chip-tool contact length or a change in the shapeof the boundary load distribution while the boundary stresses at the cutting edgeremain approximately constant.5.4.2 End mill cutting edge stresses using FEMIn the previous section cutting edge stresses using elasticity solutions were considered.In this section the elasticity solutions obtained will be verified with finite elementresults for the same boundary load distribution shown in Fig. 5.15 and wedge angleof 76°.The finite element results shown in Fig. 5.13 show that the flute stresses, andmore so the cutting edge stresses, are localized and therefore when stresses in theseregions are required, only a portion of the end mill flute near the loaded region needbe considered for the finite element model.The same procedure as described in Section 5.2.3 was used to develop the finiteelement model of the flute shown in Fig. 5.16, which is a portion of the ball end of theend mill shown schematically in Fig. 5.17. In this model because of the steep stressgradient in the loaded region finer elements were used (an average area of 0.0125mm2). The ANSYS input used in generating the model and determining the stressSTRESS RNRLYSIS IN RN ENDMILL.CHAPTER 5. STRESS ANALYSIS OF AN END MILL	120RNSYS 4.4RNOV 22 199110.00.25PLOT NO.	 1.POST! STRESSSTEP=1ITER=15163 IRVG1DMX =0.302E-03SMN =-117.428SMN8=-130.489SMX =19.225SMX8=26.562XV =0.5IV =0.5ZU =1DIST=1.666XF =1.15YE =-1ZF =-1.072CENTROID HIDDEN-117.428-102.244-87.061-71.877-56.693-41.51-26.326-11.1424.04219.225Figure 5.16: Finite element solution for the minimum principal stress a3 at the cutting edgeCHAPTER 5. STRESS ANALYSIS OF AN END MILL	1214 A 0 B =---- 4 5 . 6 °A-15 .--- 1'--E z-- 2OA =33 units in mmFigure 5.17: Schematic diagram showing the portion of the ball end mill considered forcutting edge FEM modeldistribution is given in Jemal [40] (Appendix F.2). The boundary loads were appliedon face ABC D where AB is the circular cutting edge and AD and BC are equal tothe chip-tool contact length. The applied distributed loads along any radial planeperpendicular to the cutting edge AB are taken to be identical and are shown in Fig.5.15. The same procedure were used to apply the boundary load distributions on theelement faces as described in the straight flute finite element stress analysis.Fig. 5.16 also shows the minimum principal stress distribution near the cuttingedge. The minimum principal stress occurs at the cutting edge and has the valueO3 = —117 kg/mm2. The other stress which is responsible for end mill cutting edgefailure is the maximum shear stress. Twice the maximum shear stress distributionclose to the cutting edge is shown in Fig. 5.18. From this figure the maximum shearstress in the loaded region occurs at the cutting edge and twice the maximum stressCHAPTER 5. STRESS ANALYSIS OF AN END MILL 	 122has the value 27-„,az = S/ = 112 kg/mm2A comparison of the above numerical solutions with analytical results given inTable 5.3 show less than 4% difference and therefore they are in very good agreement.Thus, the analytical solution can be used to determine cutting edge stresses, and endmill cutting edge failures could be explained by the two-dimensional cutting toolfailure analysis discussed in Section 4.5.The rake face maximum principal stress distribution of the ball end model is com-pared with the two-dimensional result in Fig. 5.19. From this figure the maximumprincipal stress in the ball end is much less than the two-dimensional solution. There-fore, two-dimensional solutions cannot be used to accurately determine the maximumprincipal stress in the loaded region of the ball end of an end mill flute. However,they can give an upper bound estimate as shown in Fig. 5.19.5.5 ConclusionsAn analytical approach for shank stress was presented and the solution obtained wascompared with finite element solution. Good agreement was obtained. Therefore,shank stresses can be determined using analytical solutions. The critical regions forshank failures obtained from these solutions were also found to reasonably predictthe location of end mill shank failures observed in practice.The possible use of two-dimensional solutions to determine maximum principalstresses in an end mill flute was studied. It was found that for the same load distri-bution and wedge angle, the maximum stress in the two-dimensional cutting tool ismuch higher than that in the end mill flute. Therefore, when accurate stress predici-tion near the end of chip-tool contact of an end mill is required, a three-dimensionalnumerical solution should be employed. If only approximate values are required,1ISTRESS RNRLYSIS IN RN ENDMILL.CHAPTER 5. STRESS ANALYSIS OF AN END MILL	 123RNSYS 4.4RNOV 22 19919.58,10PLOT NO.	 1POST1 STRESSSTEP=1ITER=1SI	 (RVG)DMX =0.302E-03TIN =4.449SMNB=2.242SMX =111.757SMX8=122.14XV =0.5YV =0.5ZV =1DIST=1.668XF =1.15yF =-1ZF =-1.072CENTROID HIDDEN4.449•16.37228.29540.21852.14164.06475.987•87.911ME 99.834111.757INMNFigure 5.18: Finite element solution for the maximum shear stress at the cutting edgeCHAPTER 5. STRESS ANALYSIS OF AN END MILL	124 11111111111150 two-dimensional wedgeball end0.011.1 0.25	 0.50	 0.75	3.10Rake distance, mmFigure 5.19: Comparison of maximum principal stress distributions (71 for a two-dimensionalwedge and a ball end millhowever, then the two-dimensional analytical solution may be used to provide upperbound estimates.The second critical region of an end mill flute is its cutting edge. Analytical andfinite element cutting edge stress solutions were compared, and it was found thatthere is good agreement. Therefore, end mill cutting edge stresses can be determinedusing analytical solutions.Chapter 6Concluding Remarks6.1 SummaryIn this study, the stresses within cutting tools (orthogonal cutting tool and an endmill) were analyzed. The stress distributions from the analytical solutions were com-pared with numerical solutions in order to investigate the possible use of analyticalsolutions in cutting tool stress analysis. The analytical and numerical results obtainedfor the location and modes of tool failure were also compared with observed cuttingtool in-service failure in order to see the correlation between predictions and actualcutting tool failures.6.2 ConclusionsFor accurate determination of the stress distribution in a cutting tool, the actualmechanical and thermal boundary loads are required. Because of the complexity ofthe cutting process, the knowledge of the boundary load distributions and thereforethe stress distribution in the cutting tool is approximate. In this study stresses dueto mechanical loads only are considered and these stresses can be superposed on the125CHAPTER 6. CONCLUDING REMARKS 	 126thermal stresses to obtain the overall stress distribution in the cutting tool.From the results obtained in the previous chapters, the following conclusions canbe made• Previous analytical stress distribution results near the end of chip tool contactwere shown to be incorrect. This was found to be due to the direct use of theinfinite wedge solution to determine stresses in the loaded region of the cuttingtool.• An analytical stress distribution in an orthogonal cutting tool which agrees withnumerical and previous experimental stress distribution results was obtained.This was done by satisfying the boundary condition beyond the loaded region.• Analytical stresses at the critical shank sections of an end mill were found toagree well with numerical solutions. This indicates the applicability of analyticalmethods in shank stress analysis.• Comparison of two-dimensional stress distribution with numerical solution ofan end mill flute show good agreement at the cutting edge, but at the end ofchip-tool contact the two-dimensional solution gave a significantly higher criticalmaximum principal stress.• The analytical and numerical prediction of cutting tool (orthogonal cutting tooland end mill) failure locations and modes of failure were found to correlate wellwith observed cutting tool in-service failure.From the results obtained in the previous chapters, the following conclusions can bemade of the stress distributions and deformations in cutting toolsCHAPTER 6. CONCLUDING REMARKS	 127• During cutting, the loaded region of a cutting tool deforms, causing shorteningof the flank face close to the cutting edge and bending of the rake face at a pointnear the end of chip-tool contact.• The shortening of the flank face results in the critical minimum principal stressto be in this face, close to the cutting edge. The rake face bending about thepoint near the end of chip-tool contact results in the critical maximum principalstress to be at this point. Within the cutting tool the principal stresses arewithin these two extreme values, increasing algebraically from the flank face tothe rake face.• The magnitude of the critical compressive stress is much higher than the criticaltensile stress. Since the maximum shear stresses on free surfaces (at the criticalpoints) is proportional to the non zero principal stress, the critical maximumshear stress occurs at the flank face near the cutting edge.• With increase in cutting forces the critical maximum principal stress resultsin initiation of cracks near the end of chip-tool contact and final fracture ofthe whole loaded region of the cutting tool. The critical minimum principalstress results in permanent deformation of the cutting edge, while the criticalmaximum shear stress results in chipping of the cutting edge.• For both low and high speed photoelastic boundary load distributions, for the ge-ometries of the cutting wedge considered, the critical maximum principal stressoccurs at the end of chip-tool contact.• The magnitude of the critical maximum principal stress, minimum principal andmaximum shear stress decrease with increase in the wedge angle of the tool.CHAPTER 6. CONCLUDING REMARKS 	 128• When prediction of stresses further from the loaded region of the cutting tool arerequired the cantilever beam equations can be used to determine the maximumprincipal stress which may cause cutting tool fracture.6.3 RecommendationsThe following recommendation are suggested for future work in order get an improvedapproximation of the stress distribution in cutting tools.• The cutting tool stress distribution due to flank loading should be included infuture cutting tool stress analysis. In this study the cutting tool was assumed tobe sharp with no flank face loads. The stresses due to flank face loads could beeasily added by using the same procedure used for the rake face loads and super-posing the results. The stress distributions due to the flank and rake face loadshave opposing effects and therefore when the flank face stresses are included themagnitude of the stresses should be lower. Thus the stress distribution obtainedin this study is the worse case. However, to get an improved solution the stressdistribution due to flank loading should also be considered. Unfortunately, ex-perimental flank face boundary load distribution data are sparse and thereforethe shape of the distribution must be assumed to be similar to the rake faceboundary load distribution but with smaller average loads and smaller contactlength.• The cutting tool stress distribution due to thermal effects should be included infuture cutting tool stress analysis. In this study the temperature effects werenot considered. The temperature effects should be considered in order to explainfailures due to thermal effects observed in cutting tools at higher cutting speeds.CHAPTER 6. CONCLUDING REMARKS	 129To accomplish this, the boundary temperature distribution on rake and flankfaces must be known or assumed. The temperature distribution along thesefaces can be estimated from the inserted thermocouple technique (Chao [30],Kusters [35]) or using the metallographic method (Smart at el. [31]) where thetemperature distributions are deduced from the structural change of the highspeed tool. Based on this temperature distribution, the thermal stresses could beestimated and then superposed on the mechanical stress distribution to enableprediction of cutting tool failure and stress levels at higher cutting speeds.• Future cutting tool stress analysis results should be verified experimentally. Inthis study, the conclusions reached for cutting tool failures was found to correlatewell with observed in-service failures of cutting tools. However, to quantitativelypredict tool failures it is necessary to compare the value of the predicted criticalstresses at cutting tool failure with the strength of the cutting tool material.For this analysis, the cutting tool strength at the temperature that the toolexperiences, as determined by Kreimer [39], should be used. It will also beimportant to compare the predicted location for initiation of cracks on the rakeface with the location observed from actual cutting tool failures. Finally, whendoing this analysis it is important to note, as discussed by Shaw [38], that thevariability in the strength of cutting tool materials is large and therefore theagreement between the predicted critical stress values and the tool materialstrength might not be that close.Bibliography[1] M. M. Frocht, Photoelasticity, John Wiley & Sons, Inc., Vol. I & II, 1948.[2] S. Kaldor, A Common Denominator for Optimal Cutting Tool Geometry, Annalsof the CIRP, Vol. 35, No. 1, pp. 41-44, 1986.[3] H. Chandrasekaran, Tool Fracture Model for Peripheral Milling, NAMRC, Vol.10, pp. 15-20, 1982.[4] J. H. Michell, proc. London Math. Soc., Vol. 31, p. 100, 1899.[5] S. P. Timoshenko and J. N. Goodier, Theory of Elasticity, McGraw-Hill, 1982.[6] F. R. Archibald, Analysis of the Stresses in a Cutting Tool, Trans. ASME, pp.1149-1154, August 1956.[7] A. I. Betaneli, A Method of Calculating Tool Strength, Russian Engineering Jour-nal, Vol. 45, pp. 71-75, 1965.[8] R. D. Cook, D. S. Malkus and M. E. Plesha, Concepts and Applications of FiniteElement Analysis, John Wiley & Sons, 1989.[9] C. J. Tranter, The Use of the Mellin Transform in Finding the Stress Distributionin an Infinite Wedge, Q. J1. Mech. appl. Math., Vol. 1, p. 125, 1948.130BIBLIOGRAPHY	 131[10] P. F. Thomason, The Stress Distribution in a Wedge-Shaped Metal-Cutting toolfor Triangular Distributions of Load on the Rake-Face Contact Length, Journalof Mechanical Engineering Science, Vol. 16, No. 6, pp. 418-424, 1974.[11] G. B. Airy, Brit. Assoc. Advan. Sci. Rept., 1862.[12] M. C. Shaw, Metal Cutting Principles, Clarendon Press, Oxford, pp. 125-129,1984.[13] E. M. Trent, Metal Cutting, Butterworths, 1977.[14] G. Boothroyd, Fundamentals of Metal Machining, Edward Arnold Ltd., 1965.[15] S. Kato, K. Yamaguchi and M. Yamada, Stress Distribution at the InterfaceBetween Tool and Chip in Machining, Journal of Engineering for Industry, Trans.ASME, pp. 683-689, May 1972.[16] G. Barrow, W. Graham, T. Kurimoto and Y. F. Leong, Determination of RakeFace Stress Distribution in Orthogonal Machining, Int. J. Mach. Tool Des. Res.,Vol. 22, No. 1, pp 75-85. 1982.[17] T. H. C. Childs and M. I. Mandi, On the Stress Distribution Between the Chipand Tool During Metal Turning, Annals of the CIRP, Vol. 38, No. 1, pp. 55-58,1989.[18] E. Usui, A Photoelastic Analysis of Machining Stresses, Journal of Engineeringfor Industry, Trans. of ASME, pp. 303-308, November 1960.BIBLIOGRAPHY	 132[19] H. Chandrasekaran and D. V. Kapoor, Photoelastic Analysis of Tool-Chip Inter-face Stresses, Journal of Engineering for Industry, Trans. of ASME, pp. 495-502,November 1965.[20] E. Amini, Photoelastic Analysis of Stresses and Forces in Steady Cutting, Journalof Strain Analysis, Vol. 3, No. 3, pp 206-213, 1968.[21] M. M. Ahmad, R. T. Derricott and W. A. Draper, A Photoelastic Analysis ofthe Stresses in Double Rake Cutting Tools, Int. J. Mach. Tools Manufact., Vol.29, No. 2, pp. 185-195, 1989.[22] T. N. Loladze, Problems of Determining Stresses at Cutting Edge of a Tool,Proceedings of the Seventh Congress on Theoretical and Applied Mechanics,Bombay, India, pp. 323-348, 1961.[23] W. Johnson and P. B. Mellor, Plasticity for Mechanical Engineers, D. Van Nos-trand Company Ltd., pp 259-264, 1962.[24] N. N. Zorev, Metal Cutting Mechanics, Pergamon Press, p. 73, 1966.[25] ANSYS, Engineering Analysis Systems Manual, Vol. I & II, Swanson AnalysisSystems, Inc., P. 0. Box 65, Houston.[26] M. E. Merchant, Journal of Applied Physics, Vol. 16, No. 5, pp. 267-324,May 1945.[27] J. Tlusty, Chipping and Breakage of Carbide Tools, Journal of Engineering forIndustry, Vol. 100, No. 4, pp. 403-412, 1978.BIBLIOGRAPHY	 133[28] W. A. Kline, R. E. DeVor, and J. R. Lindberg, The prediction of cutting forcesin End Milling with Application to cornering Cuts, Int. J. Mach. Tool Des. Res.,Vol. 22, No. 1, pp. 7-22, 1982.[29] L. Kops and D. T. Vo, Determination of The Equivalent Diameter of an EndMill Based on its Compliance, Annals of the CIRP, Vol. 39, No. 1, pp. 93-96,1990.[30] B. T. Chao and K. J. Trigger, Temperature Distribution at the Tool-Chip Inter-face in Metal Cutting, Journal of Engineering for Industry, Trans. of ASME, pp.139-151, May 1959.[31] E. F. Smart and E. M. Trent, Int. J. Prod. Res., Vol. 13, No. 3, p. 265, 1975.[32] William H. Press et al., Numerical Recipes, The Art of Scientific Computing(Fortran Version), Cambridge University Press, p. 31, 1986.[33] G. K. Bouse, Metallurgical Investigation of Several High Speed Tool Steel Failures,Cutting Tool Materials. American Society for Metals. pp. 77-92, 1981.[34] E. M. Trent, Proc. Int. Conf. M.T.D.R, Manchester, p. 629, 1967.[35] K. J. Kusters, Industrie Anzeiger, Vol. 89, p. 1337, 1956.[36] P. K. Wright and E. M. Trent, Metals Technology, Vol. 1, No. 13, 1974.[37] T. N. Loladze, Requirements of Tool Materials, Proceedings of the 9th MTDRConference, Pergamon Press, Manchester, England, 1968.[38] M. C. Shaw and T. C. Ramaraj, Brittle fracture of cutting tools, Annals of theCIRP Vol. 38, No. 1, 1989.BIBLIOGRAPHY	134[39] G. S. Kreimer, Strength of Hard Alloys, Scientific research Institute for HardAlloys, Moscow, translated from Russian, Consultants Bureau, New York, 1968.[40] G. Jemal, Investigation of Stresses in Cutting Tools, University of BritishColumbia, Department of Mechanical Engineering, Stress Analysis and Biome-chanics Laboratory (SABIL) Report 92-1, April, 1992.

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