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IntroductionIn the past decade, the need for miniaturized components in

stress. The rate of strain is proportional to the rate of dislocationformation so that an increase in strain rate has a hardening effect

Downloaded Ffields such as MEMS, biomedical devices, and micro-molding hasincreased. Advances in these fields have drawn the attention ofseveral researchers to micro-machining processes and materialproperties at the micron and submicron levels. Mechanical micro-cutting is a promising manufacturing process that is capable ofproducing micro-scale three-dimensional features with high accu-racy and precision. However, the fundamental mechanics of me-chanical micro-cutting is not well understood. For example, thereexists no consensus on a phenomenological explanation of scalingeffects in micro-cutting such as that observed in the specific cut-ting energy, where the amount of energy required for removingunit volume of material increases nonlinearly when the uncut chipthickness decreases from a few hundred microns to a few microns.

Many researchers have attempted to explain and predict thisscaling phenomenon, which is also termed the size effect. Shaw1 and Backer et al. 2 attributed the size effect to a significantlyreduced number of imperfections encountered when deformationtakes place in a small volume. Therefore, material strength wouldbe expected to increase and approach the theoretical strength. It isalso well known that the flow stress of metals depends on thestrain, strain rate, and temperature. From dislocation mechanics,material strength in plastic deformation of metallic crystals is de-termined by the motion of dislocations and their interactions. Anincrease in temperature increases the thermodynamic probabilityof the dislocations achieving sufficient energy to move past a peakin the potential, thereby producing a softening effect on the flow

on the flow stress. Kopalinsky and Oxley 3 and Marusich 4attributed the size effect in machining to an increase in the shearstrength of the workpiece material due to a decrease in the tool-chip interface temperature at small uncut chip thickness values.Larsen-Basse and Oxley 5 attributed the size effect to materialstrengthening arising from an increase in the strain rate in theprimary shear zone with decrease in uncut chip thickness. Fang6 recently presented a complex slip line model for orthogonalmachining and attributed the size effect to the material constitu-tive behavior of varying shear flow stress.

It was recently reported 79 that in nano-indentation the hard-ness of a metal is dependent on the indentation depth when it is ofthe order of a few microns. Experimental evidence from micro-twisting of thin copper wire 10 and micro-bending of thin nickelbeams 11 also display strong size effects when the characteristiclength associated with nonuniform plastic deformation is of theorder of a few microns. Dinesh et al. 12 suggested that thematerial strengthening effect in micro-cutting processes is analo-gous to that in nano-indentation because of the intense localizedinhomogeneous plastic deformation within the primary and sec-ondary deformation zones. Building on the work in 12, Joshi andMelkote 13 presented an analytical model for orthogonal cuttingthat incorporates a material constitutive law with strain gradientdependence of flow stress. Liu and Melkote 14 have recentlydeveloped a strain-gradient-based finite element model for micro/meso-scale orthogonal cutting. Their simulation results predict asizeable strain gradient strengthening effect in micro-cutting.

Besides material strengthening, other reasons have been alsoattributed to the size effect in specific cutting energy. Nakayamaand Tamura 15 conducted micro-cutting tests at a very low cut-ting speed 0.1 m/min to minimize temperature and strain rate

Contributed by the Manufacturing Engineering Division of ASME for publicationin the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedJune 9, 2005; final manuscript received November 30, 2005. Review conducted byD.-W. Cho.

730 / Vol. 128, AUGUST 2006 Copyright 2006 by ASME Transactions of the ASMEKai Liu

Shreyes N. Melkote

The George W. Woodruff School of MechanicalEngineering,

Georgia Institute of Technology,Atlanta, GA 30332-0406

MateriaMechanContribuMicro-CThe specific cuttingin uncut chip thickndependent on severaradius, and materiaeffect where the mareduced to a few mvarious factors suchthe increase in maunderstand the contrthe conditions underening factors, (i) thcutting temperaturetions to the increasFinite element (FE)-5083-H116, a work mstrain rate effects. Sand strain gradientvalidate the model iidentify the contribucutting energy. DOrom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015Strengtheningms and Theirion to Size Effect inttingrgy in machining is known to increase nonlinearly with decrease. It has been reported in the literature that this phenomenon isctors such as material strengthening, ploughing due to finite edgeparation effects. This paper examines the material strengtheningial strength increases nonlinearly as the uncut chip thickness isns. This increase in strength has been attributed in the past tostrain rate, strain gradient, and temperature effects. Given that

al strength can occur due to many factors, it is important totions of each factor to the increase in specific cutting energy andich they are dominant. This paper analyzes two material strength-ontribution of the decrease in the secondary deformation zoned (ii) strain gradient strengthening, and their relative contribu-

specific cutting energy as the uncut chip thickness is reduced.sed orthogonal cutting simulations are performed with Aluminumerial with a small strain rate hardening exponent, thus minimizingble cutting conditions are identified under which the temperatureects are dominant. Orthogonal cutting experiments are used torms of cutting forces. The simulation results are then analyzed tos of the material strengthening factors to the size effect in specific0.1115/1.2193548 Terms of Use: http://asme.org/terms

effects and still observed the size effect. They attributed the in-crease in specific cutting energy to a decrease in the shear planeangle and greater energy dissipation associated with plastic flow

The density of statistically stored dislocation, s, can be deter-mined from the uniaxial stress-strain law in the absence of strain

Downloaded Fin the workpiece subsurface. Kim et al. 16 and Lucca et al. 17considered the tool edge radius as the major cause of size effect.Armarego and Brown 18 suggested that the increase in specificcutting force with decrease in undeformed chip thickness was dueto greater relative contribution of ploughing forces arising fromfrictional rubbing and ploughing associated with material removalby a blunt tool. Komanduri 19 reported size effect due to the tooledge radius at nanometer length scales by carrying out moleculardynamics simulations of orthogonal cutting at depths of cut of0.36212.172 nm and tool edge radii of 3.6221.72 nm. Atkins20 attributed the size effect in cutting to the energy required fornew surface creation via ductile fracture. This energy is thought tobe independent of the depth of cut and, consequently, its contri-bution to the overall work increases at small depths of cut.

It is clear from the literature that material strengthening at smalluncut chip thickness values is an important contributor to the sizeeffect in specific cutting energy. While material strengthening hasbeen attributed to several factors, it is not clear what the contri-bution of each factor is and under what conditions their effect isdominant. Therefore, to partially address this need, this paper fo-cuses on two main strengthening factors: i the contribution ofthe decrease in the secondary deformation zone cutting tempera-ture with decrease in uncut chip thickness and ii strain gradientstrengthening. Finite element FE simulations of orthogonalmicro-cutting are performed using aluminum 5083-H116, a mate-rial with a small strain rate hardening exponent, thus minimizingstrain rate effects. Suitable cutting conditions are identified underwhich the temperature and strain gradient effects are dominant.Orthogonal cutting experiments are used to validate the modelusing cutting forces. The simulation results are then analyzed toidentify the contributions of these two material strengthening fac-tors to the size effect in specific cutting energy.

Model FormulationIn this paper, a strain-gradient-based finite element model for

micro/meso-scale orthogonal cutting processes developed earlierby Liu and Melkote 14 is adopted as the simulation platform.The FE model is a fully coupled thermal-mechanical model de-veloped in the commercially available software, ABAQUS/Standard version 6.4. This section reviews the following aspectsof the finite element model: a constitutive model, b modelingof tool-chip interaction, c modeling of chip separation, and dmodeling of the heat transfer. Several key techniques and theoverall simulation approach are reviewed briefly.

Constitutive Modeling. A Taylor-based nonlocal theory ofplasticity proposed by Gao and Huang 21 is used to representthe material behavior under highly localized inhomogeneous de-formation. From the viewpoint of dislocation mechanics, the ma-terial is work hardened due to the formation, motion, and interac-tion of dislocations. Statistically stored dislocations accumulate bytrapping each other in a random way while geometrically neces-sary dislocations are required for compatible deformation and arerelated to the gradient of plastic strain. This theory links Taylorsmodel of dislocation hardening to a nonlocal theory of plasticityin which the density of geometrically necessary dislocations isexpressed as a nonlocal integral of the strain field. Preserving thestructure of classical continuum theory, the balance law of thetheory, i.e., the balance of angular and linear momentum, is iden-tical to the classical theories. The Taylor dislocation model definesthe shear flow stress in terms of the dislocation density as =Gb, where the dislocation density is composed of the den-sity of statistically stored dislocations, s, and the geometricallynecessary dislocations, g.

Journal of Manufacturing Science and Engineeringrom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015gradient effects as,

= 3Gbs = ref f 1where ref is the reference stress in uniaxial tension. On the otherhand, the density of geometrically necessary dislocations, g, canbe related to the effective strain gradient as,

g = 2/b 2Based on these equations, a flow stress equation accounting for

the effect of geometrically necessary dislocations can be writtenas,

= reff2, ,T* + l 3where l is the material length scale and is given as,

l = 182 Gref

2b 4The flow stress function, f , is assumed to be of the Johnson-

Cook 22 form as follows:

f, ,T* = A + Bn1 + C ln o1 T*m 5

The stain gradient tensor is defined as,

ijk = uk,ij = ik,j + jk,i ij,k 6The effective strain gradient, which measures the density of

geometrically necessary dislocations, is defined as,

= 14ijk ijk 7where the third-order deviatoric strain gradient tensor ijk is givenby,

ijk = ijk 14 ik jpp + jkipp 8

Using Taylor series expansion of the strain components 21,the deviatoric strain gradient ijk is obtained from the followingnonlocal volume integral,

ijk =

ik j + jki ijk 14 ik j

+ jkippd

mk d1 9In the current finite element implementation of this model, the

above integral is reduced to an area integral due to orthogonalcutting assumption and is evaluated over a mesoscale cell definedaround each Gaussian integration point and contained within thecorresponding element 23.

The constitutive equations of Taylor-based nonlocal theory ofplasticity 21 are expressed in rate form as follows:

kk = 3kkk 10

ij = 2ij if e = and e 0

= 2Gij 3ij4e 6ijij ref2 l

23e + ref2 f f if e or e 0

11The key feature of the Taylor-based nonlocal theory of plastic-

ity is that it does not involve higher order terms and preserves thestructure of classical continuum mechanics. Strain gradient entersthe constitutive model as a nonlocal integral and affects the flowstress variation. Thus, it has the advantage of simpler implemen-tation compared to other gradient plasticity theories 2426.

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Choice of Finite Element PlatformThere are many commercial finite element codes that are

Downloaded FTool-Chip Interaction. Accurate representation of the interac-tion between the tool and chip is vital for obtaining a reliable andrealistic simulation. However, the friction characteristic at the toolchip interface is difficult to determine since it is influenced bymany factors such as cutting speed, contact pressure, and tempera-ture. Extensive studies have been performed on the mechanics ofinteraction along the tool-chip interface and several models havebeen developed. Of these, Zorevs model 27, which reveals thattwo distinct regions of sliding and sticking on the interface exist,is widely accepted. In the sliding region, the shear stress is afraction of the normal contact pressure, p. As the shear stressreaches a limiting shear stress value, *, sticking occurs and theshear stress equals the limiting shear stress value regardless of thenormal contact pressure. The extended Coulomb friction model,expressed in terms of the frictional shear stress see Eq. 12,appears to fit the machining problem adequately and has beenused successfully by several researchers 2833 and is chosen inthis work to model the tool-chip interaction.

s = p, when p * sliding

s = *, when p * sticking 12

Chip Separation Modeling. The pure deformation method ofchip formation 4,14,34,35 has been implemented in this workwith the help of the adaptive remeshing technique, thus avoiding achip separation criterion. The cutting process is likened to a metalforming operation, with material flowing on the two sides of thetool. There is no predefined parting line and, therefore, the shapeof the chip is not predetermined. Instead, as the tool advances,nodes in the workpiece move along the tool surface, causing theelements to deform severely near the tool tip. The severely dis-torted elements are replaced during the remeshing step by newelements that are more regular in shape. The material that overlapsthe tool is removed during the remeshing step see Fig. 1.

Heat Transfer Model. In the finite element model, heat gen-eration due to plastic deformation and friction at the tool-chipinterface is modeled as a volume heat flux. Heat conduction isassumed to be the primary mode of heat transfer, which occurswithin the workpiece material and at the tool-chip interface.

The governing equation of heat transfer is as follows:

K2Tx2

+ K2Ty2

= mCpuTx + Ty + Q 13where K is the thermal conductivity of the workpiece material, mis the mass density, Cp is the specific heat capacity, u is the ve-locity in the x direction, v is the velocity in the y direction, and Qis the volume heat flux.

The fraction of dissipated energy converted into heat due toplastic deformation and friction is assumed to be 0.9. Heat gener-ated due to friction is distributed via a weighting factor of 0.5between the two contact surfaces. Thermal boundary conditionselsewhere are set to zero.

Fig. 1 Illustration of material separation using the pure defor-mation method

732 / Vol. 128, AUGUST 2006rom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015capable of modeling the machining process. Examples includeDEFORM 2D/3D, ABAQUS, and Third Waves Advant-Edge Classic. Based on the modeling requirements noted above,the finite element package required for the present work musthave the important feature of adaptive remeshing. A few of theaforementioned packages such as DEFORM, AdvantEdge,and MARC 2D possess this feature. However, the programshould also provide the user with the flexibility to define a genu-ine strain gradient plasticity material model using higher orderelements. Higher order elements are necessary for fully evaluatingthe strain gradient tensor since the constant strain components infirst-order elements yield zero strain gradient. AlthoughABAQUS/Standard does not have true adaptive remeshing ca-pability, it is more flexible than some of the other codes as itallows user-defined subroutines and higher order elements. Con-sequently, the general FEA package ABAQUS/Standard was se-lected as the finite element platform for this work.

Choice of Element TypeA coupled temperature-displacement plane strain element is re-

quired for simulating the orthogonal metal cutting process. For afully coupled simulation, ABAQUS only allows fully integratedelements. However, as explained above, higher order elements arerequired for the evaluation of strain gradient at the element inte-gration points. Therefore, an eight-node biquadratic displacementand bilinear temperature element was chosen to approximate theworkpiece geometry.

Adaptive RemeshingAdaptive remeshing was implemented to avoid convergence

difficulties typically caused by severely distorted elements. Theremeshing module consisted of a preprocessor coded inFORTRAN 77 plus the automatic mesh generator feature of AN-SYS.

Interference depth is used as the remeshing criterion in themodel. During simulation, the amount of penetration between thetool and workpiece contact pair is checked at each time step todetermine the interference depth. Once the remeshing criterion issatisfied, the outline of the workpiece is stored and the automaticmesh generating module of ANSYS is used to create a newmesh for this region. Subsequently, the solution is mapped fromthe old mesh to the new mesh.

It is necessary to have a very fine mesh in the primary andsecondary deformation zones to resolve the relatively steep stressand strain gradients present in these zones. An element size of2 m was used in these zones. However, using the same meshsize throughout the workpiece increases the computational costsignificantly. Therefore, the mesh pattern generated in each re-meshing step was designed to be much denser in the vicinity ofthe two deformation zones, and coarser away from these zones.

A mesh density windowing technique was used for mesh refine-ment. As seen in Fig. 2, different mesh density windows weredefined in and away from the major deformation zones. This ap-proach reduces the number of elements by a factor of 10 or moreand greatly reduces the computational cost while maintaining asufficiently high resolution for the solution.

The typical workpiece dimensions used in the simulations are3 mm by 1 mm. As seen in Fig. 2, the bottom of the workpiece isfully constrained while the top and right sides of the tool are fixedin the Y direction. A velocity load is applied to the tool to simulatethe cutting speed.

Solution Mapping SchemeThe solution and state variables need to be mapped from the old

mesh to the newly created mesh after each remeshing step. Dis-continuity in the solution is inevitable because of changes in the

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Downloaded Fmesh. Therefore, it is important to keep the cumulative error un-der control so that it will not adversely affect the results afterseveral hundred remeshing steps. Remeshing before the elementsbecome excessively distorted and using a sufficiently fine meshtend to reduce the discontinuity. For micro-cutting simulation,which needs as many as several hundred remeshing steps, thesolution mapping scheme is found to be important and needs to bechosen carefully.

ABAQUS/Standard employs a standard interpolation techniquefor solution mapping. Basically, the solution variables are firstobtained at the nodes of the old mesh by extrapolating the valuesfrom the Gauss integration points to the element nodes and thenaveraging these values over all elements abutting each node. Next,the location of each integration point in the new mesh is obtainedwith respect to the old mesh and the variables are then interpo-lated from the nodes of the old element to the integration points ofthe new element. However, this technique works well only withfirst-order reduced integration elements. Problems arise from theextrapolation step when used with higher order elements. If a newGauss integration point is located near the nodes of the old mesh,the solution error at the new Gauss point will tend to be magnifieddue to extrapolation. This magnification effect can eventuallymake the cumulative error grow out of bounds.

In the present work, the diffuse approximation method 36 wasused to eliminate this error. Without extrapolating the solution tothe nodes, a weighted least squares approximation is applied di-rectly to a local window around the estimation points. The methodis as follows.

At a given gauss integration point x of the new mesh, let = px ,y be the field to be estimated; px ,y is the polyno-mial basis and is the coefficient vector depending on the co-ordinates x ,y. The coefficient vector is determined by mini-mizing the following function:

i=1

N

wxi,yipxi,yi i2 14

where xi ,yi and i are the coordinates and the field values at theintegration points of the old mesh, respectively. The approxima-tion is based on the n closest neighbors of x see Fig. 3, and

Fig. 2 Finite element model configuration

Fig. 3 The diffuse approximation method

Journal of Manufacturing Science and Engineeringrom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015wx ,y is a weighting function centered at x and is given by,

wxi,yi = exp lnXi Xrx

15By avoiding the extrapolation step and directly estimating

based on the exact solution at the old integration points, the dif-fuse approximation method yields a more accurate mapping of thesolution and minimizes the cumulative error.

The solution mapping scheme discussed above was coded inFORTRAN 77 and implemented as a postprocessor inABAQUS/Standard, thereby bypassing the default mapping rou-tine available in the software.

A flow chart summarizing the overall simulation approach isshown in Fig. 4.

Orthogonal Cutting ExperimentsAs stated earlier, the size effect in micro-cutting can arise from

various factors such as the cutting temperature, strain rate, andstrain gradient. To investigate material strengthening due to thetemperature drop in the secondary deformation zone and straingradient effects with decrease in uncut chip thickness, it is desir-able to design the experimental conditions such that a single effectis highlighted while the other effects are minimized.

In order to minimize the edge radius effect, a single-crystaldiamond SCD tool and a polycrystalline diamond PCD toolwith nominally upsharp cutting edges were chosen. Scanningelectron microscope SEM measurements of the cutting edge ofthe SCD tool shown in Fig. 5 yields a radius of 65100 nm. Traceanalysis of the PCD tool edge Kennametal NGP-3189L, KD100gives a radius that is less than 7 m. Since the edge radius is lessthan 10% of the smallest uncut chip thickness used in the experi-ments with each tool, the edge radius effect is assumed to benegligible and is not considered in the simulations.

According to a recent sensitivity study of material flow stress inmachining by Fang 37, the predominant factor governing thematerial flow stress is either strain hardening or thermal softening,depending on the specific work material employed and the vary-

Fig. 4 Overall simulation approach

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Downloaded Fing range of temperatures. Strain rate hardening is reported as theleast important factor governing the material flow stress, espe-cially when machining aluminum alloys. In this study, aluminumalloy 5083-H116, a rate insensitive material with very small strainrate hardening exponent 38,39, is chosen as the workpiecematerial.

The range of values for the cutting speed and uncut chip thick-ness were chosen as follows. At higher cutting speeds a largerdrop in cutting temperature is expected as the uncut chip thicknessis decreased. A rough calculation based on Oxleys method 40shows that the temperature rise at a cutting speed of 10 m/minproduces a negligible effect on the flow stress, while the tempera-ture rise at a cutting speed of 200 m/min is large enough to havea considerable effect on the material flow stress. Further, as can beseen from Eq. 3, the strain gradient strengthening effect isprominent only when the size of the inhomogeneous plastic flow,given by the uncut chip thickness in micro-cutting, is similar tothe intrinsic material characteristic length l, which for aluminumalloy 5083-H116 is about 5.7 m calculated using Eq. 4 and=0.3, G=26.4 GPa, ref =228 MPa, and b=0.256 nm. Hence,the strain gradient effect becomes significant when the uncut chipthickness is of the order of a few microns.

Orthogonal cutting experiments were therefore conducted attwo cutting speeds: 10 and 200 m/min. At a cutting speed of10 m/min, the uncut chip thickness was varied between 0.5 and10 m, a range where the strain gradient effect is expected to bedominant. At a cutting speed of 200 m/min, the uncut chip thick-ness was varied between 20 and 200 m in order to highlightmaterial strengthening due to a reduction in the secondary shearzone temperature with decreasing uncut chip thickness.

Orthogonal cutting experiments at the cutting speed of200 m/min were performed with the PCD tool on a HardingeT42SP super precision lathe. The workpiece was in the form oftube of 1 mm wall thickness see Fig. 6. Cutting forces weremeasured with a piezoelectric force dynamometer Kistler dyna-mometer 9257B.

Since the minimum axial feed rate of the Hardinge lathe is1 mm/min, orthogonal cutting tests at the cutting speed of10 m/min and uncut chip thickness ranging from 0.5 to 10 mcould not be conducted in the lathe. Instead, a precision two-axis

Fig. 5 SEM image of SCD tool

Fig. 6 Schematic of orthogonal micro-cutting experiment

734 / Vol. 128, AUGUST 2006rom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015motion control stage Aerotech ATS-125 was used. The work-piece with a 1 mm wide ridge was fixed to the two-axis stage andtranslated in the X and Y directions see Fig. 7. The SCD tool wasmounted on a miniature three-component piezoelectric force dy-namometer Kistler Minidyne 9256C2. The SCD tool had a 0rake and a 5 clearance angle. All components of the setup weremounted on an aluminum base plate and the entire setup wasplaced on a vibration isolation table. The depth of cut was im-parted by moving the workpiece along the Y axis and the cuttingvelocity was imparted by moving the workpiece along the X axis.

Tables 1 and 2 summarize the orthogonal cutting experimentalconditions used for studying the strain gradient and temperatureeffects, respectively.

Model VerificationOrthogonal cutting simulations were run with the full model,

which includes strain, strain gradient, and temperature effects, tovalidate against the experimental data at the two cutting speedsand two sets of uncut chip thickness. Flow stress data for Al5083-H116 derived from a hot torsion test has been reported by Zhouand Clode 41. Since strain rate dependence of the flow stress forAl5083-H116 is considered to be negligible, a modified Johnson-Cook flow stress equation Eq. 16 was used to fit the flow stressdata of Zhou and Clode, reproduced in Fig. 8, with a zero strainrate hardening exponent. The coefficients of the modifiedJohnson-Cook model are listed in Table 3 and were used in thefinite element simulation results presented in the paper see alsoTables 4 and 5.

A friction coefficient of 0.21 was used for the 200 m/min caseswhile 0.14 was used for the 10 m/min cases:

Table 1 Experimental conditions highlighting the strain gradi-ent effect

Machine tool Precision two-axis motion controlstage Aerotech ATS-125

Cutting speed m/min 10Uncut chip thickness m 0.5, 1, 2, 5, 7.5, 10

Cutting tool Single crystal diamond SCD0 rake, 5 clearance

Workpiece material Al5083-H116Dynamometer Kistler Minidyne 9256C2

Table 2 Experimental conditions highlighting the temperatureeffect

Machine tool Hardinge T42SPCutting speed m/min 200

Uncut chip thickness m 20, 50, 75, 100, 150, 200Cutting tool Polycrystalline diamond PCD

5 rake, 11 clearanceWorkpiece material Al5083-H116

Dynamometer Kistler 9257B

Fig. 7 Schematic of orthogonal micro-cutting process carriedout on the two-axis precision motion stages

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Downloaded Ff, ,T* = A + Bn1 + c1 T*m 16As seen from Figs. 9 and 10, good agreement is obtained be-

tween the predicted and measured forces at both cutting speeds.At 200 m/min, the absolute average percentage error in the cut-ting force prediction is 4.25% and 23.21% for the thrust forceprediction. At 10 m/min, the absolute average percentage error inthe cutting force prediction is 9.6% and 22.2% for the thrust forceprediction. It can be seen that a better match is obtained for thecutting force while the thrust force is generally underpredicted atsmall uncut chip thicknesses for both cutting speeds. This couldbe caused by the use of a constant friction coefficient for thewhole range of uncut chip thickness whereas the actual frictioncoefficient estimated from the measured forces shows an in-creasing trend with decrease in uncut chip thickness. Neverthe-less, the model gives fairly good predictions in terms of the abso-lute error. Thus, the finite element model is considered to be

Table 3 Modified Johnson-Cook flow stress model coeffi-cients for Al5083-H116

A MPa B MPa n C m

167 300 0.12 0 0.859

Table 4 Material properties of Al5083-H116 42

Density Kg/m3 2660Specific heat J/kg C 900Thermal conductivity

W/m K117

Coefficient of thermalexpansion m/mC

12.6

Melting temperature C 591638Yield strength MPa 228

Youngs modulus GPa 71Shear modulus GPa 26.4

Poissons ratio 0.33Burgers vector nm 0.256

Table 5 Material properties of diamond tools 42

Density Kg/m3 3500Specific heat J/kg C 471.5Thermal conductivity

W/m k1500

Coefficient of thermalexpansion m/mC

2.0

Melting temperature C 4027Youngs modulus GPa 850

Poissons ratio 0.1

Fig. 8 Flow stress data for Al5083-H116 41

Journal of Manufacturing Science and Engineeringrom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015satisfactorily validated and is used in the following section toanalyze the contribution of each material strengthening factor tothe size effect in micro-cutting.

Results And DiscussionFigure 11 shows the effective strain gradient distribution ob-

tained from the simulation at 1 m depth of cut. It is found thatvery high strain gradients are present at the tool-chip interface and

Fig. 9 Comparison of experimental and simulated cuttingforces at 200 m/min

Fig. 10 Comparison of experimental and simulated cuttingforces at 10 m/min

Fig. 11 Strain gradient contour uncut chip thickness 1 m,10 m/min cutting speed

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Downloaded Fin the workpiece surface layers. As expected, a high strain gradi-ent band is also present in the primary deformation zone. Analysisof the simulation results shows that the normal plastic strain com-ponents 11 and 22 not shown contribute to the strain gradient inthe primary deformation zone and the tool-chip interface, whilethe high strain gradient in the surface layers comes mainly fromthe distribution of shear plastic strain 12 not shown. With aneffective strain gradient of several hundred and a calculated ma-terial length scale of 5.7 m obtained by substituting the materialparameters into Eq. 4, it can be seen that the term associatedwith the strain gradient in Eq. 3 is significant compared to theother term. Consequently, the strain gradient strengthening effectis considerable. This also suggests that the effect will becomeeven more dominant at smaller depths of cut, which will producean even steeper strain gradient.

In order to examine the strain gradient effect at a low cuttingspeed of 10 m/min, two sets of orthogonal cutting simulationswere run at uncut chip thickness values ranging from0.5 to 10 m. The first set of simulations was run with all termsin Eq. 3 while the second set of simulations was run without thestrain gradient term l in Eq. 3. Figure 12 shows a plot of thespecific cutting energy versus uncut chip thickness with and with-out the strain gradient effect. The specific cutting energy wascomputed by dividing the total force acting on the tool in thecutting direction by the product of the workpiece width unityand uncut chip thickness. It can be seen that the specific cuttingenergy predicted by the model with strain gradient effect matcheswell with the experimental data and captures the size effect. Forthe model without strain gradient effect, the predicted specificcutting energy remains constant with decrease in uncut chip thick-ness. It is clear from Fig. 12 that at a low cutting speed and forsmall uncut chip thickness, strain gradient strengthening is thedominant mechanism responsible for the size effect.

Fig. 12 Variation of specific cutting energy with uncut chipthickness at 10 m/min

Fig. 13 Variation of specific cutting energy with uncut chipthickness at 200 m/min

736 / Vol. 128, AUGUST 2006rom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015Similarly, in order to examine the temperature effect at a cuttingspeed of 200 m/min, two sets of orthogonal cutting simulationswere run from 20 to 200 m uncut chip thickness. The first set ofsimulations was run with all terms in Eq. 3 while the second setof simulations was run without the temperature term in Eq. 16and hence in Eq. 3. Figure 13 shows a plot of the specificcutting energy versus uncut chip thickness with and without thetemperature effect. It can be seen that the specific cutting energypredicted by the model with the temperature effect is in goodagreement with the experimental data and captures the size effect.Simulations without the temperature effect show that the specificcutting energy remains fairly constant with reduction in uncut chipthickness. Clearly, at the higher cutting speed the temperature ef-fect is dominant compared to the strain gradient effect, especiallyat large uncut chip thickness values.

The maximum temperatures in the primary and secondary shearzones versus uncut chip thickness are shown in Fig. 14. It can beseen that the maximum temperature in the secondary shear zonedrops by nearly 200C while the maximum temperature in theprimary shear zone remains almost unchanged with a decrease inuncut chip thickness from 200 to 20 m. This directly supportsthe reasoning proposed by Kopalinsky and Oxley 3 and Maru-sich 4 that the size effect in machining at high cutting speedsand large uncut chip thickness is primarily caused by an increasein the shear strength of the workpiece material due to a decreasein the tool-chip interface temperature.

It is also of interest to examine the contributions of the twomaterial strengthening factors at cutting conditions characterizedby high cutting speed and small uncut chip thickness. Under suchconditions, both temperature and strain gradient are expected tocontribute to the size effect in micro-cutting. Therefore, orthogo-nal cutting simulations were run at a cutting speed of 240 m/minfor uncut chip thickness values ranging from 0.5 to 10 m. Fig-ure 15 shows a plot of the specific cutting energy versus uncutchip thickness with and without strain gradient effect. It can be

Fig. 14 Variation of maximum temperature in the primary andsecondary shear zones at 200 m/min cutting speed PSZ: pri-mary shear zone, SSZ: secondary shear zone

Fig. 15 Variation of specific cutting energy with uncut chipthickness at 240 m/min

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AcknowledgmentThis work was supported by the National Science Foundation

Downloaded Fseen that the specific cutting energy predicted by the model withstrain gradient almost doubles when the uncut chip thickness de-creases from 10 to 0.5 m, while only about 10% increase is cap-tured by the model without strain gradient effect. The plot showsthat at a high cutting speed and small uncut chip thickness, straingradient strengthening is more significant than material strength-ening due to a drop in the secondary shear zone temperature.

It is evident from Fig. 16 that the temperature drop at the tool-chip interface is less prominent at high cutting speeds and smalluncut chip thickness, which suggests that the temperature effectcontributes to only a small fraction of the size effect in micro-cutting under such conditions.

ConclusionsWith the aim of understanding the contributions of different

material strengthening factors to the size effect in micro-cutting,this paper focused primarily on two main strengthening factors: istrain gradient strengthening and ii decrease in the secondarydeformation zone cutting temperature, with decrease in uncut chipthickness. These factors were analyzed using a strain-gradient-plasticity-based finite element model of orthogonal micro cuttingthat was verified experimentally for aluminum alloy 5083-H116, amaterial with a small strain rate hardening exponent, thus mini-mizing strain-rate effects. The model was then used to simulatethe size effect in microcutting under conditions where the tem-perature and strain gradient effects are dominant individually. Thefollowing conclusions can be drawn from this work:

The strain-gradient-plasticity-based model of micro-cuttingis able to capture the size effect in specific cutting energy forthe aluminum alloy examined in this paper.

Strain gradient strengthening contributes significantly to thesize effect at low cutting speed and small uncut chip thick-ness 10 m.

Temperature dependence of flow stress plays a dominantrole in causing size effect at relatively high cutting speedsand large uncut chip thickness. The size effect is caused bymaterial strengthening due to a drop in the secondary shearzone temperature.

Strain gradient strengthening is more dominant than thetemperature effect at high cutting speed and small uncutchip thickness. This implies that it is necessary to considerthe strain gradient effect, especially at the micron/submicronlevels.

Although the above conclusions were obtained from the studyof a strain rate insensitive material, similar results are expected forother materials that are strain rate sensitive, since the strain gra-dient is determined by the spatial distribution of strain and not bythe strain rate. The existence and importance of strain gradientshown in this paper suggests that it is necessary to consider thiseffect when modeling micro-cutting processes.

Fig. 16 Variation of maximum temperature in the primary andsecondary shear zones at 240 m/min cutting speed

Journal of Manufacturing Science and Engineeringrom: http://dynamicsystems.asmedigitalcollection.asme.org/ on 07/17/2015through Grant No. DMI-0300457.

Nomenclatureb Burgers vector nm flow stress MPae effective stress deviatoric stressref reference yield stress MPaij Kronecker tensor dislocation densitys statistically stored dislocation densityg density of geometrically necessary dislocationsG shear modulus empirical constant effective strain gradient

ijk strain gradient tensorijk deviatoric strain gradient tensor effective true strain effective strain rateo reference strain ratep effective plastic strain

l characteristic material length

friction coefficients frictional shear stress* limiting shear stress in Coulomb friction modelp contact pressurei local coordinates in mesoscale cell volume of the mesoscale cellQ volume heat fluxT* dimensionless temperature in Johnson-Cook

equation= TT0 / TmT0T0 ambient temperatureTm melting temperatureA material constant in Johnson-Cook equationB material constant in Johnson-Cook equationc material constant in Johnson-Cook equationn material constant in Johnson-Cook equationm material constant in Johnson-Cook equationm material densityCp specific heat capacityK thermal conductivity

wx ,y weighting function in diffuse approximationmethod

px ,y polynomial basis in diffuse approximationmethod

coefficient controlling the shape of weightingfunction

X position vector at point xXi position vector at point xi

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