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Finite Element Method Simulation of Machining of AISI 1045 SteelWith A Round Edge Cutting Tool

Tuğrul Özel and Erol ZerenDepartment of Industrial and Systems Engineering

Rutgers, The State University of New JerseyPiscataway, New Jersey 08854 USA

AbstractIn this paper, FEM modeling and simulation of orthogonal cutting of AISI 1045 steel isstudied by using dynamics explicit Arbirary Lagrangian Eulerian method. The simulationmodel utilizes the advantages offered by ALE method in simulating plastic flow around theround edge of the cutting tool and eliminates the need for chip separation criteria. Johnson-Cook work material model is used for elastic plastic work deformations. A methodologydeveloped to determine friction characteristics from orthogonal cutting tests is also utilizedfor chip-tool interfacial friction modeling. The simulation results include predicted chipformation as well as temperature and stress distributions. These results are highlyessential in predicting machining induced residual stresses and other properties on the

machined surface.

1. INTRODUCTION

Finite Element Method (FEM) based modelingand simulation of machining processes iscontinuously attracting researchers for better understanding the chip formation mechanisms,heat generation in cutting zones, tool-chipinterfacial frictional characteristics and integrityon the machined surfaces. Predictions of thephysical parameters such as temperature and

stress distributions accurately play a pivotal rolefor predictive process engineering of machiningprocesses. Tool edge geometry is particularlyimportant, because its influence on obtainingmost desirable tool life and surface integrity isextremely high. Therefore, development of accurate and sound continuum-based FEMmodels are required in order to study theinfluence of the tool edge geometry, tool wear mechanisms and cutting conditions on theresidual stresses and surface integrity on themachined surfaces. This paper aims to review

the FEM modeling studies conducted in the pastand to develop FEM models for most satisfyingsimulation of the physical cutting process andmost reasonable predictions for cutting forces,temperatures and residual stresses on themachined surface.

In continuum-based FEM modeling, there aretwo types of analysis in which a continuousmedium can be described: Eulerian and

Lagrangian. In a Lagrangian analysis, thecomputational grid deforms with the materiawhere as in a Eulerian analysis it is fixed inspace. The Lagrangian calculation embeds acomputational mesh in the material domain andsolves for the position of the mesh at discretepoints in time. In those analyses, two distincmethods, the implicit and explicit time integrationtechniques can be utilized. The implicit technique

is more applicable to solving linear staticproblems while explicit method is more suitablefor nonlinear dynamic problems.

A vast majority of research has relied on theLagrangian formulation [1,2,3,4,5,6], whichallows the chip to be modeled from incipient tosteady state where as some of the studies alsoused the Eulerian formulation [7]. Howeverusing the Lagrangian formulation requires acriterion for separation of the undeformed chipfrom the work piece. Several chip separation

criteria (e.g. strain energy density, effective straincriteria) have been developed and implemented[8]. Updated Lagrangian implicit formulation withautomatic remeshing without using chipseparation criteria has also been used insimulation of continuous and segmented chipformation in machining processe[9,10,11,12,13,14].

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Arbitrary Lagrangian Eulerian (ALE) techniquecombines the features of pure Lagrangiananalysis in which the mesh follows the material,and Eulerian analysis in which the mesh is fixedspatially and the material flows through themesh. ALE formulation is utilized in simulatingmachining to avoid frequent remeshing for chip

separation [15,16].

Explicit dynamic ALEformulation is very efficient for simulating highlynon-linear problems involving large localizeddeformations and changing contact conditions asthose experienced in machining. The explicit dynamics procedure performs a large number of small time increments efficiently. The adaptivemeshing technique does not alter elements andconnectivity of the mesh. This technique allowsflow boundary conditions whereby only a smallpart of the work piece in the vicinity of the tool tipneeds to be modeled.

Friction in metal cutting plays an important role inthermo-mechanical chip flow and machined worksurface formation. Most of the approach inmodeling friction is to use an average coefficientof friction. This model consisted of the stickingregion for which the friction force is constant, andthe sliding region for which the friction forcevaries linearly according to Coulomb’s law.Interfacial friction at the tool-work contactsinfluences cutting induced residual stresses [17].

FEM simulation of machining usingrounded/blunt/worn edge tools is essential inorder to predict accurate and realistic residualstress, temperature, strain and strain rate fields.Studies focused on predicting residual stresseson machined surfaces especially on finishedmachined hardened steels [18,19].

All of the reviewed work contributed toinvestigate various aspects of fundamentalmodeling of cutting processes. After conductingthe review of the literature, the following

deficiencies of the FEM simulations in machiningare identified:a) Excessive use of non-commercial FEM

codes that makes latest developments highlydifficult to apply by end users.

b) Lack of reliable work material flow stressmodels especially those including strain andstrain history effects as well asmicrostructure-based characterization.

c) Lack of work material models inclusion of microstructure effects under the processing

conditions (strain, strain rate, andtemperature) that must be used as input toany material flow simulation program.

d) Computational difficulties associated withsize of the problems, large solution times andchallenges in frequent remeshing of locamesh densities in applying the simulations to

realistic 3-D machining operations.e) Need for extensive computer time and

engineering effort, making the techniqueuneconomical to use.

2. WORK MATERIAL MODELING

Accurate and reliable flow stress models areconsidered highly necessary to represent workmaterial constitutive behavior under high-speedcutting conditions especially for a (new) materialThe constitutive model proposed by Johnson andCook [20] describes the flow stress of a materia

with the product of strain, strain rate andtemperature effects that are individuallydetermined as given in Equation (1). In theJohnson-Cook (JC) model, the parameter A is infact the initial yield strength of the material aroom temperature and a strain rate of 1/s and εrepresents the plastic equivalent strain. The

strain rate &ε is normalized with a reference

strain rate 0ε & . Temperature term in JC mode

reduces the flow stress to zero at the meltingtemperature of the work material, leaving the

constitutive model with no temperature effect. Ingeneral, the parameters A, B, C , n and m of themodel are fitted to the data obtained by severamaterial tests conducted at low strains and strainrates and at room temperature as well as splHopkinson pressure bar (SHPB) tests at strainrates up to 1000/s and at temperatures up to 600°C. JC model provides good fit for strainhardening behavior of metals and it isnumerically robust and can easily be used inFEM simulation models. JC shear failure modeis based on a strain at fracture criteria given in

Equation 2. Many researchers used JC model aconstitutive equation for high strain rate, hightemperatures deformation behavior of steels (seeTable 1). JC shear failure model is utilized inmodeling and simulating the segmented anddiscontinuous chip formations in cutting of AIS4340 steel [21,22].

( )[ ]

−−

−

++=

m

roommelt

roomn

T T

T T C B A 1ln1

0ε

ε ε σ

&

&(1)

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−−

+

+

+=

roommelt

room

pd

f

T T

T T d ed d

eff 1ln1

0

421

3

ε

ε ε

σ

&

& (2)

Table 1. Johnson-Cook model constants for various steels

3. FRICTION MODELING

Several researchers have used Oxley’s parallel-sided shear zone in which the primary shear zone is assumed to be parallel-sided and thesecondary zone is assumed to be of constantthickness, in order to obtain work flow stressdata.

In order to successfully determine flow stress for JC material model and friction characteristics atthe tool- chip interface, Özel and Zeren [24]proposed some modifications and improvementsto Oxley’s model [25], that includes integration of Johnson-Cook constitutive model as for the flowstress and triangular shaped secondary shear zone as it was confirmed via FEM simulations(see Figure 1). The basic concept of thismethodology is the use of orthogonal cuttingexperiments and inverse solution of Oxley’smodel in order to determine the flow stress andfriction conditions experienced in the range of high-speed cutting.

As commonly accepted, in the tool-chip contacarea near the cutting edge, sticking frictionoccurs, and the frictional shearing stress, τ int iequal to average shear flow stress at tool-chipinterface in the chip, k chip. Over the remainder othe tool-chip contact area, sliding friction occursand the frictional shearing stress can be

calculated using the coefficient of friction µ e. Thenormal stress distribution on the tool rake facecan be described as:

( ) ( )max

1 /a

N N c x x l σ σ = − (3)

Normal stress distribution over the rake face ifully defined and the coefficient of friction can becomputed, once the values of the parameters

max N σ and a are found. The shear stress

distribution on the tool rake face illustrated inFigure 2 can be represented in two distincregions:a) In the sticking region:

int ( ) chip x k τ = , ( ) ,0e N chip P x k x l µ σ ≥ < ≤ (4)

b) In the sliding region:

int ( ) ( )e N x xτ µ σ = , ( ) ,e N chip P C x k l x l µ σ < < ≤ (5)

In Equations 4 and 5, k chip is the shear flow stressof the material at tool-chip interface in the chip.

Based on the methodology detailed in [26], theconstants of the JC material model for AISI 1045steel are computed as A= 451.6, B= 819.5, C=

0.0000009, n= 0.1736, m= 1.0955 for theextended ranges of strain (0.051-1.07), strainrate (1-17766 1/sec.) and temperature (20-72°C). The calculated friction characteristicsinclude parameters of the normal and frictionastress distributions on the rake face as given inTable 2.

Steel Ref. A(MPa) B(MPa) n C m

AISI4340

[21] 950.0 725.0 0.375 0.015 0.625

AISI4340

[22] 792.0 510.0 0.26 0.014 1.03

AISI1045 [23] 553.1 600.8 0.234 0.013 1.000

Tool

Workpiece

Chip

P r i m a r y z o n e

S e c o n d a r y

z o n

e

V

M

A

N

E

C

D

V c

F

t c

δ t c

l pl c

φ B

α

t u

(a) (b) Type I Slip-line field (c) Type II slip-lines field

Figure 1. Illustration of the deformation zones and simulated slip-line fields in orthogonal cutting

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Table 2. Friction characteristics determined for AISI1045 steel using carbide cutting tool

Material AISI 1045Vc (m/min) 100-400tu (mm) 0.125-0.5σ Nmax (MPa) 1312.63l p (mm) 0.639l c (mm) 3.122a 0.183k chip (MPa) 202.95µ e 0.64

4. FINITE ELEMENT MODEL AND ADAPTIVEMESHING

The essential and desired attributes of thecontinuum-based FEM models for cutting are: (1)The work material model should satisfactorilyrepresent elastic plastic and thermo-mechanicalbehavior of the work material deformationsobserved during machining process, (2) FEMmodel should not require chip separation criteriathat highly deteriorate the physical processsimulation around the tool cutting edgeespecially when there is dominant tool edgegeometry such as a round edge or a chamferededge is in present, (3) Interfacial frictioncharacteristics on the tool-chip and tool-workcontacts should be modeling highly accurately inorder to account for additional heat generationand stress developments due friction.

In this paper, a commercial software code, ABAQUS/Explicit v6.4 and ALE modelingapproach is used to conduct the FEM simulationof orthogonal cutting considering round tool edgegeometry and all of the above attributes aresuccessfully implemented in the model. The chipformation is simulated via adaptive meshing andplastic flow of work material. Therefore, no chipseparation criterion is needed.

In the ALE approach, the explicit dynamics

procedure performs a large number of small timeincrements efficiently. The general governingequations are solved both Lagrangian boundaries and Eulerian boundary approachesin same fashion. The adaptive meshingtechnique does not alter elements andconnectivity of the mesh. This techniquecombines the features of pure Lagrangiananalysis in which the mesh follows the material,and Eulerian analysis in which the mesh is fixed

spatially and the material flows through themesh. Explicit dynamic ALE formulation allowflow boundary conditions whereby only a smapart of the work piece in the vicinity of the tool tipneeds to be modeled.

The simulation model is created by including

workpiece thermal and mechanical propertiesboundary conditions, contact conditions betweentool and the workpiece as shown in Figure 2 andgiven in Table 3. The workpiece and the toomodel use four-node bilinear displacement andtemperature (CPE4RT) quadrilateral elementsand a plane strain assumption for thedeformations in orthogonal cutting.

The cutting process as a dynamic event causeslarge deformations in a few numbers oincrements resulting in massive mesh distortion

and termination of the simulation. It is highlycritical to use adaptive meshing with fine tunedparameters in order to simulate the plastic flowover the round edge of the tool. Therefore theintensity, frequency and sweeping of theadaptive meshing is adjusted to most optimumsetting for maintaining a successful mesh duringcutting.

The general equations of motion in explicdynamics analysis are integrated by usingexplicit central difference integration rule with

diagonal element mass matrices, Equation 6.[ ]{ } [ ]{ } [ ]{ } { }M C K P

N N N N u u u+ + =&& & (6)

( ) ( )1

( ) ( ) ( )M P I N NJ J J

i i iu−

= −&& (7)

Where M NJ , P

J , I J are the diagonal element mass

matrix, the applied load vector, and the internaforce vector, respectively.

( ) ( ) N

i

ii N

i

N

i ut t

uu )(

)()1(

21212

&&&&∆+∆

+= +−+ (8)

N

ii

N

i

N

i ut uu )21()1()()1( +++ ∆+= & (9)

( ) ( )1

( ) ( ) ( )C P F N NJ J J

i i iθ −

= −& (10)

N

ii

N

i

N

i t )()1()()1( θ θ θ &++ ∆+= (11)

The expressions for the nodal displacement u

velocities u& , and accelerations u&& , are given inEquations 6 and 7 where ∆t is time incrementu N is a degree of freedom and i is theincremental number in the explicit dynamics step

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[27]. In this case, we see that the systemequations become uncoupled so that eachequation can be solved for explicitly. This makesexplicit dynamic method highly efficient for non-linear dynamics problem such as metal cutting.

During metal cutting, flow stress is highlydependent on temperature fields as wediscussed earlier. Therefore, fully coupledthermal-stress analysis is required for accuratepredictions in FEM simulations. In this analysis,the equations of heat transfer are integrated

using explicit forward difference time integrationwith lumped capacitance matrix as given byEquations 10 and 11, where C

NJ is the lumpedcapacitance matrix, P

J is the applied nodalsource vector, F

J is the internal flux vector, is thetemperature θ N at node N .

In summary, the explicit dynamics method isused mainly because it has the advantages of computational efficiency for large deformationand highly non-linear problems as experience inmachining. Machining, as a coupled thermal-

mechanical process, could generate heat tocause mechanical and thermal effects influenceeach other strongly. In the mean time, workmaterial properties change dramatically as strainrate and temperature changes. Thus, the fullycoupled thermal-stress analysis, in which thetemperature solution and stress solution are alsocarried out concurrently, is applied.

Table 3. Cutting conditions, work and toomaterial properties used in the FEM simulationmodel

Orthogonal Cutting ParametersCutting speed, V c (m/min) 300Uncut chip thicknes, t u (mm) 0.1Width of cut, w (mm) 1Tool rake angle, α (degree) -5Tool clearance angle (degree) 5Tool edge radius, ρ (mm) 0.02

AISI 1045 Workpiece PropertiesCoefficient of thermal expansion(µm /m°C)

11 (at 20 °C)

Density (g/cm3) 7.8Poisson’s ratio 0.3Specific heat (J/kg/°C) 432.6Thermal conductivity (W/m°C) 47.7Young’s modulus (GPa) 200Carbide Tool PropertiesCoefficient of thermal expansion(µm/m°C)

4.7 (at 20 °C)4.9 (at 1000 °C)

Density (g/cm3) 15Poisson’s Ratio 0.2

Specific heat (J/kg/°C) 203Thermal conductivity (W/m°C) 46Young’s Modulus (GPa) 800

5. RESULTS AND DISCUSSIONS

The simulations were conducted and the chipformation process from the incipient to thesteady state was fully observed as shown inFigure 3.

CHIP

TOOL

WORKPIECE

y

x

Lagrangian

V = V

T=T

x c

U = U = 0 , T= Tx y 0

0

Figure 2. Simulation model used for ALE with Lagrangian boundary conditions

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12 µm 25 µm

50 µm 75 µm

Figure 3. FEM Simulation of orthogonal cutting with a round edge tool using ALE with Lagrangianboundary conditions

Figure 4. The stress distributions of σxx and σyy in orthogonal cutting with a round edge tool (predictedstresses are in 10-4 x MPa)

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Figure 3 also shows the temperature predictionsat increments of 12, 25, 50 and 75 µm asmaximum temperatures of 850, 910, 1090 and1120 °C respectively. The distributions of thepredicted the von Mises stress distributions aregiven in Figure 4. The von Mises stress σxx andσyy also represent the residual stress

distributions on the machined surface. From thesimulation results it was observed that there exista region of very high deformation rate around theround edge of the cutting tool. The round edge of the cutting tool and the highly deformed regionunderneath has an dominant influence on theresidual stresses of the machined surface. Thisalso signifies the current work when comparedthe earlier FEM modeling studies that usedcriteria for chip-workpiece separation [28,29].The uses of separation criteria undermine theinfluence of the cutting edge on the residual

stress on the machined surface. In this study, thework material is allowed to flow around the roundedge of the cutting tool and simulated thephysical process.

6. CONCLUSIONS

In this study we have utilized the dynamicsexplicit Arbirary Lagrangian Eulerian method anddeveloped a FEM simulation model for orthogonal cutting of AISI 1045 steel using roundedge carbide cutting tool. Johnson-Cook workmaterial model and a detailed friction model are

also used and work material flow around theround edge of the cutting tool is simulated inconjunction with an adaptive meshing scheme.The simulation of the chip formation,development of temperature distributions as wellas predictions of the stress distributions in thechip, tool and on the machined surface aresuccessfully achieved. This study establishes aframework to further study machining inducedresidual stresses accurately and process designvia optimization of cutting conditions, tool edgegeometry for high-speed machining applications.

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