Vahid Kalhori

Modelling and Simulation of Mechanical Cutting

2001:28

DOCTORAL THESIS

Doctoral thesis

Institutionen för MaskinteknikAvdelningen för Datorstödd maskinkonstruktion

2001:28 • ISSN: 1402-1544 • ISRN: LTU-DT--01/28--SE

Modelling and Simulation of Mechanical Cutting

Vahid Kalhori

Modelling and Simulation of Mechanical Cutting

Vahid Kalhori

The Polhem LaboratoryDivision of Computer Aided Design

Department of Mechanical EngineeringLuleå University of Technology

SE-971 87 Luleå, SwedenVahid.Kalhori@cad.luth.se

Akademisk avhandling som med vederbörlig tillstånd från Tekniska fakultetsnämnden vid Luleåtekniska universitet för avläggande av teknologie doktorsexamen kommer att offentligt försvaras isal E632, Studion i E-huset, fredagen den 26 oktober 2001, kl. 09:30.

Fakultetsopponent: Prof. D.R.J. Owen, Department of Civil Engineering, University of Wales,Swansea.

Academic thesis for the degree of Doctor of Philosophy, which with the due permission of theFaculty Board at Luleå University of Technology will be defended in public, in room E632, StudionE-house, Friday the 26:th of October 2001 at 09:30 am.

External examiner: Prof. D.R.J. Owen, Department of Civil Engineering, University of Wales,Swansea.

I

AbstractThe commercial success of a new product is influenced by the time to market. Shorter product lead-times are of importance in a competitive market.This can be achieved only if the productdevelopment process can be realised in a relatively small time period. New cutting inserts aredeveloped by a time consuming trial and error process guided by empirical knowledge of themechanical cutting process.

One of the state-of-the-art efforts in manufacturing engineering is the finite element simulation ofthe mechanical cutting process. These computational models would have great value in increasingthe understanding of the cutting process and in reducing the number of experiments whichtraditionally are used for tool design, process selection, machinability evaluation, and chipbreakage investigations. This thesis focuses on the development of a finite element model for thecutting process, which can predict chip formation, cutting forces, temperature and pressuredistribution on the tool-chip interface and the residual stresses of the work piece. The work isconcentrated to handle the large and localised deformations, chip formation and contact andfriction.

Two basically different modelling approaches have been used for the chip separation, geometricaland physical model. The physical model has been found to be more suitable to simulate the chipformation. The geometrical model is based on the separation of a pre-defined crack path at a certainlimit of stresses. In the physical model the chip formation occurs through the plastic deformation ofthe elements. The excessive element distortion is handled by frequently updating the finite elementmesh, using the advance front technique for generation of quadrilateral elements. The adaptivemeshing is managed using the error measures based on the stress gradients in the finite elementmesh and also the plastic work rate at each element. The automatic control of the mesh quality ismanaged by using the distortion metric. The implemented combined penalty-barrier contactalgorithm has been found to be efficient in conjunction with the adaptive remeshing.

The effect of previous cutting on chip formation and the surface residual stresses has been studied.The chip formation is not affected much. There is only a minor influence from the residual stress onthe surface from the first cutting on the second pass chip formation. Thus, it is deemed to besufficient to simulate only the first pass. The influence of the cutting speed and feed on the residualstresses has been computed and verified by the experiments. It is shown that the state of residualstresses in the work piece increases with the cutting speed.

Keywords: Adaptivity, Contact algorithm, Finite element method, Friction, Large deformation,Mechanical cutting, Paving, Residual stresses

II

DissertationThis dissertation includes an introduction and the following papers:

I. Kalhori V., Lundblad M., Lindgren L.E., Numerical and experimental analysis of orthogonal metal cutting, ASME 1997 International mechanical engineering congress & exposition, MED Vol. 6-2, Manufacturing Science and engineering, Dallas, Texas 16-21 Nov. 1997.

II. Kalhori V., Edberg J., Lindgren L. E., Coupled thermomechanical simulation of hot rolling, using an adaptive mesh, Simulation of materials processing: Theory, methods and applications, The sixth international conference on numerical methods in industrial forming process-Numiform 98, Enchede, The Netherlands 22-25 June 1998.

III. Kalhori V., Lundblad M., Finite element modelling of orthogonal metal cutting, will be published.

IV. Kalhori V., Effect of previous cutting on chip formation, Simulation of materials processing: Theory, methods and applications, The 7:th international conference on numerical methods in industrial forming process-Numiform’2001, 18-21 June 2001.

V. Lundblad M., Kalhori V., Influence of cutting speed on residual stresses in work piece, will be published.

III

Division of work between the authorsFour of the appended papers were prepared in collaboration with co-authors. The work performed in each paper has been jointly planned by the authors. Furthermore, the author of this thesis has participated in the work according to the following.

Paper IThe present author carried out the finite element simulations and wrote a minor part of the paper.

Paper II

The present author implemented the contact algorithm in conjunction of adaptive remeshing, the stick-slip friction introducing Coulomb friction and the data transfer for mesh adaptivity in the finite element code SiMPle. The author carried out the finite element simulations and compared the results with NIKE2D/PALM2D. The author wrote a minor part of the paper.

Paper IIIThe present author implemented the geometrical model and the continuous remeshing model for chip formation in the SiMPle. The advance front technique for generation of quadrilateral elements and adaptive mesh refinement based on the plastic work rate was implemented. The simulations by ABAQUS and SiMPle was carried out and evaluated by the author. A major part of the paper was written by the author.

Paper VThe present author carried out the finite element simulations by SiMPle and wrote the paper together with the co-author.

IV

AcknowledgementThe present project was started at the Polhem Laboratory on October 1996. The last five years havebeen a period of intense work and excellent learning experience. It has been highly appreciate tohave the opportunity to concentrate on an interesting area of science. There are many people to bethanked.

First of all, I would like to express my deepest appreciation and thanks to my supervisor ProfessorLars-Erik Lindgren for his support, his critical reviews of my work, his highly appreciated guidanceand his friendship during this work. Thank you Lars-Erik.

I wish also to thank Professor Lennart Karlsson, head of division of Computer Aided Design for hisdeeply engagement to bring the industry and the university around the same table. He has beeninitiating the Polhem Laboratory one of the 1VINNOVA's competence centra. There has been also aclose cooperation with SANDVIK Coromant and Volvo Aero Corporation. Their financial supportis gratefully acknowledged.

A very special thank to my co-author Mikael Lundblad at the company SANDVIK Coromant forhis stimulating comments, interesting discussions and valuable collaboration during this research.Mikael has performed the experimental part of this study and also has contributed with performingthe finite element analysis using AdvantEdge, thank you Mikael. I want also take the chance tothank Eric Tjernström at SANDVIK Coromant for his support.

I wish to thank Dr. Jonas Edberg at SUN Micro system for his highly appreciated collaborationwriting the paper II.

Dear friends and colleagues at Luleå University and especially division of CAD thank you forcontributing to a very pleasant working environment.

Finally, I would like to especially thank my dear wife Anita and my dear children Elham, Josef andEmanuel for their support, patience and encouragement during this work. A big, special thank tomy dear parents and also my dear parents-in-law who have always supported me.

Vahid Kalhori

Luleå, September 2001

1. VINNOVA is the Swedish Agency for Innovation Systems

Contents

Abstract IDissertation IIDivision of work between the authors IIIAcknowledgement IV

CHAPTER 1 Introduction 1Background 2The cutting process 2Aim and scope of the present research 4

CHAPTER 2 Metal cutting mechanics 5Cutting forces 6Metal cutting 7Influence of temperature on cutting process 8 Chip formation 10Tool-chip interface 12

CHAPTER 3 Numerical procedure 15Finite element formulations 15Contact algorithms 16Adaptivity and remeshing 17

Local refinement 19Adaptive mesh management 21

Global remeshing 23Data Transfer 26

Chip formation 27

CHAPTER 4 Results and conclusions 29Results 29

Numerical and experimental analysis of orthogonal metal cutting (paper I) 29Coupled thermomechanical simulation of hot rolling, using an adaptive mesh (paper II) 30Finite element modelling of orthogonal metal cutting (paper III) 30Effect of previous cutting on chip formation (paper IV) 30Influence of cutting speed on residual stresses in work piece (paper V) 31

Conclusions and discussion 31References 33

1

CHAPTER 1 Introduction

The study of metal cutting focuses on the features of the behaviour of tool and work materials thatinfluence the efficiency and quality of cutting operations. Development of cutting tool materials hasheld a key position. The technology of metal cutting has been improved by contributions from allthe branches of industry with an interest in machining. Productivity has been increased throughreplacement of carbon tool steel by high-speed steel and cemented carbide which allowed cuttingspeeds to be increased by many times. The special properties required for cutting machine steel athigh speed have led to the development of the most advanced tool materials. This developmentcontinues today with the use of ceramic with multiple coating technology and ultra-hard toolmaterials. Machine tool manufactures have developed machines capable of making full use of thenew tool materials, with computer control (CNC), and transfer machines greatly increase the outputper worker employed. Tool designers have optimised the shape of tool to give long tool life at highcutting speed. Lubricant manufactures have developed many new coolants and lubricants toimprove surface finish and permit increased rates of metal removal.

Today, metal cutting is a very large segment indeed of our industry. The car industry, electricalengineering, railways, shipbuilding, aircraft manufacture and the machine tool industry itself, allthese have large machine shops with many thousands of employees engaged in machining.Therefore, in order to further increase efficiency and reduce costs it is necessary to improve ourunderstanding of the metal cutting process.

Introduction

2

1.1 BackgroundThe commercial success of a new product is strongly influenced by the time to market. Shorterproduct lead-times are of importance for industry in a competitive market.This can be achievedonly if the product development process can be realised in a relatively small time period. However,the development of new cutting inserts is usually occurred through the time consuming trial anderror iterations, caused by limited, mostly empirical knowledge of the mechanical cutting process.Usually, the material removal occurs in a highly hostile environment with high temperature andpressure, in the cutting zone. this make the study of cutting process very complicated.

The objective of metal cutting studies is to establish a predictive theory that would enable us topredict cutting performance such as chip formation, cutting force, cutting temperature, tool wear,and surface finish. The ultimate objective of the science of metal cutting is to solve practicalproblems associated with efficient material removal in the metal cutting process. To achieve this,the principles governing the cutting process should be understood. A knowledge of these principlesmakes it possible to model, simulate and thereby to predict the practical results of the cuttingprocess and thus to select the optimum cutting conditions for each particular case.

One of the state-of-the-art efforts in manufacturing engineering is computer simulation of themachining process to predict power requirements, cutting forces, chip formation and residualstresses using numerical models. The finite element method has been used. These computationalmodels would have great value in increasing the understanding of the cutting process and inreducing the number of experiments which traditionally are used for tool design, process selection,machinability evaluation, and chip breakage investigations.

1.2 The cutting processMetal cutting is a process where components are arranged so that the applied external force causesthe fracture, Figure 1.1. This fracture occurs due to the combined bending stress, the component S,and the shearing stress due to compression, Q. The presence of the bending stress in thedeformation zone distinguishes the process of metal cutting from other deforming and separatingmanufacturing process. System consideration of the metal cutting process reveals that thecompetition between deformation hardening and thermal softening in the deformation zoneconstitutes a cyclical character of the chip formation process. As a result, the parameters of thecutting system vary over each chip formation cycle.

3

The cutting process

FIGURE 1.1 The interaction between tool rake face and the chip. The penetration force Pacts on the chip, causing the compressive force Q and the bending force S.

The orthogonal metal cutting process, is the focus in the present work, Figures 1.2. The chosengeometry provides a reasonably good modelling of the chip formation on the major cutting edge ofmany metal removal processes such as turning, milling, drilling, sawing, grinding, etc. One of theimportant parameters in the orthogonal metal cutting process is the rake angle, γ, between the faceof the cutting tool and the plane perpendicular to the cutting direction. The magnitude of rake anglehas significant affects on the performance of the cutting tool, chip formation and the integrity of thecut surface.

FIGURE 1.2 Schematic sketch of Orthogonal cutting with used notations

Q

P

S

γ

h2

Φ

α

h1

(a) (b)

Introduction

4

1.3 Aim and scope of the present researchThe main objective of the research presented in this thesis is the development of a finite elementtool to give a better understanding of the cutting process. The effect of the cutting parameters on thetool and the workpiece, as the process zone will be studied in detail. Additionally, the residual stressstate of the workpiece will be calculated.

Different finite element modelling and simulation approaches for the cutting process will bedeveloped and evaluated. The chosen geometry (orthogonal cutting) makes it possible to use a 2-Dmodel. Experiments will be performed in order to evaluate the numerical method.

In order to obtain an accurate prediction of mechanical cutting process, the finite element modelshould be able to deal with the following physical phenomena:

* Extremely large and localised deformations* Crack initiation and chip formation* Contact and friction* Complex material behaviour due to thermal and viscous effects* Ductile and brittle fracture

Different finite element codes are evaluated. The commercial FE code, ABAQUS/ Standard(implicit), for general analysis and special code for cutting simulation, AdvantEdge (explicit),together with a research code, SiMPle (implicit), are used. The focus has been to develop SiMPle tobecome an efficient tool for simulation of mechanical cutting process. The adopted constitutivemodel in ABAQUS [63] and SiMPle is the elasto-plastic von Mises model. In the model withAdvantEdge a power viscosity law with constant rate sensitivity [62] was used. The present thesisconcentrates to resolve the first three problems: large and localised deformations, chip formationand contact and friction.

5

CHAPTER 2 Metal cutting mechanics

The major deformations during the machining process are concentrated in two regions close to thecutting tool edge. These regions are usually called the primary and the secondary deformation zone,see Figure 2.1. The primary deformation region extends from the tip of the cutting tool to thejunction between surface of the undeformed work material and the deformed chip. The work pieceis subjected to large deformation at a high strain rate in this region. The heat generated is due to

FIGURE 2.1 Computed strain-rate (a) and heating (b) in deformation zones during orthogonal cutting of AISI 1045. The cutting speed is 198 m/min and feed is 0.25 mm.

(a) (b)

Metal cutting mechanics

6

the plastic deformation. At the secondary deformation zone, heat is generated due to the plasticdeformation and friction between the cutting tool and the chip.

The secondary deformation zone may be divided into two regions [52,61], the sticking region andthe sliding region, Figure 2.2. In the sticking region, the work piece materials adheres to the tooland shear occurs within the chip, the frictional force is high and so is also the heat generation. Thiswill be treated more in detail later in the current chapter. The highest temperature in the chip usuallyoccurs in the sliding region.

FIGURE 2.2 Definition of the primary and the secondary deformation zone and the sliding & sticking region.

2.1 Cutting forcesThe cutting forces vary with the tool angles, feed and cutting speed. Knowledge about the forcesacting on the cutting tool may help the manufacture of machining tool to estimate the powerrequirement.

The force components in lathe turning can be measured in three directions, Figure 2.3. Thecomponent of the force acting on the rake face of the tool, normal to cutting edge, in the directionOY is called the cutting force Fc. This is usually the largest force component, and acts in thedirection of the cutting velocity. The force component acting on the tool in the direction OX,parallel with the direction of feed, is referred to as the feed force, Ff. The third component, acting inOZ direction, push the cutting tool away from the work in the radial direction. This is the smallestof the force components

Chip

Slidingregion

Stickingregion

Primarydeformation zone

Secondarydeformationzone

work piece

Insert

7

Metal cutting

FIGURE 2.3 Cutting forces acting on the tool in a semi-orthogonal cutting

2.2 Metal cuttingThe major concern in dealing with fracture of engineering materials is to define conditions underwhich material should not fracture or fail in service. In contrast, in metal cutting, the opposite effectis desired, how to fracture the material with a minimum effort. Therefore, the studies of the fractureprocess in metal cutting applications should answer how the cutting parameters may be optimisedto result in crack initiation and propagation in the work piece. Fracture consists of two phases,initiation and propagation. It is usually classified as ductile or brittle fracture. Brittle fracture[12,22] is characterized by a rapid rate of crack propagation with no gross plastic deformation. Thetendency for a brittle fracture increases with decreasing temperature and increasing strain rate. It isunderstood that, in metal cutting, this is the best type of fracture, as it minimizes the consumedenergy and tool wear. Toughness should be considered of prime importance [2], among otherimportant parameters characterizing the behaviour of a material in cutting. The toughness of amaterial is defined as its ability to absorb energy in the plastic range. One way of studyingtoughness is to consider it as the total area under the stress-strain curve [12] in a tensile test. Thisarea is an indication of the amount of work per unit volume which must be done on the material tocause its fracture. The total work [2] per unit volume done in cutting Wc may be defined as follows:

Wc = τf .γ (EQ 2.1)

where, τf is the shear stress at fracture, and γ is the final true shear strain.

Y

Z

O X

F

Metal cutting mechanics

8

Two important conclusions can be drawn from this equation. First, the specific work done in cuttingdepends on the fracture true shear strain, because the shear stress at fracture of the strain-hardenedwork piece material depends also on this strain. Second, the high temperatures and strain-rates thatoccur in cutting will affect τf and must therefore affect Wc.

2.3 Influence of temperature on cutting processThe effect of temperature on the stress-strain relationship and the flow and fracture properties iswell known. In general the strength of the material decreases and ductility increases as thetemperature is increased.

In cutting operation the heat transfer is strongly dependent on the cutting velocity. At very lowcutting speeds there may be adequate time for conduction to occur. At the other extreme, at veryhigh cutting speeds there is nearly no time for heat conduction and adiabatic conditions may existwith high local temperatures in the chip. For the work piece material of AISI 1045 and the cuttingdata as presented in Table 2.1, the estimated time for a microvolume to pass the deformation zonesis about 350 µs. Since this is extremely short time, the heat conduction can be neglected. Zorev [61]and Shaw [48] assumed adiabatic conditions. This means that heat generated in the primarydeformation zone and the average temperature T in this region are proportional to the specific workfor metal removal Wc. This proportionality is shown in Figure 2.5 and 2.6. The increase oftemperature in the chip is related to the increase of plastic deformation and thereby Wc. The averagetemperature can be calculated as:

(EQ 2.2)

where ρ is the density of work piece material, c is the specific heat and T0 is the temperature prior todeformation. The computed heat generation is shown in Figure 2.4. The temperatures within theprimary deformation zone vary from about 500o C, close to the cutting tool, to about 150o C aroundthe point F.

TABLE 2.1 Cutting data for AISI 1045

Cutting speed v 3.3 ms-1

Feed f 0.25 mm

Chip thickness ratio Λ 1.98

Contact length lc 0.58 mm

Chip velocity vc = v/Λ 1.67 ms-1

TWc

ρc------- T0+=

9

Influence of temperature on cutting process

FIGURE 2.4 Computed heat generation for AISI 1045. The cutting speed is 198 m/min and the feed is 0.25 mm. P1 to P8 illustrates the path of a microvolume passing through the deformation zones.

FIGURE 2.5 The computed temperature in deformation zones. P1 to P6 are shown in Figure 2.4

Metal cutting mechanics

10

FIGURE 2.6 The computed plastic strain in deformation zones. P1 to P6 are shown in Figure 2.4

2.4 Chip formationThere are four types of chips that can be formed in metal cutting process, Figure 2.7. Thediscontinuous chip is formed by a fracture mechanism when brittle materials are cut at low cuttingspeeds. The continuous chips are formed without a built-up edge on the tool. This is characteristicfor cutting ductile materials under steady-state conditions. However, long continuous chips causehandling and removal problems in practical operations. Under conditions of low cutting speedswhere the friction between the chip and the rake face of the tool is high the chip may weld to thetool face. This accumulation of chip material is known as built-up edge. Finally the last type of thechips are macroscopically continuous chips consisting of narrow band of heavily deformed materialalternating with larger regions of relatively undeformed material. These shear localized chips canbe formed when the yield strength of the work piece decreases with temperature, under the properconditions, rapidly heated material in a narrow band in front of the tool can become much weakerthan the surrounding material, leading to localized deformation. This type of chip is obtained whencutting hardened and stainless steels and titanium alloys at high cutting speeds.

A

BC D E F

11

Chip formation

FIGURE 2.7 Four basic types of chips: (a) discontinuous, (b) continuous, (c) continuous with built-up edge, BUE (d) shear localized.

The stress in front of the cutting edge is increased when the cutting tool starting to advance into thework piece [3], see Figure 2.8-a. When this stress reaches a certain maximum limit, the followingmay happen:

* If the work piece material is brittle, a crack appears in front of the cutting edge which it finally causes fracture, Figure 2.8-b.

* If the work piece material is ductile, a certain elastoplastic zone forms in the work piece, Figure 2.8-c. The dimensions of the plastic and elastic parts of this zone depend on ductility of the work piece material.

FIGURE 2.8 Cutting tool starting to advance into the work piece.

However, two issues are important for the determination of the fracture, the first is where thefracture occur and the second what type of fracture (brittle or ductile). It is shown [2] that the

a b

c d

P P P

(a) (b) (c)

Metal cutting mechanics

12

limiting stress in the work piece material may occur in one of two regions, Figure 2.9. They areregion A, along the surface separating the work piece and the layer being removed, and region B,along the surface of maximum combined stress.

FIGURE 2.9 Zones where the limiting stress can occur

Region A is a region of high combined shear and normal stress. The shear stress stems from thedirect action of the cutting edge whereas the normal stress is due to the “tearing off” a layer of metalwhen the material of the chip is forced upwards. Consequently, the deformation level here is highdue to high intensity of the plastic flow of the material in the presence of a crack. It is believed thatfracture in metal cutting starts [2] from region A, and further development depends on a givencombination of the mechanical properties of work piece material, tool geometry, and cutting regimeused.

2.5 Tool-chip interfaceFriction along the tool-chip contact during the cutting process is a very complex phenomenon. Thefriction influences the chip formation, built-up edge formation, cutting temperature and tool wear.Therefore it is necessary to understand the friction mechanism across the faces and around the edgeof the tool, in order to be able to develop accurate models for cutting forces and temperature.

The Coulomb friction model state

τ = µ σ (EQ 2.3)

where τ is the frictional shear stress and σ is the normal stress to the surface. Usually the frictioncoefficient µ is assumed to be constant for a given interface. However, this proportionality resultsfrom the fact that real solid surface is not completely flat on a molecular scale and the contact area

B

A

13

Tool-chip interface

is much smaller then the apparent area of the sliding surfaces. The solids is in contact at the top ofthe hills, Figure 2.10.

FIGURE 2.10 The contact area at low contact pressure

In the metal cutting the contact pressure at the tool-chip interface will become several times theyield stress of the work piece material, Figure 2.11. In this extreme case, the real contact betweenthe tool and work piece is so nearly complete over a large part of the total area of the interface, thatsliding at the interface is impossible [52]. Therefore, the frictional force becomes that required toshear the weaker of the two material across the whole interface. This force is almost independent ofthe normal force, but is directly proportional to apparent area of contact, a relationship directlyopposed to that of classical friction concepts.

FIGURE 2.11 Computed pressure in orthogonal cutting of AISI 1045. The cutting speed is 198 m/min and feed is 0.25 mm.

Metal cutting mechanics

14

15

CHAPTER 3 Numerical procedure

3.1 Finite element formulationsIt is obvious that the cutting process requires a large deformation analysis. The deformations andstrains are extremely large. The use of remeshing is discussed later in this chapter. Furthermore, itrequires the simultaneous calculation of temperatures and deformations. The high rate of materialflow makes it possible to assume adiabatic heating. This assumption makes it possible to directlycompute the temperature increase in a point due to the plastic dissipation at that point. Then nofinite element solution of the field equation for the heat conduction is required. This assumption hasbeen used in [27]. This approach cannot be used if the cooling to room temperature is needed inorder to evaluate residual stresses of the work piece. Some analyses have been done using a coupledthermal-mechanical analysis. Then a heat conduction analysis is coupled with the mechanicalanalysis. A so-called staggered approach is preferred and has been used in [8,33,41,49].

The mechanical analysis can be performed using different formulations. Each formulation has itsadvantages and disadvantages. There is the possibility to treat the material as a non-Newtonianfluid. This is called a flow formulation. Thus no elastic strains are included in the material modeland residual stresses cannot be obtained directly. This approach has been used in [8,40,47]. Themore general option is the solid formulation where the material is modelled by the usualconstitutive equations for solid material.

There exists a large number of different constitutive models. There exist two main types of methodsfor the temporal discretisation. The explicit, conditionally stable, method. This is a fast method butthe limit on the time step is related to the time it takes for an acoustic wave to pass through the

Numerical procedure

16

smallest element. It is a dynamic analysis in the sense that it only uses a mass matrix and computesthe change in displacements from the current acceleration. It is an effective method as the massmatrix is always lumped and thereby there is no need to solve a set of coupled equations. It cannotbe used easily to obtain steady state results. However, it has been used with success for solvingstrongly non-linear problems where inertia is negligible by artificially increasing the density andthereby make it possible to take larger time steps [31]. Owen et al. [41], Marusich [33] andHashemi [19] have used this approach. The other, more general option is to use an implicit finiteelement code. The length of the time step is then governed only by accuracy requirements. On theother hand, a set of coupled equations must be solved. This method can be used with or without amass matrix. It has no problem in computing steady state response. This type of formulation hasbeen used in the cutting simulations in [22,27-28,56].

Different material models options have been used in simulations of cutting using the solidformulation. The material has been treated as rigid-plastic by [19,32]. This choice makes it possibleto compute cutting forces but no residual stresses can be obtained. Others have used a thermalelastic plastic material model with or without rate dependency. The plastic behaviour was assumedto be rate-dependent in [33,37,41,47].

It is preferred to have more linear elements in non-linear problems compared to fewer high orderelements as the plastic deformation implies a more rapid changing deformation behaviour. A highorder element enforces a higher degree of continuity in the solution. Furthermore, the largeincompressible plastic strains can give locking effects for low order triangular elements in 2-D andtetrahedron elements in 3-D [5,9]. Therefore, well known codes like the public domain codes NIKEand DYNA from Lawrence Livermore National Laboratory has only linear quad elements in 2-Dand hexahedron elements in 3-D. The four node quads with reduced integration for the volumetricfield are used in the present work. The four node quad element is also used in[8,22,28,32,37,41,47,49,56]. The six node triangular elements are used in [33,47].

3.2 Contact algorithmsTwo main algorithms for solving contact problems are presented; the penalty approach[11,16,20,36,43,44,55], and the method of Lagrangian multipliers [36]. Laursen and Simo [29] alsodescribe a combined method called the augmented Lagrangian technique. Other procedures, suchas the perturbed Lagrangian method [26,50], can also be applied. Other important issues areassociated with contact detection [4,16-18,39]. The latter is simpler with lower order elements and,also for this reason these low-order elements are popular. Many of the low-order contactformulations have origins in the none linear slide-line procedure presented by Hallquist [16-18]. Inrelation to these techniques, important issues are associated with the use of one-pass or two-passalgorithms. In particular, Taylor and Papadopoulos have shown [51] that a two-pass formulation isessential if the contact path test [10] is to be passed. Additional special procedures have been

17

Adaptivity and remeshing

developed for the explicit integration method [16,4,17,18], as momentum-related techniques inwhich modifications are made to the acceleration, velocities and displacements. One of the aims ofthe latter is to avoid the penalising effect on the time step of the explicit procedure, which can beintroduced by the high stiffness, associated with penalty approaches.

Normal forces are obtained by insertion of a contact stiffness when a material point on 'slave'surface penetrates into the target body 'master', in the penalty formulation. However, theimpenetrability condition of contact bodies is not exactly fulfilled in the penalty approach. Thismay cause significant changes in surface behaviour, which in turn causes divergence. A furtherdisadvantage of the penalty approach is that, in the early iterations, while a node is oscillatingbetween being in and out of contact, the convergence characteristics can be very bad. One way ofalleviating the numerical difficulties is to avoid sudden change in stiffness that occurs when apreviously contacting node moves out of contact. This can be achieved using a method proposed byZavarise et al. [57], which combines a penalty procedure with a barrier method [13].

In contrast to the penalty formulation, the Lagrangian multiplier algorithm ensures exactsatisfaction of the required contact constrains [10]. The multipliers can be interpreted as contactforces. However, there are other disadvantages. First, there is the need to increase the number ofvariables with the Lagrange multipliers, and secondly special care must be taken when solving theequations, as the diagonal terms in the matrix for these extra equations are zero. However, it ispossible to combine the penalty and Lagrangian multiplier methods, the augmented Lagrangiantechnique [13,29], with the aim of retaining the good merits of each approach. One of the mainadvantages of using this method rather than the penalty algorithm is the sensitivity of the latter tothe selection of the penalty stiffness. If the penalty parameter is too high, then a very poorconvergence rate may result. While one can reduce the penalty parameter to avoid ill conditioning,the inevitable consequence is penetration and hence a violation of the real physical situation. Thiscan be avoided by using the augmented Lagrangian method.

The augmented Lagrangian formulation, and combined penalty-barrier formulation areimplemented in this study. The penalty formulation was found most convenient. The contactalgorithm is implemented as a two-pass formulation. Each side of the contact is in turn treated asthe master surface. The stick-slip formulation [10] was implemented for contact friction. Thethermal contact uses a contact resistance when the surfaces are in contact. No radiation is includedin the heat transfer between the surfaces.

3.3 Adaptivity and remeshingThe need of adaptive strategies for large deformation finite element computations is undeniable.For many cases, adaptivity is an essential tool to obtain accurate numerical solutions. It will alsoreduce the required computational effort needed to achieve this accuracy. This is the case, for

Numerical procedure

18

instance, with problems in non-linear solid mechanics involving localised large strains. Meshadaptivity is divided into three different types of categories, h-adaptivity, p-adaptivity and r-adaptivity. H-adaptivity consists on changing the size h of the finite elements. The new mesh hasdifferent number of elements and the connectivity of the nodes is changed. In p-adaptivity thedegree of the interpolating polynomials is changed. R-adaptivity consists of relocation of the nodes,without changing the mesh connectivity.

Mesh adaptivity, for an efficient meshing, requires two essential ingredients, a tool for assessing theerror of the solution computed with a given mesh and an algorithm to define a new spatialdiscretisation. A widely used error estimator was proposed by Zienkiewiecz et al. [59-60]. McDillet al. [35] and Hyun et al. [23,46] proposed a generic error estimate for linear elements based on thegradients of displacement or heat flux. The error is associated with the gradients which arediscontinuous at interelement boundaries. This means that a mesh would be refined where largedifferences in the gradients exist between elements. Marusich and Ortiz [33] proposed to use theplastic work rate in each element in order to refine the finite element mesh in cutting simulations.This option convey to capture the plastic deformation of the work piece material. Owen et al. [41]proposed an error estimator based on the rate of fracture indicators. This should not only capture theprogression of the plastic deformation but also provide a fine mesh at regions of possible materialfailure. Owen discussed also other error criteria based on total plastic work, uncoupled integrationof Lemaitre’s damage model and total damage work. All this criteria was followed by using theprocedure originally proposed by Zienkiewicz and Zhu [60].

The delaunay triangulation for generating six-noded quadratic elements was used by Marusich et al.[33] and Sekhon et al. [47]. Refinement of the mesh in [33] was executed by adding more nodes toelements which satisfy the plastic work rate criteria. A hierarchical adaptive mesh scheme [15],used in this study, provides an efficient way to change the mesh density by refining/coarsening agiven mesh and facilitates data transfer. This so-called graded element reduces the requiredcomputational effort by locally grading the mesh where large gradients in the solution occur. Thisdoes not reduce the element distortion.

Mesh adaptivity is also important for reducing the distortion of the elements as this is a largeproblem in simulating the mechanical cutting process, using a Lagrangian finite elementformulation. Then tools for assessing the element distortion and for remeshing are needed. Afrequent updating of the mesh in which the element topology is either changed, or preserved isrequired. In the present studies, h- and r-adaptivity are combined; this is necessary as it was foundthat the application of r-adaptivity only is not sufficient to maintain the mesh quality. During the r-adaptivity or smoothing process the nodes are shifted to more favourable positions [23-24]. Themost useful smoothing technique in the current application is the so-called opti-smoothingalgorithm [24], where one tries to optimise the node positions using an element distortion metric[38]. One disadvantage for this method is the time consuming optimization procedure. A combinedLaplacian-area weighted method [23-24] has been adopted for the smoothing process, in this study.The smoothing technique is also combined with an h-adaptive remeshing using the implemented

19

Adaptivity and remeshing

hierarchical adaptive mesh scheme [15] in order to provide a dense mesh near the tip of the movingtool. These procedures are applied regularly during the simulations. However, it is also necessary tocompletely regenerate the mesh now and then. The advance front technique [6-7] for generating aquadrilateral FE-mesh was used for this.

3.3.1 Local refinement

The graded quadrilateral element proposed by McDill [34] is used in this study. The 4 to 8-nodedgrading quadrilateral is shown in Figure 3.1. Nodes 1 to 4 are mandatory while nodes 5 to 8 areoptional midedge nodes.

FIGURE 3.1 Numbering of 4- to 8-noded graded element

The midedge nodes can be added in any required combination and no constraint equations orassociated matrix operation are needed. The compatibility is inherent in the piece wise bilinearshape functions. These are formulated in a straightforward manner using the standard rules forisoparametric elements, Table 3.1.

TABLE 3.1 Shape functions for a 2-D graded element

Node P (ξ , η)

8 0.5 (1 + ξ) (1 − |η|)

7 0.5 (1 − |ξ|) (1 − η)

6 0.5 (1 − ξ) (1 − |η|)

5 0.5 (1 − |ξ|) (1 + η)

4 0.25 (1 + ξ) (1 − η) − 0.5 (P7 + P8)

3 0.25 (1 − ξ) (1 − η) − 0.5 (P6 + P7)

2 0.25 (1 − ξ) (1 + η) − 0.5 (P5 + P6)

1 0.25 (1 + ξ) (1 + η) − 0.5 (P5 + P8)

ξ

η

52 1

3 7 4

6 8

Numerical procedure

20

The presence of midedge nodes affects the basis corner nodes on that edge. Therefore, the shapefunctions should be evaluated in order from P8 to P1. If a node is not active then the value of theshape function and its derivative are zero everywhere, i.e., all subsequent to the shape function isremoved.

Propagation of refinement in a graded mesh is illustrated in Figure 3.2. If a vertex node in theelement to be refined is also a midedge node in a neighbour then the refinement must propagate intothe neighbour. The neighbour may in turn initiate propagation in its neighbours etc.

FIGURE 3.2 Propagation of refinement. a) Original mesh b) Mesh after propagation

The parent element may be refined in two direction, then four new children are created, Figure 3.3-a. However, in practical application of the finite element method the elements might be elongateddue to the locally extreme deformation. Then it might be beneficial to divide the element in such away that a better aspect ratio would be obtained, Figure 3.3-b.

FIGURE 3.3. Refinement scheme in a) x- & y-direction b) x-directions

(a) (b)

(a)

(b)x

y

21

Adaptivity and remeshing

3.3.2 Adaptive mesh management

The adaptive remeshing is managed by using two different error estimation criteria simultaneously.The plastic work rate and a generic posterior error estimate based on the gradients in finite elementmesh are used.

The plastic work rate criteria was used to capture the progression of the plastic deformation in theprimary and the secondary deformation zone, Figure 3.4. The plastic power density in eachelement, , and the averaged plastic power of the finite element mesh, , are given by

(EQ 3.1)

(EQ 3.2)

where is the effective stress and is the plastic strain rate and n is the total number of elementsin the FE-mesh. The refinement procedure is manage by comparing the with the averagedplastic power of the finite element mesh as,

(EQ 3.3)

where βe is the proportionality factor for each element. When the βe reaches a certain problem-dependent value then the element will be refined.

FIGURE 3.4 The mesh refinement using the plastic work rate criterion

Wl· p

W· avr

Wlp·

σε·p

=

W· avr

Wl· P

( )i

i 1=

n

∑n

-------------------------=

σ ε·p

Wl· p

βeWl· p

W· avr------------=

Numerical procedure

22

The generic posterior error estimate proposed by Hyun & Lindgren [23] was used. This criterionwas applied to control the mesh density at the region of cutted work piece material in order tocapture the region with high stress gradient, Figure 3.5-b.

The error measures are related to the gradients of the stresses. The local error, el, is based on thedifference between the local FE computed stresses and the exact solution, Eq. 3.4. Since the exactstresses are not available during the FE analysis the σexact is replaced by the smoothed continuousgradient field σsm, Eq. 3.5. The smoothed gradients are obtained by averaging the nodal gradientsfrom the Gauss points.

el = σexact - σFE (EQ 3.4)

el = σsm - σFE (EQ 3.5)

where σsm is the smoothed value from the finite element analysis. The local error measures arecompared with the global error measures, Eq. 3.6, to manage the adaptive process.

(EQ 3.6)

These two error measure criteria are frequently used during the simulation of mechanical cutting inthe current studies, Figure 3.5.

FIGURE 3.5 The refined finite element mesh based on a combination of the plastic work rate and the displacement gradients. a) Error based on the plastic work rate b) Refined mesh

(a) (b)

eg el2

1

n

∑=

23

Adaptivity and remeshing

3.3.3 Global remeshingThe advance front technique [6-7] for mesh generation has been found most convenient insimulating mechanical cutting, using Lagrangian finite element formulation. The need of meshregeneration is evaluated at the start of each increment by computing the elements distortion withrespect to the shear angle and elongation. This is managed by using the distortion metric proposedby Oddy [38]. It is given as,

(EQ 3.7)

where

(EQ 3.8)

n is the dimension of the problem. This metric is size and orientation independent and combineseffects of stretching and shearing of elements.

The advance front technique, so called paving, is an iterative procedure. The elements are addedone by one to the meshing area along the exterior paving boundaries, Figure 3.6.

FIGURE 3.6 A simple paving sequence and the paving boundaries

D Cij2 1

n--- Ckk

k 1=

n

∑ 2

–j 1=

n

∑i 1=

n

∑=

Cij1

det J--------------- JkiJkj

k 1=

n

∑=

Numerical procedure

24

As the elements are added, fronts may be subdivided or closed when they are filled by the lastelement. When the number of fronts is equal to zero, mesh generation is completed. The size ofelements is decided by the adjacent edge length of the exterior paving boundary, Figure 3.7 and Eq.3.9.

(EQ 3.9)

FIGURE 3.7 Calculation of desired element size

The propagation of paving boundaries occur through a number of operations which are necessary toensure the mesh quality. These operations could be summarized as below.

• Row choice. The beginning and ending node of the next sequence or row of elements to be added is found.

• Closure check. A check is made to ensure that more than six nodes remain in the paving boundary.

• Row generation. The next row of elements identified in the row choice step is incrementally added to the boundary.

• Smooth. The position of nodes is adjusted to improve mesh quality and paving boundary smoothness.

• Seam. Small interior angles in the paving boundary are seamed or closed by connecting opposing elements.

• Intersection. The paving boundary is checked for intersections with itself or other paving boundaries. Intersection are connected to form new, often separate paving boundaries.

• Clean up. The completed mesh is adjusted where element deletion and/or addition improves the overall mesh quality.

The distortion metric is used to ensure a good element quality. This is done during the rowgeneration and step 4. Typically, the average element distortion in a mesh is around 0.1-0.2 with amaximum distortion of order 1.0. There are some situations that cause problem for paving process.

hd1 d2+( )

2 α2--- sin

--------------------------=

α h

d1 d2

25

Adaptivity and remeshing

These can be transition from a denser region to a more coarsen region and also the row intersectionwhen a coarser edge is intersecting a number of finer element edge, Figure 3.8. This problem can beresolved by changing the size of elements along the exterior paving boundary, Figure 3.10. Thecoarser elements may be refined, Figure 3.9. In this case both the average distortion metric for theFE mesh and maximum distortion metric, may be increased to 1.5 respective 15.0. It may also beuseful to smooth the exterior paving boundary by the spacing function Eq. 3.10.

(EQ 3.10)

where h is the desired element size, hi is the node spacing value in Eq. 3.9, and |X-Xi| is the distancebetween the position of interest and the position of the boundary nodes.

FIGURE 3.8 Intersection between a finer and a coarser boundary section

h

hihi

X Xi–-----------------

i 1=

n

∑

hiX Xi–-----------------

i 1=

n

∑

--------------------------------------=

(a)

(b)

(b) (a)

Numerical procedure

26

FIGURE 3.9 Refinement scheme for exterior paving boundary a) One step b) Two steps

FIGURE 3.10 A paving sequence. The averaged metric for the final mesh is 1.2 and the maximum metric is 14.6 a) Seaming of small interior angles b) The exterior paving boundary refinement c) The Final mesh.

3.3.4 Data TransferThe transport of data from the old mesh to the new mesh is performed by interpolation. The nodaldata of new mesh, such as displacement and temperature, are determined by

(EQ 3.11)

where Di are the nodal value of the new node and Pj are the shape function of the element in the old

mesh. D*j are the nodal value of the element in the old mesh and n is the number of nodes for

element in the old mesh. The state variables of the integration points are transferred from the oldmesh with the same procedure. This can be outlined as follows.

Find the location (element and local isoparametric coordinates) of the new integration points in theold mesh. To obtain the state at each new integration point, extrapolate the state variables defined

(a) (b) (c)

Di PjD∗ j

j 1=

n

∑=

27

Chip formation

on the integration points of the old element, to the nodes of element. The state variables of the newintegration point are then obtained by interpolation.

3.4 Chip formationA diversity of physical phenomena, such as large plastic deformations, heat generation, friction,damage etc. exists in the process zone where the chip separation occurs. Different numericaltechniques for modelling chip separation can be used. These can be divided into two categories,geometrical or physical. The geometrical model is usually based on a tied slide-line interface [22],which debonds when a certain criterion is fulfilled. This criterion may be a certain level of stress,strain or simply when the cutting edge is close enough to the front nodes. Huang and Black [22],and Komvopoulos and Erpenbeck [28], Zang and Bagchi [58], have used this model. However,within these studies, only the mechanical behaviour of cutting was considered and no temperaturechange was computed.

The physical models are based on the physical behaviour of material, such as plastic deformation,crack initiation and crack propagation without pre-determining its path. Most of the expandedenergy during cutting is dissipated by plastic deformations. Thus, this process in the chip formationis the most important contribution to the cutting forces. Continuous remeshing is necessary in orderto simulate this. Other phenomena may also be involved in the chip formation. Marusich and Ortiz[33] used the critical stress intensity factor, KIC, as a fracture criterion, for brittle materials. Theymodelled the ductile fracture based on the Rice and Tracey solution [45]. This is based on theexistence of a critical effective plastic strain at a distance l directly ahead of the crack tip. Owen etal. [41] used a fracture criterion based on uncoupled integration of Lemaitre's ductile damage model[30]. Lemaitre postulates that damage progression is governed by void growth. Also Ceretti et al.[8] included damage mechanics in simulation of crack propagation.

Numerical procedure

28

29

CHAPTER 4 Results and conclusions

Design of the mechanical cutting process is mainly based on the empirical knowledge of theprocess. The primary objective of the research presented in this thesis is the development of a finiteelement approach to provide a better understanding of the cutting process. As a result the chipmorphology, cutting forces, heat generation and the residual stresses in the work piece may bepredicted.

4.1 Results

Different finite element modelling approaches has been evaluated and implemented in a researchFE-code called SiMPle. This is an implicit finite element code based on the Lagrangianformulation. The commercial FE-code ABAQUS/Standard was also used to simulate cuttingprocess. In this section a brief review of the appended papers will be given.

4.1.1 Numerical and experimental analysis of orthogonal metal cutting (paper I)As a first step a simplified model of chip formation was used to simulate the cutting process. Thechip formation was modelled by a pre-defined crack path. A quasi-static simulation with adiabaticheating was performed with the finite element code ABAQUS/Standard. This is an implicit FE-code using the updated Lagrangian formulation. The computational results were compared with theexperiments. The agreement between measurements and calculation was reasonable when

Results and conclusions

30

considering the simplifications introduced. The experimental results obtained in this paper werealso used in paper III.

4.1.2 Coupled thermomechanical simulation of hot rolling, using an adaptive mesh (paper II)Coupled thermo-mechanical analysis of hot rolling was performed. The efficiency and accuracywhen using an adaptive remeshing technique is compared with using a uniform, fine mesh. Theadvantages and limitations of the different techniques are discussed. The accuracy has beenconfirmed by comparison with PALM2D and by using SIMPLE without remeshing. The overheadcost associated with the remeshing is not too costly so the remeshing technique reduces the requiredcomputing time considerably. The efficiency of the implemented thermo-mechanical contactalgorithm in SiMPle was evaluated when the adaptive mesh was used. The implemented contactalgorithm and the mesh adaptivity was used in paper III-V.

4.1.3 Finite element modelling of orthogonal metal cutting (paper III)The objective of this study was to evaluate different modelling approaches for simulation ofmechanical cutting. The orthogonal cutting process has been simulated by the finite element codes,ABAQUS/Standard, the implicit in-house code SiMPle, and the explicit code AdvantEdge. Thelatter is a commercial code developed especially for simulating mechanical metal cutting. Twobasically different modelling approaches have been used for the chip separation, geometrical andphysical model. Computed and measured cutting force and chip geometry are compared. It is foundthat the use of continuous remeshing is more demanding but better than the use of a pre-definedcrack path. The advance front technique for generation of quadrilateral element was implementedin the in-house code SiMPle in order to maintain the mesh quality.

4.1.4 Effect of previous cutting on chip formation (paper IV)Simulations are performed with the purpose of investigate the chip formation process and to findthe residual stresses on the surface of the work piece. The implicit finite element code SiMPle withspecial remeshing capabilities enabling the simulations has been used in the work.

The main conclusions in this study of the effect of previous cutting on subsequent layers are asfollows. The chip formation is not affected much. There is only a minor influence from the residualstress on the surface from the first cutting on the second pass chip formation. This influence isnegligible. The residual stresses are affected more in the current model. But this influence isexpected to be smaller when an appropriate material model which includes rate-dependency anddamage effects is implemented.

31

Conclusions and discussion

4.1.5 Influence of cutting speed on residual stresses in work piece (paper V)The influence of the cutting speed and feed on the residual stresses is studied in this work. Thematerial of the work piece is the stainless steel 316L. Cutting experiments in laboratory withfollowing measurements as well as finite element simulations are used and also compared with eachother. The explicit finite element code AdvantEdge was compared with the implicit in-house finiteelement code SiMPle. The computational time was considerably shorter in simulations by SiMPle.The measured residual stress level increases with cutting speed. In the simulations by SiMPle, it isonly the heat conduction that gives a difference between different cutting speeds. There is no rate-dependency in the material model or inertia forces included. Neither is damage accounted for. Theonly softening process is thermal softening. It is assumed that the material will reach a constanthardening when the stress-strain reaches a certain value

4.2 Conclusions and discussionDifferent modelling and simulation approaches for orthogonal cutting have been developed andevaluated, using the finite element method. The use of the model can give a better understanding ofthe cutting process. The thermal and mechanical loads on the insert can be quantified. Furthermore,the chip formation process and residual stresses in the work piece can be studied. It is believed thatthe finite element simulations will improve the design of the cutting tools by reducing the testing.Thereby, it will also reduce the time to market for the new designs.

Two basically different modelling approaches have been used for the chip separation, geometricaland physical model. A pre-defined crack path has been used to model the crack initiation/propagation in the geometrical model. The continuous remeshing has been used in the physicalmodel. It is found that the use of continuous remeshing is more demanding but better than the use ofa pre-defined crack path. However, the Lagrangian finite element simulations of the cutting processusing the physical model is obstructed by excessive element distortion, caused by the large,localized deformations. The continuous remeshing can efficiently maintain well-shaped elements.

The construction of an appropriately refined mesh is a challenge, for highly non-linear, transientproblems such as the cutting process. The adaptive remeshing can be used to obtain accurateresults. This is done by using two error estimator criteria simultaneously. The error estimator basedon the stress gradients in the finite element mesh is combined with he quotient between the rate ofplastic work in each element and the averaged plastic work rate of the whole mesh. The automaticcontrol of the mesh quality is managed successfully by using the Oddy distortion metric at thebeginning of each time step. The mesh generation procedure is also utilising the distortion metric toprovide well-shaped elements.

The implemented combined penalty-barrier contact algorithm has been found to be efficient inconjunction with the adaptive remeshing. The contact algorithm is implemented as a two-pass

Results and conclusions

32

algorithm. A so called stick-slip friction model, introducing Coulomb friction was used to modelthe frictional behaviour of the tool-chip interface.

It is believed that the lack of the damage mechanics in the material model and the rate-dependentmaterial behaviour in SiMPle cause a non-lamina structured chip formation. Therefore, it isimportant to include these phenomena in the material model in order to include all effects ofvarying the cutting parameters.

The CPU time for simulation of residual stresses was considerably longer using the explicit codeAdvantEdge. The simulations with AdvantEdge required 100-130 CPU hours (about 1 million timesteps) where the simulations with SiMPle only required 6-8 hours (about 500 time steps) than usingthe implicit code SiMPle. Therefore, an implicit analysis may be preferable for two dimensionalmodels. However, in a three dimensional model of the cutting process is probably more efficient tosolve using an explicit code than using an implicit code. Furthermore, the use of parallel computingis necessary be able to simulate the cutting process using three dimensional models.

33

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55. Wriggers P., Simo J.C.,Taylor R.L., Penalty and augmented Lagrangian formulations forcontact problems, Proc. NUMETA ‘85 Conf., Swansea, 97-106, 1985.

56. Xie J.Q., Bayoumi A.E., Zbib H.M., Characterization of chip fornation and shear banding inorthogonal machining using finite element analysis, In ASME, ed. Material Instabilities: Theoryand Applications volume AMD-Vol 183/Md-vol. 50, 285-301, 1994.

57. Zavarise G., Wriggers P., Schrefler, B. A., A method for solving contact problems, Int. J. forNum. Meth. in Engng. , v42, n3, Jun 15, 1998.

58. Zhang, B., Bagchi, A., Finite element simulation of formation and compreson with MachiningExperiments, Computational methods in material processing, ASME Publication, PED-Vol. 61,61-74, 1992.

59. Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori errorestimates, Int. J. for Num. Metth in Engng, 33, 1331-1364, 1992.

60. Zienkiewicz O.C., Zhu J.Z., A simple error estimator and adaptive procedure for practicalengineering analysis, Int. J. for Num. Meth. in Engng., 24, 337-357, 1987.

61. Zorev N.N., Metal cutting mechanics, Pergamon Press, Oxford, 1966.62. Thirdwavessys, AdvantEdge User’s manual Version 3.3.63. Hibbit, Karlson, Sorensen, 1996, Inc., ABAQUS/Standard Theory Manual Version 5.5.

Paper I

NUMERICAL AND EXPERIMENTAL ANALYSIS OF ORTHOGONAL METAL CUTTING

Vahid Kalhori1

Mikael Lundblad2

Lars-Erik Lindgren1

1Luleå University of Technology, S-971 87 Luleå, Sweden2AB Sandvik Coromant, S-811 81 Sandviken, Sweden

ABSTRACTThis paper presents experimental results which can be used

for evaluating computational models to assure robust solutions.The general finite element code ABAQUS/Standard has beenused in the simulations. A quasi-static simulation with adiabaticheating was performed. The path for separating the chip fromthe work piece is predetermined. The agreement betweenmeasurements and calculation is good considering thesimplifications introduced. Still, some capabilities are lacking inABAQUS. Work is continuing in order to create and evaluatemore advanced computational models.

1.0 INTRODUCTIONNew geometries for inserts made of cemented carbide for

metal cutting are developed through extensive time and resourceconsuming prototyping and testing to assure robust designs.This is partly caused by of the difficulties in studying the cuttingprocess and the load on the insert in detail since the environmentis highly hostile due to high pressures and temperature in thecutting zone, and therefore it is not directly observable. For thatreason, indirect parameters are studied, for example cuttingforces, chip morphology and tool life. Studying only globalquantities does not reveal the details of the true mechanics of thecutting process. This hampers rapid product development intools for metal cutting for the industry. Also the not easilymeasured residual stresses in the work piece are in someinstances very interesting since they can influence the life andperformance of the work piece.

Computer simulations can reduce developing time and costof new inserts and tools and also increase the understanding ofthe metal cutting process. Many studies, where the finite

element method (FEM) has been used, have been publishedduring the last decades. See e.g. Zhang and Bagchi (1994), Shih(1995,1996), and Huang and Black (1996). There are three mainroutes, general FE-codes, more specially tailored codes or”research codes”. The last group are either in-house codes usedby research groups that are more or less available for otherusers. The degree of support and documentation is best for thegeneral FE-codes and decreasing for the more special codes andmay be non-existing for the last group of codes.

The simulation of cutting involves many difficult aspects,large deformation, thermo-mechanical coupling, how to modelthe cutting etc. Some of the phenomena are common with otherprocesses that are simulated using the FE-method. Methodsbased on flow or solid formulation are two different routes thathave been chosen. Material modelling may be simplified usingrigid-plastic or rigid-viscoplastic material models. Meshadaptivity may be included for dealing with the moving zone oflarge deformations. We will not elaborate on these aspects ofmodelling. However, the cutting process is unique for theapplication in focus. Therefore it deserves some extra attention.It has been modelled in three different ways.

A tied interface which is unzipped when a certain criteria isfulfilled, such as a certain level of stress or strain or simplywhen the cutting edge is close enough, is used in most studies.Examples of this zip method are the studies by Huang and Black(1996), and Komvopoulos and Erpenbeck(1991) where thesame formulation as in this study is used. Black used NIKE2Dfor investigate different criteria for chip separation.Komvopoulos and Erpenbeck used ABAQUS to analyse theinfluence of different levels of friction between chip and insertin orthogonal metal cutting. They included rate-dependentplasticity but no thermal effects.

2

Recently a few articles has put forward methods where acrack propagates in front of the cutting edge, a crack method.The direction of the path of the crack is not predetermined. Theycan be divided into two groups. Either fracture mechanics ordamage mechanics is used for determining where and when thecutting takes place. Ceretti et.al. (1996) use damage mechanicsfor the cracking. Fracture mechanics for crack initiation andgrowth are used by Marusich and Ortiz (1995).

The third approach is an plastic stretch method. The chipformation is not modelled by separating neighbouring elements.The element are stretched and thereafter divided by a remeshingprocedure. The plastic deformation process ”goes on forever” inthe latter case and thereby gives the chip formation. Sekhon andChenot(1993) used remeshing and visco-plastic materialallowing the chip to part from the work piece by plasticdeformations. They used a flow formulation.

The overall aim of the current project is to be able to predictthe chip forming process by FE-simulation. The possibility ofusing a general FE-code is evaluated in the first instance.ABAQUS has been chosen as it can handle large plasticdeformations very well. The code is also well supported withinSweden.

Orthogonal metal cutting in unalloyed carbon steel withcemented carbide with all material and cutting data equal hasbeen performed. The chosen geometry makes it possible to use a2D model. Experiments are performed in order to evaluate thenumerical model. A predetermined crack path was modelledwith a tied interface that releases when the stress reaches aprescribed level. A quasi-static model with adiabatic heating isused. The agreement between calculated and measuredquantities is reasonable when considering the simplificationsused. The work is continuing with fully-coupled thermo-mechanical model in ABAQUS. We will also evaluate FE-codeswithin the other two categories mention above.

2.0 EXPERIMENTAL SETUPTest pieces were cut out of the work piece of unalloyed

carbon steel and prepared for measurement. Strain vs stresscurves at different temperatures were measured in a tensiletesting machine. Thermal expansion coefficient was measuredin a dilatometer. Thermal material data were calculated withthermocalc at SANDVIK STEEL. The obtained parameters forthe used steel and cemented carbide at different temperaturesare specified in Tables 1-3. All notations and the units used aresummarized in the nomenclature section at the end of the paper.

Axial turning of the end of a tube in a support lathe wasused to imitate a orthogonal 2D metal cutting process. To reduce3D effects the work piece used was a thin walled large diametersteel tube. The set-up used is shown in Fig. 1. Used cutting dataare specified in Table 4. The cutting speed was chosen quitehigh to avoid built up edge (bue) which can be a problem atcutting speeds vc less than 60 m/min.

Table 1. Material data for SS 1672-08 (within AISI 1045)with a hardness of 200 HV. Analysis C=0.47%, Si=.24%,Mn=0.70%, P=0.007%, S=0.029%.

Table 2. Variable hardening for SS 1672-08. Yield strengthversus plastic strain.

Table 3. Material data for SANDVIK Coromant H10F witha hardness of 1700 HV. Chemical composition WC=89.5%,Co=10.0%, Cr3C2=0.5%

Temperature T 20 200 400 600Yield strength R0.2 400 340 300 160

Ultimate strength Rm 650 660 560 255

Young's modulus E 215 210 165 160Poisson’s ratio ν 0.3 0.3 0.3 0.3Density ρ 7930 7880 7820 7750Thermal expansion α [106]

10.1 12.0 13.0 15.3

Heat capacity cp 470 535 635 800

Temperature [°C]

Yield strength [MPa]Plastic strain [%]

20°C 415 470 580 640 6550.5 1.8 4.2 7.2 11.5

200°C 390 445 575 645 6650.8 1.8 4.2 7.2 9.7

400°C 360 475 525 5450.8 1.7 4.2 7.2

600°C 220 2400.9 1.9

Temperature T 20 200 400 600Ultimate strength Rm 2600 2550 2350 2000

Young's modulus E 580 570 560 540Density ρ 14500Thermal expansion α [106]

5.4 5.3 5.4 5.6

Heat capacity cp 220

3

Table 4. Tool and cutting data used for experiments andcalculations

Figure 1. Schematic sketch of orthogonal cutting with usednotations.

3.0 COMPUTATIONAL MODEL

3.1 Finite element formulationThe finite element program ABAQUS has been used in the

simulations (Hibbit et.al., 1996). The implicit version was used.A quasistatic, large deformation analysis accounting foradiabatic heating was performed. The adiabatic heating iscaused by the elastic and plastic strains. The strains give anincrease in temperature locally, i.e. there is no heat conduction.Thus it was also not possible to account for heat generated byfriction in the model. However, the temperature change wasused in a temperature dependent material model for changingmaterial properties and for generating thermal strains.

A four node element (CPE4) was used. The volumetricstrain is underintegrated in order to avoid locking due to large,incompressible plastic strains.

The updated Lagrangian formulation was used. The spatialvelocity gradient is used for computing strain increments. Thus,the strain definition can be compared to the logarithmic ornatural strain definition. An additive strain rate decomposition isobtained from the multiplicative decomposition of thedeformation gradient (Simo, 1988). The latter corresponds tothe existence of a local unloaded state to which one can reversethe elastic deformation. One such approach of extending smallstrain plasticity to finite strains is demonstrated in (Cuitino andSimo, 1992). There still remains, however, some open questionsin this respect that was addressed already by Lee, 1969. Forcases where the elastic strains are much smaller than the plasticstrains one may simplify the decomposition of the deformationgradient to an additive split of the velocity strains, that is ahypo-elastic constitutive law, where a suitable stress rate isproportional, via Hooke’s law, to the velocity strain tensor (orformally to the total velocity strain minus the plastic velocitystrain) (Simo and Pister, 1984). It is essential to employ a so

Feed f 0.25 mmCutting speed vc 198 m/min

Cutting depth ap 3 mm

Edge radius rn 50 µm

Clearance angle α 6°Rake angle γ 6°

f = h 1

v c

Φ

h 2

lc

re

α

γ

Figure 2. Finite element model.

4

called objective stress rate in the hypo-elastic constitutive lawabove (Crisfield, 1997). The Jaumann rate of the Cauchy stressis used in ABAQUS/Standard, which is the implicit version ofABAQUS.

All quantities known at the beginning of the time step arerotated to the end of the time step. The rotation matrix, ∆R, isobtained from the left polar decomposition of the deformationgradient defined over the increment, ∆F. This is

(1)

, where xn and xn+1 are coordinates at times tn and tn+1,respectively, and ∆V is the incremental left strain tensor.

A modified Newton-Raphson method was used for solvingthe system of nonlinear equations.

3.2 Finite element modelThe finite element model is shown in Fig. 2. It is a plane

strain model and consists of 1711 elements and 1954 nodes. Thelength of the work piece is 6.25 mm and its total thickness is1.25 mm. The uncut chip thickness is 0.25 mm. The analysiswas performed for 1.894 milliseconds and 1225 time steps wererequired. ABAQUS option for automatic time incrementationwas used. A contact line was included between tool and workpiece. The friction coefficient was assumed to be constantµ=0.3. A tied interface was included between the part of thework piece that is removed during the cutting and the remainingmaterial. The nodes along this line were separated when thecombined normal and shear stresses at a distance of 0.0625 mmin front of the crack exceeded the ultimate strength of thematerial. Thus one can say that the model imitates cutting. Thepath for separating the chip from the work piece ispredetermined. It was necessary to make an initial crack in orderto start the simulation.

Heat is generated due to plastic dissipation, mainly alongthe shear band across the chip. 90% of the plastic dissipatedmechanical energy was input as heat. Heat due to friction is notincluded in the model. The adiabatic model implies that there isno heat conduction in the model. This simplification is lessimportant than the neglecting of heat generated by the friction asthere is a very short time available for the heat conduction whenthe chip is formed.

The workpiece was modelled as thermo-elastoplasticmaterial with temperature dependent properties. The materialdata and geometry are given in the previous paragraph.

4.0 RESULTS

4.1 MeasurementsThe static cutting force both in cutting direction Fc and in

feeding direction Ff were measured at the toolholder with aKistler dynamometer.

To freeze the chip forming process an interrupted cut wasperformed by shooting away the toolholder with powder charge,a so-called quick-stop. A part of the work piece with the frozenchip forming process was cut out, polished and studied inmicroscope, see Fig. 3. The shear angle Φ, ratio between cut anduncut chip thickness Λ and contact length lc were measured onthe specimen.

The cutting process was also recorded using a high-speedcamera. A picture is shown in Fig. 4. Measured quantities aregiven in Table 5.

4.2 CalculationsThe computed effective stress is shown in Fig. 5. The

normal stress components are shown in Fig. 6-7. Maximumshear stress is shown in Fig. 8. The effective plastic strain isshown in Fig. 9 and the temperature in Fig. 10. The computedcutting force is 1050 N and the feed force is 200 N. The shearplane angle, contact length and chip thickness are estimated inFig. 8. They are given in Table 5.

Table 5. Measured and computed quantities

∆Fxnd

dxn 1+∆V∆R= =

Measured Computed

Cutting forc Fc 1450 N 1050 NFeed force Ff 755 N 200 N

Chip thickness ratio Λ 1.98 2.0Shear plane angle Φ 30° 30°Contact length lc 0.58 mm 0.6 mm

Figure 3. Specimen obtained by a quick stop.

5

Figure 4. Frame from high-speed photography of cutting process.

Figure 5. von Mises effective stress.

Figure 6. Normal stress in cutting, horizontal, direction.

Figure 7. Normal stress in vertical direction.

Figure 8. Maximum shear stress.

Figure 9. Effective plastic strains.

6

5.0 DISCUSSIONS AND FUTURE WORKThe chip in the experiments shows some lamina structure

which the simulated chip did not have. The overall agreementbetween calculated and measured quantities is good whenconsidering the simplifications used. The computed cuttingforces are lower than the measured forces, especially for thefeed force. It was not possible to raise the friction coefficient asthe element distortion became to severe for a successfulanalysis. However, it will not be enough raising only the frictioncoefficient in the computational model. It is believed that it isalso necessary to model the radius of the tool tip in detail toincrease the computed feed force more than the cutting force.

The work is also continuing with a fully-coupled thermo-mechanical model in ABAQUS. Furthermore, it may benecessary to include rate-dependency of the material propertiesin order to include all the effects of varying the cutting velocity.The major lacking capability in ABAQUS is adaptive meshingincluding arbitrary cracking. Therefore, we will also evaluateother FE-codes and develop the contact algorithm in our in-house code SiMPle for simulating cutting. This is an adaptive,fully-coupled thermo-mechanical finite element code with meshsmoothing facilities for handling distorted elements.

6.0 REFERENCESCeretti, E., Fallboehmer, P., Wu, W. T., and Altan, T., 1996,

”Application of 2D FEM to chip formation in orthogonalcutting”, Journal of Materials Processing Technology, vol. 59 no1-2 May 15, pp 169-180.

Crisfield, M.A., 1997, Non-Linear Finite Element Analysisof Solids and Structures, Vol. 2 Advanced topics, J Wiley &Sons.

Cuitino, A., and M. Ortiz, M., 1992, ”A material-independent method for extending stress update algorithmsfrom small strain plasticity to finite plasticity with

multiplicative kinematics”, Engineering Computations, vol. 9,pp 437.

Hibbit, Karlsson, Sorensen, 1996, Inc., ABAQUS/StandardTheory Manual (version 5.5).

Huang, J.M., and Black, J.T., 1996, ”An evaluation of chipseparation criteria for the FEM simulation of machining”,Journal of Manufacturing Science and Engineering, vol. 118, pp545-554.

Komvopoulos, K., and Erpenbeck, S. A., 1991, ”FiniteElement Modeling of Orthogonal Metal Cutting”, Journal ofEngineering for Industry, 113, pp 253-267.

Lee, E.H., 1969, ”Elastic-plastic deformation at finitestrain”, ASME Journal of Applied Mechanics, vol. 36, pp 1.

Marusich, T. D., and Ortiz, M., 1995, ”Modelling andSimulation of High-Speed Machining”, International Journal forNumerical Methods in Engineering, vol. 38, pp 3675-3694.

Sekhon, G. S., and Chenot, J. L., 1993, ”Numericalsimulation of continuous chip formation during non-steadyorthogonal cutting”, Engineering Computations, vol. 10, pp 31-48.

Shih, A.J., 1995, ”Finite element simulation of orthogonalmetal cutting”, Journal of Engineering for Industry, vol. 117, pp84-93.

Shih, A.J., 1996, ”Finite element analysis of the rake angleeffects in orthogonal metal cutting”, International Journal ofMech. Sci., vol. 38, no 1, pp 1-17.

Simo, J.C., and Pister, K.S., 1984, ”Remarks on rateconstitutive equations for finite deformation problems:Computational implications”, Computer Methods in AppliedMechanics and Engineering, vol. 46, pp 201.

Simo, J.C., 1988, ”A framework for finite strainelastoplasticity based on maximum plastic dissipation and themultiplicative decomposition”, Computer Methods in AppliedMechanics and Engineering, vol. 66, pp 199.

Zhang, B., and Bagchi, A., 1994, ”Finite elementsimulation of chip formation and comparison with machiningexperiment”, Journal of Engineering for Industry, vol. 116, pp289-297.

Figure 10. Temperatures.

7

NomenclatureThe adapted nomenclature are, where suitable, according to

the ISO standardE Young´s modulus GPa∆F relative deformation gradient Fc Cutting force N

Ff Feeding force N

∆R rotation matrix, define in Eq. (1)Rm Ultimate strength MPa

R0.2 Yield strength MPa

T Temperature °C

∆V incremental left strain tensor

ap Cutting depth mmcp Heat capacity J/kg°C

f Feed mm/rev

h1 Uncut chip thickness mm

h2 Cut chip thickness mm

lc Contact length mm

rn Radius of edge mm

tn time at increment #n

v Cutting speed m/min

xn coordinate at time tn

α Clearance angle °α Thermal expansion 1/°Cγ Rake angle °Φ Shear plane angle °Λ Ratio between cut and uncut chip

thickness-

µ friction coefficient -ρ density kg/m3

ν Poisson’s ratio -

Paper II

Coupled Thermomechanical Simulation of Hot Rolling Using an Adaptive Mesh

Vahid Kalhori

Luleå University of Technology, Division of Computer Aided Design, S-971 87, Luleå, Sweden

Jonas Edberg

MEFOS, Heating and Metal Working Dept., S-971 25, Luleå, Sweden

Lars-Erik Lindgren

Luleå University of Technology, Division of Computer Aided Design, S-971 87, Luleå, Sweden

ABSTRACT: Coupled thermo-mechanical analysis of hot rolling is performed. The efficiency and accuracy

when using an adaptive remeshing technique is compared with using a uniform, fine mesh. The advantages and

limitations of the different techniques are discussed.

1 INTRODUCTION

The Finite Element Method (FEM) is a general tool

for simulating metal working. The main drawbacks

of this general approach have been the long

CPU-times and the problem of knowing the

boundary conditions in the rolling contact. Today

most commercial finite element codes provide fairly

accurate models describing the rolling contact and

the continuous improvements in the computer

technology and in FE-techniques are reducing the

required CPU-times.

The first finite element models for simulation of

rolling were based on the so-called ¨flow

formulation¨ (Zienkiewicz 1984). Velocities are the

primary unknowns in this approach. Another

formulation for simulating metal forming problems is

a large deformation analysis where displacements are

the fundamental unknowns. The latter approach is

used in this paper. Previous simulations performed

by the authors, (Lindgren & Edberg 1990, 1992,

1993), has been iso-thermal. A finer spatial

resolution is required in this study as we perform a

coupled thermo-mechanical analysis. This will create

large thermal gradients near the rolling contact.

The objective of the investigation presented in this

paper is to compare simulations with a uniform fine

mesh with simulations using refining/coarsening

which creates a fine mesh only near the rolling

contact. We use the thermo-mechanical coupled

finite element program PALM2D, obtained from

Lawrence Livermore National Laboratory, and an

in-house code, SIMPLE. The latter utilize mesh

refinement and mesh coarsening.

It is shown that the same accuracy can be obtained

with the adaptive meshing at a reduced

computational cost. The remeshing facility also

reduces the memory requirements.

2 FINITE ELEMENT CODES

2.1 SIMPLE

The code SIMPLE solves coupled

thermo-mechanical problems. It can solve three

dimensional problems also (Lindgren 1997).

However the thermo-mechanical contact algorithm is

not implemented in three dimensions. The used

element is a four to eight node quadrilateral element

with piecewise bilinear shape functions (McDill &

Oddy 1987). The volumetric strain field is

underintegrated in order to avoid locking due to the

plastic incompressibility. The special shape functions

applied in the element makes it straightforward to

create a graded mesh as an element with three nodes

along a side can be joined to two elements, each with

two nodes along this side. No additional constraints

are required for interelement compatibility. Thus,

graded elements are specially designed for

alleviating the creation of mesh with refined regions.

The mesh regeneration scheme is described in

McDill & Oddy (1987). It has been extended with

additional smoothing facilities by Hyun & Lindgren

(1998).

Triangular elements are usually used in adaptive

meshing. However, quadrilateral elements are

preferred over triangular elements when dealing with

plasticity (Cifuentes & Kalbag 1992, Benzley et.al.

1995).

An Augumented Lagrangian technique is used in

the mechanical contact algorithm (Crisfield 1997).

The augmentation loop (Simo & Laursen 1992,

Laursen & Simo 1993, Laursen 1994, Laursen &

Maker 1995) was not used in this application. The

contact algorithm is implemented as a two-pass

algorithm. Each side of the contact is in turn treated

as the master surface by the algorithm. The thermal

contact uses a contact resistance when the surfaces

are in contact. Furthermore, heat conduction (which

is very small) through the air is accounted for when

the surfaces are separated. This is only the case when

the gap has contacts on each side of it. Then the air is

enclosed completely. No radiation is included in the

heat transfer between the surfaces.

2.2 PALM2D

The finite element programs PALM2D, obtained

from Lawrence Livermore National Laboratory,

USA, are used in the simulations of the problem

using a fine uniform mesh. PALM2D is a

thermo-mechanical coupled implicit code for the

static and dynamic response of two-dimensional

solids and structures. It lacks limits on the time step

size other than accuracy requirements as the code is

based on an unconditionally stable time integrator.

PALM2D is based on TOPAZ2D (Shapiro 1986) for

thermal analysis and NIKE2D (Hallquist 1986) for

mechanical analysis. The element is a four node

element with bilinear shape-functions and with a 2x2

Gauss quadrature rule and reduced integration

applied to the shear energy. The two thermal and

mechanical analyses are coupled together in a

staggered step approach. First the thermal time step is

calculated and then the mechanical response is

solved. Substepping of the thermal or the mechanical

analysis is possible if desired.

A penalty formulation is used in the contact

algorithm in the mechanical analysis. The thermal

contact accounts is modelled in the same way as in

SIMPLE.

3 SIMULATIONS

The simulations are based on experimental data from

a test rolling sequence at the Rautaruukki Plate mill

in Raahe, Finland. Simulations of this particular

experiment has been presented earlier (Lindgren &

Edberg 1990, 1992, 1993) for other purposes. In

those papers no thermal effects where taken into

account. When the thermal effects are taken into

consideration a much finer element mesh is needed in

order to describe the temperature gradients that are

created when the hot plate is in contact with the cold

roll. All the physical data for the experiment and the

mechanical part of the simulations can be found in

the papers by Edberg & Lindgren (1990, 1992, 1993).

In this work however, more of the work roll is

included in the model. The radial ”thickness” of the

roll is 100.0 mm in both models. The material data

for the thermal part of the simulation are shown in

Table 1. The same data is used both for the roll and

the plate. The density of the roll is 7750 kg/m3 and

the density of the plate is 7860 kg/m3. The initial

temperature of the work roll is assumed to be 20 °C

and 1000 °C in the plate. No heat generated due to

friction was included in SIMPLE as this option is not

available in PALM2D. It is assumed that 90% of the

plastic dissipated energy is converted into heat. No

thermo-elastic generated heat is included.

An elasto-plastic, temperature dependent

constitutive model with linear work hardening has

been used for the plate in the mechanical part of the

simulations. The data for this model is found in Table

2. The work roll is assumed to be linear elastic.

Only the upper roll and half of the plate is analysed

due to symmetry. The PALM2D model consists of

3993 nodes and 3674 four node elements. There are

10 elements across the half thickness of the plate and

176 elements along the plate. Thus the smallest

element is 0.64x0.5 mm2 before rolling. The model

can be seen in figure 1. The time steps are 0.1ms long

in the mechanical analysis. Thermal substepping is

done so that the length of these steps is 0.025 ms. The

simulation required 740 cpu-minutes on a an IBM

RS6000 (model 590) workstation. The process time

becomes two times longer as the code does a lot of

Table 1. Thermal material data.

Temperature

[C]

Heat

capacity

[J/kgK]

Thermal

conductivity

[W/mK]

20.0 500.0 50.0

500.0 650.0 36.0

600.0 710.0 33.0

700.0 800.0 31.0

725.0 820.0 26.0

750.0 650.0 25.0

1300.0 720.0 31.0

Table 2. Mechanical material data.

Temp.

[ C]

Young’s

Mod.

[GPa]

Poiss.’s

ratio [-]

Thermal

expansion

[10-5K-1]

Yield

stress

[MPa]

Hardening

Modulus

[MPa]

400 175 0.3 1.15 160 420

600 145 0.33 1.2 100 320

800 120 0.35 1.235 80 290

1000 100 0.35 1.292 60 270

1100 95 0.35 1.32 45 220

Figure 1. PALM2D model. The fine mesh at the right end of the contact is magnified.

Figure 2. SIMPLE model at 0.0 and 0.1 secs. Part of the mesh is magnified at 0.1 secs.

reading and writing on disk. A single user on a

workstation can therefore never extract more that

about 50% of the wall clock time as useful cpu-time.

The model used in SIMPLE consists of about 1050

nodes and 950 elements initially and raises to a

maximum of 3250 nodes and 2500 elements in the

end of the analysis. The steady increase is due to the

fact that we keep a fine mesh in the plate after the

rolling. The model at 0.1 secs can be seen in Figure

2. The remeshing is performed ten times during the

analysis. The time steps is 0.1 ms. This simulation

required 1100 cpu-minutes on the same IBM

workstation. A comparative simulation with a

uniform mesh, like the mesh used in PALM2D, was

performed. The number of nodes is 4000 and the

number of elements is 3400. This simulation required

1200 cpu-minutes.

4 COMPARISONS

The computed rolling force is 8.43 kN/mm,

according to PALM2D and 8.17 kN/mm according to

SIMPLE. This is per unit width in the direction

transverse to the rolling direction. The value is

obtained during rolling the middle part of the slab

where the force has a stationary value. The stresses in

the thickness direction at time 0.1 secs are shown in

Figure 3 for PALM2D and in Figure 4 for SIMPLE.

The shear stresses are shown for PALM2D and

SIMPLE in Figures 5 and 6, respectively. The

stresses show very good agreement between the

different models. The temperature at the surface goes

down to 665 C, according to PALM2D, and to 680 C,

according to SIMPLE, during a very short time when

contact with the roll is established. Both codes give

an increase in the temperature of 8 C in the interior of

the plate due to the plastic dissipation.

5 CONCLUSIONS

The region close to the moving roll/slab contact must

be modelled using a denser mesh in order to get an

accurate model. It has been demonstrated that this

can be efficiently achieved by using a remeshing

technique. The accuracy has been confirmed by

comparison with PALM2D and by using SIMPLE

without remeshing.

PALM2D is a very streamlined code but the version

that was used performs a lot of input/output on disk

which made it less efficient. SIMPLE does not have

this disadvantage. The overhead cost associated with

the remeshing is not too costly. The remeshing

technique reduces the required computing time

considerably. The implemented remeshing algorithm

will be extended with an error measure (McDill

1995) in order to have a fully adaptive mesh. The

smoothing algorithm implemented by Hyun &

Lindgren (1998) was not needed in this simulation

but will be useful for cases with larger reduction. The

combined remeshing and smoothing algorithms will

be powerful tools that will be valuable for such cases.

The smoothing facility is not implemented for the

three-dimensional version of SIMPLE yet.

Figure 3. Stresses in thickness direction computedby PALM2D at 0.1 secs.

Figure 5. Shear stresses computed by PALM2Dat 0.1 secs.

SYY0.00E+00-4.25E+07-8.50E+07-1.28E+08-1.70E+08-2.12E+08-2.55E+08-2.98E+08-3.40E+08

Figure 4. Stresses in thickness direction computedby SIMPLE at 0.1 secs.

SXY7.00E+075.25E+073.50E+071.75E+070.00E+00-1.75E+07-3.50E+07-5.25E+07-7.00E+07

Figure 6. Shear stresses computed by SIMPLEat 0.1 secs

REFERENCES

Benzley S.E., Perry E., Merkely K., Clark B.,

Sjaardama, G.D., 1995. A comparison of all

hexagonal and all tetrahedral finite element

meshes for elastic and elasto-plastic analysis, In:

Proc. of 14th Annual International MeshingRoundtable, Albaquerque, USA.

Cifuentes A.O., Kalbag, A., 1992. A performance

study of tetrahedral and hexahedral elements in

3-D finite element structural analysis, FiniteElements in Analysis and Design, 12:313-318.

Crisfield M.A., 1997. Non-linear Finite Element

Analysis of Solids and Structures, Vol. 2,

Advanced Topics, J Wiley, Chichester.

Edberg J., 1992. Three-dimensional simulation of

plate rolling using different friction models,

NUMIFORM 92. Proc. of the fourth InternationalConference on Numerical Methods in IndustrialForming Processes. Valbonne. France. September

14-18. A.A. Balkema. Rotterdam. pp. 713-718.

Edberg J., Lindgren L-E., 1993. Efficient

three-dimensional model of rolling using an

explicit finite-element formulation.

Communication in Applied Numerical Methods inEngineering. 9:613-627.

Hallquist J.O., 1986. NIKE2D-A vectorized implicit,

finite deformation finite element code for

analyzing the static and dynamic response of 2-D

solids with interactive rezoning and graphics.

Report UCID-19677. Rev 1. Lawrence Livermore

National Laboratory. USA.

Hyun S., Lindgren L-E., 1998. Smoothing techniques

for graded elements, Int. Conf. on Numerical

Methods in Forming Processes, June, 1998,

Twente, Netherlands.

Laursen T. A., Simo J. C., 1993. Algorithmic

symmetrization of Coulomb frictional problems

using augmented Lagrangians, Computer Methodsin Applied Mechanics and Engineering108:133-146.

Laursen T.A., 1994. Automation and Assessment of

Augmented Lagrangian Algorithms for Frictional

Contact Problems. Journal of Applied Mechanics.61: 956-963.

Laursen T. A., Maker B. N., 1995. An augmented

lagrangian quasi-newton solver for constrained

nonlinear finite element applications, Int. J. forNumerical Methods in Engineering.

38:3571-3590.

Lindgren L-E., Edberg J., 1990. Explicit versus

implicit finite element formulation in simulation of

rolling. Journal of Material Processing. 24:85-94.

Lindgren L-E., Häggblad H.-Å., McDill J.M.J., Oddy

A.S., 1997. Automatic remeshing for

three-dimensional finite element simulation of

welding, Comput. Methods Appl. Mech. Engrg.,147:401-409.

McDill J.M.J. Goldak J.A. Oddy A.S., Bibby, M.J.,

1987. Isoparametric quadrilaterals and

hexahedrons for mesh-grading algorithm.

Communications in applied numerical methods.

3:155-163.

McDill J.M.J., Johnston M. A., Oddy A.S., 1995. A

generic Error Estimate for three-dimensional

h-adaptive finite element analysis. Carleton

University, Canada

Shapiro A. B., 1986. TOPAZ2D - A two-dimensional

finite element code for heat transfer analysis,

electrostatic, and magnetostatic problems. Report

UCID-20824. Lawrence Livermore National

Laboratory. USA.

Simo J. C., Laursen T. A., 1992. An Augmented

Lagrangian Treatment of Contact Problems

Involving Friction. Computers & Structures. 42:

97-116.

Zienkiewicz O. C, 1984. Flow formulation for

numerical solution of forming processes.

Numerical analysis of forming processes. Wiley.

Paper III

Finite element modelling of orthogonal metal cutting 1

Finite element modelling of orthogonal metal cutting

Vahid Kalhori

The Polhem Laboratory, Division of Computer Aided Design,

Luleå University of Technology, S-971 87 Luleå, Sweden

Vahid.Kalhori@cad.luth.se

Mikael Lundblad

AB Sandvik Coromant, S-811 81 Sandviken, Sweden

Mikael.Lundblad@sandvik.com

Abstract

The objective of the present study is to evaluate different modelling approaches for simulation ofmechanical cutting. The orthogonal cutting process has been simulated by the finite element codes,ABAQUS (Standard), the implicit in-house code SiMPle, and the explicit code AdvantEdge. Thelatter is a commercial code developed especially for simulating mechanical metal cutting. Twobasically different modelling approaches have been used for the chip separation. Computed andmeasured cutting forces and chip geometry are compared. It is found that the use of continuousremeshing is more demanding but better than the use of a pre-defined crack path.

1. IntroductionDevelopment of new geometries for inserts occurs through, extensive time and resourceconsuming, prototyping and evaluating. Because of a highly hostile environment, high temperature,pressure, and vibration, in the cutting zone, study of the cutting process is very complicated. Thishampers rapid product development in tools for metal cutting for the industry. Cutting simulationsare developed to become an instrument for design of cutting tools. These computational willincrease the understanding of the cutting process. Simulations can reduce the number ofexperiments required during the design process. It is also possible to perform parametric studies in

Finite element modelling of orthogonal metal cutting 2

a way that is difficult to achieve by experiments. The simulations can also provide a prediction ofthe state of residual stresses in the work piece.

The current study is part of a project with the aiming at developing finite element models that canbe used to predict the chip formation process and cutting. Different modelling approaches havebeen implemented and evaluated in this study. The orthogonal cutting process has been simulatedby the finite element program ABAQUS/Standard, using a pre-defined crack path. This model hasalso been implemented into an in-house finite element code, SiMPle. A combined r-adaptivity andh-adaptivity technique for continuous remeshing has also been implemented in SiMPle to simulatethe chip formation without pre-defining the crack path. Finally, the explicit finite element codeAdvantEdge was used. This code also used a remeshing technique for the chip formation. Thecomputed chip morphology and the cutting forces are compared with experimental results. It isshown that all modelling approaches give similar cutting forces for the same material and a lowfriction coefficient, but lower than the measured values. However, it is needed to use a high frictioncoefficient to obtain better aggrement between the computed and the measured forces. This canonly be handled by the approach using continuous remeshing.

2. ExperimentsTensile tests were performed in order to obtain material properties for the work piece. Furthermore,cutting experiments were also done for comparison with simulations.

The test pieces for the tensile testing were cut out of the work piece of an unalloyed carbon steeland prepared for tensile test. The chemical composition of AISI 1045 is C=0.47%, Si= 0.24%,Mn=0.70%, P=0.007%, S=0.029%. The material has a hardness of 200 HV. Strain vs. stress curvesat different temperatures was obtained, Figure 1. The thermal expansion coefficient was measuredin a dilatometer. Thermal material data were calculated with thermocalc at SANDVIK STEEL. Theobtained parameters for the worked steel and the tool steel at different temperatures are given inTables 1-2. The tool material has a hardness of 1700 HV and a chemical composition ofWC=89.5%, Co=10.0%, Cr3C2=0.5%.

TABLE 1. Material properties for AISI 1045

Temperature T [oC] 20 200 400 600 800Yield strength R0.2 [MPa] 400 340 300 160 42

Ultimate strength Rm [MPa] 650 660 560 255 93

Young’s modulus E [GPa] 215 210 165 160 90Poisson’s ratio ν 0.3 0.3 0.3 0.3 0.3

Density ρ [Kg/m3] 7930 7880 7820 7750 7720

Thermal expansion α [10] 10.1 12.0 14.0 16.6 18Heat capacity cp 470 535 635 800

Finite element modelling of orthogonal metal cutting 3

FIGURE 1. Variable hardening for AISI 1045, yield strength versus plastic strain

The cutting experiment was performed as orthogonal cutting, Figure 2. Axial turning of the end of atube in a support lathe was performed. To reduce 3-D effects the work piece used is a thin walledlarge diameter steel tube. The tube diameter is 178.0 mm and thickness is 3.0 mm. The set-up usedis shown in Figure 2-a. The cutting data are specified in Table 3. The cutting speed is chosen quitehigh to avoid built up edge (BUE) that can be a problem at lower cutting speeds. Cutting forceswere measured during the cutting process.

TABLE 2. Material properties for SANDVIK Coromant H 10 F cement carbide

Temperature [oC] 20 200 400 600Ultimate strength Rm [MPa] 2600 2550 2350 2000

Young’s modulus E [GPa] 580 570 560 540Poisson’s ratio ν 0.22 0.22 0.22 0.22

Density ρ [Kg/m3] 14500

Thermal expansion α [10] 5.4 5.3 5.4 5.6Heat capacity cp 220

0

100

200

300

400

500

600

700

800

0 0,01 0,02 0,03 0,04 0,05 0,06 0,08 0,1 0,115

Plastic strain [%]

Stre

ss [M

Pa]

20200400600800

Finite element modelling of orthogonal metal cutting 4

The cutting forces, Fc, in cutting direction, and the feed force, Ff, were measured at the tool holderwith a Kistler dynamometer [1]. Furthermore, the chip geometry is measured by microscopicstudies [1]. The computed cutting quantities and measured parameters are presented in Table 5-6.

FIGURE 2. a) Schematic sketch of orthogonal cutting. b) Used notations.

3. Finite element modelling of cuttingSome finite element formulations and techniques that are important to consider when modellingmachining are described below. The finite element codes ABAQUS, AdvantEdge and the in-housecode SiMPle were evaluated in the current study. These codes are using different modellingoptions.

TABLE 3. Tool and cutting data used for experiments and calculations

Feed f [mm] 0.25Cutting speed vc [m/min] 198

Cutting depth ap [mm] 3

Edge radius re [µm] 50

Clearance angle α 6o

Rake angle γ 6o

( a ) ( b )

Finite element modelling of orthogonal metal cutting 5

3.1 Finite element formulations

It is obvious that the cutting process requires a large deformation analysis. The deformations andstrains are extremely large. The use of remeshing is discussed later in chapter 3.3. Furthermore, itrequires the simultaneous calculation of temperatures and deformations. The high rate of materialflow makes it possible to assume adiabatic heating. This assumption makes it possible to directlycompute the temperature increase in a point due to the plastic dissipation at that point. Then nofinite element solution of the field equation for the heat conduction is required. This assumption hasbeen used in [1]. This approach cannot be used if the cooling to room temperature is needed inorder to evaluate residual stresses of the work piece. Some analyses have been done using a coupledthermal-mechanical analysis. Then a heat conduction analysis is coupled with the mechanicalanalysis. A so-called staggered approach is preferred and has been used in [2-5].

The mechanical analysis can be performed using different formulations. Each formulation has itsadvantages and disadvantages. There is the possibility to treat the material as a non-Newtonianfluid. This is called a flow formulation. Thus no elastic strains are included in the material modeland residual stresses cannot be obtained directly. This approach has been used in [4,6-7]. The moregeneral option is the solid formulation where the material is modelled by the usual constitutiveequations for solid material.

There exists a large number of different constitutive models. There exist two main types of methodsfor the temporal discretisation. The explicit, conditionally stable, method. This is a fast method butthe limit on the time step is related to the time it takes for an acoustic wave to pass through thesmallest element. It is a dynamic analysis in the sense that it only uses a mass matrix and computesthe change in displacements from the current acceleration. It is an effective method as the massmatrix is always lumped and thereby there is no need to solve a set of coupled equations. It cannotbe used easily to obtain steady state results. However, it has been used with success for solvingstrongly non-linear problems where inertia is negligible by artificially increasing the density andthereby make it possible to take larger time steps [8]. Owen et al. [2], Marusich [3] and Hashemi [9]have used this approach. The other, more general option is to use an implicit finite element code.The length of the time step is then governed only by accuracy requirements. On the other hand, aset of coupled equations must be solved. This method can be used with or without a mass matrix. Ithas no problem in computing steady state response. This type of formulation has been used in thecutting simulations in [1,10-12].

Different material models options have been used in simulations of cutting using the solidformulation. The material has been treated as rigid-plastic by [9,13]. This choice makes it possibleto compute cutting forces but no residual stresses can be obtained. Others have used a thermalelastic plastic material model with or without rate dependency. The plastic behaviour was assumedto be rate-dependent in [2-3,7,14].

It is preferred to have more linear elements in non-linear problems compared to fewer high orderelements as the plastic deformation implies a more rapid changing deformation behaviour. A highorder element enforces a higher degree of continuity in the solution. Furthermore, the largeincompressible plastic strains can give locking effects for low order triangular elements in 2-D andtetrahedron elements in 3-D [15-16]. Therefore, well known codes like the public domain codesNIKE and DYNA from Lawrence Livermore National Laboratory has only linear quad elements in

Finite element modelling of orthogonal metal cutting 6

2-D and hexahedron elements in 3-D. The four node quads with reduced integration for thevolumetric field are used in the present work. The four node quad element is also used in [2,4-5,7,10-14]. The six node triangular elements are used in [3,7].

3.2 Contact algorithms

Two main algorithms for solving contact problems are presented; the penalty approach [17-23], andthe method of Lagrangian multipliers [20]. Laursen and Simo [24] also describe a combinedmethod called the augmented Lagrangian technique. Other procedures, such as the perturbedLagrangian method [25-26], can also be applied. Other important issues are associated with contactdetection [18,27-30]. The latter is simpler with lower order elements and, also for this reason theselow-order elements are popular. Many of the low-order contact formulations have origins in thenone linear slide-line procedure presented by Hallquist [18,28-29]. In relation to these techniques,important issues are associated with the use of one-pass or two-pass algorithms. In particular,Taylor and Papadopoulos have shown [31] that a two-pass formulation is essential if the contactpath test [32] is to be passed. Additional special procedures have been developed for the explicitintegration method [18,27-29], as momentum-related techniques in which modifications are madeto the acceleration, velocities and displacements. One of the aims of the latter is to avoid thepenalising effect on the time step of the explicit procedure, which can be introduced by the highstiffness, associated with penalty approaches.

Normal forces are obtained by insertion of a contact stiffness when a material point on 'slave'surface penetrates into the target body 'master', in the penalty formulation. However, theimpenetrability condition of contact bodies is not exactly fulfilled in the penalty approach. Thismay cause significant changes in surface behaviour, which in turn causes divergence. A furtherdisadvantage of the penalty approach is that, in the early iterations, while a node is oscillatingbetween being in and out of contact, the convergence characteristics can be very bad. One way ofalleviating the numerical difficulties is to avoid sudden change in stiffness that occurs when apreviously contacting node moves out of contact. This can be achieved using a method proposed byZavarise et al. [33], which combines a penalty procedure with a barrier method [34].

In contrast to the penalty formulation, the Lagrangian multiplier algorithm ensures exactsatisfaction of the required contact constrains [32]. The multipliers can be interpreted as contactforces. However, there are other disadvantages. First, there is the need to increase the number ofvariables with the Lagrange multipliers, and secondly special care must be taken when solving theequations, as the diagonal terms in the matrix for these extra equations are zero. However, it ispossible to combine the penalty and Lagrangian multiplier methods, the augmented Lagrangiantechnique [24,34], with the aim of retaining the good merits of each approach. One of the mainadvantages of using this method rather than the penalty algorithm is the sensitivity of the latter tothe selection of the penalty stiffness. If the penalty parameter is too high, then a very poorconvergence rate may result. While one can reduce the penalty parameter to avoid ill conditioning,the inevitable consequence is penetration and hence a violation of the real physical situation. Thiscan be avoided by using the augmented Lagrangian method.

The augmented Lagrangian formulation, and combined penalty-barrier formulation areimplemented in this study. The penalty formulation was found most convenient. The contact

Finite element modelling of orthogonal metal cutting 7

algorithm is implemented as a two-pass formulation. Each side of the contact is in turn treated asthe master surface. The stick-slip formulation [32] was implemented for contact friction. Thethermal contact uses a contact resistance when the surfaces are in contact. No radiation is includedin the heat transfer between the surfaces.

3.3 Adaptivity and remeshing

The need of adaptive strategies for large deformation finite element computations is undeniable.For many cases, adaptivity is an essential tool to obtain accurate numerical solutions. It will alsoreduce the required computational effort needed to achieve this accuracy. This is the case, forinstance, with problems in non-linear solid mechanics involving localised large strains. Meshadaptivity is divided into three different types of categories, h-adaptivity, p-adaptivity and r-adaptivity. H-adaptivity consists on changing the size h of the finite elements. The new mesh hasdifferent number of elements and the connectivity of the nodes is changed. In p-adaptivity thedegree of the interpolating polynomials is changed. R-adaptivity consists of relocation of the nodes,without changing the mesh connectivity.

Mesh adaptivity, for an efficient meshing, requires two essential ingredients, a tool for assessing theerror of the solution computed with a given mesh and an algorithm to define a new spatialdiscretisation. A widely used error estimator was proposed by Zienkiewiecz et al. [35-36]. McDillet al. [37], Hyun et al. [39] and Runnemalm [40] proposed a generic error estimate for linearelements based on the gradients of displacement or heat flux. The error is associated with thegradients which are discontinuous at interelement boundaries. This means that a mesh would berefined where large differences in the gradients exist between elements. Marusich and Ortiz [3]proposed to use the plastic work rate in each element in order to refine the finite element mesh incutting simulations. This option convey to capture the plastic deformation of the workpiecematerial. Owen et al. [2] proposed an error estimator based on the rate of fracture indicators. Thisshould not only capture the progression of the plastic deformation but also provide a fine mesh atregions of possible material failure. Owen discussed also other error criteria based on total plasticwork, uncoupled integration of Lemaitre’s damage model and total damage work. All this criteriawas followed by using the procedure originally proposed by Zienkiewicz and Zhu [35].

The delaunay triangulation for generating six-noded quadratic elements was used by Marusich et al.[3] and Sekhon et al. [7]. Refinement of the mesh in [3] was executed by adding more nodes toelements which satisfy the plastic work rate criteria. A hierarchical adaptive mesh scheme [41],used in this study, provides an efficient way to change the mesh density by refining/coarsening agiven mesh and facilitates data transfer. This so-called graded element reduces the requiredcomputational effort by locally grading the mesh where large gradients in the solution occur. Thisdoes not reduce the element distortion.

Mesh adaptivity is also important for reducing the distortion of the elements as this is a largeproblem in simulating the mechanical cutting process, using a Lagrangian finite elementformulation. Then tools for assessing the element distortion and for remeshing are needed. Afrequent updating of the mesh in which the element topology is either changed, or preserved isrequired. In the present studies, h- and r-adaptivity are combined; this is necessary as it was foundthat the application of r-adaptivity only is not sufficient to maintain the mesh quality. During the r-

Finite element modelling of orthogonal metal cutting 8

adaptivity or smoothing process the nodes are shifted to more favourable positions [40-41]. Themost useful smoothing technique in the current application is the so-called opti-smoothingalgorithm [41], where one tries to optimise the node positions using an element distortion metric[42]. One disadvantage for this method is the time consuming optimization procedure. A combinedLaplacian-area weighted method [40-41] has been adopted for the smoothing process, in this study.The smoothing technique is also combined with an h-adaptive remeshing using the implementedhierarchical adaptive mesh scheme [41] in order to provide a dense mesh near the tip of the movingtool. These procedures are applied regularly during the simulations. However, it is also necessary tocompletely regenerate the mesh now and then. The advance front technique [43-44] for generatinga quadrilateral FE-mesh was used for this.

3.4 Chip formation

A diversity of physical phenomena, such as large plastic deformations, heat generation, friction,damage etc. exists in the process zone where the chip separation occurs. Different numericaltechniques for modelling chip separation can be used. These can be divided into two categories,geometrical or physical. The geometrical model is usually based on a tied slide-line interface [10],which debonds when a certain criterion is fulfilled. This criterion may be a certain level of stress,strain or simply when the cutting edge is close enough to the front nodes. Huang and Black [10],and Komvopoulos and Erpenbeck [11], Zang and Bagchi [45], have used this model. However,within these studies, only the mechanical behaviour of cutting was considered and no temperaturechange was computed.

The physical models are based on the physical behaviour of material, such as plastic deformation,crack initiation and crack propagation without pre-determining its path. Most of the expandedenergy during cutting is dissipated by plastic deformations. Thus, this process in the chip formationis the most important contribution to the cutting forces. Continuous remeshing is necessary in orderto simulate this. Other phenomena may also be involved in the chip formation. Marusich and Ortiz[3] used the critical stress intensity factor, KIC, as a fracture criterion, for brittle materials. Theymodelled the ductile fracture based on the Rice and Tracey solution [46]. This is based on theexistence of a critical effective plastic strain at a distance l directly ahead of the crack tip. Owen etal. [2] used a fracture criterion based on uncoupled integration of Lemaitre's ductile damage model[47]. Lemaitre postulates that damage progression is governed by void growth. Also Ceretti et al.[4] included damage mechanics in simulation of crack propagation.

4. Finite element simulations

4.1 ABAQUS

The finite element program ABAQUS/Standard [48] is used for performing a quasi static, largedeformations analysis of the cutting process. It is an implicit code based on the solid formulation,and uses an updated Lagrangian mesh. In the current analysis it is assumed that heat conduction canbe ignored and therefore only adiabatic heating is accounted for. The adiabatic heating is caused byplastic strains. It is assumed that 90% of the plastic work is converted into heat. The strains give alocal increase in temperature and no finite element solution of the field equation for the heat

Finite element modelling of orthogonal metal cutting 9

conduction is required. It was also not possible to account for heat generated by friction in the usedversion of ABAQUS. However, the calculated temperature changes within the work piece are usedtogether with temperature dependent material properties and thermal strains. The work piece ismodelled as thermo-elastic-plastic material with temperature dependent properties and variablehardening, see Table 1-3 for the used data. The Lagrange multiplier technique is used in order tosolve the mechanical contact between the bodies.

The finite element model consists of 1711 four node, plane strain elements and 1954 nodes. Thevolumetric strain is under integrated in order to avoid locking due to large, incompressible plasticstrains. The length of the work piece is 6.25 mm and its total thickness is 1.25 mm. The uncut chipthickness is 0.25 mm. The analysis is performed for 1.894 milliseconds. ABAQUS option forautomatic time incrementation is used.

A tied contact interface is included in the model between the part of work piece, which will beremoved during the cutting process, and the remaining part of the work piece in order to imitatecrack propagation. The nodes along this line were separated when the combined normal and shearstresses at a specified distance in front of the crack tip exceeded the ultimate strength of thematerial. The combined normal and shear stress is calculated as [48]

(EQ 1)

where σn is the normal component of stress, τn is the shear stress component in the interface, and σfand τf are the specified normal and shear failure stresses. In the current studies σf is the final stressvalue for each temperature in Figure 1., and τf is taken as 0.5σf. The crack tip node debonds whenthe fracture criterion, f, reaches the value 1.0. This value is evaluated at a specified distance in frontof the crack tip. This is necessary to specify else the fracture will not occur near the tool but furtheraway where the stresses along the interface may become larger. The path for separating the chipfrom the work piece is thus predetermined. The friction coefficient between the insert and the workpiece material is assumed to be constant µ=0.3.

FIGURE 3. Contour of temperatures using ABAQUS

fσn

σf------ 2 τn

τ f----- 2

+=

TEMP VALUE

+1.85E+01

+5.01E+01

+8.17E+01

+1.13E+02

+1.45E+02

+1.76E+02

+2.08E+02

+2.40E+02

+2.71E+02

+3.03E+02

+3.34E+02

+3.66E+02

+3.97E+02

+4.29E+02

Finite element modelling of orthogonal metal cutting 10

Heat is generated due to plastic dissipation, mainly along the shear band across the chip, see Figure3. The highest temperatures appear along the tool-chip interface, near 400o C. The state of effectiveplastic strains is presented in Figure 4. The largest plastic strains, 250%, occur within the boundarylayer adjacent to the tool.

FIGURE 4. Contour of effective plastic strain using ABAQUS

4.2 AdvantEdge

AdvantEdge machining modelling software is an central difference explicit finite element codeusing a Lagrangian mesh. The material model accounts for elastic-plastic strains and has anisotropic power law for strain hardening. The strain rate also affects the flow stress. The materialproperties are temperature dependent and thereby it also accounts for thermal softening. Astaggered method for coupled transient mechanical and heat transfer analysis is utilized. First anisothermal mechanical step is taken followed by a rigid transient heat transfer step with constantheating from plastic work and friction. Both steps have identical meshes. The central differencescheme is also used for the time integration in the thermal analysis. A six-node quadratic triangleelement is used. The mesh, which becomes very distorted around the cutting edge, is periodicallyupdated both refining large elements and coarsening small elements. For this simulation a powerviscosity law [49] with constant rate sensitivity exponent was chosen. It is written as:

(EQ 2)

and a power hardening law with a cut-off strain at which the increase in deformation hardeningstops. A high value of m was chosen in order to eliminate the rate dependency in order to alleviatecomparisons with models for ABAQUS and SiMPle. The strain hardening and temperaturedependency of the plastic flow properties are determined via the function g (εp) as:

PEEQ VALUE

+0.00E+00

+2.20E-01

+4.40E-01

+6.60E-01

+8.80E-01

+1.10E+00

+1.32E+00

+1.54E+00

+1.76E+00

+1.98E+00

+2.20E+00

+2.42E+00

+2.64E+00

+2.86E+00

1 ε·p

ε·p

0

-------+ σ

g εp( )------------- m

=

Finite element modelling of orthogonal metal cutting 11

, if (EQ 3)

Where,

Θ(Τ) = c0 + c1T + c2T2 + ..... + c5T5 (EQ 4)

The temperature dependencies of thermal properties of the work piece material were also modelledwith polynomial functions, see Equations 5-7. The used properties are given in Table 4.

Cp(T) = Cp (Cp0 + Cp1T + ... + Cp1T5) (EQ 5)

λ(T) = λ (λ0 + λ1T + .... + λ5T5) (EQ 6)

α(T) = α (α0 + α1T + ... + α5T5) (EQ 7)

TABLE 4. Properties used in simulation by AdvantEdge

Thermal conductivity λ 45 W/mo CHeat capacity cp 540 J/Kgo CDensity 7880 Kg/m3

s0 401 MPa

εp0 ref strain 0.00191

εpcut cut-off strain 0.2

n strain hardening coeff 9m strain rate coeff 1000Young’s modulus 210 GPaPoisson’s ratio 0.3c0 coefficient thermal softening 1.001845

c1 -3.572445e-4

c2 -1.391788e-6

c3 5.953324e-10

g σ0Θ T( ) 1 εp

εp0

-------+

1n---

= εp εpcut≥ εp εp

cut=( )⇒

Finite element modelling of orthogonal metal cutting 12

The mesh consists of 1310 nodes and 2232 elements and plain strain is assumed. The highesttemperature of the tool edge is 670o C and the average is 500o C and at the secondary shear zonewhere the chip slide along the insert, see Figure 5.

FIGURE 5. Contour of temperatures, using AdvantEdge

FIGURE 6. Contour of effective plastic strain, using AdvantEdge

Finite element modelling of orthogonal metal cutting 13

4.3 SiMPle

SiMPle is an implicit finite element program, which solves coupled thermo-mechanical problems.It is a solid formulation and uses an updated Lagrangian mesh. Thus, it has the same formulation asABAQUS/Standard. The simulations are performed by a graded element [37] and assuming planestrain. This is a four to eight node quadrilateral element with piece wise bilinear shape functions[37]. The volumetric strain field is under integrated in order to avoid locking due to plasticincompressibility. The special shape functions applied in the element makes it straightforward tocreate a graded mesh as an element with three nodes along a side can be joined to two elements,each with two nodes along this side. No additional constraints are required for inter-elementcompatibility. Thus, graded elements are specially designed for alleviating the creation of meshwith refined regions.

The augmented Lagrangian formulation, the penalty method and combined penalty-barrierformulation are implemented for the mechanical contact algorithm. The penalty formulation wasfound most convenient in the current simulations. The contact algorithm is implemented as a two-pass algorithm. Each side of the contact is in turn treated as the master surface. The thermal contactuses a contact resistance when the surfaces are in contact. No radiation is included in the heattransfer between the surfaces. The full Newton-Raphson procedure is used for the solution of thenon-linear system of equations.

Two different models have been used with SiMPle. In the first model a pre-defined crack path usinga tied slide-line is used in order to simulate crack initiation and propagation in the same way as inABAQUS. The geometry and cutting parameters are the same as in the ABAQUS model. The othermodel uses continuous re-meshing, in order to form the chip. The used remeshing procedure wasdescribed in section 3.3. The material is treated as a thermo-elastic-plastic material withtemperature dependent properties in both models. This is the same model as used in ABAQUS, SeeFigure 1. and Tables 1-3 for the used data.

4.3.1 SiMPle with pre-defined crack path

This is the same approach as used by ABAQUS. The slide line is separated when the combinednormal and shear stress, Eq. 1, at a specified distance in front of the crack tip exceed the ultimatestrength of the material. The friction coefficient between the insert and the work piece material isassumed to be constant µ=0.3. Heat is generated due to plastic dissipation, and friction between theinsert and the chip, this can be seen in Figure 7. The largest temperature, near 650oC, isconcentrated within the secondary deformation zone, due to the friction. The state of effectiveplastic strains is presented in Figure 8. The largest accumulated plastic strains occur within the tool-chip interface. The average amount of plastic strains at this region is about 250%. Strains in the chipinterior remain within the range of 100-200% upon exit from the primary shear zone.

Finite element modelling of orthogonal metal cutting 14

FIGURE 7. Contour of temperatures using SiMPle with a pre-defined crack path

FIGURE 8. Contour of effective plastic strain using SiMPle with a pre-defined crack path

4.3.2 SiMPle with continuous remeshing

The plastic deformations are assumed to be the only cause of chip formation in this model. A so-called 'cut-off' plastic strain, equal to the ultimate failure strain of material at different temperatures,is introduced. This means, the stresses cannot increase any further when the cut-off plastic strain isreached; see Figure 1. In order to investigate the effect of friction coefficient on the forces andtemperature development at the secondary deformation zone, the friction coefficient on the tool-chip interface is assumed to be 0.3 in one model and 1.0 in the other model. The minimum element

Finite element modelling of orthogonal metal cutting 15

size is 10 µm. The length of the time steps is in order of 1 µs. Average temperature along the tool-chip interface is near 450oC when using the friction coefficient of 0.3, see Figure 9. The largestaccumulated plastic strains occur within the boundary layer adjacent to the tool, see Figure 10.Strains in the chip interior remain within the range of 100-200% upon exit from the primary shearzone.

FIGURE 9. Contour of temperatures using SiMPle. Friction coefficient µ=0.3

FIGURE 10. Contour of effective plastic strains, using Simple. Friction coefficient µ=0.3

Finite element modelling of orthogonal metal cutting 16

FIGURE 11. Contour of temperatures using SiMPle. Friction coefficient µ=1.0

FIGURE 12. Contour of effective plastic strains, using Simple. Friction coefficient µ=1.0

The average temperature along the tool-chip interface when µ=1.0 is near 550o C, the highesttemperature at this region is about 1100o C, see Figure 10.

Finite element modelling of orthogonal metal cutting 17

5. Results and discussionsIt can be seen in Table 5-6. that the computed cutting forces are lower than the measured. Themodels with a friction coefficient of 0.3 gave lowest forces. It is possible to estimate from themeasured forces that this coefficient is too low. However, using a higher friction coefficient in thesimulations with a pre-defined crack path caused numerical problems. A higher friction coefficientcauses the stagnation of chip in the secondary deformation zone, see Figure 13. As a consequencethe part of the model corresponding to the chip wants to buckle and separate from the work piecealong the whole tied interface. The chip was sliding upward along the tool. The same problem willnot happen in the model by using continuous remeshing and a higher friction coefficient can beused in these simulations with increased cutting forces as a result.

FIGURE 13. Chip Stagnation within a Pre-defined crack path simulation, using a friction coefficient of m = 1.0

The lamina structure of the chip formed by a thermo-plastic shear localisation at the primarydeformation zone was not obtained by any of the simulations in this study. This may be due to theneed for damage and strain-rate effects in the material model.

All the simulations predict the largest accumulated plastic strains within the secondary deformationzone. Strains at the primary deformation zone remain within the range of 100%-200%. However,the average temperature in the primary deformation zone within all simulations is near 300oC. Theaverage temperature of the secondary deformation zone is near 550oC.

TABLE 5. Measured and computed parameters using a pre-defined crack path and µ=0.3

Measured ABAQUS SiMPleCutting force Fc [N] 1450 1050 850

Feed force Ff [N] 755 200 200

Chip thickness ratio Λ 1.98 2.0 2.0Shear plane angle Φ 30o 30o 30o

Contact length lc 0.58 0.6 0.59

Chip stagnation

Crack propagation

Finite element modelling of orthogonal metal cutting 18

6. Future workThe future work will focus on crack initiation/propagation using damage mechanics, as this isexpected to give more realistic prediction of machining parameters such as, chip morphology,cutting forces and residual stresses. This will be combined with the model using continuousremeshing for chip formation, which is very useful tool to model the shear localization and crackpropagation.

Furthermore, it is necessary to include strain rate dependent material model in SiMPle in order toinclude all the effects of varying the cutting velocity. Material modelling and obtaining the materialparameters for different temperatures and strain rate by means of experiments and inversemodelling is also important for improving the accuracy of the results. The ultimate ambition of theresearch work is to develop a three dimensional finite element model of mechanical cutting. Thiscan be used as a tool to increase the understanding of the cutting process.

Acknowledgment

The present work is a project in the Polhem Laboratory at Luleå University of Technology, one of1VINNOVA's competence centra. Their financial support is gratefully acknowledged.

References1. Kalhori V., Lundblad M., Lindgren L.-E., Numerical and experimental analysis of orthogonal

metal cutting, ASME 1997 International mechanical engineering congress &exposition, MEDVol. 6-2, Manufacturing Science and engineering, Dallas, Texas 16-21 Nov. 1997

2. Owen D.R.J., M. Vaz Jr., Computational techniques applied to high-speed machining underadiabatic strain localization conditions, Computer methods in applied mechanics andengineering, 171, 445-461, 1999.

TABLE 6. Measured and computed parameters using continuous remeshing

MeasuredSiMPle µ=0.3

SiMPle µ=1.0

AdvantEdge µ=1.0

Cutting force Fc [N] 1450 850 1200 1220

Feed force Ff [N] 755 250 500 475

Chip thickness ratio Λ 1.98 1.61 1.86 1.87Shear plane angle Φ 30o 29o 30o 28o

Contact length lc 0.58 0.38 0.51 0.52

1. VINNOVA is the Swedish Agency for Innovation Systems

Finite element modelling of orthogonal metal cutting 19

3. Marusich T.D., Ortiz M., Modelling and simulation of high-speed machining, Int. J. Numer.Methods Engrg. 38, 3675-3694, 1995.

4. Ceretti E., Fallboehmer P., Wu W.T., and Altan T., Application of 2D FEM to chip formation inorthogonal cutting, Journal of Materials Processing Technology, vol. 59 no 1-2 May 15, pp 169-180, 1996.

5. Shih A.J., Finite element analysis of rake angle effects in orthogonal metal cutting, Int., Mech.Sci. 38, 1-17, 1996.

6. Olovsson L., Nilsson L., Simonsson K., An ALE formulation for the solution of two-dimensionalmetal cutting problems, Computers and Structures 72, 497-507, 1999.

7. Sekhon G.S., Chenot J.L., Numerical simulation of continuous chip formation during non-steady orthogonal cutting, Engngr Computations, vol. 10, 31-48, 1993.

8. Lindgren L.E., Edberg J., Explicit versus implicit finite element formulation in simulation ofrolling, Jornal of materials processing technology. vol. 24, pp. 85-94, 1990.

9. Hashemi J., Tseng A.A., Chou P.C., Finite element modelling of segmental chip formation inhigh-speed machining, J. Mater. Engrg. Perf., 3, 712-721, 1994.

10. Huang J.M., Black J.T., An evaluation of chip separation criteria for the FEM simulation ofmachining, J. of Manuf. Science and Engrg, vol. 118, pp 545,554, 1996.

11. Komvopoulos K., and Erpenbeck S.A., Finite element modeling of orthogonal metal cutting,Journal of Engrg. for industry, 113, pp 253-267, 1991.

12. Xie J.Q., Bayoumi A.E., Zbib H.M., Characterization of chip formation and shear banding inorthogonal machining using finite element analysis, In ASME, ed. Material Instabilities: Theoryand Applications volume AMD-Vol. 183/ Md-vol. 50, 285-301, 1994.

13. Madhavan V., Chandrasekar S., Some Observations on the uniqueness of machining, Numericaland experimental analysis of orthogonal metal cutting, ASME 1997 International mechanicalengineering congress &exposition, MED Vol. 6-2, Manufacturing Science and engineering,Dallas, Texas 16-21 Nov. 1997.

14. Obikawa T., Usui E., Computational machining of titanium alloy-finite element methods and afew results, I. Manuf. Sci. Enggr., Trans. ASME 118, 208-215, 1996.

15. Benzly S.E. et al., A comparison of all hexagonal and all tetrahedral finite element meshes forelastic and elasto-plastic analysis, Proc. of 14th Annual International Meshing roundtable,Albaquerque, USA, 1995.

16. Cifuentes A.O., Kalbag A., A performance study of tetrahedral and hexahedral elements in 3-Dfinite element structural analysis, Finite Elements in Analysis and Design, 12:313-318, 1992.

17. Curnier A. & Alart P., Generalisation of Newton type methods to contact problems with friction,Journal de Mecanique thearique et appliquee, 7, 67-82, 1988.

18. Hallquist J. O., An implicit, finite deformation, finite element code for analysing static anddynamic response of 2-D solids, Report UCID-19677, Lawrence Livermore Lab., 1983.

19. Heegard J. H., Curnier A., An augmented Lagrangian formulation for discrete large slip contactproblems, Int. J. for Num. Meth. in Engng., 36, 569-593, 1993.

Finite element modelling of orthogonal metal cutting 20

20. Michalowski R. & Mroz Z. Associated and non-associated sliding rules in contact frictionproblems, Arch. Mech., 39, 259-276, 1978.

21. Papadopoulos P., & Taylor R.L., A mixed formulation for the finite element solution of contactproblems, Comp. Meth. in Appl. Mech. & Engng., 94,373-389, 1992.

22. Peric D. & Owen D.R.J., Computational model for 3-D contact problems with friction based onthe penalty method, Int. J. for Num. Mech. in Engng., 35, 1289-1309, 1992.

23. Wriggers P., & Simo J. C., & Taylor R. L., Penalty and augmented Lagrangian formulations forcontact problems, Proc. NUMETA '85 Conf., Swansea, pp. 97-106, 1985.

24. Larsen T.A. & Simo J.C., A continuum-based finite element formulation for the implicit solutionof multibody, large-deformation, frictional, contact problems, Int. J. for Num. Meth. inEngng.,36,3451-3486, 1993.

25. Ju Jiann-Wen et al., A consistent finite element formulation of nonlinear frictional contactproblems, NUMETA 87, Vol. 1,ed. G. N. Pande & J. Middleton, Martinus Nijhoff Dordrecht,pp. D5/1-D5/13, 1987.

26. Simo J. C., Wriggers P. & Taylor R. L., A pertubed Lagrangian formulation for the finiteelement solution of contact problems, Comp. Meth. in appl. Mech. & Engng., 51,163-180, 1985.

27. Belytschko T. & Neal M.O., Contact-impact by the pinball algorithm with penalty andLagrangian methods, Int. J. for Num. Meth. in Engng., 547-572, 1991.

28. Hallquist J.O. et al., Efficiency refinements of contact strategies and algorithms in explicit FE.programming, Computational plasticity: Fundamentals & Applications, ed. D.R.J. Owen et al.,Pineridge Press, Swansea, pp. 457-481, 1992.

29. Hallquist J.O. et al., Sliding interfaces with contact-impact in large-scale Lagrangiancomputations, Comp. Meth. in Appl. Mech. & Engng., 51, 107-137, 1985.

30. Oldenburg M. & Nilsson L., The position code algorithm for contact searching, Int. J. for Num.Meth. in Engng, 37,359-386, 1994.

31. Taylor R.L. et al., On a patch test for contact problems in two dimensions, ComputationalMethods in Nonlinear Mechanics, ed. P. Wriggers and W. Wagner, Springer-Verlag, Berlin, pp-690-702, 1991.

32. Crisfield M.A., Non-linear finite element analysis of solid and structures, Vol. 2 Advancedtopics, J Wiley & Sons, 1997.

33. Zavarise G., Wriggers P., Schrefler, B.A., A method for solving contact problems, Int. J. forNum. Meth. in Engng. , v42, n3, Jun 15, 1998.

34. Fletcher R., Practical Methods of optimisation, 2nd, Wiley, New Delhi, 1989.

35. Zienkiewiecz O.C., & Zhu J.Z., A simple error estimator and adaptive procedure for practicalengineering analysis, Int. J. for Num. Meth. in Engng., 24, 337-357, 1987.

36. Zienkiewicz O.C., Zhu J.Z., The superconvergent patch recovery and a posteriori errorestimates, Int. J. for Num. Metth in Engng, 33, 1331-1364, 1992.

Finite element modelling of orthogonal metal cutting 21

37. McDill J.M., Goldak J.A., Oddy A.S., Bibby M.J., Isoparametric quadrilaterals andhexahedrons for mesh-grading algorithms, Communication in Applied Num. Methods, Vol. 3,155-163, 1987.

38. Runnemalm H., Hyun S., Three dimensional welding analysis using adaptive mesh scheme,submitted forpublication.

39. Hyun S., Lindgren L.E., Smoothing and adaptive remeshing schemes for graded element,Communications in Numerical Methods in Engineering, 17, 1-17, 2001.

40. Hyun S., Lindgren L.E., Mesh smoothing techniques for graded elements, Simulation ofMaterials processing: Theory, methods and applocations, The sixthinternational confrence onnumerical methods in industrial forming process-Numiform 98, Enchede, The Netherlands 22-25 June 1998.

41. Gupta A.K., A finite element for transition from a fine to a coarse mesh, Int. J. Num. Meth.Engng., 12, 35-45, 1978.

42. Oddy A.S., Goldak J.A., McDill J.M., Bibby M.J., A distortion metric for isoparametricelements, Engineering with computers, Aug. 1986.

43. Blacker T.B. et. al., An adaptive finite element technique using element equilibrium and Paving,ASME 1990 International mechanical engineering congress &exposition, Dallas, Texas 25-30Nov. 1990.

44. Blacker T.D., Stephenson M.B., Paving: a new approach to automated quadrilateral meshgeneration, Int. J. for Num. Meth. in Engng., vol. 32, 811-847, 1991.

45. Zhang B., Bagchi A., Finite element simulation of formation and comparison with MachiningExperiments, Computational methods in material processing, ASME Publication, PED-Vol. 61,pp. 61-74, 1992.

46. Rice J.R., Tracey D.M., On ductile enlargement of voids in triaxial stress fields, Journal ofMech. Phys. Solides, 17, 201-217, 1969.

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49. Third Wave, AdvantEdge User's Manual Version 3.3.

Paper IV

1

1 INTRODUCTION

Machining operations are widely employed in indu-stry for the production of a variety of engineered pro-ducts. The performance of these cutting operations isoften characterised by such measures as machinedsurface finish, cutting forces, tool life, etc. The resi-dual stresses and surface finish can significantly af-fect the resistance of material to failure whensubjected to high cycle fatigue loads. The fatiguecrack, in general, initiates at the surface of the compo-nents and propagates into the material. If the surfaceresidual stress is tensile and further tensile stresses areapplied, then due to the loading fatigue resistancemay be significantly reduced. Henriksen (1951)found that the main residual stress in the work mate-rial was caused by the machining, which induced theplastic deformation of a surface layer. Furthermore,the geometrical accuracy of the components will alsobe influenced by the residual stress distribution.

In this investigations the residual stresses in stain-less steel AISI 316L have been studied as a functionof cutting speed. The experimental result from an X-ray inspection of the surface was compared with theresults from the finite element simulations.

Simulation of mechanical cutting process using thefinite element method is a challenge both from nume-rical and modelling perspective. Due to extremely lo-

calised deformation, it is necessary to have a verygood adaptive meshing capability. An appropriate so-lution is a frequent updating of the FE- mesh in whichthe element topology is either changed or preserved,(Kalhori & Lundblad 2000).

Furthermore, the contact algorithm must be imple-mented in such a way that it is stable, allowing as lar-ge time steps as possible. The modelling of thematerial behaviour and the chip formation process iscomplex. The varying temperature and the high strainrates make it difficult to obtain material properties forthe corresponding conditions. The modelling of thechip formation can be done in several ways. The chipseparation can be modelled by a predefined tied con-tact interface that is untied when some criteria are ful-filled. A better approach is to use the remeshing logicitself together with plastic strains to model the chipformation, (Kalhori & Lundblad 2000).

Kalhori et al.(1997) simulated orthogonal cutting,see Figure 1, as a one layer removal. The workpiecedid not have any initial stresses. This is not true for acutting process (except at the start), as the surfacemay have residual stresses from the previous cutting.This is obvious from Figure 1, where the tube end tur-ning is continuing for more than one revolution. Thecurrent study is performed in order to check whetherthe residual stresses from previous cutting, will affectthe chip formation process or the residual stresses.

ABSTRACT: The main objective of this study is to investigate the effect of previous cutting has to be taken into account whensimulating orthogonal cutting. Simulations are performed with the purpose of investigate the chip formationprocess and to find the residual stresses on the surface of the work piece. An implicit finite element code withspecial remeshing capabilities enabling the simulations has been used in the work.

Effect of previous cutting on chip formation

Vahid KalhoriDepartment of Mechanical Engineering, Luleå University of Technology, SE-971 87 Luleå, Sweden

2

Figure 1. Orthogonal cutting, end tube turning.

Figure 2. Schematic sketch of orthogonal cutting.

2 EXPERIMENTS

The workpiece material SANMAC 316L was a mach-ine-ability improved AISI 316L stainless steel.Theproperties of work piece material at room temperatureare shown in Table 1. The strain-stress relation at dif-ferent temperatures is presented in Figure 3.

Figure 3. True strain vs true stress for SANMAC 316L attemperatures, 23oC, 200oC, 400oC, 600oC and 800o C.

The used insert was TNMG 160408-QF in grade235 produced by SANDVIK Coromant. It has an edgeradius of 45 µm and a rake angle of +6o and with 0.2mm wide chamfer with -6o rake angle.

Figure 4. Insert geometry of a TNMG 16408-QF fromSANDVIK Coromant in grade 235.

A cylindrical workpiece was first turned to removethe hardened surface and then it was turned to a pipeform with large diameter. Finally, orthogonal cuttingwas achieved by turning the end of the pipe. The cut-ting data is presented in Table 2.

Cutting forces were measured with a Kistler dyna-mometer in a lathe in three directions, cutting, feedingand passive.

Since the Kistler dynameter has a low bandwidth a

Table 1. Material properties for SANMC 316L at room temperature.

σ0 Yield stress 240 MPa

E Young’s modulus 186 GPa

ν Poisson’s ratio 0.30 -

Cp Heat capacity 445 J/kgoC

α Thermal expansion 16.5e-6 1/oC

λ Thermal conductivity 14 W/oCm

δ Density 7900 kg/m3

f = h 1

v c

Φ

h 2

lc

re

α

γ

Table 2. Used cutting data in experiments and simulations

Test noCutting speed Feed Cutting depth

1 120 0.15 3.0

2 180 0.15 3.0

vc m min( )⁄[ ] fn mm( ) rev( )⁄[ ] ap mm[ ]

3

300 Hz low pass filter was used to avoid influencesfrom resonance in the machine tool and cutting tool.Quick-stop tests were performed in order to capturethe chip morphologies and to measure residual stres-ses on surface where the insert has only passed underfull cutting. The Quick-stop specimens were cut outfrom the work pieces, grounded, polished and etchedand studied in microscope.

X-ray diffraction was used to find the residualstresses of the surface layer. X-ray diffraction measu-res change in the spacing of atomic planes in the metalcrystal from which the stress can be calculated. Themeasurements were performed using a SiemensD5000 with an Ω geometry with 11 Ψ-tilting in the re-gion of 45o to 43o, with a step length of 0.08o and atime step of 30 s, and CuKα radiation. Primary a sol-ler slit and a 1 mm divergence slit were used and se-condary a fin film attachment (0.40o) LiF-monochromator with scintillation detector was used.The scanned surface area was 2 by 15 mm with a pe-netration depth of approximate 5 µm.

3 SIMULATIONS

The finite element program SiMPle has been used inthe simulations. This is an implicit finite elementprogram, which solves coupled thermo-mechanicalproblems. It is a solid formulation and uses anupdated Lagrangian mesh. The full Newton-Raphsonprocedure is used for the solution of the nonelinearsystem of equations. A staggered method for coupled transient

mechanical and heat transfer analysis is utilised. Anisothermal mechanical step is taken followed by arigid transient heat transfer step with heating fromplastic work and friction.The material model accounts for thermo-elasto-

plastic strains using an isotropic von Mises plasticityformulation. The material properties are temperaturedependent and thereby it also accounts for thermalsoftening. The strain rate effect on the flow stress isnot taken into account in the material model. The used element is a four to eight node quadrilateral

solid element with piecewise bilinear shape functions(McDill & Oddy 1987). The volumetric strain field isunderintegrated in order to avoid locking due to theplastic incompressibility. The special shape functionsapplied in the element makes it straightforward tocreate a graded mesh as an element with three nodesalong a side can be joined to two elements, each withtwo nodes along this side. No additional constraintsare required for interelement compatibility. The meshregeneration scheme is described in McDill & Oddy(1987). It has been extended with additional r- and h-adaptive mesh regeneration, so called, advance fronttechnique. Triangular elements are usually used in adaptive

meshing. However, quadrilateral elements are

preferred over triangular elements when dealing withplasticity (Cifuentes & Kalbag 1992), (Benzley et.al.1995). Both Augmented Lagrangian technique (Simo &

Laursen 1992), (Laursen & Simo 1993), (Laursen1994), (Laursen & Maker 1995) and pure penaltyformulation (Crisfield 1997) are implemented in themechanical contact algorithm. The penaltyformulation was used in the present simulations. Thecontact algorithm is implemented as a two-passalgorithm. Each side of the contact is in turn treated asthe master surface by the algorithm. The thermalcontact uses a contact resistance when the surfaces arein contact.

4 RESULTS

The chip morphology obtained from the so calledquick-stop experiments is shown in Figure 5.

The lamina structure of the chip is formed by athermo-plastic shear localisation at the primary defor-mation zone. At this region the work material is sub-jected to large shear deformation at a high strain-rate(usually around 103 - 105). The temperature rise ismainly due to the heat generated by plastic deforma-tion. Sliding of the chip along the interface betweenthe workpiece and the insert also generates heat dueto the friction, at the secondary deformation zone.The insert/chip interface is usually divided into asticking region and a sliding region, see Figure 6. Inthe sticking region, the work piece material adheres tothe tool and shear occurs within the chip, the frictionalforce is high and so is the heat generation. The highesttemperature in the chip usually occurs in the slidingregion, see Figure 7.

Figure 5. Chip morphology from quick-stop. The feed is 0.15mm and the cutting speed is 180 m/min.

4

Figure 6. Definition of the primary and the secondary defor-mation zone and the sliding & sticking region.

Figure 7. Contour of temperature distribution at the secondlayer removal. The feed is 0.15 mm and the cutting speed is 180m/min.

Figure 8. Contour of effective plastic strain at the second lay-er removal.The feed is 0.15 mm and the cutting speed is 180 m/min.

However, the lamina structure of the chip found inthe experiments is not obtained in the simulations.This may be due to the lack of strain rate dependencyin the material model and possibly damage mecha-nics. It is also known (Vaz Jr et. al 1998) that thewidth of the shear band is strongly mesh dependent.

The shear localisation of the primary deformationzone results in a fluctuation of the cutting forces withthe same frequency as that of lamina forming. Thiscan be observed by the determined cutting force dia-gram from the simulations, Figure 9. This high fre-quency force fluctuation can not be measured by therelatively slow cutting dynamometer. Therefore, themeasured cutting forces are average forces. These to-gether with computed average cutting forces are pre-sented in Table 3.

The results of x-ray diffraction measurements ofthe specimen surface are presented in Table 4. To thelisted deviation an error of MPa shall be added.The residual stresses from simulations in Table 4, andFigure 10 are taken at positions of a typical stress dist-ribution along a line perpendicular to the surface ofthe work piece. The fluctuation of the stresses in thecutting direction can be seen in Figures 11-12.

Figure 9. Simulated force for a cutting speed of 180 m/minand a feed rate of 0.15 mm/rev. First layer is removed during 0.to s, and the second layer is removed during s to

s.

Chip

Slidingregion

Stickingregion

Primarydeformation zone

Secondarydeformationzone

Workpiece

Insert

Table 3. Measured cutting forces in cutting and feed direction compared to predicted average cutting forces from simulations.

Test no Measured Simulated

Fc (N) Ff (N) Fc (N) Ff (N)

1 1005 729 850 400

2 890 584 500 250

50±

1 3–×10 1 3–×102 3–×10

5

Figure 10. Simulated residual stress σxx vs distance from sur-face for simulated machined surface after cool down. The feed is0.15 mm and the cutting speed is 180 m/min.

Figure 11. Residual stress σxx in cutting direction of machi-ned surface after specimen cooled down to room temperature.Used cutting data are a cutting speed of 120 m/min and feed ratesff of 0.15 mm/rev.

Figure 12. Residual stress σxx in cutting direction of machi-ned surface after specimen cooled down to room temperature.Used cutting data are a cutting speed of 180 m/min and feed ratesff of 0.15 mm/rev.

Figure 13. Simulated residual effective plastic strain ε vs dis-tance from surface for simulated machined surface after cooldown. The feed is 0.15 mm and the cutting speed is 180 m/min.

5 CONCLUSIONS

The low heat transfer rate of stainless steel AISI 316Lcauses the shear deformation process in the primarydeformation zone to localise in thin layer making thecutting process non-stationary and the chip irregular.The residual strains and stresses vary on and under thesurface along the cutting direction. The residual stres-ses obtained with X-ray diffraction measurement areaverage value for the measured surface. Both measu-red and computed residual stress level increases withcutting speed, but the computed value is much higherthan the measured stress. It is only the heat conduc-tion that gives a difference between different cuttingspeeds in the current model. There is no rate-depen-dency in the material model or inertia forces included.Neither is damage accounted for. The only softeningprocess is thermal softening. It is assumed that thematerial will reach a constant hardening when thestress-strain reaches a certain value.

As it is shown in Table 4, the residual stresses forthe second pass is about 15% lower than the residualstresses for the first pass. The values for residualstresses are those taken when the material has cooledto room temperature. The measured and computed

Table 4. Residual stress on machined surface measured with X-ray diffraction compared to residual stress from simulation with SiMPle.

Specimen Measuredσ (ΜPa)

Calculatedσ (ΜPa)Pass 1

Calculatedσ (ΜPa)Pass 2

1 130 750 650

2 138 780 670

8±

9±

6

cutting forces decrease at higher cutting speed, butalso here the computed value was lower than the me-asured. A friction coefficient of 0.5 was assumed insimulation. This together with the lack of rate-depen-dent material model may explain the difference. Ho-wever, the chip formation was not affected by the firstpass. The cutting forces became lower at the secondpass. The reduced cutting forces may be explained asfollows. The first passes causes some hardening andthereby an increased yield strength of the material.This would increase the cutting forces. However, theexisting residual stresses are such that it is easier tocreate a plastic deformation during the second pass.

The main conclusions in this study of the effect ofprevious cutting on subsequent layers are as follows.The chip formation is not affected much. There isonly a minor influence from the residual stress on thesurface from the first cutting on the second pass chipformation. This influence is negligible. The residualstresses are affected more in the current model. Butthis influence is expected to be smaller when anappropriate material model which includes rate-de-pendency and damage effects is implemented. Thusone can probably ignore the effects of previous cut-ting on a surface for the studied process. This conclu-sion is limited to this 2d model where the fixture ofcutting tool is completely rigid and there is no 3d geo-metry effect causing varying chip thickness.

6 ACKNOWLEDGEMENT

The presented work is a part of an ongoing project atthe Polhem Laboratory at Luleå University of Tech-nology, one of NUTEK’s (Swedish National Board ofIndustrial and Technical Development) competencecenters. Their financial support is gratefully acknow-ledged.

REFERENCES

Benzley S.E., Perry E., Merkely K., Clark B., Sjaardama, G.D., 1995. A comparison of all hexagonal and all tetrahedral finite element meshes for elastic and elasto-plastic analysis, In: Proc. of 14th Annual International Meshing Roundtable, Albaquerque, USA.

Cifuentes A.O., Kalbag, A., 1992. A performance study of tetrahedral and hexahedral elements in 3-D finite element structural analysis, Finite Elements in Analysis and Design, 12:313-318.

Crisfield M.A., 1997. Non-linear Finite Element Analysis of Solids and Structures, Vol. 2, Advanced Topics, J Wiley, Chichester.

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Kalhori, V., Lundblad, M., Lindgren, L.-E.,1997,

Nmerical and experimental analysis of orthogonalmetal cutting”, ASME, International mechanicalengineering congress & exposition, MED Vol. 6-2, Manufacturing Science and Engineering, 29-35.

Kalhori, V., Lundblad, M., 2000, Finite element modelling of orthogonal metal cutting, Submitted for publication.

Laursen T. A., Simo J. C., 1993. Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians, Computer Methods in Applied Mechanics and Engineering 108:133-146.

Laursen T.A., 1994. Automation and Assessment of Augmented Lagrangian Algorithms for Frictional Contact Problems. Journal of Applied Mechanics. 61: 956-963.

Laursen T. A., Maker B. N., 1995. An augmented lagrangian quasi-newton solver for constrained nonlinear finite element applications, Int. J. for Numerical Methods in Engineering. 38:3571-3590.

Liu, C. R., Barash, M. M., 1976a, The mechanicalstate of thee sublayer of a surface generated bychip-removal process-Part I Cutting with a sharptool, ASME Journal of Engineering for industry,1192-1201.

Lundblad, M., Kalhori, V., 2000, Influence of cuttingspeed on residual stresses in the work piece, Submited for publiction.

McDill J.M.J. Goldak J.A. Oddy A.S., Bibby, M.J., 1987. Isoparametric quadrilaterals and hexahedrons for mesh-grading algorithm. Communications in applied numerical methods. 3:155-163.

Simo J. C., Laursen T. A., 1992. An Augmented Lagrangian Treatment of Contact Problems Involving Friction. Computers & Structures. 42: 97-116.

Vaz Jr M., Owen D.R.J., Peric D., 1998 Finite-ele-ment techniques applied to high-speed machining,Simulation of Materials Processing: Theory, Met-hods and Applications, Numiform’ 98, 973-978.

Paper V

1

Influence of cutting speed on residual stresses in work piece

Mikael Lundblad

AB Sandvik Coromant, S-811 81 Sandviken, Sweden mikael.lundblad@sandvik.com

Vahid Kalhori

The Polhem Laboratory, Division of Computer Aided Design, Luleå University of Technology, S-971 87 Luleå, Sweden

Vahid.Kalhori@cad.luth.se

Abstract Residual stresses from machining may have a great influence on component fatigue life. It is therefore important to understand how they depend on the cutting process. The influence of the cutting speed and feed on the residual stresses is studied in this work. The material of the work piece is the stainless steel. Cutting experiments in laboratory with following measurements as well as finite element simulations are used and also compared with each other. KEY WORDS: Finite element method, Metal cutting, Residual stress, X-ray diffraction

1. Introduction Stainless steel is widely used in process industry and in power plants for parts needing both good mechanical properties and high resistance to corrosion. However, during machining these good qualities may be reduced since large tensile residual stresses can be introduced in cutting operations. These residual stresses can cause severe failures due to fatigue and stress corrosion. It is therefore important to know and even better to control the residual stress state in the machined part so failures can be avoided. The cutting process is studied in order to understand and to control the residual stresses. This may be done through measurements and/or finite element simulations of the cutting process. Both of these approaches are difficult due to large plastic deformations together with high strains and strain rates in combination with rapid temperature changes.

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Measurements in the cutting zone during cutting are very difficult to perform because of the hostile environment with high temperature and pressures. The cutting insert is plowing through the work piece material compressing and shearing the material up front in the primary deformation zone, see Figure 1, before the chip is separated from the work piece. The large compressive and shearing loads deforming the material plastically in the primary deformation zone in front of the tool but also beneath the cutting edge. Instead so have indirect quantities been studied or measurements have been done after the cutting is finished in order to build know-how about the cutting process. They are for example, cutting forces, tool wear and surface integrity. Reports from experiment based work [1-2] aim at relating the surface integrity/residual stress to the used machining parameters.

Primary deformat ion zone

Work piece

Cutting insert

Chip

Figure 1. Cutting insert cutting a chip. Primary deformation zone up front of cutting edge. Finite element models of the cutting process can give more information than experiments but also require more knowledge of material behaviour, friction etc. The special conditions in the process zone require finite element codes with thermal mechanical coupling and remeshing capabilities. Models that describe deformation hardening, thermal softening and heat transfer are also needed. This is important when forecasting the cutting process with the chip forming and separating from the work piece. After the cutting operation the work piece surface cooling down to room temperature is simulated to obtain the final residual stresses of the work piece. In this study, the influence of the cutting speed and feed on the residual stresses has been computed and verified by the experiments. The explicit finite element code AdvantEdge was compared with the implicit in-house finite element code SiMPle. It is shown that the state of residual stresses in the work piece increases with the cutting speed.

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Since in most machining cases the surface of a work piece is generated by the minor cutting edge, see Figure 2, low feed rates was chosen for the machining experiments and simulations. The experimental set up is such that a 2D, plane strain finite element model can be used.

Major cutting edge

Minor cutting edge

Feed direction

Insert

Work piece

Figure 2. Major and minor cutting edges in turning operation.

2. Experiments Tensile tests at a very low strain rate were performed in order to obtain material properties for the work piece. Furthermore, the experiments were also done to decide the cutting forces, chip morphology and residual stresses on the surface of work piece. The used work piece material SANMAC 316L is a machine-ability improved AISI 316L stainless steel manufactured by SANDVIK Steel. Material properties for SANMAC 316L obtained in tensile testing are given in Figure 3 and Table 1. These data show the tensile properties of the work piece. No tests for varying strain rates were performed.

Table 1. Material properties for SANMAC 316L at room temperature

σ0 Yield stress 240 MPa E Young’s modulus 186 GPa ν Poisson’s ratio 0.30 - cp Heat capacity 445 J/kg°C α Thermal expansion 16.5e-6 1/°C λ Thermal conductivity 14 W/°Cm ρ Density 7900 Kg/m3

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0

200

400

600

800

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

True strain ε

True

stre

ss σ

[MPa

]

23 C

200 C

400 C

600 C

800 C

Figure 3. True strain ε vs. true stressσ, for SANMAC 316L for different temperatures. The used insert was TNMG 160408-QF in grade 235 produced by SANDVIK Coromant. The geometry of the insert is shown in Figure 4.

Figure 4. Insert genometry of a TNMG 160408-QF from SANDVIK Coromant in grade 235. The insert has a edge radius of 45 µm and a rake angle of +6° and with a 0.2 mm wide chamfer with -6° rake angle.

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To mimic plain-strain 2D conditions orthogonal cutting was employed. The work piece periphery was first turned to remove the hardened surface and then it was turned to a pipe form with large diameter. Finally, orthogonal cutting was achieved by turning the end of the pipe shaped work piece with the insert put into a devise for making quick stop by blasting the insert and the holder away from the cutting zone. Used cutting data are presented in Table 2.

Test no Cutting speed Feed Cutting depthv c [m/min] f n [mm/rev] a p [mm]

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

Table 2. Used cutting data in experiments and simulations. Cutting forces were measured with a Kistler dynamometer in a lathe in three directions, cutting, feeding and passive. Since the Kistler dynamometer has a low bandwidth a 300 Hz low pass filter was used to avoid influences from resonance in the machine tool and cutting tool. Quick-stop tests were done to capture chip morphologies and to measure residual stresses on surfaces. The Quick-stop specimens were cut out from the work pieces, grounded, polished and etched and studied in microscope. To find the residual stresses in the surface layer x-ray diffraction was used. The hardness was measured at a line perpendicular to the surface in order to get an estimate of the depth of the machine-affected zone. X-ray diffraction measures change in the spacing of atomic planes in the crystal. This change of spacing between the atomic planes represents elastic strain in the crystal from which the stress can be calculated. The x-ray diffraction measurements were done using a Siemens D5000 with a Ω geometry with 11 ψ-tilting in the region of -45° to 43°, with a step length of 0.08° and a time step of 30 s, and CuKα radiation. Primary a soller slit and a 1 mm divergence slit were used and secondary a fin film attachment (0.40°) LiF-monochromator with scintillation detector was used. The scanned surface area was 2 by 15 mm with a penetration depth of approximate 5 µm.

3. Finite element simulations The finite element method is a powerful technique for the numerical solution of a variety of engineering problems. This has been applied to simulate machining process with some success during the last decades. FEM seems to be suitable for analysing the state of stress and deformation in the cutting system. In the current studies, the finite element codes AdvantEdge and SiMPle were used to compute the state of residual stresses in work piece material since it is cooled down to the room temperature. The plain-strain 2D model was used. The continuous mesh updating is used to

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model the chip formation. Earlier studies [3] have used a predetermined parting line between work piece and chip.

3.1 AdvantEdge This commercial software is developed for simulation of mechanical cutting. AdvantEdge machining modelling software uses two-dimensional Lagrangian explicit finite element analysis. The material model accounts for elasto-plastic strains and has an isotropic power law for strain hardening. The strain rate also affects the flow stress. The material properties are temperature dependent and thereby it also accounts for thermal softening. A staggered method for coupled transient mechanical and heat transfer analyses is utilised. First an isothermal mechanical step is taken followed by a rigid transient heat transfer step with constant heating from plastic work and friction. Both steps have identical meshes. Central difference schemes are used for the time integration for each of the staggered steps. A six-node quadratic triangle element with three quadrature points is used. The mesh, which becomes very distorted around the cutting edge, is periodically updated both refining large elements and coarsening small elements. The coefficients for the material models for 316L and cemented carbide are included in a database of the software. The model in AdvantEdge includes strain rate dependency and the parameters are obtained from experiments performed by others [4]. For this simulation a power viscosity law [5] constant rate sensitivity exponent was chosen. It is written as

)(

1 p

_

0

m

p

p

g

=

+ .

.

εσ

ε

ε

(1)

and a power hardening law with a cut off strain εp

cut at which the increase in deformation hardening stops. The strain hardening and temperature dependency of the plastic flow properties are determined via the function g (εp) as

( ) pcut

ppcut

pp

pp Tg εεεε

εεσε =⇒≥

+Θ= 0 if, 1)(

1/n

0

(2)

Where

( ) 5510 ... TcTccT +++=Θ (3)

The temperature dependencies of thermal properties of the work piece material were also modelled with polynomial functions, see Eq. 4-6.

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( ) ( )5510 ... TCpTCpCpCpTCp +++= (4)

( ) ( )5510 ... TTT λλλλλ +++= (5)

( ) ( )5510 ... TTT ααααα +++= (6)

3.2 SiMPle SiMPle is an in-house research code with similar features as AdvantEdge. However, it is an implicit finite element program. The Newmark method is used. The simulations are performed by a graded element [6]. This is a four to eight node quadrilateral element with piecewise bilinear shape functions [6]. The volumetric strain field is under integrated in order to avoid locking due to plastic incompressibility. The special shape functions applied in the element makes it straightforward to create a graded mesh as an element with three nodes along a side can be joined to two elements, each with two nodes along this side. No additional constraints are required for inter-element compatibility. Thus, graded elements are specially designed for alleviating the creation of mesh with refined regions. The full Newton-Raphson procedure is used for the solution of the non-linear system of equations. The material is treated as a thermo-elastic-plastic material with temperature dependent properties. No strain rate dependency is included in the material model.

4. Results The morphology obtained both from machining experiments and simulations are shown in Figure 5. In the cutting of stainless steel the shear deformation in the primary shear zone localises and forms lamina structured chips. The shear localisation makes the cutting forces fluctuate with the frequency of lamina forming which are in the range of 5-15 KHz. The cutting forces from simulations, see Figure 6, shows these fluctuations in cutting forces from the lamina forming. This high frequency force fluctuations cannot be picked up by the relatively slow dynamometer. Therefore, the measured cutting forces are average forces. These cutting forces are presented in Table 3. Test no Measured Simulated

SiMPle Simulated AdvantEdge

Fc [N] Ff [N] Fc [N] Ff [N] Fc [N] Ff [N]

1 422 390 400 290 420 380 2 1005 729 780 400 860 470 3 404 360 450 330 425 375 4 890 584 780 400 840 460 5 391 357 430 320 420 375 6 857 520 770 390 845 475

Table 3. Measured cutting forces in cutting and feed direction compared to predicted average cutting forces from simulation

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( a ) ( b )

( c ) ( d )

( e ) ( f )

Figure 5. Chip morphologies from quick-stops (a-b), simulations with AdvantEdge (c-d) and simulation with SiMPle (e-f). The feed fn is 0.05 mm in a, c and e and 0.15 mm in b, d and f. The cutting speed vc is 240 m/min.

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( a )

( b )

( c )

( d )

Figure 6. Simulated cutting force Fc and feed force Ff using AdvantEdge (a-b) and SiMPle (c-d). The cutting speed is 240 m/min and the feed rate is 0.05 m/rev in a & c and 0.15 mm/rev in b & d. The hardness was measured at a line perpendicular to the surface in order to get an estimate of the depth of the machine-affected zone. The work piece material has an increased hardness down to about 0.15 mm from the surface, see Figure 7. In Figures 8-9 the strains under the surface are presented as a function of distance from the machined surface. Since stainless steel has a large deformation hardening plastic deformation extends deep down into the work piece material. At a plastic strain of 5% the yield stress has increased from 240 MPa to over 400 MPa. Plastic strain levels of 5% are found at a depth of 0.125 mm for a feed rate of 0.05 mm/rev and at a depth of 0.25 mm for a feed rate of 0.15 mm/rev.

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150

170

190

210

230

250

270

290

0.00 0.10 0.20 0.30 0.40

Depth from surface

Har

dnes

s Hv

0,3 1

23456

Figure 7. Micro hardness measured with Vickers using a load of 0,3 kg vs. depth from surface of specimen.

0,0

0,1

0,2

0,3

0,4

0,5

0,00 0,05 0,10 0,15 0,20

Distance from surface [mm]

Effe

ctiv

e pl

astic

stra

in test 1

test 2test 3test 4test 5test 6

Figure 8. Simulated residual strain ε vs. distance from surface for simulated machined surface after cool down using AdvantEdge.

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0,0

0,1

0,2

0,3

0,4

0,5

0,000 0,003 0,007 0,015 0,100

Distance from surface [mm]

Effe

ctiv

e pl

astic

stra

in test 1

test 2

test 3

test 4

test 5

Serie6

Figure 9. Simulated residual strain ε vs. distance from surface for simulated machined surface after cool down using SiMPle. The results of X-ray diffraction measurements of the specimen surfaces were evaluated using elliptical fitting. The results presented in Table 4 have the deviation from the fitting listed as ± value. To the listed deviation an error of ±50 MPa shall be added. The fluctuation of the stresses in the cutting direction can be seen in Figure 10. The averaged residual stresses from simulations in Table 4 and Figures 11-12 are taken at positions of a typical stress distribution along a line perpendicular to the surface of the work piece.

Figure 10. Computed residual stress σxx using SiMPle, on the work piece surface after specimen cooled down to room temperature. The cutting speed vc is 120 m/min and feed rates is 0.05 mm/rev

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-400

-200

0

200

400

600

800

0,00 0,05 0,10 0,15 0,20

Distance from surface [mm]

Stre

ss S

xx [M

Pa]

test 1test 2test 3test 4test 5test 6

Figure 11. Computed residual stress Sxx in cutting direction vs. distance from surface for simulated machined surface after work piece has cooled down using AdvantEdge.

-400

-200

0

200

400

600

800

0,000 0,003 0,007 0,015 0,100

Distance from surface [mm]

Stre

ss S

xx [M

Pa]

test 1test 2test 3test 4test 5test 6

Figure 12. Computed residual stress Sxx in cutting direction vs. distance from surface for simulated machined surface after work piece has cooled down using SiMPle.

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Table 4. Measured and computed residual stresses on machined.

5. Conclusions The low heat transfer rate of stainless steel 316L causes the shear deformation process in the primary shear zone to localise in thin layers and thereby making the cutting process non-stationary and the chip irregular. This shear localisation makes the residual strain and stresses vary at and under the surface along the cutting direction. This makes it difficult to determine a representative value for the residual stress level on the machined surface in the simulations. The residual stresses attained with x-ray diffraction measurement are average values for the measured surface. The high residual stress level in the cutting direction for the lower feed within simulations performed by AdvantEdge is interesting since it is contradictive to other reports [2,7]. The measured residual stress level increases with cutting speed whereas the computed value decreases. However, the measured forces decrease. The deviations between measured and computed residual stresses in AdvantEdge may be due to uncertainties in the material modelling and possible phase transformations in the work piece. However, the computed results by SiMPle indicate an increase of residual stress level when the cutting speed is increased as in the measurement. It is only the heat conduction that gives a difference between different cutting speeds in this model. There is no rate-dependency in the material model or inertia forces included. Neither is damage accounted for. The only softening process is thermal softening. It is assumed that the material will reach a constant hardening when the stress-strain reaches a certain value. Thus it may be difficult to say whether the increase in residual stress with velocity in the simulations using SiMPle will remain when its material model is improved.

Acknowledgement The present work is a project in the Polhem Laboratory at Luleå University of Technology, one of 1VINNOVA competence centra. Their financial support is gratefully acknowledged. 1. VINNOVA is the Swedish Agency for Innovation Systems

Specimen Measured Calculated AdvantEdge

Calculated SiMPle

σ [MPa] σ [MPa] σ [MPa] 1 361 ± 17 640 130 2 130 ± 9 179 200 3 629 ± 28 550 630 4 138 ± 8 164 600 5 703 ± 31 240 680 6 500 ± 25 171 690

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References 1. Jang, D. J., Watkins, T. R., Kozaczek, K., J., Hubbard, C. R., Cavin, O. B., Surface residual stresses in machined austenitic stainless steel, WEAR, vol. 196, Elsevier, p168-173, 1996. 2. Liu, C. R., Barash, M. M., Variables Governing Patterns of Mechanical Residual Stress in a Machined Surface, Journal of Engineering for Industry, vol. 104, p257-264, 1982. 3. Shih, A. J., Yang, H. T. Y., Experiments and Finite Element Predictions of Residual Stresses due to Orthogonal Metal Cutting, International Journal for Numerical Methods in Engineering, vol. 36, p1487-1507, 1993. 4. Follansbee, P. S., High-Strain-Rate Deformation of FCC Metals and Alloys, International Conference on Metallurgical Applications of Shock-Wave and High-Strain-Rate Phenomena (EXPLOMET `85), p. 451-479, 1986. 5. Third Wave Systems Inc., AdvantEdge User’s Manual version 3.3, 1999. 6. Gupta, A. K., A finite element for transition from a fine to a coarse mesh, Int. J. Num. Meth. Engng. 12,35-45,1978. 7. Okushima, K., Yoshiaki, K., The Residual stress Produced by Metal Cutting, ann. CIRP, vol. 20, p13-14, 1971.