Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 3, pp. 599-616, Warsaw 2009

AN EFFICIENT IMPLEMENTATION OF BOUNDARY
CONDITIONS IN AN ALE MODEL FOR ORTHOGONAL

CUTTING

Maŕıa H. Miguélez

Ana Muñoz-Sánchez

José L. Cantero
Universidad Carlos III de Madrid, Department of Mechanical Engineering, Madrid, Spain

e-mail: mhmiguel@ing.uc3m.es

José A. Loya
Universidad Carlos III de Madrid, Department of Continuum Mechanics and Structural Analysis,

Madrid, Spain

Prediction of machining-induced residual stresses is an interesting objective in
the field of modellingmanufacturing processes. Although Finite Element Ana-
lysis (FEA) has been widely used for this purpose, many problems are found
when the numerical model is developed. Computational cost and numerical
problems related to the extreme mesh distortion make the effort of finite ele-
mentmodelling of machining extremely time consuming. The aim of this work
is to predict machinning-induced residual stresses using a finite elementmodel
based in ALE (Arbitrary Lagrangian Eulerian) approach. The finite element
general-purpose code ABAQUS is used, modifying the previous model used
in scientific literature to predict residual stresses. Boundary conditions in the
entrance of the workpiece and in the upper border of the chip were modified
from Lagrangian boundaries in the previous model, to Eulerian boundaries in
the newmodel.
Main advantages of themodel presented in this work are low level of distortion
of the mesh, the possibility of simulate long length of machined surface and
time-efficiency. The model has been applied to calculate residual stresses in
AISI 316L during machining. Reasonable agreement with experimental results
has been found.

Key words: residual stresses, orthogonalmachining, ALE, boundary conditions

1. Introduction

FiniteElement (FE)models havebeen extensively used to simulate the cutting
process for decades. Although many authors have focused their attention on



600 M.H. Miguélez et al.

the development of numerical tools to obtain accurate results from simulation,
the problem of machining modelling can not be considered as completely so-
lved.Theanalysis of residual stresses due tomachining operations has been an
active subject of research. The reliability of structural components obtained
by machining operations is influenced by the state of residual stresses resul-
ting from processing. Tensile residual stresses in the vicinity of the machined
surface has negative effects on fatigue, fracture resistance and stress corrosion
and, therefore, can substantially reduce life of the the component (Okushima
and Kakino, 1972).

Machining is a complex process where large strains and strain rates are
produced and large temperatures are generated by dissipation of the plastic
work andby frictional heating.Residual stresses are relatedwith these coupled
thermo-mechanical phenomena (Liang and Su, 2007) and also with metallur-
gical changes related to the former (Obikawa et al., 2008).

Theprocess of creating amodel formachining could be summarised as illu-
strated inFig.1. Several blocks of parameters aredefined,beginningwith iden-
tification of the process and materials parameters, following with numerical
parameters that should be implemented and those related with the numerical
analysis. Finally, typical output results are presented.

Fig. 1. Scheme of numerical modelling of the machining process

Popular commercial FE codes,mainlyDEFORMandABAQUShave been
used to predict residual stresses in the orthogonal cutting.The commercial FE
softwareDEFORM-2D is a Lagrangian implicit codewith adaptive remeshing
(Outeiro et al., 2006a,b), meanwhile ABAQUS is one of the most versatile



An efficient implementation... 601

general-purpose code allowing both ALE and Lagrangian analysis as well as
implicit and explicit integration schemes (Nasr et al., 2007; Salio et al., 2006).

ALE formulation presents advantageswhen comparedwith theLagrangian
approach and has been mostly used to simulate residual stresses using ABA-
QUS code. Themodel commonly used in the prediction of residual stresses is
described in detail in Nasr et al. (2007), Miguélez et al. (2009) and it is sche-
matically shown inFig.2. Themain drawback of thismodel is the necessity to
precisely define the previous geometry of the chip and its mesh in a way that
allow the model to deal with the high level of distortion that appears in this
zone. An iterative work is needed to obtain precise geometry and appropriate
mesh able to deformwithout stopping the calculation due to the high level of
distortion. Another problem is the enlargement of workpiece elements located
at the interface chip-tool, due to the use of Lagrangian boundary in the upper
contour of the chip. As the calculation time increases, the number of elements
in contact with the tool decreases because they are enlarged, thus diminishing
the accuracy of the calculation. These effects are illustrated in Fig.3, sho-
wing the mesh deformed during simulation. Figure 3a presents the detail of
the mesh when the initial geometry, not appropriate to perform the calcula-
tion, is selected. It is possible to observe excessive distortion at the curvature
zone of the chip, leading to the end of the calculation. In Fig.3b, an appro-
priate initial geometry and mesh of the chip were selected. However, as the
calculation advances, the elements of theworkpiece located at the interface en-
large, and only few elements are in contact with the tool when the calculation
finishes.

Fig. 2. Model commonly used in ALE formulation in ABAQUS to predict residual
stresses. See details in Nasr et al. (2007) andMiguélez et al. (2009)



602 M.H. Miguélez et al.

Fig. 3. Details of the cutting zone when using a model with Lagrangian boundaries;
(a) deformedmesh with distorted elements in the chip curvature, (b) deformedmesh

showing enlarged elements at the interface

One of the main drawbacks of simulation of machining, especially in the
case of simulation of residual stresses, is the extremely high computational
cost of simulations. Computational time of days or even weeksmay be needed
even for simulation of only few milliseconds of orthogonal cutting using 2D
models. This very low time introduces several problems if the focus of analysis
involves thermal issues related to heat generation and diffusion into the tool
(Dogu et al., 2006; Filice et al., 2007). In fact, no steady-state conditions are
reached during the numerical simulation. Technical literature shows that it
is relatively easy to predict some process variables such as the cutting and
thrust force, chip geometry, shear angle and contact length (Mamalis et al.,
2001) however, the numerical prediction becomes poorwhen the temperatures
on the rake face and inside the tool are investigated (Umbrello et al., 2007a).
In thecase of simulationof residual stresses, stronglydependenton thermal

phenomena, it isnecessary to simulate a long lengthmachined surface to ensure
steady state conditions and stabilised level of residual stresses in themachined
surface.
It is commonly admitted that tensile residual stresses in machining result

from heating of the machined surface during the cutting operation. The large
temperature level reached in a thin thermally affected layer near the workpie-
ce surface produces thermal expansion and plastic flow. During the subsequ-
ent cooling, thermal contraction of this layer is higher than in the workpiece
in-depth, and this phenomenon is believed to be at the origin of tensile re-
sidual stresses observed in machining. Although the common understanding
is that in the absence of thermal effects the mechanical loading exerted onto



An efficient implementation... 603

the workpiece leads to compressive residual stresses, it is shown in Miguélez
et al. (2009), Lin et al. (1991) that even in the absence of thermal effects,
a substantial level of tensile residual stresses can be obtained by solely pure
mechanical effects. However, the level of tensile residual stress is also strongly
influenced by thermal effects, thus the prediction of temperature should be
as good as possible when the prediction of residual stresses is the objective.
In consequence, it is critical to achieve steady state conditions in the calcula-
tion, being necessary to perform calculations corresponding to a large cutting
time.

On the other hand, the analysis should be performed in two phases, firstly
the cutting process is simulated and secondly the workpiece is cooled down
andunloaded.Mostworks based inABAQUSused explicit integration scheme
in both phases, being the second phase extremely time consuming. In a recent
work, (Nasr et al., 2008) developed an efficient model with an explicit scheme
of integration for the cutting phase and an implicit scheme for the unloading
and cooling phase.

The conditional stability is the only concern about explicit integration,
requiring a very small time step. This time is in the order of the time required
by a dilatational wave to cross the smallest element in the mode, typically
around 10−9-10−10 s, leading to a large calculation time. The time efficiency
of this phase canbe improvedwith theuse ofmass scaling as hasbeen reported
in literature (Arrazola et al., 2008).

Implicit integration is unconditionally stable and the only limitation on
the time step is convergence of the solution, which is based on a pre-defined
criterion. The use of implicit integration in the second phase of analysis si-
gnificantly diminishes the calculation time from several days to few minutes,
when compared with that obtained with explicit integration.

The time efficiency of simulation should be understood not only as the
resultant computational time.Thepreparationof themodel is also complicated
and needs different iterations in order to complete the simulation.

The main objective of this work is presenting a FE model for prediction
of residual stresses in metal cutting using the FE commercial code ABAQUS.
The model presented is a modification of the model commonly used in lite-
rature when predicting residual stresses with ALE formulation in ABAQUS,
described in Nasr et al. (2007), Nasr et al. (2008), that proved its accuracy in
predicting residual stresses after machining different alloys and has been used
also by the authors in Miguélez et al. (2009). Although the results obtained
with this model have been satisfactory, the preparation process of the model
(initial geometry of the chip andmeshing) has been a time consuming part of



604 M.H. Miguélez et al.

the work due to the need of several iterations before obtaining amesh able to
complete the desired cutting time.

The newmodel presented in this work combines Eulerian and Lagrangian
boundaries in order to improve the preparation step during the modelling
process. The use of Eulerian boundaries in the entrance of the workpiece and
in the chip significantly diminishes thenumber of iterations needed to generate
previous geometry and its associated mesh, due to the low level of distortion
in the zone around the tip tool obtained with the Eulerian boundary in the
chip. The boundaries also allowmaintaining a constant size in the element at
the interface improving the precision of the calculation of differentmagnitudes
in this zone. It is also possible to simulate a long length of cut, reaching steady
state conditions.

In order to obtain a time efficient method, the analysis was carried out
in two phases. Firstly, using an explicit integration scheme, the cutting was
modelled and steady state conditions were reached. Following, the workpiece
was unloaded and cooled by the implicit integration scheme recently proposed
in Nasr et al. (2008).

The study is focused on the residual stresses induced after orthogonal
cutting of stainless steel AISI 316L. Results obtained through this newmodel
are compared with experimental results showing good agreement with the
simulation process and its advantages.

2. Numerical model

2.1. ALE formulation

Eulerian, Lagrangian andArbitrary Lagrangian-Eulerian (ALE) techniqu-
es have been used to simulate orthogonal cutting. In Lagrangian analysis, the
computational grid deforms with the material whereas in Eulerian analysis it
is fixed in space. The Lagrangian calculation embeds a computational mesh
in the material domain and solves for the position of the mesh at discrete
points in time (Marusich andOrtiz, 1995). TheEulerian formulation has been
used tomodel orthogonal metal cutting as a steady process of chip formation
(Dirikolu et al., 2001).

ALE is a relatively new modelling technique in machining, including a
combination of the Lagrangian and Eulerian approaches without having their
drawbacks. It was firstly introduced to model the cutting process by the end
of the last decade. Some of recent ALE cuttingmodels were presented in Özel



An efficient implementation... 605

and Zeren (2005), Miguélez et al. (2006), Pantalé et al. (2004). This approach
became popular due to its implementation in commercial finite element codes.
Explicit dynamic ALE formulation is very efficient for simulating highly non-
linear problems involving large localised strains. Detailed formulation of the
ALE approach can be found in different papers, see for instance (Pantalé et
al., 2004). A brief summary is presented below.

As is well known, the ALE approach is an extension of both Lagrangian
and Eulerian descriptions, since the mesh does not remain fixed in space ne-
ithermoves attached tomaterial points. Then, the grid points have their own
governing equations of motion.

In the ALE description, material points are represented by a set of La-
grangian coordinates X, spatial points with a set of Eulerian coordinates x,
and a reference or grid points with a set of arbitrary coordinates ξ. At time t,
a spatial point x is obtained by mapping a material point X with material
motion x = Ψ(X, t) or by mapping the reference point ξ with grid motion
x = Ψ(ζ, t). The material velocity v of the particles is obtained using the
classical material derivative

v= ẋ=
∂x

∂t

∣∣∣
X=cte

(2.1)

The grid velocity v̂ is obtained after the introduction of a mixed derivative,
which represents the time variation of a physical quantity for a given grid
point

v̂= ẋ=
∂x

∂t

∣∣∣
ζ=cte

(2.2)

Mass, momentum and energy conservation laws in the ALE description are
used in a similar form as in the Eulerian description. Taking into account the
definition of the convective velocity c = v− v̂, conservation laws for ALE
approach are written as follows

ρ̇+c∇ρ+ρdivv=0 ρv̇+ρc∇v=f+ divσ
(2.3)

ρė+ρc∇e=σ :D− divq+r

where ∇ is the gradient operator, ρ is themass density, f are the body forces,
σ is theCauchy stress tensor, e is the specific internal energy, D is the rate of
deformation tensor, r is the bodyheat generation (ohmic resistance, inductive
heating...) and q is the heat flux vector. According to characteristics of the
machiningproblem, temperature is consideredasuniquecomponentof internal
energy, bodyheat generation is neglected, mechanical dissipation is associated



606 M.H. Miguélez et al.

to plastic deformation, and Fourier’s heat conduction law rules the heat flow.
Then, the First Law of Thermodynamics, Eq. (2.3)3, may be simplified

ρcV θ̇+ρcVc∇θ=βσ :D
P +k∆θ (2.4)

θ being the temperature, cV being the specific heat, k – thermal conductivity
coefficient, DP – plastic rate of deformation tensor, ∆ – Laplacian operator,
and β – fraction of plastic work converted into heat (Taylor-Quinney coeffi-
cient) included to match experimental results.

2.2. Model description

A plane strain ALE model was developed using the commercial Finite
Element code ABAQUS/Explicit. One of the requirements of the model is
to avoid mesh distortion from low cutting speed to elevated cutting speed
conditions. The residual stress distribution in themachined surface should be
stabilised and steady state conditions shouldbe reached.A thermo-mechanical
coupled analysis was developed by CPE4RT element type, see ABAQUS ma-
nual (2003), that are plane strain, quadrilateral, linearly interpolated, and
thermally coupled elements with reduced integration and automatic hourglass
control, for ALE formulation.

As explained before, the analysis was carried out in two steps: cutting,
using an explicit integration scheme and cooling and unloading, using the im-
plicit integration scheme proposed in Nasr et al. (2008). The residual stress
distribution was obtained in a section of the workpiece corresponding to sta-
tionary conditions during cutting.

Geometry of the numerical model is shown in Fig.4. The tool is fixed and
the cutting speed is applied to the workpiece. The cutting takes place in 1-2
plane under plane strain conditions. Continuous chip formation is assumed.
The geometry shown in Fig.4 corresponds with an uncut chip thickness equal
to 0.1mm and cutting edge radius of the tool equals 0.02mm. The depth of
the workpiece was selected taking into account the typical profile ofmachined
induced residual stresses. On the other hand, it is important to note that the
dimensions of the undeformed chip (curvature radius andwidth) are not criti-
cal, and thus, changes in thesemagnitudes lead to successful calculations with
the model. The sensibility of the model to changes in these variables are not
so accused as in the case of the previousmodel, inwhich lots of iterationswere
needed before obtaining a geometry able to perform the calculation during the
desired cutting time.



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Fig. 4. Geometry of the undeformed numerical model (dimensions in mm)

TheALEmodel uses sliding, Lagrangian andEulerian contours (seeABA-
QUS user manual) allowing the material to flow across an internal Eulerian
zone surrounding the tool tip. This approach avoids extreme distortion of the
mesh, allowing the simulation of a longmachined surface (larger than 15mm,
150 times the value of the uncut thickness of the chip equal to 0.1mm).

Themodel was divided in several zones allowing meshmotion or material
flowacross the fixedmesh, dependingonwhich zone is considered in themodel
(see Fig.5). Zones 1, 2 and 3 combine Lagrangian/Eulerian boundaries with
sliding boundaries (where the material is allowed to flow tangentially to the
contour and not allowed to go across this boundary). Eulerian boundaries
in the entrance of zone 1 and in zone 2 (chip) avoid distortion that usually
appears as the calculation advances.

Zone 4 is an Eulerian region, with themesh fixed, allowing the flow of the
material across this region.

Fig. 5. Boundary conditions implemented in the model



608 M.H. Miguélez et al.

The main advantage of this technique is that the distortion is avoided
in the region surrounding the tool tip and it is possible to simulate a long
machined surface (that is necessary to obtain stabilised residual stresses). Fi-
gure 6a and 6b shows respectively the initial mesh of themodel and deformed
configuration during the simulation.

Fig. 6. (a) Undeformedmesh; (b) deformedmesh during cutting

As is widely accepted in literature (Barrow, 1973), a value of theQuinney-
Taylor coefficient equal to 0.9 is assumed. An initial temperature of 293K has
been imposed.Conduction and convection to air (only in the freshlymachined
surface) were taken into account, radiation was neglected. The coefficient of
heat convectionwas 20W/m2Kand sink temperaturewas 293K.Thermal flux
was allowed in the attached contour of the tool.
Friction is one of the hardest phenomena to simulate in machining. In

the present work, a constant coefficient of friction along the tool/workpiece
contact length is assumed.Although it is the simplest formulation, it has been
widely used in numerical simulation of machining (Moufki et al., 1998).The
value of the friction coefficient equal to 0.4 reproduces accurately the resultant
residual stress observed in experimental tests, as will be shown later. The heat
partition between the tool andworkpiece was assumed to be 50-50% (Lin and
Lin, 1992).
The workpiece material is modelled using the Johnson-Cook (JC) consti-

tutive model (Johnson and Cook, 1983)

σY = [A+B(ε
pl)n]
(
1+C ln

ε̇pl

ε̇0

)[
1−
( θ−θref
θmelt −θref

)m]
(2.5)



An efficient implementation... 609

σY being the flow stress, ε
pl – equivalent plastic strain, θ – temperature, and

all remainingparameters beingmaterial constants. This constitutive equation,
available in ABAQUS/Explicit, is useful to simulate mechanical processes in-
volving high strain and strain rates, and thermal softening, and it has been
widely used to model cutting processes (Umbrello et al., 2007b).

Characterisation of the workpiece material AISI 316L has received consi-
derable attention in the literature. It is possible to found JC parameters in
a wide range of values, associated with thermomechanical behaviour of this
alloy. In this work, parameters presented in Table 1, proposed in Tounsi et al.
(2002) have been used.

Table 1.Tool material properties (Tounsi et al., 2002)

A [MPa] B [MPa] n C m ε̇0 [s
−1]

514 514 0.508 0.042 0.533 10−3

Physical properties of the cutting material (carbide) and workpiece mate-
rial were obtained from scientific literature (Jang et al., 1996; Umbrello et al.,
2007) and are presented in Table 2 and Table 3.

Table 2.Tool material properties (Jang et al., 1996)

Properties Carbide tool

Density [kg/m3] 14900

Specific heat [J/kg◦C] 138

Thermal conductivity [W/m◦C] 79

Table 3.Workpiece material properties (Umbrello et al., 2007b)

Properties AISI 316L

Youngmodulus [GPa] 202

Poisson coefficient 0.3

Density [kg/m3] 7800

Specific heat J/kg◦C] 542

Thermal expansion coefficient 1.99 ·10−5

Thermal conductivity [W/m◦C] 20



610 M.H. Miguélez et al.

3. Results and discussion

3.1. Numerical results

The previous preparation of the model with the new implementation of
boundaryconditions is easier than thatneededwith thepreviousmodel.When
the model based in Lagrangian boundaries was used, the elements located in
the chip curvature interface and in the interface experienced great distortion
as the calculation advanced, aborting numerical simulation in the case of large
cutting time problems.

Therefore, new attempts were needed to modify previous geometry of the
chip and/or the mesh of the workpiece had to be meshed again in order to
avoid distortion. This iterative process had to be repeated several times be-
fore obtaining an optimal geometry andmesh configuration able to finish the
calculation.

The statement ofEulerianboundariesat the chip andat the entrance of the
workpiece, leads to a deformedmesh shown in detail in Fig.7. It is possible to
appreciate the low level of distortion in theelements in theworkpiece compared
to the deformedmesh shown in Fig.3.

Fig. 7. (a) Detail of the deformedmesh close to the tool tip showing low distortion
achieved with the model. Temperature field is shown during cutting, once steady
state conditions have been reached. (b) Paths of nodes belonging to the tool and

chip where the temperature in the interface is recorded at the interface

Steady state conditionswere evaluated in terms of the temperature output.
In Fig.8, the evolution of temperature with cutting time in three elements of



An efficient implementation... 611

the workpiece located just beneath the tool tip is shown. Temperature incre-
ases with time and stabilises around the cutting time of 3ms corresponding
to 6mm of the machined length.

Fig. 8. Evolution of temperature of workpiece elements 1-3 located just beneath the
tool tip

On the other hand, it has been experimentally shown (Shaw et al., 1960)
that during cutting operation very high levels of pressures are reached at the
interface, leading us to consider that the hypothesis of temperature continu-
ity at the interface, should be verified. The evolution of temperature in the
interface in both tool and chip is shown in Fig.9. The contact length and chip
morphology during cutting is presented in Fig.7b, showing the path of nodes
where the temperature in the interface has been obtained.

Fig. 9. Evolution of temperature distribution at the interface



612 M.H. Miguélez et al.

It should be noted that temperature continuity in the interface is reached
around the cutting time of 5ms. As the calculation advances, up to the value
of cutting time equal to 10ms (corresponding to a length of machined surface
of 20mm), the differences in the temperature distribution in both chip and
tool at the interface are negligible. Although the level of temperature in the
interface does not directly influence the residual stresses, it is very important
whenstudyingtoolwear evolution.Thenewconfigurationof themodel allowed
simulation of large cutting times, thus reaching steady state conditions, also
fromthepointofviewof theanalysis of temperaturecontinuity in the interface.

3.2. Experimental validation

The model was validated comparing the predicted residual stresses obta-
ined with the model through simulation and experimentally measured. The
predicted residual stress distribution was obtained in depth in the workpiece,
in a section corresponding with steady state conditions.

Orthogonal dry cutting tests with air cooling were carried out in a lathe
(PINACHO model Smart-Turn 6/165) with the following parameters: cut-
ting speed 120m/min, feed rate 0.1mm/rev, cutting width 2mm (correspon-
ding with those used in simulations). The workpiece material was a tube of
AISI 316L steel with wall thickness 2mm. The tool geometry was generated
by electro-discharge machining in a hard metal. The cutting edge radius was
20µm. The distribution of residual stress in the circumferential direction was
measured in the Technological Centre IDEKO (see http://www.ideko.es/) in
a specimen previously machined in the same conditions as those imposed in
simulations.

Figure 10 shows reasonable accuracy of themodelwhenpredicting residual
stresses. Experimental tensile values in the machined surface are larger than
the predicted ones. This behaviour could be related with the uncertainty of
the measurement which value is around 120MPa.

4. Concluding remarks

In this work, a finite elementmodel for calculation of themachinning-induced
residual stresses has been presented.Themodelwas developed in the commer-
cial code ABAQUS using the ALE approach and both explicit and implicit
integration schemes. The model was developed from the basis of well known



An efficient implementation... 613

Fig. 10. Experimental and numerical residual stresses in the circumferential
direction, cutting speed equal to 120m/min

models used in the scientific literature and also by the authors. Boundary con-
ditions in the entrance of the work piece and in the upper border of the chip
were modified from Lagrangian boundaries in the previous model to Eulerian
boundaries in the newmodel. This change significantly diminishes the prepa-
ration time of the model that is usually extremely time consuming, although
this fact is not commonly referred in the literature. This approach, not only
decreases the work time, it also allows simulation of longer machined surfaces
and thus increased cutting times. The simulation of a large cutting time is
necessarywhen a study of thermal issues should be performed due to the long
cutting time corresponding to steady state conditions of temperature. The
model has been satisfactorily applied to calculation of residual stresses when
machiningAISI 316L steel. Orthogonal cutting testswere performed in similar
conditions with those imposed in the simulations. Resultant residual stresses
predicted by the numerical simulation have been comparedwith experimental
measurements showing reasonable accuracy of the model.

Acknowledgements

The authors are indebted for the financial support of this work to theMinistry of

ScienceandEducationofSpain (underprojectDPI2008-06746)andto theComunidad

Autónoma deMadrid and Universidad Carlos III deMadrid (under project CCG07-

UC3M/DPI-3396).



614 M.H. Miguélez et al.

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Efektywne sformułowanie warunków brzegowych w modelowaniu procesu
skrawania we współrzędnych Lagrange-Eulera (ALE)

Streszczenie

Przewidywanieszczątkowychnaprężeńwywoływanychobróbkąskrawaniemwma-
teriale próbki stanowi interesujące zagadnieniemodelowania tego typu technologiiwy-
twarzania.Mimo, że analiza oparta na elementach skończonych znalazła szerokie za-
stosowaniew tej dziedzinie, jej skuteczność jest problematyczna na poziomie budowy
modelu numerycznego.Koszt symulacji i kłopoty obliczeniowe związane z ogromnym
zniekształceniem siatki elementów skończonych czynią ten rodzaj analizy wyjątko-
wo czasochłonnym. Celem tej pracy jest opis metody określania naprężeń szczątko-
wych indukowanychobróbką skrawaniemza pomocąmodelu z elementami skończony-
mi w opisie eulerowsko-lagrange’owskimALE (ang. Arbitrary Lagrangian Eulerian).
W obliczeniach zastosowanowielozadaniowypakiet ABAQUS, który pozwolił namo-
dyfikację dotychczas stosowanegomodelu, opisanegow literaturze.Warunki brzegowe
w obszarze wejścia noża w obrabiany materiał i górnej strefie wióra zmodyfikowa-
no z typu Lagrange’a na Eulera w nowej wersji. Zaletą tego rozwiązania okazało się
małe zniekształcenie siatki elementów skończonych, możliwość symulacji długich po-
wierzchni obróbczych i znacznie krótszy czas obliczeń.Model wykorzystano do okre-
ślenia naprężeń szczątkowych w stali AISI 316L podczas skrawania. Potwierdzono
satysfakcjonującą zgodność symulacji z wynikami pomiarów doświadczalnych.

Manuscript received February 4, 2009; accepted for print March 19, 2009