modelling and simulation of machining processes

Arch Comput Methods Eng (2007) 14: 173204DOI 10.1007/s11831-007-9005-7

Modelling and Simulation of Machining ProcessesM. Vaz Jr. D.R.J. Owen V. Kalhori M. Lundblad L.-E. Lindgren

Published online: 8 June 2007 CIMNE, Barcelona, Spain 2007

Abstract The modelling of metal cutting has proved to beparticularly complex due to the diversity of physical phe-nomena involved, including thermo-mechanical coupling,contact/friction and material failure. The present work out-lines the wide range of complex physical phenomena in-volved in the chip formation in a descriptive manner. Inorder to improve and understand the process different nu-merical strategies have been used for simulation. Several ofthese numerical strategies are reviewed and a short discus-sion of their relative merits and drawbacks is presented. Bymeans of several examples, where a combined experimen-tal/numerical effort was undertaken, we try to illustrate whatnumerical techniques, models and pertinent parameters areneeded for successful simulations.

M. Vaz Jr. ()Department of Mechanical Engineering, State University of SantaCatarina, 89223-100 Joinville, Brazile-mail: [email protected]

D.R.J. OwenCivil and Computational Engineering Centre, University of WalesSwansea, SA2 8PP Swansea, UKe-mail: [email protected]

V. Kalhori M. LundbladSandvik Coromant, USE-811 81 Sandviken, Sweden

V. Kalhorie-mail: [email protected]

M. Lundblade-mail: [email protected]

L.-E. LindgrenDivision of Material Mechanics, Lule University of Technology,S-971 87 Lule, Swedene-mail: [email protected]

1 Introduction

Metal cutting is a process in which, by action of a cuttingedge (or edges) of a tool, unnecessary material is removed. Itis one of the most common manufacturing processes for pro-ducing parts and obtaining specified geometrical dimensionsand surface finish. Turning, drilling, and milling are exam-ples of different industrial applications that use this principlewith different geometry and number of cutting edges. Nev-ertheless, in the current work, the analysis is restricted tomodels that describe the local behaviour due to one cuttingedge. These models can be used to increase the knowledgeof the cutting process and improve it. Understanding of thematerial removal process in metal cutting is important in se-lecting tool material and design and in assuring consistentdimensional accuracy and surface integrity of the finishedproduct.

Although mechanical cutting is one of the most wide-spread processes, the modelling and simulation of this phe-nomenon is by no means trivial. It has proved to be particu-larly complex due to the diversity of physical phenomena in-volved, including large elasto-plastic deformation, compli-cated contact/friction conditions, thermo-mechanical cou-pling and chip separation mechanisms. One factor that hascaused considerable difficulty and frustration to researchersinvestigating the chip formation is the fact that the phenom-ena occurring in the vicinity of the cutting edge are highlylocalized and not directly observable. After more than a hun-dred years of research, the study of metal machining stillconstitutes a current challenging task. Although the first the-oretical models were able to describe the problem only qual-itatively, their principles and assumptions laid foundationsfor further advancements. The main shortcoming of exist-ing solutions for chip formation mechanisms perhaps lies in

174 M. Vaz Jr. et al.

the oversimplification and consequent disregard of the com-plex interplay of the different parameters. Numerical simu-lation by the finite element method has proven to be a reli-able alternative to analyse several metal forming operations.Nevertheless, only recently, with the advent of high-speedcomputers and robust large-strain/large-displacement proce-dures, contact/fracture algorithms, adaptive re-meshing pro-cedures for inelastic problems and robust finite/discrete al-gorithms, have numerical simulations of forming operations,which involve material removal become possible.

The purpose of the current paper is to illustrate whatnumerical techniques, models and pertinent parameters areneeded for successful simulations. The presentation is orga-nized as follows. After this introduction, Sect. 2 presents ashort review of books dedicated to the subject. The chip for-mation fundamentals based on mechanical, thermal and tri-bological principles are presented in Sect. 3 in a descriptivemanner. In Sect. 4, individual aspects of simulation tech-niques and numerical strategies are described. Numericaland experimental examples are provided in Sect. 5, that il-lustrate the performance of the formulations used in a vari-ety of conditions. Finally, in Sect. 6, some conclusions arepresented and discussion provided on the need for improve-ments.

2 Further Reading

Several books have been written about machining processesand manufacturing productivity. They are mainly concernedwith how mechanical and materials engineering sciencecan be applied to understand the process and support fu-ture developments. Metal Cutting Principles by M.C. Shaw(1984) [1] is one of the first books to be published on thesubject and carefully discusses, in a descriptive way, the dif-ferent parameters that influence the chip formation mech-anism. Nevertheless, there have been many developmentssince this first publication. Metal Cutting by E.M. Trent(1991) [2] is another major work, but written more fromthe point of view of a materials engineer. Fundamentalsof Machining and Machine Tools by G. Boothroyd andW.A. Knight (1989) [3] covers mechanical and productionengineering perspectives. Metal Machining: Theory and Ap-plications by T.H.C. Childs et al. (2000) [4] provides a dis-cussion of the theory and application of metal machining.The underlying mechanics are analysed and discussion ofsimulation results and process control is undertaken.

3 Chip Formation Fundamentals

Cutting processes involve a wide range of physical phenom-ena. This section introduces the mechanical, thermal and tri-bological principles on which understanding of the process

is based. First, Sect. 3.1 reports the types of chips that can beformed, depending on the material and cutting conditions.Section 3.2 describes the main regions of plastic flow anddiscusses how the dissipation of the inelastic work gener-ates energy resulting in a subsequent temperature rise. Theforces generated on a tool during cutting are described inSect. 3.3. Section 3.4 emphasizes the importance of know-ing how much of the heat generated is convected into thechip and how the cutting speed affects the heat transfer. Therole of friction along the toolchip contact is of paramountimportance to the cutting process. Section 3.5 draws atten-tion to the interdependence of different phenomena that takeplace at the toolchip interface.

3.1 Chip Geometry and Influencing Factors

The type of chip produced can characterize the various cut-ting processes. Although there exist many individual typesor combinations thereof, a general classification is widelyaccepted today. In general, the chips are classified as dis-continuous, continuous, continuous with built-up-edge andshear-localized, as shown in Fig. 1. The discontinuous chipis commonly observed when brittle materials are cut at lowcutting speeds. The chip separation mechanisms have notbeen fully explained and several factors are said to affect theprocess. The phenomenon is frequently described throughplastic shear strain, shear stress and shear instability models.The continuous chip is commonly produced when cuttingductile materials and the operation can be regarded as steadystate. However, long continuous chips cause handling andremoval problems in practical operations. Under conditionsof low cutting speeds where the friction between the chipand the rake face of the tool is high the chip may weld ontothe tool face. This accumulation of chip material is knownas a built-up edge (BUE).

Finally, the last type of chips are macroscopically contin-uous chips consisting of narrow bands of heavily deformedmaterial alternating with larger regions of relatively unde-formed material. These shear-localized chips can be formedwhen the yield strength of the workpiece decreases withtemperature. Under the proper conditions, rapidly heatedmaterial in a narrow band in front of the tool can becomemuch weaker than the surrounding material, leading to lo-calized deformation. This type of chip is obtained when cut-ting hardened and stainless steels and titanium alloys at highcutting speeds.

The characteristics of crack formation have a significantinfluence on the chip formation pattern as the cutting processinvolves the separation of a chip from the workpiece. Fur-thermore, Atkins (1985) [5] postulates that fracture is in-herent in material removal processes, including continuouschip formation. When the cutting tool movement towardsthe workpiece starts, the stress concentration in front of the

Modelling and Simulation of Machining Processes 175Fig. 1 Four basic types ofchips: (a) discontinuous, (b)continuous, (c) continuous withbuilt-up edge (BUE), (d) shearlocalized

Fig. 2 Cutting tool starting toadvance into the workpiece

cutting edge is increased [6] (see Fig. 2(a)). When this stressreaches a certain maximum limit, the following may happen:

I. If the workpiece material is brittle, then a crack appearsin front of the cutting edge, which finally causes fracture,Fig. 2(b).

II. If the workpiece material is ductile, then a certain elasto-plastic zone forms in the workpiece, Fig. 2(c). The di-mensions of the plastic and elastic parts of this zone de-pend on the ductility of the workpiece material.

3.2 Process Zones

The major deformations during the machining process areconcentrated in two regions close to the cutting tool edge.These regions are usually called the primary and the sec-ondary deformation zones, Fig. 3. The primary deformationregion extends from the tip of the cutting tool to the junctionbetween the undeformed work material and the deformedchip. The workpiece is subjected to large deformation at ahigh strain rate in this region. The heating is due to energydissipation from the plastic deformation. At the secondarydeformation zone, heat is generated due to the plastic defor-mation and friction between the cutting tool and the chip.Figure 4 shows the computed plastic strain rate and heatingrate in these regions. The secondary deformation zone maybe divided into two regions [2, 7], the sticking region and thesliding region, Fig. 3. In the sticking region, the workpiece

Fig. 3 Locations of the primary and secondary deformation zones andthe sliding and sticking regions

material adheres to the tool and shear occurs within the chip.The heat generation per unit volume is large in this regiondue to the highly localized plastic deformation near the sur-face of the chip. The highest temperature usually occurs inthe sliding region close to the sticking region.This will betreated in more detail later in the paper.

The location of fracture onset is of great importance inthe understanding of the physics of chip formation. It isshown in [8] that the limiting stress in the workpiece ma-


Fig. 4 Strain-rate and heatingin deformation zones duringorthogonal cutting of AISI 1045.The cutting speed is 198 m/minand the feed is 0.25 mm

Fig. 5 Zones where the limiting shear stress can occur

terial may occur in one of two regions, Fig. 5. They areregion A, along the surface separating the workpiece andthe layer being removed, and region B, along the surface ofmaximum combined stress. Region A is characterized witha high combined shear and normal stress. The shear stressstems from the direct action of the cutting edge whereas thenormal stress is due to the tearing off of a layer of metalwhen the chip material is forced upwards. The cracks usu-ally start at the surface in region B. However, there may beexceptions as in the cases of cast iron and sintered materials.

3.3 Mechanical Effects

The forces involved in chip formation, in orthogonal cutting,are depicted in Fig. 6. The fracture in the chip formation oc-curs due to the combined bending stress, the component S,and the shearing stress due to compression Q. The pres-ence of the bending stress in the deformation zone distin-guishes the processes of metal cutting from other deformingand separating manufacturing process. The competition be-tween deformation hardening and thermal softening in thedeformation zone constitutes a cyclical character of the chipformation process. As a result, the parameters of the cut-ting system vary over each chip formation cycle. The cuttingforce depends on several parameters, such as the tool angles,feed and cutting speed.

Fig. 6 The interaction between tool rake face and the chip. The pene-tration force P acts on the chip, causing the compressive force Q andthe bending force S

The more general three-dimensional case occurs, for in-stance, in lathe turning. In this case the resultant force hasthree components, Fig. 7. The component of the force act-ing on the rake face of the tool, normal to the cutting edge,in the direction OY is called cutting force Fc. This is usuallythe largest force component, and acts in the direction of thecutting velocity. The force component acting on the tool inthe direction OX, parallel with the direction of feed, is re-ferred to as the feed force Ff . The third component, actingin the OZ direction, pushes the cutting tool away from thework in the radial direction. This is the smallest of the forcecomponents.

The specific work done in cutting, Wc, depends mainlyon two factors: the fracture shear strain and the tempera-ture. The former changes because the shear stress at fractureof the workpiece material depends on the strain. The lattercombined with high strain rates that occur in cutting willaffect the frictional shear stress f and must therefore af-fect Wc.

3.4 Thermal Effects

The effect of temperature on the stressstrain relationshipand fracture properties is well known but difficult to quan-tify. In general the strength of the material decreases andductility increases as the temperature increases. In cutting

Modelling and Simulation of Machining Processes 177

Fig. 7 Cutting forces acting on the tool in a semi-orthogonal cutting

operations the heat transfer is strongly dependent on the cut-ting velocity. At very low cutting speeds there may be ade-quate time for conduction to occur. At the other extreme,at very high cutting speeds there is nearly no time for heatconduction and adiabatic conditions may exist with high lo-cal temperatures in the chip. Zorev (1966) [7] and Shaw(1984) [1] assumed adiabatic conditions. This means thatheat generated in the primary deformation zone and the av-erage temperature T in this region are proportional to thespecific work for metal removal Wc. The increase of tem-perature in the chip is related to the increase of plastic de-formation and thereby Wc. The average temperature can beestimated by

T = Wcc

+ T0 (1)

where is the density of workpiece material, c is the spe-cific heat and T0 is the temperature prior to deformation. Thecomputed heat generation due to plastic dissipation and fric-tion is confirmed in the numerical analysis shown in Fig. 4.

3.5 ToolChip Interface

Friction along the toolchip contact interface, during thecutting process, is a very complex phenomenon. It influ-ences the chip geometry, built-up edge formation, cuttingtemperature and tool wear. Therefore it is necessary to un-derstand the friction mechanism across the faces and aroundthe edge of the tool, in order to be able to develop accuratemodels for cutting forces and temperature. The most simplefriction model is Coulomb friction,

= (2)

where is the frictional shear stress and is the normalstress to the surface. Usually the friction coefficient is as-sumed to be constant for a given interface. There exist ad-vanced models that are more relevant for the cutting processwhere rate, pressure and temperature dependency are ac-counted for. However, it is not possible to perform directmeasurements of these for the extreme conditions that existin the contact region.

4 Numerical Simulation of Metal Machining

The importance of the subject is evinced by the increasingnumber of research works which have been published in thelast thirty five years since the pioneering studies by Usuiand Shirakashi (1974, 1982) [9, 10] and Klamecki (1973)[11]. A review of modelling methods, including early nu-merical works, discussed by Ehmann et al. (1997) [12], andthe extensive bibliography presented by Mackerle (1999,2003) [13, 14] further highlight the efforts being made todevelop new approaches to solve this class of problems.In this section, a discussion on the application of numeri-cal models to metal machining is presented, which includesmechanical and thermo-mechanical simulations. Individualaspects of simulation techniques and numerical strategiesare presented, such as solution methods, constitutive mod-els, thermo-mechanical coupling strategies, time integrationschemes, chip morphology, friction models, element tech-nology, mesh and re-meshing procedures, contact and frac-ture. The finite element model should incorporate some ofthese numerical strategies in order to accurately simulate thecomplex physical phenomena.

4.1 Solution Methods

Metal cutting simulations have been modelled using up-dated Lagrangian, Eulerian and Arbitrary LagrangianEulerian approaches. Lagrangian formulations assume thatthe finite element mesh is attached to material and followsits deformation, which brings the following advantages tomachining simulation: the chip geometry is the result ofthe simulation and provides simpler schemes to simulatetransient processes and discontinuous chip formation. How-ever, element distortion has been a matter of concern andhas restricted the analysis to incipient chip formation ormachining ductile materials using larger rake angles and/orlow-friction conditions [11, 1533]. Pre-distorted meshes[3451] or re-meshing [5284] have been used to minimizethe problem. An alternative approach to simulate steadystate chip formation using a Lagrangian formulation wasproposed by Usui and Shirakashi (1974, 1982) [9, 10]. Thestrategy, known as iteration convergence scheme, com-putes the chip final geometry and corresponding variables


based on an initial assumption and on a combination of asmall tool advance and an iterative evaluation of the stressfield, velocity distribution and cutting forces [85, 86].

In Eulerian formulations, the mesh is fixed in space andmaterial flows through the element faces allowing largestrains without causing numerical problems. Moreover, thisstrategy is not affected by element distortion and allowssteady state machining to be simulated. However, Eulerianapproaches do not permit element separation or chip break-age and require a proper modelling of the convection termsassociated with the material properties. In addition, suchformulations also require the prior knowledge of the chipgeometry and chiptool contact length, thereby restrictingthe application range. In order to overcome this shortcom-ing, various authors have adopted iterative procedures to ad-just the chip geometry and/or chip/tool contact length [8797].

In an attempt to combine advantages of both Lagrangianand Eulerian formulations, a mixed approach, known as Ar-bitrary LagrangianEulerian formulation (ALE) has beenproposed to model machining operations. This method ap-plies Lagrangian and Eulerian steps sequentially and usesthe so-called operator split, illustrated in Fig. 8. The firststep assumes that the mesh follows the material flow, inwhich a Lagrangian problem is solved for displacements.Subsequently, the reference system is moved (the mesh isrepositioned) and an advection problem is solved (Eulerianstep) for velocities. Despite the fact that ALE methods re-duce the element distortion problem typical of Lagrangianapproaches, a careful numerical treatment of the advectionterms is required. More elaborate discussions on use of ALEformulations in modelling metal machining are presented byRakotomalala et al. (1993) [98], Olovsson et al. (1999) [99],Movahhedy et al. (2000) [100], Benson and Okazawa (2004)[101], Pantal et al. (2004) [102] and Madhavan and Adibi-Sedeh (2005) [103].

4.2 Workpiece and Tool: Constitutive Models and Analysis

A wide range of constitutive models have been employedfor the workpiece, such as rigid-plastic, rigid-viscoplastic,elasto-perfectly-plastic, elasto-plastic and elasto-viscoplas-tic. Elas-to-plastic materials have been the most commonlyused in simulations, with the plastic strain-rate dependencybeing included in some studies [9, 10, 15, 16, 22, 24, 2628,

39, 40, 50, 51, 61, 68]. The elasto-viscoplastic model hasbeen adopted by Shih and others [29, 34, 37, 38, 43, 57].Some authors, aiming at simplifying the analysis, have usedrigid-plastic [59, 65, 87] and rigid-viscoplastic [30, 52, 73,75, 76, 78, 8890, 9396, 99] materials; however, the ther-mal strains and residual stresses could not be evaluated.

Rigid tooling has been assumed by most authors exceptfor Carrol III and Strenkowski (1988) [88], Eldridge et al.(1991) [90], Xie et al. (1994) [17], Kalhori et al. (1997)[104], Monaghan and MacGinley (1999) [58], MacGin-ley and Monaghan (2001) [63], Klocke et al. (2001) [65],Kishawy et al. (2002) [29] and Madhavan and Adibi-Sedeh(2005) [103].

4.3 Thermo-mechanical Coupling

In cutting processes, energy is generated due to the dissipa-tion of both inelastic work and frictional work, being trans-ferred through the workpiece/chip and tool and lost to thesurrounding environment by convection and radiation. Tem-perature rise causes thermal strains and affects the materialproperties. Most finite element approximations use the con-cept of the weak form of the governing equation, as

CT Tn + KT Tn = Q(tn) (3)in which T are the nodal temperatures at time tn, CT andKT are, respectively, the heat capacity and heat conductionmatrices and Q is the heat flux and heat generation due toinelastic deformation.

In adiabatic processes no heat transfer takes place (KT isassumed zero in (3)), i.e., the heat generated due to inelasticdeformation and friction is kept inside the element causingthe temperature to rise [19, 21, 25, 28, 33, 46, 48, 56, 64, 83,88, 90, 104]. The solution of the field equation for the heatconduction is not required when this assumption is made,but this approximation can only be safely adopted for low-diffusivity materials or in high-speed processes. If the cool-ing to room temperature is needed in order to evaluate resid-ual stresses of the workpiece, it is not advisable to adopt thissimplification. One possible technique is to perform a cou-pled thermal-mechanical analysis by the so-called staggeredapproach, as has been used in references [38, 53, 78]. Themost common staggered approach combines a rigid ther-mal step with a mechanical step. This algorithm is condi-tionally stable [106108]. Nevertheless, the results obtained

Fig. 8 The ALE operatorsplit [101]


by such simulations are reliable, provided the time step limitis adhered to.

4.4 Time Integration Schemes

In general, there are two strategies for time integration,namely quasi-static implicit and dynamic explicit schemes.The former requires convergence at every time step or loadincrement and the latter predicts stresses and strains by solv-ing an uncoupled equation system based on informationfrom the previous time steps.

Quasi-static implicit time integration schemes have beenlargely used in metal cutting simulations for either La-grangian or Eulerian formulations, as inertia forces canusually be ignored when compared with the deformationforces. The equilibrium equations and constitutive relationare highly non-linear (both material and geometrically),thereby requiring a proper discretization and solution pro-cedure. In the first case, the NewtonRaphson scheme forthe finite element equations can be generally expressed as

Kn+1(un+1)un+1 = Rn+1(un+1),

Rn+1(un+1) = Fintn+1 Fextn+1 and(4)

un+1 = un + un+1 (5)in which Kn+1 is the tangent stiffness matrix, Fintn+1 andFextn+1 are, respectively, internal and external global forcevectors, un+1 is the vector of unknown incremental dis-placements, and un+1 and un are the current and previoustotal nodal displacements.

The general form of the finite element equations forEulerian formulations is represented as follows:

Kn+1un+1 = Fn+1 (6)where Kn+1 is the stiffness matrix, Fn+1 is the load vectorand un+1 are the nodal velocities. Difficulties dealing withdiscontinuous chip formation and restrictive contact condi-tions are the main drawbacks of implicit schemes. This typeof approach has been used by most authors.

Dynamic explicit time integration schemes have been em-ployed in metal forming problems that involve high non-linearity, complex friction-contact conditions and fragmen-tation. The finite element discretization can be derived fromthe weak form of the mechanical equilibrium equations as

Mun + Cun + Fintn (un) = Fextn (tn) (7)where u, u and u are the nodal acceleration, velocity and dis-placement at time tn, M and C are mass and damping matri-ces (the latter is disregarded in some explicit formulations)and Fint and Fext are internal and external forces. Nodal

velocity and acceleration can be approximated in terms ofthe displacements using central differences, which enablesderivation of the following recurrence equation,

un+1 =(

M + Ct2

)1+

[2Mun t

(Fintn + Fextn

)

(

M Ct2

)un1

]. (8)

Although no iteration procedure is required, the stabil-ity of the algorithm depends strongly on the time step size,which is invariably much smaller then its implicit counter-part. However, this restriction makes possible use of masslumping, which, with the assumption of mass proportionaldamping, leads to an uncoupled equation system easily par-allelised. Pioneered by Hashemi et al. (1994) [18], explicitschemes have been extensively applied to chip removalprocesses [19, 21, 25, 26, 28, 29, 33, 47, 51, 53, 56, 64,83, 103], in special those involving chip breakage and re-meshing procedures.

4.5 Chip Morphology

For a long time, continuous chip formation had been theonly cutting condition analysed. This is primarily due toissues, such as restricted computer capacity, limitation ofstress update and contact algorithms, convergence problemsor even the lack of complete understanding of chip separa-tion/fragmentation mechanisms.

The use of explicit algorithms by Hashemi et al. (1994)[18] and Marusich and Ortiz (1995) [53], combined withrobust contact algorithms and brittle/ductile fracture cri-teria, made possible the simulation of discontinuous chipformation. In both works the path of chip separation andbreakage is not assigned in advance, which constitutes asignificant advance in metal cutting simulation. AlthoughObikawa and Usui (1996) [40] use a pre-defined cuttingline, the chip is allowed to fracture, forming a serratedgeometry which is characteristic when adiabatic shear ispresent. It can be observed in recent years a steady in-crease in the number of simulations using chip breakagestrategies, some of which combining re-meshing procedures[33, 54, 56, 59, 64, 67, 76]. Further details on chip separa-tion and breakage are presented in Sect. 4.10.

The great improvement in computer capacity in the lastyears and further refinement in modelling strategies madepossible attempts to simulate transient 3D chip formation.One of the first simulations of material removal processeswas presented by Guo, Dornfeld and co-workers for incip-ient cutting in drilling [19, 21] and hard turning [25] usingan elasticplastic material model associated with an explicittime integration scheme. Lin and Lin (1999, 2001) [20, 23]used a structured mesh and a pre-defined parting line to sim-ulate a 3D machining operation. Similar approach was also


utilized by Soo et al. (2004a) [32] and Ng et al. (2002) [26].An ALE formulation was used by Pantal et al. (2004) [102]to simulate 2D and 3D oblique machining and milling. Re-meshing techniques in association with unstructured mesheswere adopted by Ceretti et al. (2000) [62] and Fang andZeng (2005) [77] to simulate oblique cutting and Soo et al.(2004b) [74] to simulate a milling process. Further aspectsof 3D machining simulation are presented in Sect. 5.4.

4.6 Friction Models

Friction between chip and tool constitutes one of most im-portant and complex aspects of machining processes. It candetermine, not only the tool wear and quality of the ma-chined surface, but also structural loads and power to re-move a certain volume of metal. Zorevs (1963) [109] tem-perature independent, stickslip friction model has been oneof the most commonly used approximations to frictionalcontact between chip and tool. Zorev advocated the exis-tence of two distinct chip/tool contact regions: near the tooltip, shear stresses f are assumed to be equal to the shearstrength of the material being machined, Y , whereas, in thesliding region, the frictional stress is proportional to the nor-mal stress, n,

f ={Y , 0 c (n Y ) stick,n, > c (n < Y ) slid (9)

in which is commonly associated with the Coulombsfriction coefficient and c is the transitional zone, assumedto be known in advance in most cases (see Fig. 9). Sim-ilar approaches have been applied to simulate machiningprocesses, such as defining an average friction coefficientover the rake face, separate coefficients for each region, dif-ferent lengths for the sticking region, or even neglecting al-together the low stress variation of shear and normal stressesand simply assuming f = mY (m < 1) along the rake face[17, 1927, 2932, 3439, 43, 4548, 5761, 65, 68, 73, 75,7779, 83, 9496, 98, 103105].

Other researchers attempted to use more realistic frictionmodels. Experimental models were introduced by Usui andco-workers [9, 10, 4042, 50], who used a non-linear stressexpression to relate the normal stress and frictional stress as

f = Y[

1 exp(n

Y

)](10)

where is a tool/chip material constant and Y is the max-imum shear stress of the chip surface layer in contact withthe rake face of the tool. Equation (10) approaches f = Yfor large normal stresses (sticking region) and the classicalCoulombs law, f = n, for smaller values of n (slidingregion), as illustrated by the dashed line shown in Fig. 9.

Fig. 9 Frictional and normal stresses along the rake face

Childs and co-workers (2000, 2001) [4, 85] introduced fur-ther modifications in (10) so that

f = mY[

1 exp(n

mY

)n]1/n(11)

in which n controls the transition from the sticking to thesliding region and m accounts for a lubrication effect. zel(2006) [84], by comparing experimental results with severalfriction models based on (9, 10) and (11), concluded thatpredictions are more accurate when utilising friction mod-els based on the measured normal and frictional stresses onthe tool rake face and when implemented as variable frictionmodels at the toolchip contact in the finite element simula-tions.

Based on the experiment where a bar-shaped tool slidesover the inner surface of a ring specimen, Iwata et al. (1984)[87] proposed an expression for frictional stress dependenceon Coulombs friction coefficient, normal stress and Vickershardness of the workpiece material, Hv , as

f = Hv0.07 tanh(

n

Hv/0.07

) f = mY tanh

(n

mY

).

(12)

which was later approximated using the shear flow stressto guarantee that f < Y in the elements immediatelyin contact with the rake face. Noticeably, (12) yields aclose approximation of Usuis friction model when definingHv/0.07 mY , as illustrated by the dotted line depictedin Fig. 9.

Eldridge et al. (1991) [90] used an experimental curvethat relates shear flow stress and yield stress in shear andthe position along the chiptool interface, f (x,T0). Thetemperature dependency is accounted for by an exponentialfunction, so that

f = f (x,T0) exp(

A

T

). (13)


Sekhon and Chenot (1993) [52] adopted Nortons frictionlaw which assumes that frictional stresses are proportional tothe relative sliding velocity between the chip and tool, vf ,as

f = Kvf p1vf (14)where is the friction coefficient, K is a material constantand p is a constant dependent upon the nature of the toolchip contact.

Yang and Liu (2002) [28] proposed a friction law basedupon a polynomial series to relate frictional stress and nor-mal stress as

f =4

k=0k(n)

k

= 0 + 1n + 2(n)2 + 3(n)3 + 4(n)4 (15)in an attempt to reproduce a non-linear behaviour of thefrictional stresses. The fourth-order polynomial can also ap-proximate Usuis and Iwatas models with good accuracy bysetting the series coefficients accordingly.

Finally, frictionless contact has also been assumed byseveral authors [11, 33, 43, 69, 71, 72, 80, 81]. Despite theimportance of friction in machining simulation, most au-thors agree that the existing models present limitations andfurther experimentalnumerical efforts are still required todescribe interaction between tool and workpiece.

4.7 Contact

Contact modelling is of great importance in metal machin-ing due to the important effects associated with the toolchip interface. The two most common algorithms for solv-ing contact problems are the penalty approach [110116]and the method of Lagrangian multipliers [113]. Laursenand Simo (1993) [117] also describe a combined methodcalled the augmented Lagrangian technique. Other proce-dures, such as the perturbed Lagrangian method [118, 119],can also be applied. Other important issues are associatedwith contact detection [111, 120122]. The latter is simplerwith lower order elements and, also for this reason theselow-order elements are popular. Many of the low-order con-tact formulations have origins in the non-linear slide-lineprocedure presented by Hallquist [111, 121, 123]. In rela-tion to these techniques, important issues are associated withthe use of one-pass or two-pass algorithms. In particular,Papadopoulos and Taylor (1992) [114] have shown that atwo-pass formulation is essential if the contact patch test[124, 125] is to be passed. Additional special procedureshave been developed for the explicit integration method[111, 120, 123], such as momentum-related techniques inwhich modifications are made to the acceleration, velocities

and displacements. One of the aims of the latter is to avoidthe penalizing effect on the time step of the explicit proce-dure, which can be introduced by the high stiffness, associ-ated with penalty approaches.

The contact condition is not fulfilled exactly in thepenalty approach. A method proposed by Zavarise et al.(1998) [126], which combines a penalty procedure with abarrier method [127], may also be applied to improve con-vergence when solving the equation of motion. Regularizingthe friction does also improve the convergence in presenceof contacts.

In contrast to the penalty formulation, the Lagrangianmultiplier method ensures exact satisfaction of the requiredcontact constraints [124]. The multipliers can be interpretedas contact forces. However, it increases the number of vari-ables with the introduction of the Lagrange multipliers. It isalso possible to combine the penalty and Lagrangian mul-tiplier methods, as in the augmented Lagrangian technique[117, 127], with the aim of retaining the merits of each ap-proach.

4.8 Elements

One aspect of paramount importance in the present contextis the fact that any adopted finite elements must be able tocope with the range of techniques employed in the simula-tion of cutting. As the incompressible limit is approached,elements with low order shape functions are known to per-form poorly, showing the typical volumetric locking behav-iour which, in many circumstances, invalidates the finite el-ement solution. In spite of such a major drawback, low orderelements, often together with special formulations to avoidthis locking problem, are preferred due to their inherent sim-plicity. This is particularly true in cutting simulations wherea number of complex interacting phenomena, such as theones discussed in this paper, may be present. Barge et al.(2005) [33], well aware of such difficulties, presented a dis-cussion on the influence of the numerical parameters, in spe-cial the hourglass effect upon the equivalent stress, equiva-lent plastic strain and cutting forces.

Linear, plane strain, isoparametric quadrilateral ele-ments and structured meshes have been generally adoptedthroughout previous simulations. Quadrilateral, 8-noddedelements were employed by Joshi et al. (1994) [92], Mc-Clain et al. (2002) [47] and Fihri Fassi et al. (2003) [31],whereas quadrilateral, 9-nodded elements were used by El-dridge et al. (1991) [90] and Tyan and Yang (1992) [91].Enhanced four-nodded elements have also been used by sev-eral authors [17, 28, 29, 33, 35, 3840, 43, 50, 52, 5456,61, 64, 6872]. Linear, triangular elements have been usedwidely in the works of Usui and Shirakashi (1974, 1982)[9, 10] and Lin and co-workers (1992,1993) [15, 16], and re-cently by Ng et al. (1999) [57] and Borouchaki et al. (2002)


Fig. 10 Pre-distorted meshes

[67]. Sekhon and Chenot (1993) [52] and Marusich and Or-tiz (1995) [53] have used 6-nodded triangular elements andre-meshing strategies.

The approaches previously outlined use the classical fi-nite element approximation. Recently, Lorong et al. (2006)[128] presented preliminary results using a meshless method.The authors used the so-called constrained natural elementmethod (CNEM), in which the shape functions are com-puted based on the constrained Voronoi diagram. Resultsfor incipient chip formation illustrated application of themethod.

4.9 Large-Deformations and Adaptive Meshing

Numerical simulation of metal cutting operations present anadditional degree of complexity caused by the necessity ofa proper modelling of both plastic deformation and materialremoval. Large plastic deformations not only bring compli-cations from the mathematical point of view, but can alsocause rapid solution degradation due to element distortion.

The problem was firstly tackled using pre-distortedmeshes, most of which defined as illustrated in Fig. 10[3451]. This strategy has been extensively used since theearly nineties, e.g. Shih (1990) et al., to this date, such as theworks of Ohbuchi and Obikawa (2005) [50] and Mabroukiand Rigal (2006) [51]. In spite of this approach, applicationof adaptive re-meshing techniques seems to be essential toobtain reliable numerical solutions. In general, adaptive re-meshing techniques encompass three main aspects, as rep-resented in Fig. 11:

(i) Meshing techniques;(ii) Error and distortion metrics guiding the re-meshing;

(iii) Data transfer from old to new mesh.

4.9.1 Meshing

There are several methods employed to improve the mesh,namely: h-adaptivity, p-adaptivity and r-adaptivity. The first

Fig. 11 Adaptive meshing procedure

method, h-adaptivity, consists of changing the element sizeh. This general procedure can be used to generate well-shaped elements. It is also an efficient way to change themesh density by refining/coarsening the domain accordingto the error estimator/indicator. The new mesh has a differ-ent number of elements and the connectivity of the nodesis changed. A subset of the h-method is a method whereexisting elements are split or merged based on the originalmesh topology. Therefore, there exists a hierarchy betweenold and new elements and the approach may be called a hi-erarchical h-method. In p-adaptivity the degree of the in-terpolating polynomials is changed in order to increase theaccuracy of the solution.

Another technique, r-adaptivity, consists of relocationof the nodes without changing the mesh connectivity. It iscalled smoothing when used to improve the shape of ele-


Table 1 Comparison of re-meshing methods

Methods Advantages Disadvantages

General h-method The most general and flexible method Requires more elaborate algorithmsHierarchical h-method Fast method. Easy to implement for regular mesh Difficult to implement for irregular mesh. Will not re-

duce element distortionr-Method Easy to implement. Inexpensive Limited capacity to improve meshp-Method Fast and simple The degree in the shape functions cannot be increased

too much so the improvement is limited. Cannot reduceelement distortion

ments. A summary of the advantages and disadvantages ofthe different methods is shown in Table 1.

There exist several algorithms for mesh generation in h-adaptivity. The Delaunay triangulation method for generat-ing six-nodded triangular elements with quadratic interpola-tion functions was used by Marusich and Ortiz (1995) [53].A local re-meshing in which the crack tip was surroundingwith a rosette of elements has also been used to increasethe angular resolution when computing the crack direction.Sekhon and Chenot (1993) [52] employed an automatic re-meshing technique performed through a DelauneyVoronoitype of algorithm.

The hierarchical adaptive meshing scheme proposed byMcDill et al. (2001) [129], and used in [66], provides an effi-cient way to change the mesh density by refining/coarseninga given mesh and facilitates data transfer. However, this doesnot reduce the element distortion. The advancing front tech-nique, so-called paving, is another procedure for generatinga mesh. The elements are added one by one to the mesh-ing area along the exterior paving boundaries, as illustratedin Fig. 12. As the elements are added, fronts may be subdi-vided or closed when the enclosed volume is filled by thelast element. The mesh generation is completed when nomore fronts exist. This method was also used by Kalhori(2001) [66] in combination with the hierarchical adaptivemeshing.

During the r-adaptivity or smoothing process, the nodesare shifted to more favourable positions. Using an optimi-sation algorithm to determine the node positions by min-imizing an element distortion metric [130] is an effectivebut time consuming approach [131]. Some algorithms wereevaluated by Hyun and Lindgren (2001) [131].

4.9.2 Error and Distortion Metrics

One aspect of great importance in adaptive re-meshing isthe assessment of the quality of the solution. In recent years,several error estimates have been proposed for inelasticproblems based not only on mathematical aspects, but alsoon physical considerations. In the late seventies Babukaand Rheinboldt (1978) [132] proposed an a posteriori error

estimate procedure using Sobolev spaces. The topic gainedmomentum when Zienkiewicz and Zhu (1987) [133] intro-duced error estimates based on post-processing techniquesof the finite element solutions, which could easily be usedin conjunction with mesh refinement strategies. The adventof Zienkiewicz and Zhus estimator instigated a healthy dis-cussion on the best approach to a posteriori error estima-tion. From that point forward, two general strategies havebeen established: the so-called residual estimators, basedon Babuka-type strategies [132, 134], and indicators basedon projection/smoothing, after Zienkiewicz and Zhus tech-nique [133, 135138].

Marusich and Ortiz (1995) [53] used the plastic work ratein each element in order to refine the finite element mesh incutting simulations. Owen and Vaz Jr. (1999) [56] proposedan error indicator based on rate of fracture indicators I . Themesh refinement should not only capture the progressionof the plastic deformation but also provide a fine mesh inregions of possible material failure. Further investigationswere carried out by Vaz Jr. and Owen (2001) [64] for errorindicators based on plastic work rate,

IW = Y p, (16)uncoupled integration of Lemaitres [139] damage model,

ID = 2sY

(2Er)s

[23(1 + ) + 3(1 2)

(H

Y

)2]sp, (17)

and total damage work,

IWD = (Y) D

= 2Y

2E(1 D)

[23(1 + ) + 3(1 2)

(H

Y

)2]D,

(18)in which (Y) is the damage strain energy release rate, Dis the damage variable, p is the equivalent plastic strain, Eand are the Youngs modulus and Poissons ratio, and rand s are damage parameters. These criteria were based onthe procedure originally proposed by Zienkiewicz and Zhu[133] and extended to elasto-plasticity at finite strains [138].


Fig. 12 Paving method formesh generation

Other measures that complement error indicators aresometimes needed to detect excessive element distortion.There exist several measures of how the element deviatesfrom ideal shape. The distortion metric proposed by Oddyet al. (1988) [130] is formulated as

Dm =n

i=1

nj=1

C2ij 1n

(n

k=1Ckk

)2and (19)

Cij = 1det [J ]n

k=1JkiJkj (20)

where n is the dimension of the problem. This metric issize and orientation independent and combines the effects ofstretching and shearing of elements. However, the measurefails in case of a negative determinant of the Jacobean J .

4.9.3 Transfer Operators

After assessing the quality of the solution it is necessary totransfer variables from the old mesh to the new mesh. Thenecessity for reliable and efficient transfer operators washighlighted with the development of adaptive re-meshingtechniques for history dependent problems. In this class ofproblems, not only nodal displacements or nodal velocitieshave to be transferred, but also Gauss point variables, whichcreate a complex problem of compatibility and consistency.

Ortiz and Quigley (1991) [140], in the context of strainlocalization, proposed a transfer operator based on the weakform of the equilibrium equations in conjunction with inter-polation of the internal variables. An adaptive strategy wasintroduced by Lee and Bathe (1994) [141] for large-strainelasto-plasticity, which adopts separate transfer strategiesfor nodal displacements and history-dependent variables.The former uses the so-called inverse isoparametric map-ping technique. The internal variables are transferred (andsmoothed) to nodes of the old mesh followed by transfer to

the nodes of the new mesh and, finally interpolation to theGauss points.

Peric et al. (1996) [142] presented further considera-tions on transfer operators for small-strain elasto-plasticity,in which aspects of reliability and consistency were ad-dressed. The transfer strategy resembles Lee and Bathes(1994) [141] technique, however the basic mesh generationalgorithm and implementation strategy are markedly differ-ent.

The basic principles described in reference [142] forimplicit time integration schemes were applied to explicitschemes by Dutko et al. (1997) [143] and Peric et al. (1999)[138]. This strategy constituted the basis of the Owen andVaz Jr. (1999) [56] algorithm for the numerical simulationsof high speed machining.

4.10 Chip Separation and Breakage

A diversity of physical phenomena, such as large plastic de-formations, heat generation, friction, damage, etc., exists inthe process zone where the chip separation occurs. Most au-thors acknowledge that the physics of chip separation is notyet completely understood but is a key issue in the com-putation of the cutting forces. The ideal numerical modelshould be able to simulate continuous, discontinuous and in-termediate states of chip formation, in which material failureshould not be restricted to either a pre-defined parting planeor progression of a simple crack. In general, simulationsfollow three basic strategies (i) continuous chip separationalong a pre-defined cutting plane (ii) chip separation andbreakage (iii) no chip separation. The last option is based onlarge plastic deformation and continuous re-meshing, as de-scribed in Kalhori (2001) [66]. The options for the two firstprinciples are summarised in Table 2 and described below.

4.10.1 Chip Separation

The most common approach to machining simulation pre-defines a chip separation line (or plane), along which a sep-

Modelling and Simulation of Machining Processes 185Table 2 Chip separation and breakage criteria

Criterion Definition

Nodal distance [9, 10, 22, 24, 27, 3438, 4043, 47, 68, 105]a d = dcrEquivalent plastic strain [17, 18, 26, 28, 29, 32, 88]a [19, 21, 83]b Icr = pEnergy density [15, 16, 20]a [23, 44]b Icr =

: d

Tensile plastic work [87]b [54, 59, 76]c [30]a Icr =

1Y

dp

Brozzo et al. [75]c Icr = 21

3(1H ) dpOskada et al. [87]b Icr =

p + b1H + b2dpStress index [4548, 104]a f =

(nf

)2 + ( f

)2Maximum principal stress [18]c f = 1Toughness [53]d f = KIC2Rice and Tracey [53]c f = 2.48 exp

( 32 HY)

Obikawa et al. [42]c f = 0 p VcObikawa and Usui [40]c f =

0.075 ln

( p100

) H37.8 + 0.09 exp( T293 )JohnsonCook [26, 101]c [33, 51]a f =

[D1 + D2 exp

(D3

HY

)] [1 + D4 ln( p0p

)][1 + D5

(T TTmT

)]Damage considerations [56, 64]e f = A

{ 2Y2Er

[ 23 (1 + ) + 3(1 2)

(HY

)2]}s+ f

T(T T0)

a Chip separation along a pre-defined parting line/planeb Chip breakagec Steady-state analysis without actual chip separation or breakaged Fracturing material and brittle-type fracturee Multi-fracturing materials and chip breakage Accounts for positive values only

aration indicator is computed. There are basically two typesof indicators: those based on either geometrical or physicalconsiderations. Usui and Shirakashi (1974, 1982) [9, 10] pi-oneered the discussion by proposing a chip separation cri-terion based on the distance between the tool tip and thenearest node along a pre-defined cutting direction, as illus-trated in Fig. 13(a). The criterion is purely based on geomet-rical considerations and does not account for possible chipbreakage outside the cutting line. On the other hand chipseparation can be easily controlled. In simulations using thisstrategy, as the tool advances, the distance between the nodeFW,C and the tool tip decreases and, at a critical distance dcr ,either a new node is created or a restriction in superimposednodes are removed, which makes possible for the materialto separate.

Equivalent plastic strain, advocated by Carrol III andStrenkowski (1988) [88] and others (see Table 2), gave riseto a whole family of chip separation criteria (physical indi-cators). In this case, the chip is said to separate when p ,calculated at the nearest node to the cutting edge, reachesa critical value. This criterion has been frequently criticizeddue to the fact that, if uncontrolled, node separation prop-agates faster then the cutting speed forming a large opencrack ahead of the tool tip. The process is illustrated inFig. 13(b), in which the chip separation indicator Icr rep-

resents the equivalent plastic strain. Furthermore, the criti-cal value was found to affect the magnitude of the residualstresses [15]. A chip separation criterion based on the totalstrain energy density, introduced by Lin and co-workers [15,16], is said to overcome the previous shortcomings.

The use of ductile fracture concepts was first introducedby Iwata et al. (1984) [87], who suggested a version of Cock-roft and Latham (1968) [144] (tensile plastic work) and Os-akada et al. (1984) [145] as possible chip separation criteria.Although the initial consideration had been quite tentative,the model simulates steady state machining and, therefore,no actual chip separation takes place. The fracture criteriaare computed a posteriori based on the final stress-strainstate. According to the authors, Osakadas criterion is saidto be the best chip separation indicator once it producesthe largest fractured area (the total area where the indica-tor reaches its critical value). Actual chip separation usingCockroft and Lathams (1968) [144] criterion was presentedby Ko et al. (2002) [30].

In recent years, a chip separation criterion based on astress index parameter has been used by several authors [4548, 104]. The indicator assumes that the chip separates whena parametric measure of normal and shear stresses with re-spect to the normal and shear failure stresses, respectively,reaches a critical value.


Fig. 13 Chip separation along apre-defined parting line/plane

A separation criterion based on a critical strain to frac-ture was used by Barge et al. (2005) [33] and Mabrouki andRigal (2006) [51] based on the general Johnson and Cooks[146] yield stress equation. The criterion accounts for equiv-alent plastic strain rate, hydrostatic and yield stresses androom and melting temperatures.

In all the previous cases chip separates along a pre-defined cutting plane. It is worth mentioning that the firstsystematic analysis of material separation in orthogonal ma-chining was presented by Huang and Black (1996) [39],who evaluated chip separation along a pre-defined cuttingplane using criteria based on geometrical (nodal distance)and physical (equivalent plastic strain, energy density andstresses) considerations. The authors conclude that neither ageometrical nor a physical criterion simulate incipient cut-ting correctly.

4.10.2 Chip Breakage

A combination of equivalent plastic strain, to model chipseparation, and maximum principal stress, to simulate chipbreakage, was used by Hashemi et al. (1994) [18] to modelhigh-speed machining. The authors pioneered use of explicit

finite element schemes in association with fracturing mate-rials.

Marusich and Ortiz (1995) [53] proposed use of eitherbrittle or ductile fracture criteria depending on the machin-ing conditions. The former is formulated in terms of thetoughness, KIC , being used in conjunction with a multi-fracturing algorithm. The latter is expressed in terms of thefracture strain, derived from Rice and Traceys (1969) [147]void growth criterion.

Obikawa et al. (1996, 1997) [40, 42], based on experi-ments, obtained general expressions for the strain to frac-ture, which account for the equivalent plastic strain, equiv-alent plastic strain rate, hydrostatic stress and absolute tem-perature.

Owen and Vaz Jr. (1999) [56], using a combined fi-nite/discrete element algorithm and multi-fracturing mate-rials, adopted a chip breakage criterion based on Lemaitresductile damage model. Borouchaki et al. (2002) [67] also in-cluded damage mechanics in the simulation of crack prop-agation based on Lemaitres model. The formulation pro-posed by Lemaitre (1985) [139] postulates that damage pro-gression is governed by void growth.


The effect of hydrostatic stress on chip segmentation dur-ing orthogonal cutting is emphasised by Umbrello et al.(2004) [75] who adopted a chip breakage criterion based onBrozzo et al. (1972) [148].

Some criteria previously used to indicate chip separa-tion were also used as chip breakage criteria. Ceretti et al.(1996, 1999) [54, 59] adopted a chip breakage criterionbased on the combination of the effective stress and Cocroftand Lathams (1968) [144] maximum tensile plastic work.The latter was also used by Hua and Shivpuri (2004) [76] tosimulate chip breakage in orthogonal cutting of Ti6Al4Vtitanium alloy. Lin and co-workers et al. (2001) [23, 44] ex-tended use of the maximum strain energy density, and Ng etal. (2002) [26] and Benson and Okazawa et al. (2004) [101]proposed use of the strain to fracture based on the modifiedJohnson and Cooks [146] yield stress equation to simulatediscontinuous chip formation.

5 Applications

This section presents three examples aiming to illustrate ba-sic aspects of the techniques described previously. Differentnumerical techniques have been assembled in each of theprograms used to numerically simulate the problem at hand.Comparing the model predictions with experimental data al-lowed an assessment of the algorithms.

5.1 Orthogonal Cutting and Residual Stresses

The machining process evokes a residual stress in the sur-face layer of the component. This unwanted phenomenonis usually the outcome of aggressive machining conditions.The residual stresses on the machined surface are an im-portant factor in determining the performance and fatiguestrength of the workpiece. It is therefore important to under-stand and control the residual stress state in the machinedpart so that undesired failure can be avoided.

A combined numerical/experimental study of the behav-iour of different cutting parameters on cutting forces andresidual stresses was undertaken by Kalhori (2001) [66].This study is by no means trivial, even in orthogonal ma-chining, where a number of complex interacting phenom-ena, such as the ones discussed in this paper, may be present.Measurements in the cutting zone are particularly difficult toperform due to the hostile environment and numerical simu-lations require the use of complex algorithms. Several labo-ratory experiments were performed in order to simulate dif-ferent cutting conditions frequently encountered in practice.Relevant measurements on the component were undertaken.The experimental procedure was then reproduced numeri-cally using two finite element codes: AdvantEdge [149] andSiMPle.

5.1.1 Experimental Procedure

In order to reproduce orthogonal cutting (see Fig. 14), theworkpiece periphery was first turned to remove the hardenedsurface and then it was turned to the shape of a pipe with alarge diameter. Finally, orthogonal cutting was achieved byturning the end of the pipe. The material of the workpiecewas SANMAC 316L. It is a machine-ability improved AISI316L stainless steel manufactured by SANDVIK Steel. Thematerial properties of the workpiece, at room temperature,obtained from tensile specimens are depicted in Table 3.

Although the tensile tests were preformed for constantstrain rate, the stress-strain relations at different tempera-tures were obtained and are presented in Fig. 15. The toolinsert was TNMG 160408-QF in grade 235 produced bySANDVIK Coromant. It has an edge radius of 45 mm and arake angle of +6 and with a 0.2 mm wide chamfer with 6rake angle. The geometry of the insert is shown in Fig. 14(b).In order to analyse the process under the conditions usuallyfound in practice, several cutting tests were undertaken. Thedifferent cutting parameters used are given in Table 4.

A Kistler dynamometer was used to measure the cuttingforces in a lathe in three directions: cutting, feeding and pas-sive. Since this device has a low bandwidth it was necessaryto use a 300 Hz low pass filter to avoid the influence of res-onance in the machine tool and cutting tool.

Quick-stop tests were done to measure the residualstresses on the surface. The specimens were cut out from

Table 3 Material data for SANMC 316L at room temperature

Description Symbol Value

Yield stress 0 240 MPaYoungs modulus E 186 GPaPoissons ratio 0.3Specific mass 7900 kg/m3

Specific heat c 445 J/kg KThermal conductivity k 14 W/mKThermal expansion 16.5e-6 1/K

Table 4 Cutting parameters used in the experimental procedure andnumerical simulations

Test Cutting speed Feed Cutting depth# [m/min] [mm/rev] [mm]

1 120 0.05 3.02 120 0.15 3.03 180 0.05 3.04 180 0.15 3.05 240 0.05 3.06 240 0.15 3.0


Fig. 14 Orthogonal cuttingusing an insert TNMG160408-QF from SANDVIKCoromant in grade 235

Fig. 15 True strainstress forSANMAC 316L attemperatures: 23C, 200C,400C, 600C and 800C

the workpieces, ground polished, etched and studied by mi-croscope. The residual stresses in the surface layer were de-termined using X-ray diffraction showing the change in thespacing of atomic planes in the crystal. This change of spac-ing between atomic planes represents the elastic strain in thecrystal from which the stress can be calculated. The X-raydiffraction measurements were done with a Siemens D5000with a geometry with 11 -tilting in the region of 45to 43, with a step length of 0.08, a time step of 30 s andCuK radiation. Primarily, a soler slit and a 1 mm diver-gence slit were used and secondary, a fin film attachment(0.40) LiF-monochromator with scintillation detector wasused. The scanned surface area was 2 mm by 15 mm with apenetration depth of approximate 5 m. Hardness measure-ments were performed at different depths in order to get anestimate of the depth of the machine-affected zone.

5.1.2 Simulations

Two different finite element codes were used in the simu-lations to compute the state of residual stress in the work-piece after being cooled down to room temperature. Orthog-onal cutting was simulated using a 2D plane-strain model.

In these simulations both codes model the chip formationmechanism by continuously updating the mesh.

AdvantEdge [149] is a commercial computational pack-age specially developed for simulation of mechanical cut-ting. The process is modelled using an updated Lagrangianformulation combined with a dynamic explicit time integra-tion scheme. A staggered method for coupled transient me-chanical and heat transfer analysis is used. First an isother-mal mechanical step is taken followed by a rigid transientheat transfer step with constant heating from plastic workand friction. A six-node quadratic triangle element withthree quadrature points is used. The material model accountsfor elasto-plastic strains and has an isotropic power lawfor strain hardening. The strain rate also affects the flowstress. The material properties are temperature dependentand therefore also account for thermal softening. The fol-lowing power viscosity law with [149] constant rate expo-nent was chosen,

(1 + p

0p

)=

(

g(p)

)(21)


and a power hardening law with a cut-off strain at which theincrease in deformation hardening stops. The strain harden-ing and temperature dependency of the plastic flow proper-ties are determined by the function g(p) as:

g(p) = 0(T )(

1 + p0p

)1/n

if p cutp p = cutp (22)where

(T ) = c0 + c1T + c2T 2 + + c5T 5. (23)The temperature dependency of thermal properties of the

workpiece were modelled with polynomial functions,

cp (T ) = cp0 + cp1T + cp2T 2 + + cp5T 5, (24)

(T ) = 0 + 1T + 2T 2 + + 5T 5, and (25)

(T ) = 0 + 1T + 2T 2 + + 5T 5. (26)The coefficients for the material models for 316L and ce-

mented carbide are included in a database of the software,which is not open for the user. This is a serious drawbackwhen using this kind of software.

SiMPle is an in-house research code with similar featuresto AdvantEdge. However, it models the process using an up-dated Lagrangian formulation combined with a quasi-staticimplicit time integration scheme. The full NewtonRaphsonprocedure is used for the solution of the non-linear systemof equations. The material of the workpiece accounts forthermo-elasto-plastic strains with a von Mises yield surfaceand associative flow rule. The material properties are consid-ered temperature dependent, which allows thermal softeningto be reproduced. Nevertheless, the effect of strain rate onthe yield stress is not taken into account.

The simulations are performed by a graded finite element[129]. It is a four to eight node quadrilateral element withpiecewise bilinear shape functions. The element volumet-ric strain field is under-integrated in order to avoid the phe-nomenon of volumetric locking that arises under constitutiveconstraints, such as the near incompressibility that charac-terizes ductile materials. The nature of the shape functionsused by the element, allows a straightforward mesh genera-tion where an element with three nodes on one side can becombined with two elements, each with two nodes along thisside. This technique does not require additional constraintsfor interelement compatibility, which enables a simple gen-eration of a graded mesh. The re-meshing strategy, origi-nally proposed by McDill et al. (2001) [129], has been ex-tended to include r- and h-adaptivity. The contact constraint

is enforced by means of the classical penalty method and athermal resistance is used when the surfaces are in contactto model the heat transfer.

5.1.3 Results

The chip morphology obtained by the so-called quick-stopmethod can be classified as shear localized (see Sect. 3.1).This type of chip is continuous with narrow bands of heavilydeformed material alternating with larger regions of relativeundeformed material.

This morphology results from the shear localization onthe primary deformation zone and provokes a fluctuationof the measured cutting forces. These high frequency forcefluctuations cannot be measured by the relatively slow cut-ting dynamometer. Therefore, the measured cutting forcesare obtained in an average sense. In Table 5, it is possibleto compare the forces in the cutting, Fc, and feed directions,Ff , obtained by the two finite element codes with the exper-imental values.

The hardness was measured in order to get an estimateof the depth of the machine-affected zone. The workpiecematerial has an increased hardness down to about 0.15 mmfrom the surface, see Fig. 16.

The results of X-ray diffraction measurements of thespecimen surfaces were evaluated using elliptical fitting.The averaged residual stresses from simulations are givenin Table 6. The results presented have the uncertainty fromthe determination of the peak in the diffraction pattern listedas value. It should be noted that this is not the total er-ror as there are also other experimental uncertainties. Thesimulations showed that the stresses fluctuated in the cuttingdirection.

The variations of the residual stresses in the depth direc-tion are shown in Fig. 17 for AdvantEdge and in Fig. 18 forSiMPle. They are evaluated at positions of a typical stressdistribution along a line perpendicular to the surface of theworkpiece.

Table 5 Measured and computed average cutting forces in cutting andfeed direction

Test Measured SiMPle AdvantEdgeFc[N] Ff [N] Fc[N] Ff [N] Fc[N] Ff [N]

1 422 390 400 290 420 3802 1005 729 780 400 860 4703 404 360 450 330 425 3754 890 584 780 400 840 4605 391 357 430 320 420 3756 857 520 770 390 845 475


Fig. 16 Measured Vickersmicro-hardness using a load of0.3 kg vs. depth from surface ofspecimen

Fig. 17 Computed residualstress xx in cutting directionvs. distance from surface forsimulated machined surfaceafter workpiece has cooleddown using AdvantEdge

Table 6 Measured and computed average cutting forces in cutting andfeed direction

Specimen Measured AdvantEdge SiMPle# [MPa] [MPa] [MPa]

1 361 17 640 1302 130 9 179 2003 629 28 550 6304 138 8 164 6005 703 31 240 6806 500 25 171 690

5.1.4 Comments on Results

In this example a combined numerical/experimental studyof the behaviour of different cutting parameters on cuttingforces and residual stresses has been undertaken. Experi-mental evidence demonstrated that the stainless steel 316Lunder the cutting conditions analysed produces a shear lo-calized chip. However, the chip morphology predicted by

the numerical simulations was not consistent with observa-tion, perhaps the most severe test for any machining model.The programs used assemble several numerical techniquesand the analysis of the results obtained for the cutting forces(Table 5) and residual stresses (Table 6) by AdvantEdge andSiMPle enables us to conclude that the values predicted forthe cutting forces are within the same order of magnitude,whereas the values obtained for the residual stresses revealsubstantial discrepancies.

The problem of chip formation and its control is also akey issue because metal cutting is a chip-formation process.The deviations between measured and computed values maybe due to several factors:

The numerical strategy used to simulate the chip forma-tion by continuously updating the mesh without account-ing for fracture, severely limits the type of chip morpholo-gies which can be predicted.

The material model used does not account for the materialdegradation caused by large and localized plastic defor-


Fig. 18 Computed residualstress xx in cutting directionvs. distance from surface forsimulated machined surfaceafter workpiece has cooleddown using SiMPle

Fig. 19 Problem geometry andinitial mesh

mation. The inclusion of damage in the behaviour closeto collapse could improve the numerical model.

The friction law for the tool/chip interface.

5.2 Orthogonal Cutting: Adiabatic Shear Localization

The present example focuses on the simulation of machiningprocesses involving material failure due to adiabatic strainlocalization and addresses important issues such as adaptivemesh refinement and material failure. Solution is achievedusing a dynamic explicit time integration scheme in con-junction with a combined finite/discrete element algorithm.The thermal solution is approximated using an adiabaticthermo-mechanical model since heat conduction within thechip and workpiece is very low in high-speed machining.

Titanium alloys are particularly susceptible to adiabaticshear localization due to their very low thermal diffusivityand associated high sensitivity of the yield stress to tem-perature. High-speed machining, therefore, amplifies theseeffects due the high rate of energy generation. The presentexample simulates machining of Ti6Al4V titanium alloy,for which the following aspects are addressed:

Evaluation of the mesh refinement procedure.

Evaluation of the strain localization process. Evaluation of the failure process.

The geometry of the problem and initial mesh for a rakeangle, , of 3 are depicted in Fig. 19, whereas materialdata and other simulation parameters are presented in Ta-ble 7. The yield stress law adopted in the simulations is

Y = 0(0 + p)n + YT

(T T0), (27)

in which the component Y /T represents the thermalsoftening and T and T0 correspond to the current and initialtemperature respectively. An enhanced 4-node one-Gausspoint element is used in the simulations [150].

5.2.1 Thermal Softening

Chip breakage in high-speed machining of Ti6Al4V ti-tanium alloy takes place primarily due to adiabatic (local-ized heating) shear strain localization (see Sect. 5.2.4). Theprocess can be summarised as follows: large plastic defor-mation associated with high stresses cause localized heat-ing, which in turn, prompts thermal softening and eventualmaterial failure. Therefore, computational approaches to er-ror estimation, mesh adaption and material failure must be


Fig. 20 Thermal softeningeffect

Table 7 Material data for Ti6Al4V and other simulation parameters

Description Symbol Value

Specific mass 4420 kg/m3

Specific heat c 582.2 J/kgKThermal conductivity k 5.86 W/mKYoungs modulus E 115.7 GPaPoissons ratio 0.321Yield stress data 0 1231.5 MPa

0 0.008n 0.059

Softening data T0 298 KY /T 2.3 and 7.0 MPa/KW0 33 to 1339 J/m2

rt 0.236Damage data s 1.0Dissipation factor 0.85Coupling interval Every time stepTarget error I 8%Maximum element size hmax 0.1 mmMinimum element size hmin 0.007 mmCutting speed v 4 to 20 m/sCutting depth t 0.5 mmRake angle 9 to +9Flank angle 5Coulomb friction 0.1Tool tip radius rtip 25 m

able to account for such effects. The thermo-mechanicalbinding variable is a temperature-dependent yield stress,which, in present simulation, is expressed by the parame-ter Y /T .

It is interesting to note that (27) accounts for two op-posing effects: strain hardening and thermal softening. Fig-

ure 20 illustrates the typical behaviour of such materi-als using a simple adiabatic model (material propertiesare presented in Table 7). It has been found that, for thepresent yield stress definition, the equivalent plastic strainat instability onset decreases exponentially with increasing|Y /T |. Furthermore, investigations reported by Bker(2005) [80] for titanium alloys have shown that an increaseof the thermal softening effect causes an increase of chipsegmentation and reduction of the cutting forces. Some is-sues that relate thermal softening to mesh adaption and otherphysical aspects are addressed in the following sections.

5.2.2 Error Estimation and Adaptive Re-meshing

The simulations employ the error indicators based on the un-coupled integration of Lemaitres damage model, proposedin reference [56], and are performed for cutting speeds be-tween 5 and 20 m/s and rake angles between 9 and +9.Separate tests are undertaken to evaluate the effects of thecutting speed, rake angle and thermal softening on the ca-pacity of the adaptive re-meshing procedure to describe theprocess evolution.

The material properties of titanium alloys createfavourable conditions for strain localization at cutting speedsas low as 0.0051 m/s [151]. Therefore, an efficient errorindicator should be able to translate the strain localizationphenomena into refined meshes over the critical zones fora wide range of cutting conditions. Figures 21(a)(c) showtypical meshes for cutting speeds v = 5 m/s, v =15 m/s andv = 20 m/s and thermal softening Y /T = 2.3 MPa/K,in which no substantial difference in the refinement patternhas been found.

The rake angle directly affects the configuration of theshear band, which in turn, influences the distribution of themesh density over the workpiece. Figures 22(a)(c) presentmeshes for Y /T = 2.3 MPa/K, a cutting speed of


Fig. 21 Mesh refinement forrake angle 3 andY /T = 2.3 MPa/K

Fig. 22 Mesh refinement forcutting speed of 10 m/s, depthof 0.5 mm and rake angles from9 to +9

10 m/s, rake angles 9, 0 and +9 and a tool advanceU 0.15 mm. As in the previous example, the distributionof mesh density follows the same pattern, which shows re-fined meshes over the localization zone and near the tool tip.It is worth noting that the small tool tip radius requires a re-fined mesh to ensure a proper contact between the cuttingedge of the tool and workpiece. An error indicator based onthe rate of fracture criteria has been proposed [56] based onthe principle that mesh refinement should be able not onlyto capture the progression of the plastic deformation but alsoto produce efficiently refined meshes at regions of possiblematerial failure.

This concept is clearly illustrated in Figs. 23(a)(d)which shows the evolution of the mesh refinement processfor both Y /T = 2.3 MPa/K and 7.0 MPa/K. It isworth mentioning that, in this example, material failure isinhibited, otherwise the chip would have separated beforereaching the stage shown in Figs. 23(b) and (d).

5.2.3 Adiabatic Strain Localization in High-SpeedMachining

The numerical simulations show that, during early stages ofthe development of the shear band, the stresses decrease out-side the localization zone, which may fall within the elasticdomain. The unloading of the elements causes a small move-ment of the material towards the localization zone, whichis clearly visible at the opposite side of the tool. Figure 24shows the velocity vector at an early stage of deformationfor a region near the free surface of the workpiece. Not onlycan the plastic flow direction along the shear band be seen,but also the material movement during the elastic unloading.

The very essence of thermoplastic shear localization ispresented in Figs. 25(a)(d), which show the distribution ofthe temperature and yield stress for thermal softening para-meters Y /T = 2.3 and 7.0 MPa/K. Highly localizedplastic deformation causes the temperature to rise sharply,well above the neighbouring regions, which in turn, causesthe yield stress to decrease. Furthermore, the regions of lowyield stress are more susceptible to larger plastic deforma-tions, which favour the cyclical response localized plasticdeformation temperature rise yield stress reduction unloading in the neighbourhood large and local-ized deformation. Thereafter localized plastic deformationis built up again repeating this cycle. This shear localizationinstigates a significant upward movement of the portion ofthe chip above the shear band, which can eventually lead toits separation.

An increase of |Y /T | noticeably causes a significantreduction of both yield stress and temperature rise. It is ob-served in Figs. 25(c) and (d) an average of 45% decreaseof the yield stress along the shear band. Such behaviour canbe readily explained from (27). Temperature distribution isdictated by heat generation due to inelastic deformation asfollows: the heat generated between two consecutive timesteps is estimated as Q = Yp , where is the dissi-pation factor, Y is the current yield stress and p is theequivalent plastic strain increment. Therefore, the reductionof yield stresses causes heat generation to decrease, therebyleading to a smaller temperature rise.

In the present case, localization was found to be con-fined, in average, to a 35 m zone, corresponding to a regionapproximately five-element wide, as depicted in Fig. 26(a)for Y /T = 2.3 MPa/K. The distribution pattern of theequivalent plastic strain rate can change during the process


Fig. 23 Typical meshes fornegative rake angles. Cuttingspeed v = 10 m/s and rakeangle 3 (a) U = 0.18 mm(Y /T = 2.3 MPa/K),(b) U = 0.38 mm(Y /T = 7.0, MPa/K),(c) U = 0.16 mm(Y /T = 2.3 MPa/K),(d) U = 0.34 mm(Y /T = 7.0 MPa/K)

Fig. 24 Velocity vector at an early stage of plastic deformation for = 3, v = 10 m/s and t = 0.5 mm

due to the cyclical character of the mechanisms of chip for-mation. However, even at very early stages, p was foundto be significantly high, reaching values up to 1.2 106 1/sfor this particular example. A high strain rate causes local-ized plastic deformation, which is reflected by the distrib-ution of the equivalent plastic strain shown in Fig. 26(b).The shear band geometry in high-speed machining is nota straight line, and, therefore, its characterisation based onthe shear angle is not particularly appropriate [56]. More-

over, the simulations show that initial inclination of the shearband increases with the rake angle and tool advance. Someof these issues were also addressed by Bker et al. (2002)[69] using a different computational approach.

The analysis of progression of the strain localizationprocess provides useful information on when a fracturedchip is formed. Therefore, chip breakage is inhibited byde-activating the algorithm responsible for failure soften-ing. Figures 27(a)(c) shows the shear band progression asa function of the tool advance for a cutting speed of 10 m/s,a rake angle of = 3 and Y /T = 2.3 MPa/K.When the tool tip touches the workpiece a well-definedshear band is rapidly developed, as shown in Fig. 27(a).A further tool progression gives rise to a second shear band,as shown in Fig. 27(b). The importance of the present as-sessment is highlighted by the fact that the chip breaks be-fore the development of a new shear band. Therefore, forthe present cutting conditions, the critical fracture indicatoris estimated from values corresponding to a tool advance be-tween 0.04 mm and 0.25 mm.

5.2.4 Material Failure

Material failure in high-speed machining is essentially dueto phenomena associated with adiabatic strain localization.The present section illustrates the mechanisms of the failure

Modelling and Simulation of Machining Processes 195Fig. 25 Thermal softening:temperature and yield stress forU = 0.16 mm, = 3 andv = 10 m/s

Fig. 26 Thermoplastic strainlocalization: U = 0.16 mm,v = 10 m/s andY /T = 2.3 MPa/K

process associated with adiabatic strain localization. Mater-ial failure is simulated using an element erosion technique.Material failure in high-speed machining takes place overthe plastic strain localization zone. Therefore, the dominantquestion is when material failure initiates. According to theprevious analysis of shear band progression, chip forma-tion is of a cyclical character when machining Ti6Al4V.Therefore, in this example, fracture is assumed to initiatebefore the second cycle begins, corresponding to a tool ad-vance U = 0.1037 mm.

It is worth noting that a definitive fracture criterion is,as yet, unavailable, however, experimental observations sug-gest that a fracture strain, f , may indicate failure onset dueto void growth in problems involving plastic strain local-ization [152, 153]. Therefore, in the present study, fractureindicators based on the equivalent plastic strain and uncou-pled integration of Lemaitres damage model are assessed inassociation with Y /T = 2.3 MPa/K [56].

The chip breakage process for a fracture strain based onthe equivalent plastic strain is illustrated in Figs. 28(a)(c),which show the elements undergoing a failure softening and


Fig. 27 Progression of theshear band: = 3,v = 10 m/s andY /T = 2.3 MPa/K

Fig. 28 Failure process basedon the equivalent plastic strainfor W0 = 670 J/m2 andrt = 0.236

Fig. 29 Failure process basedon the Lemaitres damagemodel for W0 = 670 J/m2 andrt = 0.236

fracture propagation. This criterion assumes that fractureinitiates when f = p 1.0. The use of this criterion im-plies that fracture initiates near the tool tip and progressesrapidly towards the free surface of the chip. The incrementaltool advance, U , which promotes a complete chip sepa-ration from the workpiece depends directly on the energyrelease factor. For 33 J/m2 W0 1339 J/m2 no substan-tial difference in the breakage pattern is observed, howeverlarger values of W0 may inhibit the fracture from progress-ing the full length of the shear band. For W0 = 670 J/m2and rt = 0.236, complete chip separation takes place for atool advance of U = 0.0089 mm.

The fracture strain based on Lemaitres damage model,promotes a different breakage pattern, as illustrated inFigs. 29(a)(c) for s = 1, Ars = 3.0 MPa and f /T =0.001325 1/K. Fracture onset is assumed when f 1.0. Inthis case, the failure process initiates at the free surface ofthe chip and propagates towards the tool tip. The effect ofthe temperature is accounted for f /T so that the ductilityincreases with increasing temperature, i.e., higher tempera-tures cause a complete material failure at later stages. This

approach appears to provide a better physical description ofchip separation for this class of problems [154].

5.2.5 Comments on Results

In high-speed machining, the combined effects of high strainrate and low thermal diffusivity of some materials give riseto adiabatic strain localization (and eventual material fail-ure), which requires an efficient adaptive mesh refinementprocedure. An error indicator based on the uncoupled in-tegration of Lemaitres damage model is found appropri-ate for this class of problems due to its capacity to captureshear band formation efficiently. Machining simulation ofTi6Al4V for rake angles 9 +9 cutting speeds5 m/s v 20 m/s and cutting depth t = 0.5 mm illustratethe adaptive procedure.

This study assumes that a fracture strain is the governingparameter of material failure in high-speed machining. Fur-thermore, two different indicators are assessed: equivalentplastic strain and a fracture strain based on Lemaitres dam-age model. The former indicates fracture onset near the tool


Fig. 30 Effect of edgereinforcements on shear angle

Fig. 31 Effect of edgereinforcements on shear angle

tip whereas the latter indicates the free surface of the chip asa possible failure initiation site.

5.3 Application of Two-Dimensional Model in ToolDesign

Some of the vital demands on the metal cutting tools are(i) high productivity high speeds and feeds, (ii) low cutting

forces and power consumption, (iii) low residual stresses be-neath machined surface, (iv) long and predictable tool life,and (v) a smooth surface of the work piece.

Simulations may be used to investigate how alterations inthe cutting geometry influences these factors without havingto first manufacture prototype-cutting tools and then per-form the testing. The latter is a rather expensive and time-consuming method, but still is the most commonly used ap-


Fig. 32 Pressure on surface ofcutting tool

Fig. 33 Insert temperature

proach today. The simulations have been performed usingAdvantEdge [149], previously described in Sect. 5.1.

Figure 30 shows two-dimensional models, utilizing dif-ferent cutting edge geometry, with the shear plane anglemarked. The shear plane angle is a good measurement ofthe effectiveness of a cutting geometry. An increasing of theshear angle reduces volume of deformed workpiece mater-ial and the influence from the machining on the generatedsurface. A large shear plane angle implies also a low cut-ting force, especially a low feed force. It is also of interestto reduce the feed force to get a freer flowing chip. The feedforce is also reduced by increased cutting speed, more posi-tive rake angle, decrease in friction and contact area betweenchip rake face and chip. Increasing the chip rake will, on theother hand, increase the risk for rupture of the cutting edge.The strength can be improved by a reinforcement of the cut-

ting edge geometry with a negative chamfer which then willcounteract some of the advantages by having a sharp tool.

The effect of different rake angles and reinforcements onthe shear plane angles is presented in Fig. 30. The effecton the cutting forces can be investigates for these geome-tries, Fig. 31. There are small variations between the cut-ting and feed forces for the different cutting tool designs.Furthermore, the stresses on the tool were investigated (notshown here). The final outcome was that the intermediatecase in Fig. 30, having a small reinforcement chamfer with achip rake angle of zero which was attaining the largest shearplane angle, was chosen. Wear can also be investigated bythe use of simulations. Figure 32 shows an example wherethis is done. This pressure distribution in combination withtemperature and relative sliding velocity affect the wear rate.


Fig. 34 Workpiece temperaturein turning

Fig. 35 Temperature duringmilling

5.4 Three-Dimensional Cutting

Most of the industrial applications that remove material byaction of a cutting edge (or edges) cannot be reproduced bya two-dimensional model. The simulation of cutting usingthree-dimensional models is naturally less common than us-ing plane strain models, as in the previous examples, be-cause the degree of complexity is increased. We includesome examples of simulation of turning and milling below

using the finite element code SuperForm [155]. There is nocomputation and measurement of residual stresses availablefor these cases. SuperForm [155] is an implicit finite ele-ment program. It is based on a solid formulation and uses anupdated Lagrangian mesh. The full NewtonRaphson pro-cedure is used for the solution of the non-linear system ofequations. The quasi-static analyses are performed usingeight-node brick elements with constant dilatation. The lat-ter is applied in order to relieve the plastic incompressibil-


Fig. 36 Workpiece temperaturein milling with the tool removed

ity constraint. A re-meshing procedure is used to update themesh in order to avoid excessive element distortions.

The turning is performed with a feed of 0.15 mm and aspeed of 4 m/s. The insert geometry has been created byrotating the shape in Fig. 14 by an angle of 80. The temper-ature of the insert is shown in Fig. 33 and the temperature ofthe workpiece, with the insert removed from the picture, isshown in Fig. 34.

An example of simulation of milling is shown in Fig. 35.The feed is 0.15 mm and the speed is 4 m/s, as in the previ-ous turning example. Thus the dif

modelling and simulation of machining processes

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