Analysis of adiabatic shear banding in orthogonal cutting of Ti alloy

International Journal of Mechanical Sciences 75 (2013) 212–222

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International Journal of Mechanical Sciences

0020-74http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/ijmecsci

Analysis of adiabatic shear banding in orthogonal cutting of Ti alloy

M.H. Miguélez a,n, X. Soldani a, A. Molinari a,b

a Department of Mechanical Engineering, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911 Leganés, Madrid, Spainb Laboratoire d’Etude des Microstructures et de Mécanique des Matériaux, LEM3, Labex Damas, Université de Lorraine, Ile du Saulcy, Metz Cedex, France

a r t i c l e i n f o

Article history:Received 27 February 2013Received in revised form2 May 2013Accepted 11 June 2013Available online 28 June 2013

Keywords:Adiabatic shear bandingTi6Al4VNumerical modelingOrthogonal cuttingChip segmentation

03/$ - see front matter & 2013 Elsevier Ltd. Ax.doi.org/10.1016/j.ijmecsci.2013.06.011

esponding author. Tel.: +34 916 2494 02.ail address: mhmiguel@ing.uc3m.es (M.H. Mig

a b s t r a c t

This work is focused on the numerical analysis of adiabatic shear banding in orthogonal cutting ofTi6Al4V alloy. Segmented chip results from adiabatic shear banding, depending on the competition ofthermal softening and strain and strain rate hardening. The influence of cutting velocity and feed in thechip segmentation is studied. Also the role of friction at the tool-chip interface and the effect ofrheological parameters of the constitutive equation are analyzed. Experimental tests obtained fromprevious work of the authors [Molinari A, Musquar C, Sutter G, Adiabatic shear banding in high speedmachining of Ti–6Al–4V experiments and modeling, Int J Plast, vol. 18, 2002, p. 443–459] and others wereused as a reference to validate the models. Cutting forces and the mechanism of plastic flow localizationare analyzed in terms of frequency of segmentation and shear band width and compared toexperimental data.

1. Introduction

Thermo-resistant alloys such as Ti alloys and nickel-basedalloys are widely used in high responsibility applications, forinstance in aero engine components, due to high corrosionstrength and good mechanical properties at high temperature.Mechanical properties of these alloys and their low thermalconductivity are related with low machinability during cuttingprocesses. High temperatures are produced at the interface result-ing in reduced tool life and could also affect surface integrity of theworkpiece [1].

Finite element (FE) method has been widely used to modelmetal cutting and to improve the understanding of mechanisms ofchip generation, contact phenomena [2] and the effect in surfaceintegrity [3]. This approach allows gathering information ondifficult to measure variables during the machining process. Highspeed cutting of thermo-resistant alloys commonly producesserrated and segmented chip due to adiabatic shear banding atthe primary shear zone. Numerical models should be able toreproduce localization phenomena [4].

The interest for industrial applications of Ti alloys, has moti-vated a great number of scientific works focusing on machining ofthese materials. The alloy Ti6Al4V is commonly used in a variety ofstructural engineering applications and has been considered asone of the best examples to study chip segmentation [5].

ll rights reserved.

uélez).

Molinari et al. [6] presented an experimental analysis oforthogonal cutting of a Ti6Al4V ranging from 0.01 to 73 m/s usinga high-speed testing machine and a ballistic set-up. The shearband width and the distance between bands were determined bymicrographic observations, analyzing their dependence upon cut-ting velocity and feed.

Sun et al. [7] experimentally analyzed chip formation duringdry turning of Ti6Al4V alloy under different cutting speeds, feedrates and depths of cut. High frequency of the cyclic force wascaused by the segmented chip formation process. The frequency ofsegmentation was found to be proportional to the cutting speedand inversely proportional to the feed rate. Similar behavior wasobserved by Cotterell and Byrne [8] using image analysis of therecorded video sequences.

Some processing routes vary the elastic limit or the strainhardening capacity of the material. Analyzing how machiningoperations are affected by values of material parameters is aprerequisite for understanding the effects of material processingprior to machining. The influence of material behavior on theformation of shear bands in cutting has been analyzed by differentauthors with Finite Element modeling. Umbrello [9] analyzed theeffect of plastic flow parameters on the cutting process of TiAl6V4alloy. The Johnson–Cook's constitutive equation (with three differ-ent sets of material constants) was implemented in a numericalmachining model and the results were compared with experi-mental data, establishing that fairly good numerical results couldbe obtained. Naturally, the goodness of results is only insured forcutting conditions for which the loading rates and stresses under-gone by the work-material are in the range explored during the

M.H. Miguélez et al. / International Journal of Mechanical Sciences 75 (2013) 212–222 213

experimental characterization of the work-material constitutivelaw as shown for instance by Muñoz-Sanchez et al. [10].

Calamaz et al. [11] implemented a new material constitutivelaw in a 2D finite element model to analyze the chip formationand shear localization when machining titanium alloys. The modelaccounted for the influence of strain, strain rate and temperatureon the flow stress and also introduced a strain softening effect. Thehypothesis of a material strain softening phenomenon enablesprediction of a segmented chip under low cutting speeds andfeeds. The influence of strain softening on chip generation and theevolution of temperature and cutting forces, have been alsoanalyzed in the work of Sima and Özel [12]. It was found thatflow softening increases the degree of chip serration producingmore curved chips.

Karpat [13] proposed various temperature-dependent flowsoftening scenarios which were tested using finite element simu-lations. The results were compared with experimental data fromthe literature: flow softening initiating around 350–500 1C com-bined with appropriate softening parameters yields simulationoutputs in good agreement with the experimental measurements.

Chen et al. [14] analyzed with a FE code the distribution ofstresses, temperature, and strains during the process of chipsegmentation in orthogonal cutting. It was shown that thermalsoftening predominates over strain hardening along adiabaticshear band during segmented chip formation while strain hard-ening predominates over thermal softening between two adjacentshear bands

Bäker et al. [15] and Bäker [16], developed a two dimensionalfinite element model of orthogonal cutting process and analyzedthe influence of material parameters on cutting forces and on thechip formation. General trends concerning chip segmentationwere deduced related with workpiece material behavior: thetendency to chip segmentation should be enhanced in materialswith greater strength; materials with small yield strength and alarge amount of hardening should show a transition to segmentedchips only at very high cutting speeds or not at all; materials inwhich strong strain softening occurs should form segmented chipsat all cutting speeds.

Bäker et al. [17] studied the influence of thermal conductivityon the chip segmentation process. Increase in thermal conductiv-ity leads to a decreasing degree of segmentation and increasing inthe cutting force, showing the importance of heat localization inthe formation of segmented chips.

The current demands of productivity and workpiece qualityhave led to the increase of cutting speed even in the case ofdifficult to cut materials. Despite of the efforts developed in thestudy of machining of Ti alloys the understanding of the thermo-mechanical phenomenon involved during cutting is still an impor-tant objective.

U = U = R = 0

L = 2mm

Clearance angle= 7o

Rake angle= 0o

WORKPIECE

CUTTING VELOCITY

Zone A

Zone C

Zone Bf

Fig. 1. Scheme of the numerical model including boundary conditions (U displace-ment, R rotation).

In this paper, we perform numerical simulations of orthogonalcutting of Ti6Al4V in a wide range of cutting speeds and feed ratesin order to investigate the phenomena involved during chipsegmentation. A finite element model is developed and validatedin terms of cutting forces, frequency of segmentation and chipmorphology by comparison with experimental results obtainedfrom literature including previous work of the authors [6]. Thedependence of cutting forces and chip morphology with respect tocutting parameters, friction coefficient and material rheologicalparameters is analyzed. The paper is organized as follows. Thenumerical model is described in the second section. In thirdsection cutting forces and overall friction coefficient are analyzedwhile chip morphology is presented in sections fourth and fifth.Main contributions of the paper are summarized in theconclusions.

2. Numerical modeling

As it has been commented in the previous section, FiniteElement (FE) modeling of cutting processes has been widely usedto improve the understanding of mechanisms of chip generation.Numerical simulations permit to gather information on difficult tomeasure variables during machining processes and also diminishthe need of experimental work which is commonly time consum-ing and expensive. These approaches have been used for decadesfocusing on different aspects of the metal cutting process such as,interface contact [2,18], constitutive model of the workpiece [19],surface integrity [3] or influence of tool wear [20]. The simulationof oblique cutting commonly performed in real industrial applica-tions needs three dimensional (3D) modeling. The detail of thetool nose and cutting surfaces geometry can only be reproducedwith three dimensional approaches. However two dimensional(2D) models of orthogonal cutting, widely used in the scientificliterature, are very useful to simulate phenomena such as adiabaticshearing at the primary shear zone (being the objective of thispaper) or contact phenomena at the secondary zone (see forinstance, previous works of the authors [2,21]). Moreover a veryfine mesh is required in order to predict shear band width; thisfact increases computational cost and makes really difficult the useof 3D models involving very large number of elements and longcalculation time.

Thus 2D approach has been used in this work in order to modelorthogonal cutting of Ti alloy. A plane strain model was developedusing the commercial Finite Element code ABAQUS/Explicit withLagrangian formulation. A thermo-mechanical coupled analysiswas developed by using CPE4RT element type [22]. Those areplane strain, quadrilateral, linearly interpolated, and thermallycoupled elements with reduced integration and automatic hour-glass control.

The basic geometry, dimensions and boundary conditions ofthe numerical model are shown in Fig. 1. The workpiece is fixed insome contours and the cutting speed is applied to the tool. Planestrain condition is assumed. In this study the rake angle is zero.

The workpiece is divided into three different zones where themesh has different characteristics. The layer of material which willbe removed by cutting is composed of Zone A (main part) andZone B (thin bottom layer of 4 μm thick). The upper limit of Zone Ccorresponds to the machined surface. The mesh at Zone B and C isparallel to horizontal and vertical directions, while the mesh at theZone A is characterized by an inclination angle θ with thehorizontal direction. The aim of this mesh configuration is tofacilitate the formation of segmented chip during machining of Tialloys [4]. As occurs in other localization problems the formationof adiabatic band is easier when the mesh is oriented parallel tothe band. Thus the inclination angle of the mesh should be close to

50 μm

Fig. 2. Formation of shear bands: equivalent plastic strain contours obtainedduring formation of segmented chip (cutting speed 6 m/s, feed 50 microns, frictioncoefficient¼0.4).

Table 1Parameters of constitutive equation for Ti alloy used in simulations [26].

A (MPa) B (MPa) n C m _ε0(1/s)

Ti6Al4V 782 498 0.28 0.028 1.0 10–5

Table 2Physical properties of Ti6Al4V from websiteAZOM, 2013.

Density (kg/m3) 4420Thermal conductivity (W/m K) 7.2Specific heat capacity (J/kg K) 560Thermal expansion (1/K) 9.2�10–6

Modulus of elasticity (GPa) 114

M.H. Miguélez et al. / International Journal of Mechanical Sciences 75 (2013) 212–222214

the direction of shear banding. Formation of shear bands isillustrated in Fig. 2 where the field of equivalent Mises plasticdeformation has been displayed.

The sensibility of the model to the element size was checked bycomparing results obtained with different values of element size(1, 2 and 6 μm). The smallest element size (1 μm) was used to getsufficient accuracy in the results.

Also the influence of mesh inclination θ was analyzed. Thevalue was stated equal to 401 but other mesh configurations with θequal to 351, 451, and 501 were also considered. Only smallvariations in terms of chip frequency were observed (around 8%)while the rest of parameters did not change significantly.

There are different possibilities for selecting an erosion criter-ion for highly distorted elements elimination when simulatingmechanical processes involving elevated strain, such as machiningor impact problems for instance. Analysis and calibration of sevenfracture models such as constant equivalent strain, maximumshear stress or Johnson–Cook fracture model can be found in [23].

In this paper a simple erosion criterion was stated as themaximum level of the equivalent plastic strain εcritp above whichmesh elements are erased. This criterion available in most com-mercial FE codes has been widely used for simulation of chipseparation. It should be fixed in a range representative of mechan-ical phenomena involved during cutting. It is recommendable toselect a higher value in the primary zone than that imposed at thezone of separation line due to the different values of triaxialitybetween both zones. At the separation zone and for the zero valueof the rake angle considered here, the material is subjected topositive values of triaxiality just before rupture. However, thecombination of compressive and shear loading exerted at theprimary zone leads to negative or small values of triaxiality. It isaccepted that the level of rupture deformation is lower as thetriaxiality increases [23]. The level of εcritp have been stated equal to4 in both Zones A and C (in order to remove just few highlydistorted elements, most elements remained active in the model),and equal to 3 in Zone B. The erosion criterion is not critical inZone C since most of the elements experiences low deformationfar from erosion values, only few element at the beginning of thecalculation are distorted and the erosion is applied in order toavoid calculation ending. The value εcritp ¼ 3 is close to those usedwhen modeling material failure at high strain rates such as in ringexpansion experiments [24]. The geometrical and numerical

characteristics of the model allow for performing simulations oforthogonal cutting in a large range of cutting speeds. Interruptionof calculations due to excessive mesh distortion could be avoided.

It should be noted that in the present modeling the separationof the chip from the workpiece occurs by deletion of a layer ofelements along the workpiece surface. Let us denote by Gc theenergy consumed by the separation process per unit machinedsurface. As the separation process is similar to mode 1 ductilecrack propagation, Gc is related to the fracture toughness KIc by therelationship Gc ¼ ðK2

Ic=2EÞ, where E is the Young elastic modulus.For a titanium alloy with KIc ¼ 60 MPa m1=2 and E¼ 100 GPa, wehave Gc ¼ 18;000 J=m2. It is checked in Appendix A that: (i) fortitanium alloys and for the values of the feed considered here, theseparation energy is small as compared to the total cutting energy,(ii) the separation energy involved in the present model iscomparable to the theoretical value Gc ¼ 18;000 J=m2. However,there is no need in the numerical model to return precisely thetheoretical value Gc ¼ 18;000 J=m2 since in any case the separationenergy is playing a minor role in our problem.

The tool was assumed to be rigid. The workpiece material wastaken to be elastic–viscoplastic, isotropic, and to obey the J2 flowtheory (Mises yield criterion). The viscoplastic constitutiveresponse was described by the Johnson–Cook law [25]

sY ¼ ðAþ BεnpÞ 1þ Cln_εp_ε0

� �1−

T−T0

Tm−T0

� �m� �ð1Þ

where sY is the tensile flow stress, εp is the accumulated plasticstrain, _εp is the equivalent Mises plastic strain rate and T is theabsolute temperature. For Ti6Al4V alloy, the parameters of theconstitutive equation were obtained from [26] and are shown inTable 1. Table 2 summarizes the physical properties of Ti6Al4Vobtained from the website AZOM [27].

Concerning heat generation due to plastic work the value of theQuinney–Taylor coefficient was taken as β¼0.9. The initial tem-perature T0 for both the tool and the workpiece was equal to293 K, the melting temperature was 1900 K.

The contact at the tool-chip interface is simulated with thesimple Coulomb law. Constant values of the friction coefficient areassumed (ranging from 0 to 0.8). The ability of this formulation toreproduce complex phenomena including sticking at the tool-chipinterface due to thermal softening has been shown in a previouswork of the authors [21].

3. Cutting forces and overall friction coefficient

Fig. 3a shows the evolution with time of the specific cuttingforce FC=ðbf Þ (b is the width of cut and f is the feed, see Fig. 1) forthe low cutting speed V¼1 m/s (continuous chip) and the high

3000V=20m/sV=1m/s

bfF stCmean /

V=1 m/s

Time period= 40µs

0 100 200 3000

V=20 m/s

bfF stTmean /

0 100 200 300Cutting time (µs) Cutting time (µs)

V=1 m/sV=20 m/s

t fric

0 100 200 300

Cutting time (µs)

Fig. 3. Evolution with time of (a) the specific cutting force, (b) the specific thrust force and (c) the overall friction coefficient. Low and high cutting speeds are considered(respectively 1 and 20 m/s), corresponding to continuous and segmented chip. The feed is 100 microns and the sliding friction coefficient is 0.4.

speed V¼20 m/s (segmented chip). In the case of continuous chipformation (V¼1 m/s), after a transient regime of about 0.1 ms, thecutting force reaches a stationary value which is denoted as FstC .

For V¼20 m/s, chip segmentation occurs and a steady stateperiodic regime is rapidly realized after about 0.07 ms. In thesteady state regime the cutting force has periodic oscillations witha well defined time period. Each drop of the force coincides withthe formation of an adiabatic shear band. Adiabatic shearing is theconsequence of thermal softening which triggers instability of theplastic flow. The mean cutting force FstCmean is defined as the timeaverage of the force over several time periods. The associatedspecific cutting force is FstCmean=ðbf Þ, see Fig. 3a.

Fig. 3b shows the evolution with time of the specific thrustforce FT=ðbf Þ for low and high cutting speeds. When periodicoscillations occur, the mean specific thrust force FstTmean=ðbf Þ isdefined as was done for the cutting force. Identical trends areobserved in Fig. 3a and b.

The apparent friction coefficient is defined as the ratio betweenthrust and cutting force μap ¼ ðFT=FCÞ. The evolution of μap withtime is shown in Fig. 3c for low and large values of the cuttingspeed. Oscillations are seen when segmentation occurs. Then, atime average value ~μap is defined in the steady regime, see Fig. 3c.

Cutting forces obtained with the model are compared in Fig. 4to experimental results referring to orthogonal cutting of Ti6Al4Valloy. The characteristics of these experimental works are brieflydescribed in the following.

Molinari et al. [6] analyzed the development of adiabatic shearbanding on awide range of cutting speeds (from 0.01 m/s to 73 m/s)in orthogonal cutting with a tool with rake angle equal to zero. Twodevices were used for the experimental tests. A universal high-speed testing machine for the low cutting speeds (from 0.01 to 1 m/s). The highest velocities (from 10 to 73 m/s) were achieved with anairgun set-up. In both cases, the cutting force was measured for twovalues of the feed (120 and 250 μm). The authors studied the chipmorphology: frequency of segmentation and width of shear band.

Armendia et al. [28] measured the cutting forces in orthogonalcutting process for feed 100 μm and depth of cut 2 mm. Cuttingvelocity ranged from 0.83 to 1.5 m/s. The clearance angle of thetool was equal to 61, and the rake angle was equal to 71.

Cotterell and Byrne [29], performed experimental tests inorthogonal cutting conditions (rake angle equal to 6.51) andcutting velocity ranging from 0.067 to 2 m/s. Cutting forces,temperature distribution, tool-chip contact length and the shearangle, were obtained for different values of the feed (50, 75 and100 μm).

Fang and Wu [30] studied high speed finishing operation inwhich the tool edge radius plays a significant role. The tool chosenin the experiments was cemented carbide with a TiC/TiN/TiCNcoating with rake angle 51 and edge radius 60 μm. The authorshave limited the study to low velocity (from 0.97 to 2.9 m/s) and

four values of the feed (75, 90, 105 and 120 μm). The large value ofthe ratio between edge radius and feed led to high specific cuttingforces, the highest between the cases reviewed.

Hoffmeister [31] and Gente and Hoffmeister [32], studied the chipformation of Ti6Al4V in orthogonal cutting in a wide range of cuttingspeed from 5 m/s to 100 m/s. The cutting experiments were per-formed with two values of the uncut chip thickness (40 and 80 μm)and two rake angles (01 and 301). The width of cut was 5 mm.

Fig. 4a and b shows the evolution of the specific cutting andthrust forces in terms of the cutting speed. Considering specificforces allows for comparing results at different values of the feed(in the interval 80–127 microns).

Various values of the sliding friction coefficient μ (ranging from0 to 0.8) are accounted. For discontinuous chip the mean sta-tionary values of the cutting and feed forces are considered,respectively ðFstCmean=f bÞ and ðFstTmean=f bÞ.

We note that, in agreement with [21], the effect of μ is gettingnegligible at high cutting speeds and for μ≥0:4 due to the sticking ofthe chip to the tool generated by the enhanced thermal softening.

Fig. 4a and b show that the correlation of the modeling withexperimental data is rather satisfactory. The variability in experi-mental data between the various works considered may be due todifferences in the cutting tools and to different thermomechanicalhistories experienced by the Ti alloys during their manufacture.

The influence of cutting speed on cutting and thrust forcescould be summarized in two aspects. At relatively low cuttingspeed (lower than 5 m/s) a large decrease of the cutting force withthe velocity is observed. At higher cutting speeds the value ofcutting force tends to saturate and no significant force drop is seenwhen the cutting speed is increased.

Fig. 4c shows the dependence of the average apparent frictioncoefficient ~μap with respect to the cutting speed for various valuesof the sliding friction coefficient μ (entering into the Coulombfriction law). It is seen that ~μap decreases significantly with thecutting speed for large values of the sliding friction coefficient(μ≥0:4). The apparent friction coefficient follows trends similar tothose of the thrust force displayed in Fig. 4b. These trends havebeen discussed in [21] and are essentially related to the highercontribution of sticking at the tool chip interface when the cuttingspeed is increased. The expansion of the sticking zone is aconsequence of thermal softening of the work-material along therake face. For μ≥0:4 and at high cutting speeds the tool-chipcontact is mostly governed by the shear flow stress of the workmaterial along the sticking zone and the contribution of the slidingfriction coefficient μ to the apparent friction coefficient becomesnegligible. This feature is clearly seen in Fig. 4c.

On the contrary, for low values of μ the tool-chip contact ismostly governed by sliding friction, and the effect of μ on theapparent friction coefficient ~μap remains significant at high cuttingspeeds.

0 5 10 15 20 25 30Cutting speed (m/s)

ifc th

0 5 10 15 20 25 30Cutting speed (m/s)

0 5 10 15 20 25 30

Cutting speed (m/s)

Fig. 4. Evolution with cutting speed of the (time averaged): (a) specific cutting force FstCmean=ðbf Þ, (b) specific thrust force FstTmean=ðbf Þ and (c) apparent friction coefficient ~μap .Numerical results are compared to experimental data from literature.

Fig. 5. Characterization of chip segmentation : Ls is the mean shear band spacing; tþ2 and t−2 are the peak and valley levels respectively. (a) schematic view of a regular chipand (b) irregular chip morphology obtained for feed 50 microns and cutting speed 6 m/s. In this case average values of Ls, tþ2 and t−2, and along the chip are considered.

00 5 10 15 20 25 30

Cutting speed (m/s)

Fig. 6. Evolution of peak and valley levels, respectivel tþ2 and t−2, with cutting speed(feed 100 microns and sliding friction coefficient 0.4).

4. Chip morphology: effect of cutting velocity and feed

4.1. Characterization of chip serration

The parameters used to characterize chip serration are pre-sented in Fig. 5. The mean thickness of the deformed chip ðt2Þ isobtained as t2 ¼ ðt−2 þ tþ2 Þ=2.

tþ2 and t−2 represent respectively the level of peaks and valleyswith respect to the base of the chip, see Fig. 5a. The shear bandspacing Ls is the distance between two consecutive bands. If thechip morphology is irregular as in Fig. 5b, tþ2 , t

−2 and Ls are meant as

average values along the chip.The evolution of tþ2 and t−2 with the cutting speed V is shown in

Fig. 6 for the feed f¼100 microns. A transition from continuous toserrated chip is observed and the evolution of tþ2 and t−2 is rapid in therange of relatively low cutting speed (o5 m/s). At higher velocitiesthe values of tþ2 and t−2 do not experience significant changes.

Fig. 7. Evolution of the chip morphology of Ti6Al4V alloy with the cutting speed (sliding friction coefficient 0.1; feed 50 and 100 microns). Equivalent plastic strain levelcontours are shown.

Friction coefficient = 0.1

00.1 1 10 100

Cutting speed (m/s)

Molinari et al.

Coterell and Byrne

(2002)

(2008b)

Fig. 8. Influence of the cutting speed and of the sliding friction coefficient on thefrequency of segmentation. Simulations are performed with the feed of f¼0.1 mmand a sharp tool. Experimental data of Molinari et al. [6] and of Cotterell and Byrne[8] are performed with f¼120 microns and f¼100 microns, respectively.

35000Feed = 50

Feed = 100

Feed = 150

00 10 20 30

Cutting speed (m/s)

Fig. 9. Frequency of segmentation multiplied by the feed f vs cutting speed(simulation results). The frequency of segmentation appears to be proportional tothe cutting speed and inversely proportional to the feed.

The evolution of the chip morphology with cutting speed isillustrated in Fig. 7 for the feeds 50–100 microns. Continuous chipformation is observed at low cutting speed (2 m/s). Adiabaticshearing appears between 2 m/s and 4 m/s. As the cutting speedincreases, the trend to segmentation is enhanced and at highcutting speed the chip is completely segmented. In the experi-ments by Molinari et al. [6] saw-tooth chip was observed forcutting speeds, V, lower than 1.2 m/s while continuous chipformation is predicted by the present simulations for these lowcutting speeds. However, for Vo1.2 m/s the experiments did notshow evidence of well formed shear band. The shear band widthwhich was clearly defined at larger cutting speeds (413 m/s)could not be measured at Vo1.2 m/s. Adiabatic heating andpossibly phase transformation are activated at high cutting speeds.At lower cutting speeds, dynamic recrystallization might beinvolved in the mechanism of shear flow instability [33]. Thematerial softening due to dynamic recrystallization of Ti6Al4V isnot simulated with the present FE model.

4.2. Frequency of segmentation and shear band spacing

The frequency of segmentation (the inverse of the time intervalbetween two consecutive drops of the cutting force) is illustrated

in Fig. 8, together with experimental results obtained for Ti 6Al4Vby Molinari et al. [6] and Cotterell and Byrne [29]. The frequency ofsegmentation exhibits a linear dependence with cutting speed V(logarithmic scale) for V41 m/s. There is no friction dependenceof the frequency of segmentation (four values of the slidingfriction coefficient were considered, 0, 0.1, 0.4 and 0.8).

The influence of the feed f is illustrated in Fig. 9, showing thefrequency of segmentation multiplied by the feed, for: f¼50, 100and 150 microns. The segmentation frequency varies linearly withthe cutting speed irrespective of the feed considered. From theresults shown in Fig. 9, it appears that the frequency of segmenta-tion is inversely proportional to the feed. It will be shown laterthat this feature is due to the fact that the shear band spacing Ls isproportional to the feed in the range of cutting speedsconsidered here.

Fig. 10a and b provides the band spacing Ls as a function of thecutting speed. These results were deduced from chip morphologiesobtained with numerical simulations as those displayed in Fig. 7.

Fig. 10a indicates that the band spacing is approximatelyconstant with respect to the cutting speed (in the range ofvelocities explored here). This trend is in agreement with experi-mental results obtained by Calamaz et al. [11], Sun et al. [7], Simaand Özel [12] and Ye et al. [34] which are presented together withnumerical results in Fig. 10a.

Feed = 100 µmFeed = 50 µm

Feed = 150 µm

00 10 20 30

0.5Ban

Cutting speed (m/s)

0.1 1 10 100

Cutting speed (m/s)

Fig. 10. (a) Influence of the cutting speed V on the band spacing Ls. Numerical simulations are performed with a sharp tool and the feed 100 microns. Experimental data areobtained at various feeds and cutting edge radii. For a given feed f, Ls appears as independent of V (in the range of cutting speeds considered here) and (b) Ls/f in terms of V. Itappears that the band spacing Ls is nearly proportional to the feed. The value of the friction coefficient is 0.4.

1.5Stra

Relative position in the chip ( m)5 10 15 20 25 30

0 20 40 60 80 100 120 140Relative position in the chip (μm)

50 μm

Fig. 11. (a) Distribution of the equivalent plastic strain in the chip (left) and along the path AB (dashed line). Cutting conditions are V¼6 m/s and f¼50 microns and thefriction coefficient is 0.4 and (b) Schematic view showing the definition of w20%.

From the theoretical and numerical results displayed in Fig. 10ait appears that the shear band spacing Ls is proportional to thefeed. This is clearly confirmed by Fig. 10b showing Ls/f in terms ofthe cutting speed for various values of the feed.

It should be noted that the agreement between theoreticalpredictions and experimental data in Fig. 10a is satisfactory despiteslight variations in experimental configurations. Variations incutting angles of the inserts could influence the results shown inFig. 10a. Molinari et al. [6], Sima and Ozel [12] and Ye et al. [34]used a null rake angle. For Sun et al. [7] and Calamaz et al. [11] therake angle was equal to 151 and −41, respectively. It is worthmentioning that, according to Atlati et al. [35] the shear bandspacing increases when the rake angle α is reduced. Thus, for azero rake angle the values of Ls of Calamaz et al. [11] shown inFig. 10a would be lowered and therefore would be in betteragreement with the results of the simulations made for α¼0 andother experimental results also obtained for α¼0.

The results could be also affected by different tool edge radii. Tocheck this effect, some simulations have been performed with atool edge radius of 25 μm and compared to results obtained with asharp tool. It was observed that the shear band spacing increaseswith the edge radius. However this effect is moderate and does notcompromise the good correlation observed in Fig. 10a betweentheoretical and experimental results.

The fact that the shear band spacing is independent fromcutting speed and proportional to the feed (see Fig. 10b showingthe ratio band spacing over feed versus cutting speed) leads to a

segmentation frequency that is proportional to the cutting speedand inversely proportional to the feed as observed in Fig. 9.

4.3. Shear band width

Few authors have theoretically analyzed the thickness of adiabaticshear bands occurring during cutting operations. Thus, a procedureto characterize the shear band width from numerical simulationsshould be firstly developed. In this work the shear band width ismeasured from strain fields. The interest of using a strain measure-ment in place of strain rate comes from the fact that, during thecutting process, the strain profile associated to a shear band remainsunaltered as soon as the band is formed. Typical strain profiles alonga line perpendicular to shear bands are displayed in Fig. 11a.

The shear band width cannot be defined in an absolute way.However, an operational way to characterize the width of a shearband can be developed as follows. For a given band, consider thepeak strain εmax accumulated in the band and define by εbase thestrain level outside the band, see Fig. 11b. We define w20% as beingthe width of the shear zone where the strain is higher than thereference strain level εbase+0.2(εmax−εbase), see Fig. 11b (20% refersto the factor 0.2 in this relationship). w20% represents somehow anarbitrary definition of the shear band width, but this definition issufficient to analyze trends, i.e. the effects of loading conditions onthe distribution and intensity of shear localization. The definitionw10% corresponding to strains higher than εbase+0.1(εmax−εbase) willbe also used in order to show the consistency of results when

100 100

1100101

Cutting speed (m/s)100101

Cutting speed (m/s)

Fig. 12. (a) Evolution of the shear band width with the cutting speed. Numerical results, based on the definitions w10% and w20% of the shear band width, are compared toexperimental data of Molinari et al. [6] and (b) numerical results based on w20% for several values of the feed, compared to experimental data. These results show that theshear band width increases with the feed.

0 10 20 30

Cutting speed (m/s)

Fig. 13. Evolution of the average chip thickness with cutting speed for variousvalues of the sliding friction coefficient (feed¼100 μm).

150Friction coefficient = 0Friction coefficient = 0.1Friction coefficient = 0.4Friction coefficient = 0.8

00 10 20 30

Cutting speed (m/s)

Fig. 14. Shear band spacing Ls in terms of the cutting speed for various values of thesliding friction coefficient (feed¼100 μm).

changing the level of the reference strain chosen in the definitionof the shear band width.

The theoretical evolution of the shear band width with cuttingspeed is displayed in Fig. 12a (log–log diagram) for the feedf¼0.1 mm. The two characterizations w20% and w10% show similartrends, i.e. a decreasing of the shear band width when the cuttingspeed is increased. The shear band width was measured in [6] forexperiments at high cutting speeds (>10 m/s). Measures wereeasily made in this range of velocities since shear band boundarieswere quite well defined (probably due to phase transformationwithin the shear zones).

The experimental data are compared against simulation resultsin Fig. 12a, and the agreement appears as reasonably good. In thislog–log diagram, showing the shear band width versus cuttingspeed, a trend to the slope −1 is observed for experimental dataand a similar tendency is seen for numerical simulations but to alower extent.

The slope −1 corresponds to a shear band width being inverselyproportional to the cutting speed (width∼1/V). It should be notedthat the later trend was theoretically predicted for adiabatic shearbands by Wright and Ockendon [36] and Dinzart and Molinari [37].

The decreasing of the width of adiabatic shear bands at highershear rates can be explained as follows [36–38]. At higher strainrates, the heat produced by plastic flow within a given shear bandhas less time to be transferred to the surrounding by heat diffusioneffects. This leads to a narrowing of the heat affected zone. Finally,noting that strain localization is controlled by thermal softening, it

can be concluded that the shear band width is directly related tothe thickness of the heat affected zone and is therefore decreasingat higher cutting speeds.

The influence of feed on band width is illustrated in Fig. 12b,where theoretical results obtained with the criterion w20% arecompared against experimental data. It is clearly observed fromboth numerical and experimental results, that the band widthincreases with feed.

5. Chip morphology: effect of friction and material parameters

5.1. Influence of the sliding friction coefficient

Fig. 13 shows the mean chip thickness t2 in terms of the cuttingspeed for the feed f¼100 μm and for various values of the slidingfriction coefficient. It is seen that the chip thickness decreases withcutting speed.

Fig. 14 shows that the influence of friction coefficient on bandspacing is negligible.

5.2. Influence of rheological parameters

The influence of material properties on shear flow instability iswell established. Heat conductivity, strain hardening, material rate

400Strain hardening exponent = 0

Strain hardening exponent = 0100

00 5 10 15 20 25

Cutting speed (m/s)

0 5 10 15 20 25

Cutting speed (m/s)

Fig. 15. (a) Shear band spacing Ls and (b) frequency of segmentation in terms of the cutting speed for various values of the strain hardening exponent n. Ls increases with nwhile the segmentation frequency decreases (stabilizing effect of strain hardening). The feed is equal to 100 μm.

Initial yield stress = 547 Mpa

782 Mpa

1016 MPa

Initial yield stress = 547 Mpa

782 Mpa

1016MPa

00 5 10 15 20 25

Cutting speed (m/s)

00 5 10 15 20 25

Cutting speed (m/s)

Fig. 16. (a) Shear band spacing Ls and (b) frequency of segmentation in terms of the cutting speed for various values of the initial yield stress A, Eq. (1). Ls decreases for highervalues of A while the segmentation frequency increases. The feed is equal to 100 μm.

sensitivity and inertia have a stabilizing effect, while heat produc-tion and thermal softening have a destabilizing role as revealed byperturbation approaches and finite difference simulations, see forinstance Molinari [39]. The effect on shear localization of strainsoftening (due to damage or phase transformation) and the role ofinitial geometrical defects, material heterogeneities and initialtemperature fluctuations was investigated by Molinari and Clifton[40,41] based on a non-linear analytical approach. An overview ofthe role of material parameters and defects on adiabatic shearbanding can be found in [42,43].

The effect of material parameters on the propagation ofadiabatic shear bands was also investigated by analytical meansby Mercier and Molinari [44] and with numerical modeling byBonnet-Lebouvier et al. [45]. Similar effects were analyzed in [16]in the context of chip segmentation by adiabatic shearing.

In this section, we analyze the influence of the initial yieldstress A and of the strain hardening exponent n of the constitutivelaw, Eq. (1). Fig. 15a and b shows the effect of the hardeningexponent (n¼0, 0.28, 0.56) on shear band spacing and segmenta-tion frequency. Increasing the value of n leads to larger bandspacing. This is an expected outcome of the stabilizing effect ofenhanced strain hardening [46]. The decreasing of the segmenta-tion frequency for higher values of the hardening exponentobserved in Fig. 15b is simply a consequence of the enhancedshear band spacing. Otherwise, Fig. 15a and b confirms theinsensitivity of the band spacing with respect to the cutting speedV and the linear dependence of the segmentation frequency withrespect to V previously observed in Figs. 9 and 10.

In Fig. 16, the yield stress A is varied by +/−30% (resp. 547 MPaand 1016 MPa)) with respect to the reference value A¼782 MPa

given in Table 1. Increasing the level of the elastic limit, results inhigher level of plastic work and of dissipated heat energy. Thus,the effect of thermal softening is enhanced and so is flowinstability. Therefore, the increasing of A should lead to smallershear band spacing and higher segmentation frequency. Thesefeatures are indeed observed in the results displayed inFig. 16a and b.

6. Conclusions

Adiabatic shear banding was analyzed with a numerical modeldeveloped in ABAQUS/explicit, able to reproduce shear localizationphenomena involved in high speed machining of Ti6Al4V. The modelwas applied to the simulation of orthogonal cutting and wasvalidated with theoretical and experimental works of the literature.

The conclusions that can be derived from this numerical studyare four-fold.

Firstly, the effects of cutting conditions on chip segmentationwere analyzed and it was demonstrated that:

Secondly, effects of some material parameters on shear flowstability and chip morphology have been investigated

Thirdly, the effect of the sliding friction coefficient on chipsegmentation and overall cutting forces has been analyzed

Lastly, the correlation of the theoretical results with experi-mental data was found to be satisfactory.

Acknowledgments

The authors acknowledge the financial support of this work tothe Ministry of Economy and Competitiveness of Spain with theproject DPI2011-25999 and to the Universidad Carlos III de Madrid—Banco de Santander for the program ⪡Chair of Excellence⪢. Thiswork was also supported by the French State through the program“Investment in the future” operated by the National ResearchAgency (ANR) and referenced by LabEx DAMAS.

Appendix A. Separation energy compared to cutting energy

The separation energy per unit time is given by:Wseparation ¼ GcbV where Gc is the separation energy per unitsurface, b is the width of cut and V the cutting speed. The cuttingenergy is given by Wcutting ¼ Fspecif icbf 1V , where Fspecif ic is thespecific cutting force. Thus, Wseparation=Wcutting ¼ Gc=ðFspecif icf 1Þ.A typical value of the specific cutting force for titanium isFspecif ic ¼ 1500 N=mm2 and the separation energy was estimatedin Section 2 as Gc ¼ 18;000 J=m2. Thus, for the feedf 1 ¼ 100 microns we have Wseparation=Wcutting ¼ 0:12.

Consequently, the separation energy plays a minor role as itappears to be small compared to the total cutting energy for valuesof the feed of about 100 microns and larger. It follows, that there isno necessity to precisely characterize the separation energy. Avalue of Gc broadly approaching 18;000 J=m2 would be sufficient.

Let us check now that this condition is fulfilled. A roughestimate (lower bound estimate) of the separation energy involvedin the proposed cutting model can be obtained from the work ofplastic deformation within highly distorted elements in theprocess zone at the tip of the crack propagating along themachined surface. It appears that a layer of one element thickness

is deleted in Zone B along the machined surface, according to thecriterion εcrit¼3. This corresponds to the path of the ductile crackthat separates the chip from the workpiece. Above the machinedsurface, a layer with thickness of about three elements is highlydeformed to a value of the strain close to 3. Below the crack a layerof two elements is less deformed to a level of about ε¼0.4. Theplastic work per unit volume in the highly deformed elements canbe estimated as sYεcrit ¼ 2400 MJ=m3 if we consider that the yieldstress for Ti alloys is about sY ¼ 800 MPa and that the accumulatedplastic strain is close to the value ε¼3. Considering that theelement size is δ¼ 1 μm, that a layer of 4 adjacent elements ishighly deformed at the crack tip and neglecting the work ofdeformation in the elements below the crack, a lower bound ofthe plastic work associated to the propagation of the ductile crackis obtained as: Wlayer ¼ 4δsYεcrit ¼ 9600 J=m2. This quantity corre-sponds to the separation energy per unit cutting width and perunit cutting length involved in the numerical model. It is seen thatWlayer ¼ 9600 J=m2 is smaller (lower bound) than the value of theseparation energy for titanium alloys that was estimated asGc ¼ 18;000 J=m2. It can be noted that these values are of compar-able magnitude.

The value of Wlayer could be adjusted by tuning the elementsize δ. However, this is of no importance, since, as previouslydiscussed, the contribution of the separation energy to the totalcutting work remains small for titanium alloys and for the valuesof the feed considered in this work.

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Analysis of adiabatic shear banding in orthogonal cutting of Ti alloy

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