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Journal of Materials Processing Technology 186 (2007) 66–76

Simulation of chip formation during high-speed cutting

Christian Hortig ∗, Bob SvendsenChair of Mechanics, University of Dortmund, 44227 Dortmund, Germany

Received 4 July 2006; accepted 5 December 2006

bstract

The purpose of this work is the modeling and simulation of shear banding and chip formation during high-speed cutting. During this process, shear
ands develop where thermal softening dominates strain- and strain-rate-dependent hardening. This occurs in regions where mechanical dissipationominates heat conduction. On the numerical side, we carry out a systematic investigation of size- and orientation-based mesh-dependence of theumerical solution. The consequences of this dependence for the simulation of cutting forces and other technological aspects are briefly discussed. 2006 Elsevier B.V. All rights reserved.
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eywords: High-speed cutting; Adiabatic shear-banding; Finite-element analys

. Introduction

High-speed cutting is a process of great interest in mod-rn production engineering. In order to take advantage of itsotential, a knowledge of the material and structural behaviorn combination with the technological conditions is essential.o this end, investigations based on the modeling and simu-

ation of the process are necessary. Initial such investigationsere analytical in nature and focused on the process of machin-

ng (e.g., [11,12]). For the significantly more complex processesnd geometries of today, approaches based on numerical and inarticular finite-element simulation represent the state of thert [1–4,10,13,14,17]. In order to account for the effects ofigh strain-rates and temperature on the material behavior, mostf these approaches are based on thermoviscoplastic materialodeling. For example, the Johnson-Cook model [9] is used in

4,14,17] and in the current work.Experimental results [6,7,17,20] show that shear banding

epresents the main mechanism of chip formation and resultsn reduced cutting forces. In the context of a finite-elementnalysis, such shear banding can be modeled using thermo-
isco-plastic material models including in particular the effectf thermal softening (and in general damage as well: e.g., [17]).s is well-known, this results in a loss of solution uniqueness,
∗ Corresponding author. Tel.: +49 231 755 6347; fax: +49 231 755 2688.E-mail address: [email protected] (C. Hortig).URL: www.mech.mb.uni-dortmund.de (C. Hortig).

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924-0136/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.jmatprotec.2006.12.018

esh dependence

esulting in so-called pathological mesh-dependence of the sim-lation results. Usually, this dependence is expressed in termsf the size of the elements used, i.e., the element edge-length.owever, it is not restricted to this property of the elements.

ndeed, as investigated in the current work, other properties,.g., element orientation, or interpolation order, are just as, ifot more, influential in this regard. All such element propertiesre relevant in the context of, e.g., the use of adaptive remesh-ng techniques [1–3,13,14] to deal with large element distortion,esulting almost invariably in unstructured meshes. In the litera-ure, remeshing techniques using structured meshes can also beound. In [1,3], for example, an arbitrary Lagrangian-Eulerian-ike approach is used to rearrange and refine a structured mesh.s will be shown in the current work, the influence of the meshrientation becomes significant in the context of adiabatic shearanding, especially in connection with structured meshes.

The use of remeshing techniques may lead to a reduction ofesh-dependence, but of course cannot eliminate it. This can

e achieved only by working with models based on additionalriteria (e.g., penalization of “vanishingly thin” shear-bands viaegularization). As a first step in the direction of developingdaptive remeshing techniques for such regularized modeling,he purpose of the current work is an investigation of the effectsf variable element properties such as orientation on simulationesults for shear-band development and chip formation in the
ontext of metal cutting processes. This is done here primarilyor structured meshes. In future work, the issue of unstructuredeshes relevant to the issue of adaptive mesh refinement will
lso be investigated.

mailto:[email protected]

dx.doi.org/10.1016/j.jmatprotec.2006.12.018

ials Processing Technology 186 (2007) 66–76 67

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C. Hortig, B. Svendsen / Journal of Mater

The paper begins with a brief discussion of the thermoelastic,iscoplastic material model used as based on the Johnson-Cookodel (§2). Since chip formation begins with shear banding,

his model is then used (§3) to investigate the mesh-dependencef shear-band formation, in particular with respect to mesh ori-ntation. Building on this, a finite-element model is developed,n particular to examine the dependence of the simulation resultsor chip formation on these properties of the mesh (§4). Finally,he work ends with a brief summary (§5).

. Material modeling

As is well-known, metal cutting is influenced by a number of competinghysical processes in the material, in particular heat conduction and mechanicalissipation. Consider for example the cutting of the material X20Crl2 at differentutting speeds as shown in Fig. 1.

At lower cutting speeds (left) and so strain-rates, heat conduction is fastnough to prevent the temperature increase due to mechanical dissipation, result-ng in thermal softening. At higher speeds (right) and so strain-rates, however,eat conduction is too slow to prevent the temperature from increasing to theoint where thermal softening occurs, leading to shear banding and chip forma-ion.

The strong dependence of this process on strain-rate and temperature implieshat the material behavior of the metallic work piece is fundamentally ther-

oelastic, viscoplastic in nature. For simplicity, isotropic material behavior isssumed here. Restricting attention then to metals and to small elastic strain, theodel for the stress can be based on the thermoelastic Hooke form

= {λ0(I · ln V E) − (3λ0 + 2μ0)α0θ}I + 2μ0 ln V E (1)

or the Kirchhoff stress K in terms of the elastic left logarthmic stretch ln VE

nd temperature θ. Here, λ0 and μ0 represent the elastic longitudinal and shearoduli, and α0 the thermal expansion, all at a reference temperature θ0. The

volution of ln VE is given by the objective associated flow rule

ln∗V E = ∂KφP (2)

s based on an inelastic flow potential φP given by

P = εP0C0σYd exp{ 〈σP − σYd〉

(C0σYd)

}, (3)

ith

Yd = (A0 + B0εn0P )

{1 −

[ 〈θ − θ0〉(θM0 − θ0)

]m0}, (4)

he quasi-static yield stress. In the context of the corresponding evolution relation

˙P = ∂σPφP (5)

or the equivalent inelastic strain rate εP, these are consistent with the Johnson-ook model [9]. Here, εP represents a characteristic inelastic strain-rate. Further,

0 represents the initial quasi-static yield stress, while B0 and n0 govern quasi-tatic isotropic hardening. In addition, θM0 represents the melting temperature,nd m0 the thermal softening exponent. Further, C0 influences the dynamic

tti

s

able 1aterial parameter values for the Johnson-Cook model of Inconel 718 taken from [1

0 (MPa) μ0 (MPa)10476 80000

0 (MPa) B0 (MPa) C0 m0

50 1700 0.017 1.3

ig. 1. Metal cutting experiments with X20Crl3 at cutting speeds vc of 8 m/minleft) and 200 m/min (right) (courtesy of S. Hesterberg, Department of Machin-ng Technology, University of Dortmund).

ardening behavior and strain-rate dependence, and 〈x〉 = (1/2)(x + |x|). The ther-odynamic force driving the evolution of εP is given by

P = σvM(K) − ∂εPψP, (6)

epresenting the effective flow stress in terms of the von Mises effective stress

vM (K) determined by the Kirchhoff stress K and inelastic part ψP of theree energy density. Together with εP, σvM (K) determines the rate of inelasticechanical dissipation

P = {σvM(K) − ∂εPεP}εP (7)

f inelastic heating. Lacking a model for the cold-work term ∂εPεP (e.g., [16]),ne often works with the alternative form

P = {βσvM(K)εP (8)

f this last relation in terms of the Taylor-Quinney coefficient β. In [16], it haseen shown that β is in fact not a constant but rather depends on strain andtrain rate to varying degrees. In the following, this coefficient will be treated asonstant as there is no experimental data concerning β for the material (Inconel18) considered in this study.

The material parameters for the material Inconel 718 used in the currenttudy have been identified in [18] and are summarized in Table 1.

Give such parameter values, one can investigate the model behavior. In partic-lar, when the metal is deformed plastically, the part of the inelastic mechanicalissipation transformed into heat (as determined by β) results in a temperatureise. In particular, this temperature increase is shown as a function of equivalenttrain in Fig. 2 (left). In contrast to accumulated inelastic strain, an increase ofemperature results in softening. At points of maximal mechanical dissipation in
he material, softening effects may dominate hardening (Fig. 2, right), resultingn material instability, deformation localization and shear-band formation.
This completes the summary of the model. Next, we turn to the finite-elementimulations and the issue of mesh-dependence.

8]

α0 (K−1) β0

4.3 × 10−6 0.9

n0 εP0(s−1) θ0 (K) θM0 (K)

0.65 0.001 300 1570

68 C. Hortig, B. Svendsen / Journal of Materials Processing Technology 186 (2007) 66–76

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ig. 2. Temperature (left) and yield stress (right) as a function of equivalent strainodel.

. Finite-element simulation of thermal shear-banding

Since chip formation begins with the onset of shear-banding,e begin by looking at this process. For comparison with the

ollowing results, consider the cutting process idealized as aimple shear of the material in the shear zone as shown in Fig. 3.utting of the region enclosed in the dashed box is assumed to

ake place at a shear angle of φ = 40◦ and a cutting tool angle of= 0◦ to a depth of 0.25 mm. The deformation is assumed to be

lane strain. The applied shear velocity vshear corresponds to autting velocity vc in the shear zone of about 1000 m/min.

Roughly speaking, a shear band begins to form in the mate-ial at a point where the behavior changes from hardening tooftening. In particular, in a material deforming initially homo-eneously, this will occur in regions of stress concentration, i.e.,t geometric or material inhomogenities. In the technologicalontext of chip formation, the contact of the tool edge with theork piece and the subsequent loading results in such an inho-ogeneity. In the context of real materials, of course, material

eterogeneity often plays a role as well.As we have seen in Fig. 1 for the material X20Crl3, the cut-

ing speed plays a role in whether or not shear-band and chip
ormation occurs during cutting. To look at this briefly in the con-ext of the simulation, consider the idealized notched specimenhown in Fig. 4. Except where otherwise indicted, the simula-ions in this work have been carried out in ABAQUS/Explicit
ig. 3. Cutting zone idealized as a shear zone in the material undergoing simplehear.

trtsobt

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namic uniaxial tension at two different strain rates as based on the Johnson-Cook

sing bilinear quadrilateral elements with reduced integrationCPE4R). The reduced integration scheme is based on the uni-orm strain formulation [8]. In this method, the element strains assumed to be given by the average strain over the element.

The notch in the idealized specimen represents a geometricnhomogenity where stress concentrates upon loading. Conse-uently, the material yields there first, inelastic deformationccumulates there the fastest, and the temperature increase dueo inelastic dissipation is the greatest. From the point of viewf the material behavior as shown in Fig. 2, the notch regionsill consequently be the first to soften, concentrating further

nelastic deformation there and resulting in band formation.Using this geometry, consider first the influence of strain-rate

n shear-band formation. In the technological context, the strain-ate is correlated with the cutting speed. Consider now the sheareformation of the structure in Fig. 4 at rates representing cut-ing speeds of 10 m/min and 1000 m/min, respectively. As shownn Fig. 5, at 10 m/min, heat conduction in Inconel 718 (whichas a thermal conductivity k of 50 (Wm−1 K−1)) has sufficientime to prevent any significant temperature rise in the mate-ial due to mechanical dissipation which could result in thermalnd shear-band formation. On the other hand, at 1000 m/min,hermal conductivity is simply too slow in comparison to theate of mechanical dissipation to prevent a sufficient tempera-ure rise for thermal softening and shear-banding to occur, ashown in Fig. 6. For the case of Inconel 718, in the contextf the Johnson-Cook model, note that the temperature variesetween the melting temperature (1570 K) and room tempera-
ure (300 K).
Clearly, for Inconel 718 subject to a shearing rate correspond-ng to a cutting speed of 1000 m/min, mechanical dissipationominates thermal conduction leading to thermal softening,

ig. 4. Idealized notched structure discretized with bilinear elements orientedn the predicted shear-band direction. Average element edge-length here is.005 mm.

C. Hortig, B. Svendsen / Journal of Materials Processing Technology 186 (2007) 66–76 69

Fig. 5. Temperature distribution inside the notched structure from Fig. 4 subject to a shearing rate equivalent to a cutting speed vc of 10 m/min. At this “slow” speed,thermal conduction is sufficiently fast to prevent any increase of temperature in the structure to the point where thermal softening begins and leads to shear-bandformation.

Fig. 6. Temperature distribution inside the notched structure in Fig. 4 subject to a sheaat low cutting speeds in Fig. 5, here thermal conduction is too slow to prevent therma

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ig. 7. Idealized structure with elements oriented at 45◦ to the direction ofhearing. As before, the average element edge length here is 0.005 mm.

hear-band development and chip formation. Restricting nowttention to this “high” cutting speed, one can reasonably assumediabatic conditions for simplicity. In this case, the spatial dis-ribution of the temperature and the (equivalent) strain-rate areorrelated, and either can be used to display shear-band devel-pment.

Now, as discussed in the introduction, shear-band formationue to thermal softening in the context of the local materialodel being used here is inherently dependent on the properties

f the mesh. To take a first step in our look at this, consider ahange of element orientation at constant element edge lengthor the structure from Fig. 4. In particular, rotation of all ele-ents in the corresponding mesh to an angle 45◦ to the expected

i.e., horizontal) shear-band orientation yields the alternative dis-retization shown in Fig. 7. For simplicity, we will refer in whatollows to the discretization parallel to the shear direction (Fig. 4)s being “parallel”, and that in Fig. 7 as “rotated”.

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ig. 8. Temperature distribution in the mesh from Fig. 7 after shearing at a rate equivn Fig. 6. See text for an explanation and details.

ring rate equivalent to a cutting speed vc of 1000 m/min. In contrast to the casel softening and shear-band formation.

Corresponding to the case shown in Fig. 6, the structuren Fig. 7 is sheared at a rate equivalent to a cutting speed of000 m/min. The resulting temperature field is shown in Fig. 8.n contrast to the case of the mesh parallel to the direction ofhearing in Fig. 6, the rotated mesh shows no shear-band forma-ion in the expected direction. The material instability proceeds,ontrary to physical expectations, slanted across the structure.

To understand why the orientation of the mesh influenceshear-band development in this fashion, consider the situationhown in Fig. 9.

In a coarse, rotated mesh such as that in Fig. 8, any nucleatinghear-band, which physically “wants” to form in the directionf shearing, would have to cross the element interior. Since thelements involved are constant-strain elements, however, theyre unable to resolve the corresponding strain gradient in theirnteriors (see Fig. 9). In contrast, the strain field can vary fromne to the next across the element boundary, facilitating theesolution of strain gradients associated with shear-band forma-ion in the case of the parallel mesh. This is also reflected inhe development of the respective shear stresses as shown inig. 10. The inability of the rotated coarse mesh to resolve thehear-band leads to a stiffer behavior than in the parallel case
ith shear band. On the other hand, if we increase the number of
lements (in the process decreasing the element edge-size downo 0.0025 mm), a sufficient number of elements becomes avail-ble for the shear-band to form over multiple rotated elements

alent to a cutting speed of 1000 m/min. Temperature contours are the same as


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Fig. 10. Averaged shear stress along upper, sheared edge of structure in thepa

dt

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ig. 9. Lack of shear-band formation for rotated elements due to the incapacityf the element elements.

hich together can resolve the strain gradient. This is shown inetail in Fig. 11.

Since in the rotated case, many more elements are requiredo resolve the same strain-gradient, the shear-band in this case is

uch wider and “smeared-out” than in the parallel case. Becausef this, the development of the shear band in the rotated meshs also much more sensitive to a change of element edge-lengthhan in the parallel mesh.

Up to this point, we have fixed the average element edge-
ength to 0.005 mm. Reducing this size to 0.0025 mm, onebtains the results shown in Fig. 12 for the parallel case andn Fig. 13 for the rotated case. In the case of the parallel meshith constant strain elements, the usual pathological mesh-
otlt

Fig. 11. Shear-band development in a finely-discretized parallel (above) and

ig. 12. Temperature distribution in the notched structure discretized parallel to the shebelow). Temperature contours are the same as in Fig. 6.

arallel mesh (Fig. 6; continuous curve) and rotated mesh (Fig. 8; dashed curve)s a function of the displacement of the upper edge of the structure.

ependence is evident. In this case, the shear-band volume tendso zero and as the number of elements tends to infinity. On thether hand, in the case of the rotated mesh with such elements,
he constant-strain constraint clearly prohibits this and wouldead to the attainment of a minimum shear-band width. Again,hese tendencies are also reflected in the corresponding ones
rotated (below) mesh at constant element edge-length of 0.0025 mm.

ar direction using different element edge lengths: 0.005 mm (above), 0.0025 mm


Fig. 13. Temperature distribution in the notched structure discretized at a 45◦ angle to the shear direction using different element edge lengths: 0.005 mm (above),0.0025 mm (below). Temperature contours are the same as in Fig. 6.

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ig. 14. Averaged shear stress along upper, sheared edge of structure as a functhe rotated mesh (right) with average element edge-lengths of 0.005 mm (contin

or the shear stress as a function of upper-edge displacement ashown in Fig. 14. As expected, the coarser mesh in both cases,nd the rotated mesh in general, behave more stiffly, resultingn “delayed” shear-band development.

Up to this point, we have worked with a fixed element for-ulation. For completeness, consider now the use of (i) 4-node

ilinear elements, and (ii) 8-node biquadratic elements, both
aving an average element edge-length of 0.005 mm. Sinceuch elements are not available in ABAQUS/Explicit, we usedBAQUS-/Standard to carry out the corresponding simulations.ig. 15 displays the results obtained for the average shear-stress
igfd

ig. 15. Averaged shear stress along the sheared (upper) edge of structure as a functioase (left) and for the 8-node biquadratic element case (right). The continuous line resh. Average element edge-length for all meshes is 0.005 mm.

the displacement of the top of the structure for the parallel mesh (left) and forcurve) and 0.0025 mm (dashed curve).

s a function of displacement. These can be compared with thenalogous results for the 4-node reduced integration elementsrom Fig. 10.

In both cases, the average element edge length remainsonstant. For the parallel mesh, simulations yield comparableesults, as again, the shear-band localizes on aligned elementoundaries. As expected for the rotated mesh, an increase
n interpolation order and the possibility of resolving strain-radients within the element results in an accelerated shear-bandormation and a slightly faster drop of the shear stress withisplacement.
n of the displacement of the top of the structure for the 4-node bilinear elementepresents the case of the parallel mesh, and the dashed line that of the rotated


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ig. 16. Finite-element model for the work-piece/tool system used for the cuttingimulation. Mesh orientation relative to the cutting plane is represented here byhe angle δ.

. Finite-element simulation of chip formation

On the basis of the insight gained into the mesh-dependencef shear-band formation from the previous section, we now turno the modeling and simulation of the cutting process and chipormation. To this end, we work with the finite-element idealiza-ion of the tool/work-piece system shown shown schematicallyn Fig. 16.

For the simulations, the work piece (in blue) is dis-retized using 4-node bilinear elements with reduced integrationCPE4R). Further, plane strain conditions are assumed. Theesh is oriented at an angle δ to the cutting plane. Initially, weork with a 60 × 10 element mesh for the work piece. As for the

ool, it is treated here for simplicity as an analytical rigid body.lso indicated in Fig. 16 are the contact pairs between the tool

nd work piece surfaces as well as the fixed nodes. The frictionoefficient between tool and workpiece has been estimated ands fixed at μ= 0.1. All simulations to follow have been carriedut using ABAQUS/Explicit.

The separation of the chip from the work piece is modeledith the help of a failure zone (Fig. 17). Up to failure, this zoneehaves according to the current Johnson-Cook-based modelescribed in §2. The failure of this zone takes place at a critical
alue εPf of the accumulated equivalent inelastic deformationP set to a value of 2. Between failure zone and the rest of theork piece, a rigid contact layer is used to avoid penetration of
he work piece into the area of the failure zone. This ensures

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ig. 18. Chip formation and temperature field development for different mesh orieontours here, and in the following are the same as in Fig. 6.

ig. 17. Failure zone in the work piece defined by the critical value εPf ofhe accumulated equivalent inelastic deformation. The result is a controlledeparation of the chip at a defined distance from the tool tip.

continuous shear deformation of the failure zone and thus aontrolled separation of the chip from the surrounding workiece in a defined distance from the tool tip.

On this basis, consider now the simulation of cutting and chipormation in relation to the discretization. To begin, attention isocused on the relation between the shear angle φ and the meshiscretization angle δ. Since there is no information about thealue of φ, we begin by considering the well-known models oferchant [11] and Lee and Schaffer [12]. In the context of theerchant model, the relation

= π

4− 1

2(arctanμ− γ) (9)

etween φ and the chip angle γ holds, with μ the coefficient ofriction between chip and tool. Assuming for example γ = −5◦again with μ= 0.1), for example, one obtains φ = 40◦. Alterna-ively, in the model of Lee and Shaffer, one derives the relation

= π

4arctan μ+ γ (10)

or φ. For the same values of γ and μ, it predicts a smaller shearngle φ = 35◦ than the Merchant model.

In the current adiabatic context, the shear angle is determinedn the simulations on the basis of the “orientation” of the tem-erature field within the chip (i.e., normal to the temperatureradient). Determined in this fashion, φ varies between 30◦ and5◦. For discretization angles δ equal to φ, one has in effect thease of the parallel discretization from the last section. Analo-
ously, for δ larger than φ, the case of the rotated discretizationolds. Indeed, as shown by the simulation results in Fig. 18,hip formation becomes increasingly inhibited and diffuse as δncreases beyond φ.
ntation angles δ:δ= 20◦ (left), δ= 40◦ (middle), δ= 60◦ (right). Temperature


Fig. 19. Shear-band development during chip formation. Left: δ= 20◦. Right: δ= 35◦.

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ig. 20. Chip formation with γ = −5◦ and δ= 30◦ for different discretizations.he mesh-dependence of segmentation, i.e., an increase in segmentation frequen

Also shown in Fig. 18 (left), as well as close-up in Fig. 19left), is the case δ<φ. In essence, this also represents the casef the rotate mesh in that shear-band formation is retarded andore diffuse. This is in contrast to the case shown in Fig. 19

right), representing in essence the case of the parallel mesh.he shear band is resolved by exactly one layer of elementsnd there is practically no rotation of the elements. The primary
eformation is shear parallel to the element edges.
Turning next to the issue of element edge-length, consider theesults shown in Fig. 20. As discussed above, a reduction of the

tam

ig. 21. Influence of the discretization on the specific cutting force for γ = −5◦; left: 6050 × 30 elements. Note that the specific cutting force is defined as the cutting forceote the increase in segmentation frequency with mesh refinement.

60 × 10 elements; middle: 150 × 20 elements; right: 250 × 30 elements. Noteith mesh refinement.

haracteristic element length causes an accelerated formationf the shear band with, globally considered, smaller deforma-ion. For the chip formation process, this implies that shear-bandormation takes place at a smaller total deformation. As usual,ncreasing the fineness of the mesh also makes it softer, leadingo the tendency shown for the cutting forces in Fig. 21.

A reduction of the element size results in a reduction in cut-
ing forces with increased segmenting frequency. As discussedbove, a reduction of the element size causes stronger defor-ation localization. This results in an increase of the local
× 10 elements, 150 × 20 elements (dashed); right: 150 × 20 elements (dashed),divided by the cutting cross section (cutting depth times cutting feed). Again,


Fig. 22. Effect of a change of tool angle γ on chip formation for a 150 × 20 element mesh. Left: γ = −5◦, δ= 30◦. Right: γ = 10◦, δ= 45◦.

F ing. LR

db

esetibiaf

(cif(affti

i(Fig. 23), which reduces the amount of work needed to deformthe material in the chip.

Consider next the effect of varying the amount of frictionbetween the work piece and tool. As shown in Fig. 24, the cut-

ig. 23. Effect of a change of tool angle γ on the cutting force and work of cuttight: work of cutting for γ = −5◦ (solid curve) and γ = 10◦ (dashed curve).

eformation-rate to the point where the numerical simulationecomes unstable, as shown in Fig. 21 (right; solid curve).

As has been detailed in the previous sections, the finite-lement simulation of shear banding and chip formation istrongly dependent not only on mesh size but also on mesh ori-ntation. In light of this, the practice of using the mesh to fithe orientation and thickness of simulated shear bands to exper-mental results is somewhat questionable and in any case muste done with great care. Keeping this in mind, consider now thenfluence of technological parameters like cutting speed, toolngle, and friction between the tool and work piece on chipormation.

The tool angle has a main influence on the forming of the chipFig. 22) and on the cutting force (Fig. 23). The analytical (i.e.,onstant strain) models of Lee [12] and Merchant [11] predictncreasing cutting forces with decreasing tool angle. This differsrom the behavior in the simulation. Indeed, as shown in Fig. 23left), the maximum specific force increases with decreasing toolngle. As the specific cutting forces oscillate due to shearing chip
ormation, the work of cutting (Fig. 23, right) is more reasonableor comparing processes with different tool angles γ . Indeed,his work, and so the averaged specific cutting force, is nearlyndependent of the tool angle γ . This behavior is due to the
Ffc

eft: specific cutting force for γ = −5◦ (solid curve) and γ = 10◦ (dashed curve).

ncrease of segmentation frequency with decreasing tool angle

ig. 24. Influence of the friction between tool an chip on the specific cutting forceor tool angle γ = −5◦ discretized with 60 × 10 elements: μ= 0.1 (continuousurve) and μ= 0.3 (dashed curve).


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Fig. 25. Simulated chip formatio

ing force increases with an increase in friction between theool and work piece. This is in agreement with engineering intu-tion as well as with the analytical models discussed above. Alson agreement with engineering intuition is the idea that, as theutting speeds increase further, inertial forces should begin toominate the deformation process. Indeed, as shown in Fig. 25or an (unrealistically high) cutting speed of 20,000 m/min,nertial forces are dominant and inhibit shear-band formation.onsequently, the deformation remains homogeneous (left) andutting forces remain high (right).

. Summary

A major issue in the numerical modeling of shear-bandingnd chip formation during high-speed cutting is the strongependence of the results on the choice of element size andrientation. As shown in the current work, this choice can havemajor influence on the prediction of, for example, chip geome-

ry and cutting forces. The common practice of using the choicef element geometry to adjust simulation results to be in agree-ent with experimental results (e.g., [17]) simply sidesteps the

roblem. Indeed, doing this simply ignores the fact that an addi-ional physical criterion is missing in the model, the applicationf which would result in a unique solution to the boundary-valueroblem in the softening regime. Various possibilities exist,ncluding variational (e.g., [21]) and non-local (i.e., regulariz-ng) approaches also involving damage (e.g., [15]). In addition,rror control and adaptive mesh-refinement methods (also forariational and non-local models) are being implemented forfficient and robust finite-element simulations of deformationocalization (e.g., [5]).

eferences

[1] M. Baker, J. Rosier, C. Siemers, A finite element model of high speed metal
cutting with adiabatic shearing, Comput. Struct. 80 (2002) 495–513.
[2] M. Baker, An investigation of the chip segmentation process using finiteelements, Tech. Mech. 23 (2003) 1–9.

[3] M. Baker, Finite element simulation of high speed cutting forces, J. Mater.Process. Technol. 176 (2006) 117–126.

[

cutting speed of 20000 m/min.

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jmpt 186 spanbildunghsc simulation

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