Simulation of fatigue crack growth in coated cemented ...
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INOM EXAMENSARBETE MATERIALTEKNIK, AVANCERAD NIVÅ, 30 HP , STOCKHOLM SVERIGE 2017 Simulation of fatigue crack growth in coated cemented carbide milling inserts ANDRÉ TENGSTRAND KTH SKOLAN FÖR INDUSTRIELL TEKNIK OCH MANAGEMENT
Transcript of Simulation of fatigue crack growth in coated cemented ...
INOM EXAMENSARBETE MATERIALTEKNIK,AVANCERAD NIVÅ, 30 HP
, STOCKHOLM SVERIGE 2017
Simulation of fatigue crack growth in coated cemented carbide milling inserts
ANDRÉ TENGSTRAND
KTHSKOLAN FÖR INDUSTRIELL TEKNIK OCH MANAGEMENT
Acknowledgement
This work was performed at AB Sandvik Coromant in Vastberga, Stockholm be-tween January and June 2017 as a master thesis project in the Solid Mechanicsmaster track in Materials Design and Engineering programme at the Royal Instituteof Technology (KTH).
I would first like to give my sincere thanks to my supervisor Dr. Jose Garcıa atSandvik Coromant for the wonderful time I have had at the company. It has beena very interesting and challenging process of bringing an idea into an actual model.I would also like to thank Jonas Ostby at Sandvik Coromant for his expertise incemented carbide FEM modelling, as well as the rest of the Comb crack researchteam for the many innovative hours we have spent together. Further would I alsolike to thank my supervisor at KTH Prof. Per-Lennart Larsson for his guidance andsupport.
I would also like to give warm thanks to the wonderful people working at SandvikCoromant in Vastberga for the welcoming atmosphere they have created.
Lastly I would like to bring very special thanks to my wonderful Amanda for hersupport throughout the project.
Sincerely,Andre Tengstrand
Abstract
The aim of this work is to create a finite element model that simulates fatigue crackgrowth in coated cemented carbide milling inserts. A typical fatigue wear of coatedmilling tools are comb crack formation on the cutting edges of the tool inserts.Comb cracks initiate from pre-existent cooling cracks in the coating layers. Severetemperature changes occur during the intermittent machining and when using liquidcoolant. As a result thermal stresses cause the initial cooling cracks to propagate.With the model this work also aim to model a possible explanation for lateral combcrack formation by introducing a material alteration zone. Chemical attack occurswhen wet-milling as the cooling liquid enters the principal comb crack and reactswith the substrate.
The presented model is based on WC–Co substrate with TiCN and Al2O3 coat-ings. The model allow for a quick and comprehensive understanding of fatiguecrack growth in milling tools due to simplified loading conditions, which includethermo-mechanical and mechanical stresses. Thermo-mechanical stresses are calcu-lated corresponding to the temperature change during one milling cycle as well asstress contribution corresponding to residual stresses from the coating applicationprocess. The mechanical stresses are approximated and applied as pressures. Thefatigue crack growth simulation is governed by Paris law and is performed usingExtended Finite Element Method technique in ANSYS Mechanical APDL.
The result of the simulations indicates that the residual stress state is more accurateusing non-linear instead of linear material models, when comparing to experimentallymeasured stress states. The calculated fatigue cracks path were found to be similarin shape and length to that of a comb crack in early growth stage. The stresslevels in the substrate are found to be compressive except for close to the crack tipwhere the stresses are tensile. The stress level in the proximity of the crack tip areseveral magnitudes larger than in the surroundings. The crack paths change in anglewere due to the applied mechanical pressure. The length of the crack is found tobe influenced by the thermo-mechanical stresses. The chemical attack zone modelwas introduced as a circular zone in the WC–Co layer. The zone had its stiffnessaltered to 80 and 120 % of the initial stiffness during fatigue crack growth phase.The proposed model did not provide any indication that lateral comb crack formsdue to such a stiffness change.
Sammanfattning
Malet med detta arbete var att skapa en finit element modell for simulering av ut-mattningsspricktillvaxt i belagda frasskar av hardmetall. Kamsprickbildning ar envanlig typ av utmattningslitage som sker pa skareggen pa skaret. Kamsprickor ini-tieras fran tidigare existerande kylningsprickor i belaggningslagrena. Stora tempera-turskillnader uppstar under intermittent bearbetning och da vatske-baserad kylninganvands. Som ett resultat av de termiska spanningarna propagerar kylningssprickor-na.
Den modell som presenteras ar baserad pa WC–Co som substrat med TiCN ochAl2O3 som ytbelaggning. Modellen mojliggor for en snabb och mangsidig forstaelseav utmattningssprickor i skarverktyg pa grund av forenklade lastvillkor, vilka inklu-derar termomekaniska och mekaniska spanningar. De termomekaniska spanningarnamotsvarar de som uppkommer pa grund av temperaturskillnader under en frascykelsamt spanningsbidraget pa grund av restspanningar fran belaggningsproceduren. Demekaniska spanningarna approximeras och appliceras som tryck. Simuleringen av ut-mattningspricktillvaxten beraknas enligt Paris lag och med hjalp av Extended FiniteElement Method teknik i ANSYS Mechanical APDL. Med modellen amnar dettaarbete aven att modellera en mojlig forklaring till uppkomsten av laterala kamspric-kor genom att att introducera en forandrad materialzon. Kemisk attack sker underfrasning med vatske-baserad kylning da kylmedia tranger in i huvudsprickan ochreagerar med substratet.
Resultatet av simuleringarna pavisar att restspanningen ar mer riktig vid anvandandetav icke-linjara an linjara materialmodeller, om de jamfors med experimentellt uppmattavarden. Den beraknade sprickan visade sig vara liknande i form och langd till enkamspricka i tidig tillvaxtfas. Spanningsnivaerna i substratet ar kompressiva overalltutom i sprickspetsen dar de ar i dragtillstand. Spanningsnivaerna i sprickspetsen ab-soluta narhet ar manga magnituder hogre an i omkringliggande omraden. Sprickanskrokning beror pa den applicerade mekaniska spanningen. Sprickans langd fannsvara influerad av den termiska spanningen.
Den kemiska attackzonen introducerades som en cirkel i WC–Co substratet. Underutmattningspricktillvaxt-fasen andrades cirkelns styvhet till 80 och 120 % av initialstyvhet. Den presenterade modellen pavisade inga indikationer av att bilda lateralakamsprickor.
List of Figures
1 Example of milling application: Milling of an engine block. The cutteris equipped with multiple cutting inserts. Picture taken from [5].Image courtesy of AB Sandvik Coromant . . . . . . . . . . . . . . . . 3
2 Illustration of the cutting process. . . . . . . . . . . . . . . . . . . . . 43 Milling parameters. Images taken from [7]. Image courtesy of AB
Sandvik Coromant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Microstructure of a CVD-coated milling insert. Image courtesy of AB
Sandvik Coromant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Fractography of a CVD milling insert with wear damage [2]. Mark
(a) points to typical coolings cracks. Mark (b) points to wear causedby comb crack formation. Image courtesy of AB Sandvik Coromant . 6
6 Macroscopic detail of comb crack wear on the cutting edge. Imagecourtesy of AB Sandvik Coromant . . . . . . . . . . . . . . . . . . . . 7
7 Microscopic detail of comb cracks. Figure (a) shows a comb crackwith an extensive crack growth and with an upcoming tooth-loss.Figure (b) shows a close-up of several comb cracks in early stages.Image courtesy of AB Sandvik Coromant . . . . . . . . . . . . . . . . 7
8 Illustration of a comb crack. . . . . . . . . . . . . . . . . . . . . . . . 89 Tomography of a comb crack [10]. Cyan color indicates the crack and
violet the coating. The substrate is not visible. Image courtesy of ABSandvik Coromant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
10 Fractography of a comb crack. Region (b) indicates the comb cracksurface, region (a) indicates a normal cemented carbide fracture sur-face. Image courtesy of AB Sandvik Coromant. . . . . . . . . . . . . . 9
11 Fractographies corresponding to region (a) and (b) in Fig. 10. Thecarbides are roughly in the size of 1 [µm]. Image courtesy of ABSandvik Coromant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
12 Crack formed during dry milling. Image courtesy of AB SandvikCoromant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
13 Force components acting on the cutting edge according to [17]. Imagecourtesy of AB Sandvik Coromant. . . . . . . . . . . . . . . . . . . . 11
14 Illustration of instantaneous and secant definition of thermal expan-sion coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
15 Material models for stress-strain relation: Ideal plasticity (left), bi-linear hardening (middle) and multi-linear hardening (right). . . . . . 16
16 Illustration of a two-dimensional element with four nodes (left) anda mesh consisting of nine of such elements joined together at theircommon nodes. (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 18
17 Illustration of the field variables φ approximative solution over theelements length for different types of approximation techniques [29]. . 18
18 A body in equilibrium containing a traction-free crack [24, 31]. . . . . 1919 Crack termination occurs inside the element in singularity-based method
[20]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
20 Illustration of phantom-node method as illustrated in [20]. Cracktermination occurs at the element edge. . . . . . . . . . . . . . . . . . 22
21 The three loading modes defined in fracture mechanics correspondingto KI , KII and KIII [38]. The dark gray planes indicates the initialcrack plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
22 Illustration of fatigue crack growth graph. . . . . . . . . . . . . . . . 2423 Illustration of evaluation of fracture criteria for MCSC. . . . . . . . . 2524 Geometry of the model. Figure (a) illustrates a mock-up of the actual
microstructure and (b) an idealization. . . . . . . . . . . . . . . . . . 2925 Geometry of the chemical attack zone model. . . . . . . . . . . . . . . 3026 Mesh of normal model with 36450 elements . . . . . . . . . . . . . . . 3127 Mesh of chemical attack zone model with 53252 elements. . . . . . . . 3228 Illustration of boundary condition used in TM calculations. . . . . . . 3229 Coating deposition time-line used in CVD deposition simulation. Each
dot indicates a load step. A coating layer is applied at each temper-ature plateau. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
30 Initial crack enrichment. . . . . . . . . . . . . . . . . . . . . . . . . . 3631 Possible translation of Fr into applied load in the model. . . . . . . . 3732 Illustration of boundary condition setup for fatigue crack growth. . . 3833 Temperature response during CVD deposition. The result are taken
for the nodal solution of the substrate. Each dot indicates a load step.A coating layer is applied at each temperature plateau. . . . . . . . . 40
34 Calculated temperature response for one cutting cycle. The result aretaken for the nodal solution of the substrate. . . . . . . . . . . . . . . 41
35 Calculated ∆σtherm as a function of time. . . . . . . . . . . . . . . . . 4136 Fatigue crack growth simulation. Plot of crack path at end time. . . . 4237 Fatigue crack growth simulation. Plot of first principal stress σ1 at
initial time. Units are in [GPa]. . . . . . . . . . . . . . . . . . . . . . 4238 Fatigue crack growth simulation. Plot of first principal stress σ1 at
full time. Units are in [GPa]. . . . . . . . . . . . . . . . . . . . . . . 4339 Crack path at end time for FCG simulation where ECAZ = 1.2 ·
EWC−Co. The plot is cropped. . . . . . . . . . . . . . . . . . . . . . . 4440 Crack path at end time for FCG simulation where ECAZ = 1.0 ·
EWC−Co. The plot is cropped. . . . . . . . . . . . . . . . . . . . . . . 4441 Crack path at end time for FCG simulation where ECAZ = 0.8 ·
EWC−Co. The plot is cropped. . . . . . . . . . . . . . . . . . . . . . . 4542 Result plot of principal stress σ1 in [GPa] . . . . . . . . . . . . . . . i43 Result plot of principal stress σ2 . . . . . . . . . . . . . . . . . . . . . ii44 Result plot of principal stress σ3 . . . . . . . . . . . . . . . . . . . . . iii45 Result plot of normal stress σx . . . . . . . . . . . . . . . . . . . . . . iv46 Result plot of normal stress σy . . . . . . . . . . . . . . . . . . . . . . v47 Result plot of normal stress σz . . . . . . . . . . . . . . . . . . . . . . vi48 Result plot of normal stress τxy . . . . . . . . . . . . . . . . . . . . . vii49 Result plot of principal stress σ2 in [GPa] . . . . . . . . . . . . . . . viii50 Result plot of principal stress σ2 . . . . . . . . . . . . . . . . . . . . . ix51 Result plot of principal stress σ3 . . . . . . . . . . . . . . . . . . . . . x
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52 Result plot of principal stress σ2 in [GPa] . . . . . . . . . . . . . . . xi53 Result plot of principal stress σ2 . . . . . . . . . . . . . . . . . . . . . xii54 Result plot of principal stress σ3 . . . . . . . . . . . . . . . . . . . . . xiii55 Result plot of principal stress σ2 in [GPa] . . . . . . . . . . . . . . . xiv56 Result plot of principal stress σ2 . . . . . . . . . . . . . . . . . . . . . xv57 Result plot of principal stress σ3 . . . . . . . . . . . . . . . . . . . . . xvi
List of Tables
1 Stress state at room temperature for sample Ti(C, N) 554U accordingto [45]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2 Thickness of each model layer. . . . . . . . . . . . . . . . . . . . . . 303 Element settings for PLANE55 and PLANE182 elements. . . . . . . . 314 Time-table with temperature and corresponding layer activation. . . . 335 Fatigue related parameters based on Grade 10F according to [40]. All
parameters are converted from their original state to match Eq. 39and the [GPa]-[mm] consistent system. . . . . . . . . . . . . . . . . . 35
6 σres comparison. σelast.res,x are results from using linear material model.
σplast.res,x are results from using non-linear material model. Stresses are
stated in [MPa]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Material data related to WC–Co based on values obtained from [6, 14]xvii8 Material data related to TiCN based on values obtained from [14, 46,
47, 55] and internal Sandvik Coromant databases. . . . . . . . . . . . xvii9 Material data related to Al2O3 based on values obtained from [14]
and internal Sandvik Coromant databases. . . . . . . . . . . . . . . . xvii
Contents
1 Background 1
2 Aim 1
3 Introduction 33.1 Milling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Materials design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Wear mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3.1 Comb cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.4 Possible influence of chemical attack . . . . . . . . . . . . . . . . . . 83.5 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5.1 Continuum approach . . . . . . . . . . . . . . . . . . . . . . . 103.5.2 Mechanical loads in a milling insert . . . . . . . . . . . . . . . 10
4 Theory 124.1 Stress and strain relation . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.1 Thermal strain . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1.2 Plane conditions . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.3 Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Heat transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.1 Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.2 Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2.3 Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . 17
4.3 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3.2 Solving procedure . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Extended Finite Element Method . . . . . . . . . . . . . . . . . . . . 204.4.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4.2 Implementation in ANSYS Mechanical APDL . . . . . . . . . 21
4.5 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.5.1 Stress-intensity factors . . . . . . . . . . . . . . . . . . . . . . 234.5.2 Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5.3 Fracture criteria . . . . . . . . . . . . . . . . . . . . . . . . . . 244.5.4 Linear elastic fracture mechanics . . . . . . . . . . . . . . . . 26
5 Theoretical reference frame 27
6 Method 286.1 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2 FEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2.1 Material data . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.2.3 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306.2.4 Boundary condition . . . . . . . . . . . . . . . . . . . . . . . . 326.2.5 Thermal-stress calculation from CVD deposition procedure . . 33
6.2.6 Thermal-stress calculation from cutting operation . . . . . . . 346.3 XFEM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
6.3.1 Milling related mechanical forces . . . . . . . . . . . . . . . . 376.3.2 LEFM applicability . . . . . . . . . . . . . . . . . . . . . . . . 38
6.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
7 Results 407.1 Residual stress state after CVD deposition . . . . . . . . . . . . . . . 407.2 Thermal stress during cutting operation . . . . . . . . . . . . . . . . 407.3 Fatigue crack growth . . . . . . . . . . . . . . . . . . . . . . . . . . . 417.4 Chemical attack zone . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
8 Result analyses 46
9 Discussion 479.1 Further research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
10 Conclusion 49
A Stress plots iA.1 Stress state in fatigue crack growth . . . . . . . . . . . . . . . . . . . i
A.1.1 Principal stress σ1 . . . . . . . . . . . . . . . . . . . . . . . . iA.1.2 Principal stress σ2 . . . . . . . . . . . . . . . . . . . . . . . . iiA.1.3 Principal stress σ3 . . . . . . . . . . . . . . . . . . . . . . . . iiiA.1.4 Normal stress σx . . . . . . . . . . . . . . . . . . . . . . . . . ivA.1.5 Normal stress σy . . . . . . . . . . . . . . . . . . . . . . . . . vA.1.6 Normal stress σz . . . . . . . . . . . . . . . . . . . . . . . . . viA.1.7 Shear stress τxy . . . . . . . . . . . . . . . . . . . . . . . . . . vii
A.2 Chemical attack zone model . . . . . . . . . . . . . . . . . . . . . . . viiiA.2.1 80% of initial stiffness - Principal stress σ1 . . . . . . . . . . . viiiA.2.2 80% of initial stiffness - Principal stress σ2 . . . . . . . . . . . ixA.2.3 80% of initial stiffness - Principal stress σ3 . . . . . . . . . . . xA.2.4 100% of initial stiffness - Principal stress σ1 . . . . . . . . . . xiA.2.5 100% of initial stiffness - Principal stress σ2 . . . . . . . . . . xiiA.2.6 100% of initial stiffness - Principal stress σ3 . . . . . . . . . . xiiiA.2.7 120% of initial stiffness - Principal stress σ1 . . . . . . . . . . xivA.2.8 120% of initial stiffness - Principal stress σ2 . . . . . . . . . . xvA.2.9 120% of initial stiffness - Principal stress σ3 . . . . . . . . . . xvi
B Material data xvii
Nomenclature
[D]−1 Compliance matrix
α Thermal expansion coefficient
αIN Instantaneous thermal expansioncoefficient
αSE Secant thermal expansion coeffi-cient
u Displacement vector
β Paris law exponent
aj Displacement jump enriched nodaldegree of freedom
f b Body force vector
f t Traction force vect
f Total applied force vector
n Normal vector
uh Total degree of freedom vector
ui Enriched degree of freedom vectorfor i
∆σentry/exit Intermittent cutting relatedstress
∆σmachining Sum of cutting related stress
∆σmech Mechanical stress
∆σtherm Cutting related thermal-stress
∆Keqv Equivalent stress intensity factorrange
∆KI Stress intensity factor range, modeI
∆Kth Threshold for fatigue crack growth
∆T Temperature difference
γ0 Chip angle
Γc Crack boundary
Γt Traction boundary
Γu Displacement boundary
γij Shear strain tensor
κ Entering/attack angle
Ωi Domain i
φ Field function
φLSM Level-set method function
ψLSM Level-set method function
σ Stress
σB Fracture strength
σe Effective stress function
σy Yield strength
σ1 First principal stress
σ2 Second principal stress
σ3 Third principal stress
σθθ Hoop stress
σcycl,s Yield strength at cyclic loading
σvMe Effective von Mises stress
σij Stress tensor
σres Residual stress
σtotal Total stress contribution
τ Shear stress
τij Shear stress tensor
θ Search angle
ε Strain
εij Strain tensor
σ Stress vector
ε Total strain vector
εel Elastic strain vector
εth Thermal strain vector
a Crack length
ap Axial depth of cut
at Transition crack length
bj Crack tip enriched nodal degree offreedom
C Paris law constant
d Signed distance
da Crack growth increment
dN Cycle increment
E Young’s modulus of elasticity
ET Hardening modulus
F Crack-tip function
Fa Cutting force component (axial)
Fc Cutting force component (tangen-tial)
Fr Cutting force component (radial)
fz Feed depth per tooth
G Shear modulus of elasticity
H Heaviside function
h Film coefficient
hex Chip thickness
HV Vickers hardness value
K Stress intensity factor
kc Specific cutting force
kc1 Specific cutting force constant
KIC Fracture toughness
KIII Stress intensity factor, mode III
KII Stress intensity factor, mode II
KI Stress intensity factor, mode I
L Length
l Characteristic length
L0 Initial/undeformed length
mc Chip formation related constant
Nj Element shape function
q Heat flow
qconduction Heat flow by conduction
qconvection Heat flow by convection
qradiation Heat flow by radiation
R Load ratio
T Temperature
t Time
Tref Strain-free reference temperature
u Displacement
v Poisson’s ratio
Y Form factor
Al2O3 Aluminum oxide
Co Cobalt
TiCN Titanium Carbo-nitride
WC Wolfram carbide
CAZ Chemical Attack Zone
CVD Chemical Vapor Deposition
DOF Degree of Freedom
FCG Fatigue crack growth
FEM Finite Element Method
LEFM Linear Elastic Fracture Mechan-ics
LSM Level set method
MCSC Maximum circumferential stresscriterion
PU Partition of Unity
PVD Physical Vapor Deposition
SIF Stress intensity factor
XFEM Extended Finite ElementMethod
1 Background
Cemented carbides such as the WC-Co system are common composite materialsused in machining tools. In the WC-Co system tungsten carbide (WC) act as a hardparticle phase in a cobalt (Co) matrix [1]. WC-Co composites are also commonlycoated using other materials in a few micrometers thin film-like layers that togetheroffer properties that are suitable for machining applications [1].
Machining is the process where a desired shape of a workpiece material is obtainedby material removal. Milling is a common machining method. In milling the cuttingtool is rotated at high velocity during operation. A cutting tool consists of severalcutting inserts. The cutting inserts are sheared into the workpiece and cause materialremoval by chip formation in the workpiece.
Milling is an intermittent machining operation. This means that each cutting insertonly is in contact with the workpiece for a fraction of each milling revolution. Thiscauses steep changes in both temperature and mechanical stresses. The temperaturechanges are even larger when using liquid coolant. Temperature change causes thecutting tool to expand and contract. This inflicts large thermal strains and stressesas the cutting insert consists of several different materials. The large stress cyclesinitiate crack growth in pre-existing short cracks found in the coating layers fromthe production stage.
Thermal cracks become critical after a few thousand milling revolution, causingmaterial loss on the cutting edge [2]. This type of fatigue wear is known as combcracks, as the material loss occur on a regular distance [2–4].
In order to prevent or minimize wear damage knowledge of the cause of e.g. fatiguecrack formation is vital. Stress fields governs crack propagation and can be examinedusing numerical models. Such models are advantageous for manufacturing companiesas they are considerably cheap and facilitate understanding of complex problems.
In this work a numerical model of fatigue crack growth have been developed usingfinite element method and linear elastic fracture mechanics. The loads are basedon residual stress state from production, thermo-mechanical stress from milling andmechanical cutting loads.
2 Aim
The aim of this report is to present a simplified engineering model that simulatesfatigue crack growth in coated milling tool inserts. This engineering model allowsfor a quick and comprehensive understanding of crack growth due to simplified loadconditions. The model is created by using the commercially available finite elementmethod software ANSYS Mechanical APDL using XFEM -technique (Extended Fi-nite Element Method). The purpose of the model is to make an initial contributionto a better understanding of how stress fields spans during fatigue crack growth.More specifically the model is created to test if it is possible to model the effect of
1
crack deflection path due to chemical attack at the crack surface by introducing achemical attack zone.
2
3 Introduction
3.1 Milling
Machining is the process where a desired shape is obtained through material removal.Machining can be done through several different methods. One of the more commonmethods is milling. In milling a rotating multi-tooth cutter as seen in Fig. 1 is usedto remove material from the workpiece. The cutter is rotated at high velocity andis equipped with three up to a hundred cutting inserts [2]. As the cutting insertsenters the workpiece segments of material are sheared off into chips.
Figure 1: Example of milling application: Milling of an engine block. The cutteris equipped with multiple cutting inserts. Picture taken from [5]. Image courtesy ofAB Sandvik Coromant
Large frictional forces are induced in the contact zone between workpiece and cuttinginsert, causing an increase in temperature [6]. Most of the heat generated duringmilling leaves with the chip, but some heat also transfer into the cutting edge of thecutting insert. Estimations of this indicate that the temperature on the rake andflank face of the milling insert as seen in Fig. 2, can reach well above 1000 C [6].
3
Flank face
Rak
efa
ce
New surface
Cutting insert
Chip
Old surface
Shear def.
Cutting direction
Figure 2: Illustration of the cutting process.
Several parameters control and describe the milling process. In Fig. 3 milling pa-rameters are defined and illustrated, including entering angle κ, axial depth ap, feeddepth fz and maximum chip thickness hex [2]. Other parameters that may alsoeffect the milling process are the rotational speed, material properties of the tool,workpiece material and the choice of cooling media. Furthermore, a typical cuttinginsert measured 1− 2 [cm] in diameter and 0.5 [cm] in thickness.
(a) Cutting parameters (b) Cooling media
Figure 3: Milling parameters. Images taken from [7]. Image courtesy of AB SandvikCoromant.
3.2 Materials design
Cemented carbides, such as the WC–Co system, are powder metallurgical materialscommonly used in cutting tools [5, 8]. In WC–Co composites tungsten-carbides(WC) act as hard particles and cobalt (Co) as ductile matrix phase. These materialscombined offers properties which are desired for cutting tools [1]. Such propertiesinclude
• high wear resistance
4
• high toughness
• high hardness at service temperature (i.e. elevated temperatures).
The use of cemented carbides in machining began in the 20th century. In extension tocemented carbides, film-like thin layers of other materials can be added. Such coatinglayers were commercially introduced in the 1970’s. Coatings are usually added dueto their property improving ability, leading to better wear and abrasive resistance ofthe cutting tool [1]. The outermost layer can also function as a identification layerto indicate the inserts type and model. Some of the materials that can be used ascoatings are titanium nitride (TiN), titanium carbide (TiC), titanium carbonitride(Ti(C, N)), aluminum oxide (Al2O3), hafnium-nitride (HfN), titanium aluminumnitride ((Ti, Al)N) and aluminum chromium nitride ((Al, Cr)N) [1].
Two methods used today to apply coatings are CVD (abbr. for Chemical VaporDeposition) and PVD (abbr. for Physical Vapor Deposition) [1].
CVD is the most commonly used method [1]. In the CVD method coating layersare added by having the coating material in a gaseous form react with the solidsubstrate. This is often done at high temperatures between 700 to 1050 C andyields excellent adhesion between the layers [5]. An example of a CVD coatedmilling inserts microstructure is presented in Fig. 4.
Al2O3
Ti(C, N)
WC–Co
Figure 4: Microstructure of a CVD-coated milling insert. Image courtesy of ABSandvik Coromant.
PVD is another deposition method used to coat cemented carbides. In PVD thecoating material is deposited on the substrate by evaporation at temperatures be-tween 400 and 600 C [5]. An issue encountered using CVD, which do not occurin PVD, is that cracks are formed in the coating when cooling it down to roomtemperature after deposition. During heating and cooling each layer will expandand contract differently. This causes crack formation called cooling cracks. In Fig.5 cooling cracks can be observed in region (a). Cooling cracks are believed to bea result of the thermal expansion mismatch of the different materials [2]. Previousresearch [2, 4] have concluded that coolings cracks can be assumed to be initiatedbefore milling or during the initial milling. Furthermore, cooling cracks have been
5
found to grow perpendicular to the rake face of the milling insert and are generallyfound completely fracturing the coating layers but not propagated into the substrate[2, 4].
(a)
(b)
Figure 5: Fractography of a CVD milling insert with wear damage [2]. Mark (a)points to typical coolings cracks. Mark (b) points to wear caused by comb crackformation. Image courtesy of AB Sandvik Coromant
3.3 Wear mechanisms
3.3.1 Comb cracks
Due to the nature of the rotational cutting movement of the cutting tool, the cuttinginsert will be operating at intermediate circumstances [6]. For each revolution thecutting insert will be alternating between being engaged to the workpiece and beingcontact free. This shifting behavior results in alternating stresses and temperaturesduring each revolution. Quick temperature alternation will cause thermal shock onthe cutting edge of the tool insert. The thermal shock will lead to an unmatchedexpansion and contraction between the material layers, similarly to what occursduring the CVD deposition process. In cases where cooling media is used an evenmore severe thermal shock occur. Cooling cracks formed during the CVD depositionacts as initiation points for severe thermal-mechanical fatigue wear. In Fig. 5 point(a) initial cooling cracks in the coating can be seen acting as initiation points forcrack growth in the substrate. This is a mechanism typically seen in cementedcarbide milling inserts. The mechanism is known as comb cracks and results inmaterial loss on the cutting edge. The loss of material occur at a regular distance,hence the name ’comb cracks’, as seen in Fig. 6-7. [4, 9].
6
(a) Illustration of a comb crack (b) Photograph of comb crack
Figure 6: Macroscopic detail of comb crack wear on the cutting edge. Image courtesyof AB Sandvik Coromant
(a) Highly developed comb cracks (b) Comb cracks in different stages
Figure 7: Microscopic detail of comb cracks. Figure (a) shows a comb crack with anextensive crack growth and with an upcoming tooth-loss. Figure (b) shows a close-upof several comb cracks in early stages. Image courtesy of AB Sandvik Coromant
Fig. 8 illustrates a comb crack, where two types of cracks are distinguished. Theprinciple comb crack are initiated and grows from the cooling crack in the coatings.The lateral comb crack is a second branch which initiates from the principle combcrack at a certain depth [2]. Furthermore, the lateral crack propagates in an initiallyparallel direction to the cutting surface and then deflects towards the surface, causinga loss of material at the cutting edge.
7
Cooling crack Lateral comb crack
Principle comb crack
Figure 8: Illustration of a comb crack.
In Fig. 9 the actual crack planes have been visualized by Focused Ion Beam (FIB)tomography and can be seen in a slightly semi-elliptical form [10].
Figure 9: Tomography of a comb crack [10]. Cyan color indicates the crack andviolet the coating. The substrate is not visible. Image courtesy of AB SandvikCoromant.
3.4 Possible influence of chemical attack
Chemical attack occurs at the crack surfaces in the propagated comb crack. Herecooling media and adhered workpiece material reacts with the substrates constituent.The binder phase (Co) is mainly affected by the chemical attack. The result of the
8
chemical attack can be seen in Fig. 11a, in contrast to a chemically unattackedmicrostructure as seen in Fig. 11b.
(b)
(a)
Figure 10: Fractography of a comb crack. Region (b) indicates the comb cracksurface, region (a) indicates a normal cemented carbide fracture surface. Imagecourtesy of AB Sandvik Coromant.
WC-grain
Co-binder phase
(a) Normal fracture surface
WC-grain
Reacted Co-binder phase
(b) Chemically attacked fracture surface
Figure 11: Fractographies corresponding to region (a) and (b) in Fig. 10. Thecarbides are roughly in the size of 1 [µm]. Image courtesy of AB Sandvik Coromant.
Previous studies on comb cracks found that the lateral comb crack only initiates inwet milling and not when using dry milling [2, 11, 12]. Fig. 12 shows a crack thathas propagated without being in contact with liquid cooling media.
9
Figure 12: Crack formed during dry milling. Image courtesy of AB Sandvik Coro-mant.
A possible explanation for lateral comb crack initiation is that they occur due tothe chemical attack in the principle crack [3]. It is possible that the dissolution ofbinder phase causes the substrates mechanical properties to be different than in itsinitial state and therefor cause lateral crack formation.
3.5 Modeling
3.5.1 Continuum approach
Previous modeling studies on WC-Co systems that have used a continuum-approachto describe the materials mechanical behavior include [6, 9, 13, 14]. Examples ofstudies that have discretized the microstructural features include [15]. In a contin-uum approach the material is approximated to be homogeneous even though themicrostructure have distinct constituents (i.e. WC-grains in a Co-matrix). Thus,individual properties of the grains and the matrix are not considered but insteadtheir combined effective properties are used.
3.5.2 Mechanical loads in a milling insert
The total load that act as driving force for a crack in a milling inserts can accordingto [6] be estimated as
σtotal = ∆σmachining + σres (1)
where ∆σmachining is the sum of stresses which arise during the cutting operationand σres are residual thermal stresses from the CVD deposition.
10
The residual stress state in WC-Co depends on several factors such as the con-stituents thermal expansion coefficients α, the binder content, the particles sizesand their distribution. The magnitude of the stress varies locally in the microstruc-ture. The binder phase is generally in a tensile state while the carbides are incompression [16].
The cutting related stresses in ∆σmachining consists of both thermal and mechanicalrelated stresses with contributions stated as:
∆σmachining = ∆σentry/exit + ∆σtherm + ∆σmech (2)
and where ∆σentry/exit are stresses that occur due to the intermittent operation,∆σtherm are thermal stress related to the thermal changes on the cutting edge and∆σmech are mechanical stresses related to the contact between workpiece and cuttinginsert. The cutting operations force components are illustrated in Fig. 13 accordingto [17].
Figure 13: Force components acting on the cutting edge according to [17]. Imagecourtesy of AB Sandvik Coromant.
In Fig. 13 Fc is the cutting direction force component and is calculated accordingKienzles equation [17] as:
Fc = fz · ap · kc (3)
where kc is the specific cutting force. The magnitude of kc is compensated for thechip angle γ0 as
kc = kc1 · a−mcp · (1− γ0100
) (4)
where kc1 and mc are constants. For e.g. cast iron workpiece material the specificcutting force constant kc1 is listed in the magnitude of 750− 1650 [MPa] accordingto [5].
The radial component Fr and the axial component Fa are approximated accordingto [17] as
Fr ≈Fc2
and Fa ≈Fc4
(5)
11
4 Theory
4.1 Stress and strain relation
Stresses (σ) are internal forces in a body which are defined as force per unit area[ Nm2 = Pa]. These inner body forces results from external mechanical loading, ther-
mal loading, phase transformations, plasticity, load history etc [18].
A body is defined to be in an undeformed state when the stresses equals zero and indeformed state when the stresses are positive or negative. Negative stress (σ < 0) iscommonly known as compressive stress and positive stress (σ > 0) as tensile stress.
The deformation of a body is quantified in terms of strain. Strain (ε) is generally de-fined as the length change in the deformed state compared to the initial undeformedlength of the body. Strain theory is divided into small and large strain theory. Smallstrain theory is defined as
ε =du(x)
dx(6)
where u(x) is the displacement at coordinate point x [18, 19].
Large strain theory is used when the bodies deformed length L and undeformedlength L0 are significantly different. Large strain theory is according to [19] definedas
ε = lnL
L0
(7)
Hooke’s law relates stresses and strains as
σ = Eε (8)
where E is Young’s modulus and is a material-dependent coefficient that define thematerials stiffness in terms of pressure [Pa] [19].
In the FEM software ANSYS Mechanical APDL [20] stresses and strains are relatedas
σ = [D]εel (9)
where [D] is the stiffness matrix. The inverse of stiffness is known as compliance[21]. The compliance matrix [D]−1 is defined as
[D]−1 =
1Ex
−vxyEx
−vxzEx
0 0 0−vyxEy
1Ey
−vyzEy
0 0 0−vzxEz
−vzyEz
1Ez
0 0 0
0 0 0 1Gxy
0 0
0 0 0 0 1Gyz
0
0 0 0 0 0 1Gxz
(10)
12
where v is the Poisson’s ratio which relates the transverse strain to axial strain andG is the materials shear stiffness coefficient [20]. Furthermore, the stress vectorcomponent in Eq. 9 is defined as
σ = [σx σy σz τxy τyz τxz]T (11)
and the elastic strain vector component is defined as
εel = [εx εy εz εxy εyz εxz]T (12)
The elastic strain vector also relates the total strain vector and the thermal strainvector as
εel = ε − εth (13)
where the total strain vector is defined as
ε = εth+ [D]−1σ (14)
and the thermal strain vector is defined as in Eq. 17.
4.1.1 Thermal strain
Thermal strain is the strain caused by dimensional change due to change in tem-perature [19]. The dimensional change is proportional to the material parameterthermal expansion coefficient α which has the dimension [ 1
K] and can be defined as
αSE =L− L0
L0(T − Tref )(15)
where L is the expanded length and L0 is the initial length. T is the current tem-perature and Tref is the strain-free reference temperature.
The thermal expansion coefficient and strain are related to the temperature changeaccording to [19] as
εth = (T − Tref )α = ∆Tα (16)
Thermal strain in ANSYS Mechanical APDL [20] is expressed equivalent to Eq. 16in three-dimensions as
εth = ∆T [αx αy αz 0 0 0]T (17)
The thermal expansion coefficient is either defined as the secant coefficient αSE or asthe instantaneous coefficient of thermal expansion αIN . ANSYS Mechanical APDLconverts the input to an internal used effective thermal expansion coefficient [20,22]. The difference between the two definitions is illustrated in Fig. 14, where α isa function of temperature.
13
T
ε
αSE
αIN
Figure 14: Illustration of instantaneous and secant definition of thermal expansioncoefficient.
The secant coefficient is the average thermal strain resulting from a change in tem-perature from a strain-free temperature. Thermal strain εth as a function of αSE iscalculated as
εth = αSE(T ) · (T − Tref ) (18)
The instantaneous coefficient is based on the thermal strain caused by an infinitesmall change in temperature. Thermal strain εth as a function of αIN is calculatedas
εth =
∫ T
Tref
(αIN(T ))dT (19)
4.1.2 Plane conditions
In continuum mechanics two general types of approaches exists to describe a three-dimensional space (x, y, z) ∈ R3 in two-dimensional calculations (x, y) ∈ R2, namely
• plane stress assumption
• plane strain assumption
The two assumptions differ in the way of treating the third dimension (z). Planestress is commonly applied for geometries where one dimension is much smaller thanthe others. Plane stress defines one of the principle stresses as zero [19]. This resultsin that the out-of-plane stresses are stated as
σz = τxz = τyz = 0 (20)
Plane strain is commonly used for geometries where one dimension is consideredmuch larger than the others [23]. Plane strain defines one of the principle strains aszero [19]. This results in that the out-of-plane strains are stated as
εz = γxz = γzy = 0 (21)
The much smaller or larger dimension will be treated as the out-of-plane directionin the calculation.
14
Furthermore, a third variant called generalized plane strain exists in ANSYS Me-chanical APDL. Generalized plane strain is similar to plane strain but instead ofsetting the out-of-plane strains at zero is it prescribed at a constant non-zero value.This is done by treating the z-direction as a finite length instead of infinite as it isdone in the plane strain assumption [20].
4.1.3 Plasticity
Plastic deformation is defined as an irreversible deformation of a body. The oppositeto plastic deformation is elastic deformation where a body recovers to its initial shapewhen unloading. Plasticity occurs when a body is subjected to stresses that exceedsa local yield criteria [24]. The criterion can be defined in different ways, e.g. as afunction of state variables such as stress tensor or strain components. A commonstress criteria is the material parameter yield strength σy. The yield strength of amaterial is commonly an uniaxially measured property. Real case loading involvesmultiaxial loading. To relate uniaxial and multiaxial stress state a yield criterionis introduced. Plastic deformation is generally assumed to occur when the yieldstrength limit σy is reached [25], i.e. when σ ≥ σy. Crack initiation generally occurin the plastic zone and where the stresses are maximum [24].
On a microstructural level plasticity is an important energy-consuming mechanism.The deformation mechanisms differs between materials but do generally for solidcrystalline materials involves slip planes and dislocation rearrangement in the crystallattice when the atomic bonds are broken and reformed [25, 26].
A materials response to plastic deformation can be dependent of the loading rate.If a materials plastic response is not dependent on the load rate it is said to berate-independent plasticity. Most metals experience rate-independent plasticity [26].Creep plasticity is an example of time-dependent deformation as the material do notdirectly respond to the load [19, 25].
The stress-strain behavior for a material can be modeled through different mathe-matical models in ANSYS Mechanical APDL. For some materials hardening effectsoccurs after loading beyond the yield point [19]. In ANSYS Mechanical APDL thiscan be modeled as e.g. linear strain hardening [21] as seen in Fig. 15 where ET gov-erns the hardening effect. ET is defined as the slope of the curve after yield point,similarly to that of E in the elastic region. A material may be better represented byseveral ET -parameters (i.e. ET1, ET2, ...). This is due to the fact that the measuredhardening behavior can be better represented using several line divisions. This isknown as multi-linear strain hardening. Furthermore, if a material experience nohardening after yield point then it is ideal-plastic. A material that do not show anyevident plastic behavior is regarded as brittle [25]. Ceramics are commonly foundto be brittle [25].
15
ε
σ
σy
E
ε
σ
σy
σmax
E
ET
ε
σ
σy
σmax
E
ET1ET2
ET3
Figure 15: Material models for stress-strain relation: Ideal plasticity (left), bi-linearhardening (middle) and multi-linear hardening (right).
4.2 Heat transfer
Temperature is on a microscopic level a measurement of molecules vibrational mo-tions. An increase in temperature results in an increase in motion of the molecules.Heat transfer occurs when a body do not have a uniform temperature [25]. Heatflow q is the exchange rate of thermal energy which will occur between the bod-ies temperature regions. The magnitude of heat flow is material dependent. Themechanisms which heat can be transfered by is generally divided into
• conduction
• convection
• radiation
The total heat flow is ideally the sum of the contributions from the different heatflow mechanisms, i.e. as
q = qconduction + qconvection + qradiation (22)
4.2.1 Conduction
Conduction is a heat transfer mechanism which can be described as heat transferwithin a solid body. Regions with larger kinetic energy, i.e. motion energy, willexchange energy to regions with lower kinetic energy [25]. Heat flux by conductionis defined as
qconduction = −kdTdx
(23)
where k is the thermal conductivity in [ Wm·K ] and dT
dxis the temperature gradient [25].
4.2.2 Convection
Convection describes heat energy transfer between a moving fluid and a solid surface.If there is a temperature difference between a fluid and a solid heat will be transfered
16
according to
qconvection = h∆T (24)
where h is the film coefficient which represent the heat transferability in [ Wm2·K ] and
∆T is the temperature difference between the two bodies.
Convection can be free or forced. Cooling of cutting inserts by cutting fluid is atypical forced convection and its heat transfer coefficient in the magnitude of 10000[ Wm2K
] [27].
4.2.3 Thermal radiation
Thermal radiation is generated by charged particles that cause collisions betweenthe atoms. This causes the atoms to exchange kinetic energy. Heat transfer bythermal radiation qradiation is calculated according to Stefan-Boltzmann law [14].
4.3 Finite Element Method
Finite Element Method (abbr. FEM) is a numerical method developed to solvedifferential equations approximatively. Such equations are commonly found whenmathematically stating the physical reality through boundary value problems (alsoknown as field problems) and are commonly found in engineering problems [28].These problems can be too complex to be solved using common analytical toolswhich is why FEM is used.
4.3.1 Discretization
The general method of FEM is to discretize the body into smaller parts. Theseparts are called finite elements and allow for local approximations of the governingequations. The elements are given identification numbers and are joined togetherin what is known as a mesh. When elements are assembled into a larger systemand solved they represent an approximative solution of the full structure. Theelements can be in different shapes and dimensions e.g. as one-dimensional lines,two-dimensional areas or three dimensional volumes [29]. In Fig. 16 such two-dimensional elements can be seen. PLANE55 and PLANE182 are examples oftwo-dimensional quadrilateral area elements available in ANSYS Mechanical APDL[20]. Each element consists of several points that defines the location of the element.These points are known as nodes and are generally defined at the corners of theelements, as seen in Fig. 16.
17
i j
kl
1 2
34
Figure 16: Illustration of a two-dimensional element with four nodes (left) and amesh consisting of nine of such elements joined together at their common nodes.(right).
Boundary conditions of the physical problem are defined at the nodes. The allowedmovements of each node are known as degree of freedom (DOF). The DOFs allow forreactions from the applied conditions to be transfered from element to element [29,30]. The type of DOF depends on the discipline of the physical problem, i.e. whichfield variable the differential equations depends on. In structural analysis the DOFare seen as displacements and the forces are mechanical forces. In thermal analysisthe DOFs are seen as temperatures and the forces as heat fluxes [29].
To solve the field variable at non-nodal points (i.e. anywhere inside the element),FEM introduces approximative functions called interpolations functions (also knownas shape functions) [28]. These functions are typically chosen as polynomials andallow for interpolation between the nodes. As the mesh, i.e. sub-part division, be-comes more dense the element solution monotonically converge to the exact solution,as seen in Fig. 17.
x
φ
x1 x2
Exact solution
Quadratic approx.
Linear approx.
Figure 17: Illustration of the field variables φ approximative solution over the ele-ments length for different types of approximation techniques [29].
A requirement in order for the approximative function to converge towards the ex-act solution is that there are no kinks or jumps over the element i.e. they must becontinuous in the element and no gaps can exist between elements [29, 31].
18
4.3.2 Solving procedure
The principle of virtual work is a fundamental basis in structural mechanics andthus in FEM. The principle states that in order for a body as seen in Fig. 18, toremain in equilibrium the external virtual work must equal the internal virtual workdue to applied load [20].
Γ
Γt
ΓuΓc
f b
n
ΩA
ΩB
f t
u
Figure 18: A body in equilibrium containing a traction-free crack [24, 31].
The boundary conditions for such a body can be defined accordingly [24]
• σ · n = f t, on external traction boundary Γt
• u = u, on prescribed displacement boundary Γu
• σ · n = 0, on traction free crack boundary Γc
The discretization of such a system into finite element yields a system of equationsaccording to
Kuh = f (25)
where K is the coefficient matrix (also known as the stiffness matrix), uh is thetotal degree of freedom vector (containing the unknowns) and f is the applied forcevector [20, 24, 29].
If the coefficient matrix consists of unknowns then Eq. (25) is non-linear. ANSYSMechanical APDL utilize the Newton-Raphson method to solve such non-linearequations [20]. Newton-Raphson is a iterative solving process where the load isdivided into increments in order to obtain a solution. Each increment yields asolution for which it is evaluated if equilibrium is achieved. If the incrementalsolution do not yield equilibrium the procedure returns to the last increment whichdid yield equilibrium and initiates a smaller increment. This procedure is repeateduntil all the increments result in convergence of the full solution.
Furthermore, even though the Newton-Raphson iteration method allows for a non-linear equations to be solved the requirement of continuity between elements stillexists. Such discontinuities are commonly found for e.g. cracks as seen in Fig. 18.
19
In a crack growth simulation, a crack will propagate based on some fracture criterion[20]. As the domain is discretized into elements the crack can pass either betweenor through elements. This is a violation of the previously stated requirement whereno gaps or jumps between or in an elements field function is allowed. In order tosolve a crack growth simulation using FEM, each crack growth increment must befollowed by an updated discretization so that the mesh conforms to the propagatingcrack tip [20].
4.4 Extended Finite Element Method
The Extended Finite Element Method (abbr. XFEM) is a modification of the con-ventional FEM that was introduced in the 1950’s [20, 29]. XFEM was first formu-lated in 1999 [24] and in 2016 fully introduced in the commercial software ANSYSMechanical APDL [32]. Cracks, voids, inclusions and bi-materials are examples ofdiscontinuities [31]. XFEM offers a convenient method to solve discontinuities thatconvenient FEM formulation can not solve without the need for e.g. re-meshing [24,31, 33].
4.4.1 Theory
XFEM theory is based on the theory of partition of unity (abbr. PU). PU is definedas m set of functions fk(x) within a domain ΩPU [24] such that
m∑k=1
fk(x) = 1 ∀x ∈ ΩPU (26)
PU is used to enrich solutions in XFEM by applying the PU condition to elementshape functions Nj [24, 34]
n∑j=1
Nj(x) = 1 (27)
Enrichments of solution is done to increase the accuracy of the solution. This isdone by including information obtained from the analytical solution. In XFEMenrichments are done locally and extrinsic. Extrinsic enrichment means that extraenrichment functions are added to the standard approximation. A local enrichmentmeans that only nodal points in the proximity of the discontinuity. This results inkeeping the additional unknown constants at a minimum [24, 31]. In the case ofenriched displacement field u(x) extrinsic enrichment is done as [31]
uh(x) = uFEM + uEnriched =n∑j=1
Nj(x)uj +m∑k=1
Nk(x)ψ(x)ak (28)
where ψ are discontinuous enrichment function and ak unknown parameters thatadjusts the enrichment so that it best best approximates the ψ solution [24, 34].
20
uEnriched can account for several types of discontinuities simultaneously, e.g. as
uEnriched = uH(x) + utip(x) + umat(x) (29)
where uH is the crack-splitting displacement, utip is the crack-tip displacement andumat is the material interface displacement in the case of a bi-material interface.
4.4.2 Implementation in ANSYS Mechanical APDL
ANSYS Mechanical APDL allows for two different XFEM technique options [20],namely
• singularity-based method
• phantom-node method
The singularity-based method accounts for crack-tip singularities and displacementjumps across crack surfaces as
uh(x) = Nj(x)uj︸ ︷︷ ︸Regular
+H(x)Nj(x)aj︸ ︷︷ ︸Displ. jump enr.
+Nj(x)∑
F (x)bj︸ ︷︷ ︸Crack-tip sing. enr.
(30)
where H(x) is a Heaviside function and takes the values ±1 depending on whichside of the crack the current analyzing point is and aj is an enriched nodal degree offreedom that accounts for the displacement jumps. F (x) is a crack-tip function andbj is an enriched nodal degree of freedom that accounts for crack tip singularities[20]. Furthermore, crack termination occurs inside an element in the singularity-based method, as seen in Fig. 19.
Figure 19: Crack termination occurs inside the element in singularity-based method[20].
The phantom-node method only accounts for displacement jumps across crack sur-faces
uh(x) = Nj(x)uj︸ ︷︷ ︸Regular
+H(x)Nj(x)aj︸ ︷︷ ︸Displ. jump enr.
(31)
Phantom nodes are introduced and superposed on conventional element nodes asseen in Fig. 20 [20]. The crack terminates at the elements edge using phantom-based enrichment [20].
21
= +
Cracked elem. Sub-elem. 1 Sub-elem. 2
Conventional FEM-defined nodes
XFEM-defined phantom nodes
Figure 20: Illustration of phantom-node method as illustrated in [20]. Crack termi-nation occurs at the element edge.
Using the phantom nodes the displacement approximation can be expressed as afunction of real and phantom nodes [20] as
uh(X,x) = uj,1(x)Nj(X) ·H(−f(X)) + uj,2(x)NI(X) ·H(f(X)) (32)
where X are the material coordinates and x are spatial coordinates related to Xand time [35]. ui,1(x) and ui,2(x) are displacement vectors in sub-elements 1 and 2in Fig. 20. H is the Heaviside step function given according to [20, 34, 35] as:
H(x) =
= 1, if x > 0
= 0, if x ≤ 0(33)
Furthermore, f is implicit function that describes the signed distance to the discon-tinuity boundary where f(X) = 0 is the crack surface [20, 34, 35]. Similarly to fthe Level-Set method (abbr. LSM) is a signed distance function. LSM complementXFEM with information of where and how to enrich the elements [31, 36, 37]. Inorder to reduce computational costs only regions close to the discontinuity are eval-uated [31, 36]. LSM captures information of the signed distance to the discontinuityboundary through the functions φLSM(x, t) and ψLSM(x, t) according to [20, 36] as:
φLSM(x, t) =
= 0, if x ∈ Γc , i.e. x is behind crack tip
> 0, if x ∈ ΩA , i.e. x is in front of crack tip
< 0, if x ∈ ΩB , i.e. x is at the crack tip
(34)
ψLSM(x, t) =
= 0, if x ∈ Γc , i.e. x is along crack path
> 0, if x ∈ ΩA , i.e. x is above crack path
< 0, if x ∈ ΩB , i.e. x is below crack path
(35)
4.5 Fracture mechanics
Fracture mechanics is the study of crack propagation within a structure. Fracturesare consequences of cracks and are defined as the generation of new boundary sur-faces within a structure. Fatigue cracks are material defects caused by the servicecondition that the structure is subjected to [38].
22
4.5.1 Stress-intensity factors
Stress-intensity factor (abbr. SIF) is a common method used to relate applied loadon a body to the stress state in the crack tip. Stress-intensity factors K are linearrelations between load and stress, expressed as
KI = σgen√πafgeom (36)
where σgen is a measure of the applied load, a is the crack length and fgeom is ageometry-related function. Three types of SIFs (KI , KII and KIII) are defined inorder to characterize the stress state in a three dimensional body [38]. Each SIFcorresponds to a load mode as illustrated in Fig. 21.
(a) Mode I (b) Mode II (c) Mode III
Figure 21: The three loading modes defined in fracture mechanics corresponding toKI , KII and KIII [38]. The dark gray planes indicates the initial crack plane.
4.5.2 Fatigue
Fatigue is when fracture occurs for low loads that occur in a cyclic manner of sev-eral repetitions. The loads are defined as low as they do not instantaneously yieldcatastrophic fracture. Tensile stresses act in a crack opening manner. If the stressesare compressive then the crack propagation halts. Load cycles in milling machiningcan be pure tensile or compressive, or vary between compressive and tensile statewithin the cycle. The cyclic behavior will result in a minimum and maximum SIF.The stress-intensity factor range is defined as
∆KI = KI,max −KI,min (37)
where the maximum and the minimum SIF also are related through the load ratioR according to [38] as
R =KI,max
KI,min
(38)
Experimentally measured fatigue crack growth (abbr. FCG) exhibits three distinctstages when plotted in a log-log diagram against ∆KI .
23
log(∆KI)
log( dadN
)
1 2 3
∆Kth KIC
Figure 22: Illustration of fatigue crack growth graph.
In stage one in Fig. 22 no significant crack growth occur for loads below the thresholdfatigue growth range ∆Kth [38], which is generally dependent on R [38–41].
In stage two in Fig. 22 stable crack growth occur. Stable crack growth is charac-terized by relatively slow growth [25]. A linear relation can for most materials beobserved in the log-log diagram in stage two. This linear relation is described byParis law according to [20, 38] as
da/dN = C(∆KI)β (39)
where C and β are parameters obtained from curve-fitting stage two in Fig. 22. The
unit of C is dependent on β as [ m/cycleMPa
√m
β] [20, 38].
In stage three in Fig. 22 the crack growth becomes unstable and results in catas-trophic fracture, i.e. instantaneously. Unstable crack growth are characterized byvery rapid growth with little plastic deformation [25].
Lifetime-wise the three stages corresponds can be summarized to infinite lifetime instage one, finite lifetime in stage two and instant fracture in stage three.
4.5.3 Fracture criteria
A fracture criterion governs if a structure will fracture due to the applied load. Anexample of a stress-based criterion is Mohr’s criterion, which is defined as
|τ | = f(σ) (40)
and states that fracture occurs if the shear stress τ on any plane equals or is greaterthan a function f of the normal stress σ on the same plane [38].
In fracture mechanics models no mechanistic basis for crack propagation are gener-ally made as stated in Eq. (40). Instead crack propagation is based on experimentswhere certain critical limits in terms of material parameters are measured and re-lated to mathematical functions that relates loading and fracture as
g(σij(x)) ≤ σcr (41)
24
where g is a function of the stress tensor σij in a certain point x [38]. This impliesthat fracture occurs when the function is greater than the critical value σcr, whichis a material related parameter.
The fracture criteria is based on the circumferential stress state in XFEM-basedcrack growth modeling in ANSYS Mechanical APDL [20]. The maximum circum-ferential stress criterion (abbr. MCSC) defines an arc region in front of the cracktip. The arc boundary is used as stress evaluation points, illustrated in Fig. 23where r is the radius from the crack tip and θ is the search angles of the arc.
Crack tip r
θ
Figure 23: Illustration of evaluation of fracture criteria for MCSC.
MCSC states that the crack propagates in the direction where the hoop stress σθθis at maximum [42, 43]. The hoop stress is calculated as
σθθ =1√2πr
[KI
4
(3cos(
θ
3) + cos(
3θ
2))
+KII
4
(− 3sin(
θ
2)− 3sin(
3θ
2))]
(42)
and only considers load mode I and II from Fig. 21 to influence crack growth. TheMCSC criterion can also be based on shear stress τrθ [20]. The fracture direction isthen determined from
τrθ = 0 (43)
Paris law is used as FCG criteria in ANSYS Mechanical APDL [20]. If the FCGinvolves mixed load modes then the SIF range ∆KI in Eq. (39) is calculated as theequivalent stress-intensity factor range ∆Keqv according to
∆Keqv =1
2cos(θ
2
)[∆KI(1 + cos(θ))− 3∆KIIsin(θ)
](44)
where θ is the angle of the arc defined as in Fig. 23 [20]. Similar to Eq. (41) aFCG criterion is expressed in terms of ∆Keqv so that crack growth initiates andpropagates when
∆Keqv > ∆Kth (45)
and catastrophic fracture occurs when
∆Keqv = KIC (46)
25
The use of Paris law require that either the crack growth increment per cycle (da)or the cycle increment (dN) is defined. In ANSYS Mechanical APDL this is definedas two methods, namely [20]
• Life-Cycle (abbr. LC)
• Cycle-by-cycle (abbr. CBC)
In LC method the crack-increment per cycle da is defined. In CBC method thecycle increment dN is defined. Furthermore is fatigue crack growth in ANSYSMechanical APDL based on singularity-based XFEM [20]. This results in that thecrack propagates one element at a time. Crack propagation is only modified afteran element is fully fractured[20].
4.5.4 Linear elastic fracture mechanics
Linear elastic fracture mechanics (abbr. LEFM) is a concept where the plasticdeformation region at the crack tip is assumed to be very small compared to therest of the body. Plastic deformation occurs in the crack tip regions as stresseswhich are generally many magnitudes larger than in the surroundings. Paris law inANSYS Mechanical APDL is based on LEFM [20].
LEFM assumes that a semi-infinite crack within an infinite body is considered selfsimilar to that of a crack tip in a finite body [38]. A criteria for when LEFM isapplicable is proposed when the following conditions are satisfied
l > 2.5 ·( ∆KI
2σcycl,s
)2(47)
∆KI > 1.1 ·∆Kth (48)
KI,max < 0.9 ·KIC (49)
where l is the shortest characteristic length and σcycl,s is the yield strength at cyclicloading [18]. Furthermore, LEFM is generally not applied to short cracks. Shortcracks are in the same lengths as the microstructural features. Therefor they areprone to interact with the microstructure. The transition of short crack to longcrack is according to [6] defined as
∆Kth = Y σcycl,s√at (50)
where at is the transition crack length, Y is a geometrical form factor and σcycl,s isthe cyclic yield strength [6].
Previous research [6, 44] in the field have utilized LEFM.
26
5 Theoretical reference frame
Crack analysis of WC–Co in milling inserts have previously reported to satisfy theLEFM conditions according to [6]. In this work a two-dimensional FEM model isused to evaluate thermal-stress and temperature distribution during cutting oper-ation. Furthermore, LEFM has been applied for crack growth in similar cementedcarbide systems in [41].
FEM based residual and thermal-stress related to cutting as well as mechanicalstresses have been previously been reported by [14] and [9]. The calculated residualstress state will be evaluated against values obtained from experimentally measuredstresses and from analytically calculated stresses.
The analytical stress levels are based on one dimensional linear-elastic assumptionbased on [2, 6, 9] and stated as
σanres,coating =∆TElayer1− vlayer
(αsubst. − αlayer) (51)
and
σanres,subst. =∆TElayerhsubst.
∑ hlayerElayer1− vlayer
(αsubst. − αlayer) (52)
where h is the layers thickness. The results in Tab. 1 are based on ∆T = 1030 andmeasurement at room-temperature.
The experimentally measured stress levels are synchrotron measured at BESSYBerlin for AB Sandvik Coromant. The experimental stresses are reported in [45].
The analytical solution and the experimental stresses are stated in Tab. 1.
Layer Thickness [µm] σexpres [MPa] σanres [MPa]
Al2O3 5.5 613 801.3TiCN 4 615 1003.6WC–Co (60) -10 45.84
Table 1: Stress state at room temperature for sample Ti(C, N) 554U according to[45].
Material data related to WC–Co, TiCN and Al2O3 are limited as previously reported[9]. It is especially the plasticity material data that is scarce why assumptions onthe data are made as in [9]. Material data have been obtained from [6, 9, 46, 47].Fatigue-related material data related to WC–Co have been reported by [39, 41, 48,49]. The fatigue data is reported in the form of Paris law parameters.
27
6 Method
6.1 General procedure
The finite element model is performed in the following order:
Stage 1: Initialization of model
1. Analysis options - defining parameters which governs the simulation
2. Material database - defining material datasets
3. Geometry - generating substrate and coating layers
4. Meshing - defining element settings and generating thermal elements
Stage 2: Residual stresses calculation (σres) after CVD deposition procedure
5. Thermal analysis - calculate temperature distribution
6. Structural analysis - calculate corresponding thermal stresses
Stage 3: Thermal-stress calculation (∆σtherm) for one milling cycle
7. Thermal analysis - calculate temperature distribution
8. Structural analysis - calculate corresponding thermal stresses
Stage 4: Crack growth
9. Analysis options - define enrichment region, crack growth governing settingsand XFEM technique
10. Structural analysis - calculate crack growth due to applied cutting forces,thermal stresses and residual stresses
Stage 5: Result analysis
11. Post-processing of results
All material data and input parameters presented are converted to the GPa-consistentunit system according to [50] where pressure is stated in [GPa], force in [N ], massin [kg], length in [mm], energy in [J ], temperature in [K] and time in [ms].
6.2 FEM model
Conventional FEM were used to initiate the model as well as calculating the thermo-mechanical (abbr. TM) σres and ∆σtherm calculations.
28
6.2.1 Material data
The material data are taken from various literature sources [6, 9, 46, 47] and frominternal databases at AB Sandvik Coromant. The used material properties aregenerally temperature dependent. All materials have non-temperature dependentplasticity data. The plasticity is modelled according to bi-linear isotropic materialmodel. Extrapolation of the temperature dependent data have been done whensufficient data have not been found for the full temperature range.
The material data is presented Tab. 7-9 in App. B.
6.2.2 Geometry
The model geometry is based on two-dimensional rectangles that are idealizationof the material layers in Fig. 24. In Fig. 24b the geometry is idealized such thateach rectangle represent a material layer corresponding to the layers in Fig. 4. Allareas are glued together through a Boolean operation. This is equivalent to assumeperfectly bonding between the layers.
(a) Mock-up of Fig. 4
WC–Co
TiCN
Al2O3
(b) Idealized geometry
Figure 24: Geometry of the model. Figure (a) illustrates a mock-up of the actualmicrostructure and (b) an idealization.
A chemical attack zone (abbr. CAZ) is modelled by introducing a circular zone, asseen in blue colour in Fig. 25. The extra lines seen in Fig. 25 are added to controlthe mesh procedure.
29
(a) Chemical attack zone model
Help lines
CAZ
(b) Detail of (a)
Figure 25: Geometry of the chemical attack zone model.
The total width of the model is set to 50 [µm] and the thickness of each layeraccording to Tab. 2. The CAZ regions radius is set to 6 [µm] and is defined at 20[µm] depth measured from the surface.
Layer Thickness [µm]3. Coating layer (Al2O3) 42. Coating layer (TiCN) 5.51. Substrate (WC–Co) 60
Table 2: Thickness of each model layer.
6.2.3 Mesh
The elements are set to be four-node quadrilateral and consideration have beenmade to keep the aspect ratios close to 1:1. As the CAZ is modelled as a circleand quadrilateral elements are used a special meshing techniques is used to keepthe elements quadrilateral shapes from being violated. The element size of the CAZregion is set to be 1
6of the general element size which is set to 0.31 [µm]. The total
number of elements is 36450 in the normal model and 53252 in the CAZ model.
Thermal solid PLANE55 elements are used in the thermal analyses. Structural solidPLANE182 elements are used in the structural analyses. The element settings foreach element type is stated in Tab. 3. Other element options were set to theirdefault settings if nothing else is stated.
30
Thermal solid PLANE55Option SettingElement behaviour Plane strainFilm coefficient evaluation Differential temperature between bulk
and surface
Structural solid PLANE182Option SettingElement behaviour Plane strainElement technology Full integration
Table 3: Element settings for PLANE55 and PLANE182 elements.
Fig. 26 show the normal models mesh.
(a) Mesh of normal model (b) Detail of (a)
Figure 26: Mesh of normal model with 36450 elements
Fig. 27 show the chemical attack zone models mesh.
31
(a) Mesh of CAZ model (b) Detail of (a)
Figure 27: Mesh of chemical attack zone model with 53252 elements.
6.2.4 Boundary condition
Fig. 28 illustrates the boundary condition used for σres and ∆σtherm calculation. Allsides except for the top are set to roller condition. This means that the bottom ofthe model is restrained from movement in the vertical direction. The sides of themodel are set to be free to move in vertical direction but constrained from movementin the horizontal direction.
Figure 28: Illustration of boundary condition used in TM calculations.
32
6.2.5 Thermal-stress calculation from CVD deposition procedure
The residual stress state σres which is related to the CVD deposition procedure werecalculated using coupled thermal-structural analysis.
Coating layer creation is done using element kill and birth function of PLANE55and PLANE182. All elements were initially assigned an uniform temperature as wellas initial condition corresponding to room-temperature. All layers are de-activatedbefore simulation initiation. Each coating layer is re-activated according to a vectorwhich flags in an algorithm if the layer is supposed to be active and/or activated,corresponding to a certain time position. Temperature, layer activations and corre-sponding time point can be seen in Tab. 4. Fig. 29 show Tab. 4 in plotted form.The time steps are calculated assuming a heating rate of 5 [K] per minute, a holdingrate where 1 [µm] coating is created per hour and a cooling rate of 1 [K] per minute.
Each mark in Fig. 29 corresponds to a load step, i.e. a total of nine load-steps isused here. Each load step is divided into 50 sub-load steps. At the first load step ofeach temperature plateau in Fig. 29 a coating layer is activated.
Elapsed time [ms] Elapsed time [h] Applied tempera-ture [K]
Layer activation
1000 2.77 · 10−7 293 WC–Co10165000 2.82 1143 TiCN29965000 8.32 114332125000 8.92 1323 Al2O3
46525000 12.9 1323108145000 30.04 293108205000 30.05 293
Table 4: Time-table with temperature and corresponding layer activation.
33
TiCN
Al2O3
0 5 10 15 20 25 30200
400
600
800
1000
1200
1400
Time [h]
Tem
per
ature
[K]
Figure 29: Coating deposition time-line used in CVD deposition simulation. Eachdot indicates a load step. A coating layer is applied at each temperature plateau.
Temperature loads are defined on the current top surface nodes by thermal DOFconstraint. A new DOF constraint is applied at the activation of each new coatinglayer. The previous DOF constraint is simultaneously removed. Convection orradiation load is not included in this stage.
Temperature distribution is calculated in a thermal transient analysis with PLANE55elements. The results are saved for each load step. A corresponding structural tran-sient analysis is performed with PLANE182 elements. In the structural analysistemperature results from the thermal analysis are applied as body forces. Largedeformations and time effects are included in both analysis steps.
The strain-free temperature are set to deposition temperature for the coating layersand 880 [C] for WC–Co in accordance with [14].
6.2.6 Thermal-stress calculation from cutting operation
The thermo-mechanical influence ∆σtherm of one cutting cycle is calculated using acoupled thermal-structural analysis.
Temperature loads are defined at the top surface. The temperature loads consistof convection cooling caused by use of cooling media as well as a thermal DOFconstraint. The convection load is active during all time and is set to 10.587 · 10−6
[J ·mm−2 · K−1]. The thermal DOF constraint represents the temperature in thecutting zone. This temperature is set to 1373 [K] [6].
Tool engagement and free-running time are set to tcontact = 13 [ms] and tfree = 200[ms] according to [6].
The transfer of loads between thermal and structural analysis as well as analysissettings are performed as described in Ch. 6.2.5.
34
6.3 XFEM model
Fatigue crack growth is performed using XFEM technique. The XFEM-dependentsimulations are incorporated in the initial FEM model.
The fracture related material parameters used are listed in Tab. 5. The materialparameters are based on fine-grained WC − 10.2 vol%Co composition Grade 10Faccording to [40]. The corresponding hardness is estimated from [1].
Parameter Description Value Source
C Paris law constant 2.106 · 1010 [ mm/cycle(GPa
√mm)β
] [40]
β Paris law exponent 21 [40]damax Maximum allowed crack increment 10−4 [mm] [40]damin Minimum allowed crack increment 10−6 [mm] [40]Kth Threshold SIF 0.1866 [GPa
√mm] [40]
KIC Fracture toughness 0.23717 [GPa√mm] [40]
R Load ratio 0.1 [40]dmean Mean grain size 0.39 [µm] [40]vCo Volume fraction Co 10.2 [%] [40]HV Hardness 1300 [1]
Table 5: Fatigue related parameters based on Grade 10F according to [40]. Allparameters are converted from their original state to match Eq. 39 and the [GPa]-[mm] consistent system.
Definition of initial crack and enrichment zone as well as choice of governing crackgrowth model are made in this stage. The full model domain is set as an enrichmentapplicable zone. This allows for the crack to propagate anywhere in the model.The initial crack is positioned vertically from the middle of the top surface to theinterface between substrate and the first coating layer, as seen in Fig. 30. This is arepresentation of the cooling crack found after CVD deposition. The initial cracksLSM enrichment is manually performed through a LSM calculating algorithm. LSMvalues of the propagating crack are calculated automatically [20].
σres is applied as initial state condition using results from the last load-step of theσres calculation. ∆σtherm is applied as nodal reaction forces from the last load-step, i.e. ∆σtherm = σtherm(tend). The second (last) load-step was used, as the fulltransient results could not be transferred into the FCG simulation. Furthermore,only the last load-step showed tensile stresses.
35
Figure 30: Initial crack enrichment.
The following optional settings are used in the XFEM FCG simulation model
• Plane strain approximation
• Life-cycle method
• Stress/load ratio is set to 0.1
• Sub-step division is set to 100 steps.
The following settings are non-optional in FCG simulation using XFEM accordingto [20]
• Quasi-static analysis
• No large deformation effects
• Step-applied loads
• Crack growth criterion based on SIF-calculation i.e. Eq. 42
• Singularity-based enrichment
• The cracks surfaces are traction-free
36
6.3.1 Milling related mechanical forces
The mechanical loads presented in Eq. 2 are approximated to only consist of ∆σmechand ∆σtherm. ∆σentry/exit is for simplicity not strictly modelled but its contributionis assumed to be included in ∆σmech.
Fc and Fr are applied as element pressures. The following methods are evaluatedfor how to interpret the lateral load Fr in the model
• Shear pressure along the top surface
• Slope-decreasing normal pressure along the side
• Constant normal pressure along the side
and are illustrated in Fig. 31.
(a) Shear (b) Slope-decreasing (c) Constant
Figure 31: Possible translation of Fr into applied load in the model.
Applying the load as a constant pressure rests on the assumption that the depth ofthe model is so small that the physical shear load can be translated into a normalpressure along the side in the depth direction - even though the actual load is onthe surface.
Applying the load as a slope-decreasing pressure is based on the same assumptionbut where the stress decrease further into the model, i.e. close to the surface thepressure magnitude is maximum and at the bottom of the model equal to zero.
The pressure corresponding to Fc is assumed to be a compressive pressure of −1[GPa] based on the magnitude of kc1 for a generic cast iron workpiece. The radialcomponent Fr is defined as a tensile pressure of 0.5 [GPa] based on Eq. 5.
Fig. 32 illustrates the boundary condition used for the fatigue crack growth calcula-tion. The boundary conditions are similar to that of σres and ∆σtherm but the sidewhere Fr is applied is set to be unconstrained.
37
Figure 32: Illustration of boundary condition setup for fatigue crack growth.
6.3.2 LEFM applicability
The LEFM applicability conditions are evaluated using parameters stated in Tab.5. The LEFM applicability according to Eq. 47 - 49 is tested under the assumptionthat the substrate is the most influential of all layers. Therefore only substratematerial properties according to Tab. 5 are evaluated.
σcycl,s for WC–Co materials can according to [6] be approximated as:
• σB = 3 ·HV
• σs,cycl = 0.5 · σBat in Eq.50 is calculated to 2.323 [µm]. The pre-defined initial crack is set to 8 [µm]and therefore falls within the long crack regime.
∆KI in Eq. 47 will be evaluated as a long shallow surface crack:
∆KI = Y∆σ√a (53)
where the geometrical form factor Y is set to 1.985 according to [6]. ∆σ is the tensilestress range. Tensile stresses in a milling tool are estimated to be in the magnitudefrom 700 to 1300 [MPa] [6]. A tensile stress value of 1200 [MPa] is assumed. Theseassumptions leads to that:
• Eq. 47 is satisfied as the initial crack length the width and height of the modelall satisfies l > 7.461 [µm].
• Eq. 48 is satisfied as ∆KI = 6.737 [MPa√m] and 1.1 ·Kth = 6.49 [MPa
√m].
• Eq. 49 is satisfied if KI,max = ∆KI , i.e. KI,max < 0.9 ·KIC = 6.75 [MPa√m].
38
6.4 Limitations
The current implementation of fatigue-based crack growth analysis in ANSYS Me-chanical APDL is restricted to two-dimensional geometries [20]. A two-dimensionalanalysis is a limitation as the actual crack growth and especially comb crack growthin coated milling inserts occur in three dimensions [4]. Concerning plane approxima-tions only plane stress and plane strain conditions are allowed in the FCG analysis,why the use of generalized plane strain is limited.
Thermal convection was not included in the CVD deposition simulation. This couldnot be included as (coating) layer creation were done using element kill/birth func-tion in ANSYS Mechanical APDL. This function de-actiavtes and re-activates ele-ments [20]. Convection condition must be applied to free edges [20]. De-activatedcoating layers are still defined in the model - why the free edges of the model arealways the outermost exterior even when all coating layers are de-activated usingelement kill command.
A mesh sensitivity test was performed, where it was found that the stress levels inthe crack tip varied with coarser mesh sizes (> 0.51 [µm]). However, the stresses inthe crack tip are reported to be high at the crack tip due to singularity effects [51].Furthermore, due to hardware limitations the presented model were not able to beevaluated for finer meshes than element sizes of 0.31 [µm].
Furthermore, the crack deflection angle is not evaluated before an element is con-sidered fractured [20]. This means that coarser meshes will have fewer crack angleevaluation points. This leads to that a coarser mesh have less-accurate crack paths.It was reasoned that the element size should be in the magnitude of the physicalmeasured crack increment length [20]. Comb cracks generally grow intergranularly[6] , why it is reasonable that the crack angle evaluation should be roughly everylength equivalent to the mean grain diameter.
A technical restriction using older ANSYS Mechanical APDL versions (e.g. 17.0)was found for calculations using finer mesh sizes (< 0.5 [µm]) where the solver wouldcrash. This restrict the possibility to evaluate the model for very fine mesh sizesin this software version. ANSYS Mechanical APDL version 18.0 did not have thisissue. Furthermore, ANSYS Mechanical APDL is restricted for using small numbersin the magnitude of e.g. 10−42. For WC–Co compositions C are often found in themagnitudes of 10−11 to 10−42 [39, 40, 48, 49, 52] when stated in the [MPa]− [mm]consistent unit system. A numerical related issue was encountered when defining theC parameter in Paris law for such small numbers. ANSYS Mechanical APDL treatssuch small values as zero [53]. However, rescaling to the somewhat unconventional[GPa]− [mm] consistent unit system solved this issue.
39
7 Results
7.1 Residual stress state after CVD deposition
Simulation of temperature changes during CVD deposition can be seen in Fig.33.
0 2 4 6 8 10200
400
600
800
1000
1200
1400
Time [106 ms]
Tem
per
ature
[K]
Figure 33: Temperature response during CVD deposition. The result are taken forthe nodal solution of the substrate. Each dot indicates a load step. A coating layeris applied at each temperature plateau.
The stress state in terms of σx in each layer are listed in Tab. 6 where they arecompared to reference values taken from Tab. 1.
Layer σelast.res,x σplast.
res,x σanres Experimental
Al2O3 5089 516.76 801.3 615TiCN 3920 1010.4 1003.6 613WC–Co ≈ 0 127.85 45.84 -10
Table 6: σres comparison. σelast.res,x are results from using linear material model. σplast.
res,x
are results from using non-linear material model. Stresses are stated in [MPa].
7.2 Thermal stress during cutting operation
Simulation of temperature changes during one milling cycle is plotted in Fig. 34.The corresponding thermo-mechanical stress analysis is plotted in Fig. 35 for σx.
40
0 50 100 150 200200
400
600
800
1000
1200
1400
Time [ms]
Tem
per
ature
[K]
Figure 34: Calculated temperature response for one cutting cycle. The result aretaken for the nodal solution of the substrate.
0 50 100 150 200
−1
−0.5
0
0.5
1
Time [ms]
Str
essσx
[GPa]
WC–CoTiCNAl2O3
Figure 35: Calculated ∆σtherm as a function of time.
7.3 Fatigue crack growth
Crack path results for the fatigue crack growth simulation can be seen in Fig 36.
41
Figure 36: Fatigue crack growth simulation. Plot of crack path at end time.
Stress field result for the fatigue crack growth simulation can be seen in Fig. 37 - 38.All stress field plots are found in App. A Fig. 42-48. The corresponding number ofcycles are calculated to approximatively 50 cycles.
Figure 37: Fatigue crack growth simulation. Plot of first principal stress σ1 at initialtime. Units are in [GPa].
42
Figure 38: Fatigue crack growth simulation. Plot of first principal stress σ1 at fulltime. Units are in [GPa].
7.4 Chemical attack zone
Calculated crack path for simulation runs using chemical attack zone (CAZ) modelcan be seen in Fig. 39-41. Corresponding stress plots are found in App. A Fig.49-57.
43
Figure 39: Crack path at end time for FCG simulation where ECAZ = 1.2 ·EWC−Co.The plot is cropped.
Figure 40: Crack path at end time for FCG simulation where ECAZ = 1.0 ·EWC−Co.The plot is cropped.
44
Figure 41: Crack path at end time for FCG simulation where ECAZ = 0.8 ·EWC−Co.The plot is cropped.
45
8 Result analyses
The residual stress state σres show similar behaviour as previous FEM-based studiesconcerning the introduction of plasticity [9]. The stress state however deviates fromthe measured stress levels. This indicates that Tref and/or the estimated materialdata could be misrepresented. Especially the linear-based material models show veryhigh stresses and did not agree with similar studies or with the analytical solution[9]. The reasons to this was not found.
The thermo-mechanical stress state for one milling cycle ∆σtherm also show similarstress levels to previous FEM model based studies [6]. The ∆σtherm stress level isvery similar to σres. This can be related to the fact that the cutting temperatureset to 1100 [C] is similar to the maximum deposition temperature. There wereno distinct temperature gradients during heating and cooling of the model. Thehomogeneous temperature field can be related to the models small dimensions incombination with an effective cooling (high film coefficient).
The stress field analysis of the FCG simulation indicates that a compressive stressstate dominates in the cutting tool. The stress levels are generally compressiveexpect for close to the the crack tip where they are tensile. The tensile stressesin the absolute proximity of the crack tip are very high. This is expected due tothe singularity-based method [51]. LEFM is also based on the assumption that theregion near the crack tip have infinite stress levels [38].
The FCG simulation indicates that the crack propagates with an initially straightpath. The crack declines at a certain depth where after the crack growth terminates.This agrees well with the observed crack growth behaviour seen in cutting tools, ifcompared to Fig. 7. The crack path declination is directly related to the appliedlateral pressure Fr. The total crack extension is approximately 20 [µm], resultingin a total crack length of around 30 [µm]. This and the corresponding number ofcycles are in the range for where crack growth can be expected to have initiated[2]. Furthermore, early comb cracks as seen in Fig. 7 are in this length. The modeldid however not produce crack lengths which are found for fully propagated combcracks.
The presented CAZ model gives no indication of that a stiffer or softer region inthe substrate would cause crack declination similar to lateral comb crack, as seenin Fig. 7. The crack appears to follow the same crack path that the normal FCGmodel predicts but with earlier crack terminations.
46
9 Discussion
The aim of this work was to create a simple engineering model to simulate thermal-related fatigue crack growth. Therefore assumptions had to be made regardingwhich factors that are most relevant for the comb crack growth. The cause of combcracks are mainly thermo-mechanical related [2]. Therefore the main aspect of thismodel is not to model mechanical loads but thermo-mechanical loads for the FCGsimulation. However mechanical forces still contribute to the crack propagation.This is indicated in the model as the mechanical pressure Fr causes the crack topropagate with an inclination. The model indicates that even though Fr cause theinclination σres and ∆σtherm are the major contributors to the FCG driving force.
Furthermore, the thermo-mechanical residual stresses that was calculated did showdifference to the experimentally measured. Generalized plane strain have previouslybeen used for similar thermo-mechanial models [9]. The use of generalized planestrain and in combination with different boundary conditions that include similarto Fig. 32 and also where only the bottom is constrained - did not yield moreaccurate results in this model even though so could be expected. The use of non-symmetric boundary conditions in the thermo-mechanical models caused stress fieldgradients and very large stresses that are not expected when compared to analyticalsolution σanres. Generally the linear elastic FEM and analytical solution should agreewell [6, 9]. The reason for this remains unknown, but it is believed that the causecan be due to misrepresented material data and/or erroneous thermal expansionre-calculation done internally in ANSYS Mechanical APDL. Estimation of materialdata had to be done due to the limited available data in literature [9]. Using plasticitymaterial models and symmetrical boundary conditions as in Fig. 28 the calculatedstresses were closer to the experimental magnitudes. ∆σcutting is similar to σres asthe temperature levels are similar. The model can not be said to be a true reflectionas the calculated thermo-mechanical stresses include some sort of error, but as theyare in the same magnitude as experimental measurements using plasticity the modelcan be as an estimation.
The presented CAZ model was not able to reproduce the lateral comb crack be-haviour. This could be due to several reasons. Firstly the material data of the chemi-cally attacked zone was not based on any actual measurements but only assumptionsof the percentage change. Secondly, comb crack growth show substantially three-dimensional dependency [4], which is difficult to account for in a two-dimensionalsimulation.
In this work only CAZ regions centred in the model have been evaluated. Severalradiuses was evaluated in order to observe dependency of the zones depth. Theproposed CAZ model did not possess any difference in the crack path other thanthat the crack was terminated at an earlier point compared to the non-CAZ model.Several reasons could exist to that lateral comb crack formation was not obtained.It could indicate that the chemical attack zone can not be modelled as presentedin this model, that the lateral crack formation is expected to occur on a deeperdepth according to [4] than the current models crack growth occur at or that other
47
mechanisms cause lateral comb crack formation
Even though the presented results are not considered a true representation of realitythey could still be useful as initial FCG model that gives an initial understandingof the general stress field behaviour during cyclic loading.
9.1 Further research
This model is to be regarded as an initial simplified engineering model. A milling in-sert is loaded in a complex manner. Several assumptions on thermal and mechanicalloading are made in this model. For future work a step into the three-dimensionalspace is recommended. XFEM in ANSYS Mechanical APDL is somewhat limitedwhen using Paris law, as it only allows two-dimensional analysis and linear elasticmaterial models. A recommendation would be to use a non-fatigue based crackgrowth criteria. STTMAX or PSMAX are such non-fatigue crack growth criterion[20]. As comb cracks appear early in service e.g. as early as after 2000 cycles, simu-lations based on STTMAX/PSMAX could be rather easily run for several iterationsto represent the real amount of cycles.
Material data and especially plasticity data is scarce for coated cemented carbidesystems. For future research it is recommended to introduce more appropriate andalloy-consistent material data. It appears that the model or the material data con-tains an error that renders erroneous stress results. In the further research this mustbe taken into account.
A more sophisticated CAZ model could in the future include that only elements inthe crack tips proximity are altered. Furthermore, a type of contact mechanism caus-ing lateral vent cracks have previously been reported for ceramic materials accordingto [54], where it has been proposed that lateral cracks are formed upon unloadingbetween plastic and elastic stress fields. It could be of interest to apply this theoryto the case of comb cracks in milling inserts. For further research on comb cracksimulation contact and non-elastic material behaviour could be introduced.
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10 Conclusion
• A fatigue crack growth model based on Extended Finite Element Methodtechnique in ANSYS Mechanical APDL was presented.
• The calculated fatigue crack path agrees well with that of a comb crack in anearly growth stage.
• An estimation of the residual stress after CVD deposition are calculated usingFEM. Simulation with non-linear material models show more accurate resultsthan linear elastic material model when compared to experimentally measuredstresses.
• An estimation of thermal stresses induced during one milling cycle are calcu-lated using FEM.
• Lateral comb crack behaviour was not possible to reproduce using a weak orstiff circular zone in the substrate
49
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A Stress plots
A.1 Stress state in fatigue crack growth
A.1.1 Principal stress σ1
(a) Initial time
(b) End time
Figure 42: Result plot of principal stress σ1 in [GPa]i
A.1.2 Principal stress σ2
(a) Initial time
(b) End time
Figure 43: Result plot of principal stress σ2
ii
A.1.3 Principal stress σ3
(a) Initial time
(b) End time
Figure 44: Result plot of principal stress σ3
iii
A.1.4 Normal stress σx
(a) Initial time
(b) End time
Figure 45: Result plot of normal stress σx
iv
A.1.5 Normal stress σy
(a) Initial time
(b) End time
Figure 46: Result plot of normal stress σy
v
A.1.6 Normal stress σz
(a) Initial time
(b) End time
Figure 47: Result plot of normal stress σz
vi
A.1.7 Shear stress τxy
(a) Initial time
(b) End time
Figure 48: Result plot of normal stress τxy
vii
A.2 Chemical attack zone model
A.2.1 80% of initial stiffness - Principal stress σ1
(a) Initial time
(b) End time
Figure 49: Result plot of principal stress σ2 in [GPa]
viii
A.2.2 80% of initial stiffness - Principal stress σ2
(a) Initial time
(b) End time
Figure 50: Result plot of principal stress σ2
ix
A.2.3 80% of initial stiffness - Principal stress σ3
(a) Initial time
(b) End time
Figure 51: Result plot of principal stress σ3
x
A.2.4 100% of initial stiffness - Principal stress σ1
(a) Initial time
(b) End time
Figure 52: Result plot of principal stress σ2 in [GPa]
xi
A.2.5 100% of initial stiffness - Principal stress σ2
(a) Initial time
(b) End time
Figure 53: Result plot of principal stress σ2
xii
A.2.6 100% of initial stiffness - Principal stress σ3
(a) Initial time
(b) End time
Figure 54: Result plot of principal stress σ3
xiii
A.2.7 120% of initial stiffness - Principal stress σ1
(a) Initial time
(b) End time
Figure 55: Result plot of principal stress σ2 in [GPa]
xiv
A.2.8 120% of initial stiffness - Principal stress σ2
(a) Initial time
(b) End time
Figure 56: Result plot of principal stress σ2
xv
A.2.9 120% of initial stiffness - Principal stress σ3
(a) Initial time
(b) End time
Figure 57: Result plot of principal stress σ3
xvi
B Material data
Tab. 7-9 presents material data used in the model. Numbers in parenthesis representcorresponding temperature in [K]. All units are converted to the [GPa] − [mm]consistent unit system. Numbers presented in bold font are modified from theirsources and have had their temperature range and values altered and may thus notbe realistic. All data presented are reserved to not be correct.
Entry 1 2 3 4 5 6 7 8 9
E 577 (293) 562.4 (473) 555.1 (573) 549.5 (773) 533.2 (873) 525.9 (973) 518.6 (1073) 511.3 (1173) 489.4 (1673)v 0.22 (293) 0.22 (473) 0.22 (573) 0.22 (773) 0.22 (873) 0.22 (973) 0.22 (1073) 0.22 (1173) 0.22 (1673)c 300 (293) 310 (500) 360 (900) 400 (1300) 400 (1673)k 60E-6 (293) 60E-6 (500) 60E-6 (900) 55E-6 (1300) 55E-6 (1673)αSE 6.04E-6 (293) 6.32E-6 (473) 6.46E-6 (573) 6.58E-6 (673) 6.67E-6 (773) 6.73E-6 (873) 6.80E-6 (973) 6.81E-6 (1673)σy 1ET 0.1
Table 7: Material data related to WC–Co based on values obtained from [6, 14]
Entry 1 2 3 4 5 6 7 8 9
E 575 (293) 475 (1673)v 0.23c 352 (250) 596 (500) 650 (750) 684 (1000) 704 (1250) 717 (1500)k 14E-6αSE 2.877E-6 (200) 5.676E-6 (400) 6.1755E-6 (600) 6.7733E-6 (800) 6.949E-6 (1000) 7.043E-6 (1200) 7.0987E-6 (1400) 7.1389E-6 (1600)σy 1ET 1
Table 8: Material data related to TiCN based on values obtained from [14, 46, 47,55] and internal Sandvik Coromant databases.
Entry 1 2 3 4 5 6 7 8 9
E 416 (293) 390 (773) 364 (1273) 354 (1473) 343 (1673) 338 (1773)v 0.231 (293) 0.237 (773) 0.244 (1273) 0.247 (1473) 0.250 (1673) 0.252 (1773)c 775 (293) 1165 (773) 1255 (1273) 1285 (1473) 1315 (1673) 1330 (1673)k 32E-6 (293) 11.4E-6 (773) 7.22E-6 (1273) 6.67E-6 (1473) 6.34E-6 (1673) 6.23E-6 (1773)αSE 4.6E-6 (293) 7.1E-6 (773) 8.1E-6 (1273) 8.3E-6 (1473) 8.5E-6 (1673) 8.6E-6 (1773)σy 0.5ET 1
Table 9: Material data related to Al2O3 based on values obtained from [14] andinternal Sandvik Coromant databases.
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