Simulation of fatigue crack growth in coated cemented carbide milling inserts

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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

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Acknowledgement

This work was performed at AB Sandvik Coromant in V¨astberga, Stockholm be-tween January and June 2017 as a master thesis project in the Solid Mechanics master track in Materials Design and Engineering programme at the Royal Institute of Technology (KTH).

I would first like to give my sincere thanks to my supervisor Dr. Jos´e Garc´ıa at Sandvik Coromant for the wonderful time I have had at the company. It has been a 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 in cemented carbide FEM modelling, as well as the rest of the Comb crack research team for the many innovative hours we have spent together. Further would I also like to thank my supervisor at KTH Prof. Per-Lennart Larsson for his guidance and support.

I would also like to give warm thanks to the wonderful people working at Sandvik Coromant in V¨astberga for the welcoming atmosphere they have created.

Lastly I would like to bring very special thanks to my wonderful Amanda for her support throughout the project.

Sincerely,

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Abstract

The aim of this work is to create a finite element model that simulates fatigue crack growth in coated cemented carbide milling inserts. A typical fatigue wear of coated milling 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. Severe temperature changes occur during the intermittent machining and when using liquid coolant. 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 comb crack formation by introducing a material alteration zone. Chemical attack occurs when wet-milling as the cooling liquid enters the principal comb crack and reacts with 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 fatigue crack growth in milling tools due to simplified loading conditions, which include thermo-mechanical and mechanical stresses. Thermo-mechanical stresses are calcu-lated corresponding to the temperature change during one milling cycle as well as stress contribution corresponding to residual stresses from the coating application process. The mechanical stresses are approximated and applied as pressures. The fatigue crack growth simulation is governed by Paris law and is performed using Extended Finite Element Method technique in ANSYS Mechanical APDL.

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Sammanfattning

M˚alet med detta arbete var att skapa en finit element modell för simulering av ut-mattningsspricktillväxt i belagda frässkär av h˚ardmetall. Kamsprickbildning är en vanlig typ av utmattningslitage som sker p˚a skäreggen p˚a skäret. Kamsprickor ini-tieras fr˚an tidigare existerande kylningsprickor i beläggningslagrena. Stora tempera-turskillnader uppst˚ar under intermittent bearbetning och d˚a vätske-baserad kylning används. Som ett resultat av de termiska spänningarna propagerar kylningssprickor-na.

Den modell som presenteras är baserad p˚a WC – Co som substrat med TiCN och Al2O3 som ytbeläggning. Modellen möjliggör för en snabb och m˚angsidig först˚aelse

av utmattningssprickor i skärverktyg p˚a grund av förenklade lastvillkor, vilka inklu-derar termomekaniska och mekaniska spänningar. De termomekaniska spänningarna motsvarar de som uppkommer p˚a grund av temperaturskillnader under en fräscykel samt spänningsbidraget p˚a grund av restspänningar fr˚an beläggningsproceduren. De mekaniska spänningarna approximeras och appliceras som tryck. Simuleringen av ut-mattningspricktillväxten beräknas enligt Paris lag och med hjälp av Extended Finite Element Method teknik i ANSYS Mechanical APDL. Med modellen ämnar detta arbete även att modellera en möjlig förklaring till uppkomsten av laterala kamspric-kor genom att att introducera en förändrad materialzon. Kemisk attack sker under fräsning med vätske-baserad kylning d˚a kylmedia tränger in i huvudsprickan och reagerar med substratet.

Resultatet av simuleringarna p˚avisar att restspänningen är mer riktig vid användandet av icke-linjära än linjära materialmodeller, om de jämförs med experimentellt uppmätta värden. Den beräknade sprickan visade sig vara liknande i form och längd till en kamspricka i tidig tillväxtfas. Spänningsniv˚aerna i substratet är kompressiva överallt utom i sprickspetsen där de är i dragtillst˚and. Spänningsniv˚aerna i sprickspetsen ab-soluta närhet är m˚anga magnituder högre än i omkringliggande omr˚aden. Sprickans krökning beror p˚a den applicerade mekaniska spänningen. Sprickans längd fanns vara influerad av den termiska spänningen.

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List of Figures

1 Example of milling application: Milling of an engine block. The cutter is equipped with multiple cutting inserts. Picture taken from [5]. Image courtesy of AB Sandvik Coromant . . . 3 2 Illustration of the cutting process. . . 4 3 Milling parameters. Images taken from [7]. Image courtesy of AB

Sandvik Coromant. . . 4 4 Microstructure of a CVD-coated milling insert. Image courtesy of AB

Sandvik Coromant. . . 5 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 crack formation. Image courtesy of AB Sandvik Coromant . 6 6 Macroscopic detail of comb crack wear on the cutting edge. Image

courtesy of AB Sandvik Coromant . . . 7 7 Microscopic detail of comb cracks. Figure (a) shows a comb crack

with 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. . . 8 9 Tomography of a comb crack [10]. Cyan color indicates the crack and

violet the coating. The substrate is not visible. Image courtesy of AB Sandvik Coromant. . . 8 10 Fractography of a comb crack. Region (b) indicates the comb crack

surface, 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. The

carbides are roughly in the size of 1 [µm]. Image courtesy of AB Sandvik Coromant. . . 9 12 Crack formed during dry milling. Image courtesy of AB Sandvik

Coromant. . . 10 13 Force components acting on the cutting edge according to [17]. Image

courtesy 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-bi-linear hardening (right ). . . 16 16 Illustration of a two-dimensional element with four nodes (left ) and

a mesh consisting of nine of such elements joined together at their common nodes. (right ). . . 18 17 Illustration of the field variables φ approximative solution over the

elements length for different types of approximation techniques [29]. . 18 18 A body in equilibrium containing a traction-free crack [24, 31]. . . 19 19 Crack termination occurs inside the element in singularity-based method

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20 Illustration of phantom-node method as illustrated in [20]. Crack

termination occurs at the element edge. . . 22

21 The three loading modes defined in fracture mechanics corresponding to KI, KII and KIII [38]. The dark gray planes indicates the initial crack plane. . . 23

22 Illustration of fatigue crack growth graph. . . 24

23 Illustration of evaluation of fracture criteria for MCSC. . . 25

24 Geometry of the model. Figure (a) illustrates a mock-up of the actual microstructure and (b) an idealization. . . 29

25 Geometry of the chemical attack zone model. . . 30

26 Mesh of normal model with 36450 elements . . . 31

27 Mesh of chemical attack zone model with 53252 elements. . . 32

28 Illustration of boundary condition used in TM calculations. . . 32

29 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. . . 36

31 Possible translation of Fr into applied load in the model. . . 37

32 Illustration of boundary condition setup for fatigue crack growth. . . 38

33 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 are taken for the nodal solution of the substrate. . . 41

35 Calculated ∆σtherm as a function of time. . . 41

36 Fatigue crack growth simulation. Plot of crack path at end time. . . . 42

37 Fatigue crack growth simulation. Plot of first principal stress σ1 at initial time. Units are in [GP a]. . . 42

38 Fatigue crack growth simulation. Plot of first principal stress σ1 at full time. Units are in [GP a]. . . 43

39 Crack path at end time for FCG simulation where ECAZ = 1.2 · EWC−Co. The plot is cropped. . . 44

42 Result plot of principal stress σ1 in [GP a] . . . i

43 Result plot of principal stress σ2 . . . ii

44 Result plot of principal stress σ3 . . . iii

45 Result plot of normal stress σx . . . iv

46 Result plot of normal stress σy . . . v

47 Result plot of normal stress σz . . . vi

48 Result plot of normal stress τxy . . . vii

49 Result plot of principal stress σ2 in [GP a] . . . viii

50 Result plot of principal stress σ2 . . . ix

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52 Result plot of principal stress σ2 in [GP a] . . . xi

53 Result plot of principal stress σ2 . . . xii

54 Result plot of principal stress σ3 . . . xiii

55 Result plot of principal stress σ2 in [GP a] . . . xiv

56 Result plot of principal stress σ2 . . . xv

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List of Tables

1 Stress state at room temperature for sample Ti(C, N) 554U according to [45]. . . 27 2 Thickness of each model layer. . . 30 3 Element settings for PLANE55 and PLANE182 elements. . . 31 4 Time-table with temperature and corresponding layer activation. . . . 33 5 Fatigue related parameters based on Grade 10F according to [40]. All

parameters are converted from their original state to match Eq. 39 and the [GP a]-[mm] consistent system. . . 35 6 σres comparison. σres,xelast. are results from using linear material model.

σplast._res,x are results from using non-linear material model. Stresses are stated in [M P a]. . . 40 7 Material data related to WC – Co based on values obtained from [6, 14]xvii 8 Material data related to TiCN based on values obtained from [14, 46,

47, 55] and internal Sandvik Coromant databases. . . xvii 9 Material data related to Al2O3 based on values obtained from [14]

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Nomenclature

[D]−1 Compliance matrix

α Thermal expansion coefficient

αIN Instantaneous thermal expansion coefficient

αSE _{Secant thermal expansion}

coeffi-cient ¯

u Displacement vector

β Paris law exponent

aj Displacement jump enriched nodal

degree of freedom

fb _{Body force vector}

ft Traction force vect

f Total applied force vector

n Normal vector

uh Total degree of freedom vector ui _{Enriched degree of freedom vector}

for i

∆σentry/exit Intermittent cutting related

stress

∆σmachining Sum of cutting related stress

∆σmech Mechanical stress

∆σtherm Cutting related thermal-stress

∆Keqv Equivalent stress intensity factor

range

∆KI Stress intensity factor range, mode

I

∆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

σvM_e 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}

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a Crack length ap Axial depth of cut

at Transition crack length

bj Crack tip enriched nodal degree of

freedom

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 stress criterion

PU Partition of Unity

PVD Physical Vapor Deposition SIF Stress intensity factor

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1 Background

Cemented carbides such as the WC-Co system are common composite materials used in machining tools. In the WC-Co system tungsten carbide (WC) act as a hard particle phase in a cobalt (Co) matrix [1]. WC-Co composites are also commonly coated using other materials in a few micrometers thin film-like layers that together offer properties that are suitable for machining applications [1].

Machining is the process where a desired shape of a workpiece material is obtained by material removal. Milling is a common machining method. In milling the cutting tool is rotated at high velocity during operation. A cutting tool consists of several cutting inserts. The cutting inserts are sheared into the workpiece and cause material removal by chip formation in the workpiece.

Milling is an intermittent machining operation. This means that each cutting insert only is in contact with the workpiece for a fraction of each milling revolution. This causes steep changes in both temperature and mechanical stresses. The temperature changes are even larger when using liquid coolant. Temperature change causes the cutting tool to expand and contract. This inflicts large thermal strains and stresses as the cutting insert consists of several different materials. The large stress cycles initiate crack growth in pre-existing short cracks found in the coating layers from the production stage.

Thermal cracks become critical after a few thousand milling revolution, causing material loss on the cutting edge [2]. This type of fatigue wear is known as comb cracks, 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. fatigue crack formation is vital. Stress fields governs crack propagation and can be examined using numerical models. Such models are advantageous for manufacturing companies as they are considerably cheap and facilitate understanding of complex problems. In this work a numerical model of fatigue crack growth have been developed using finite element method and linear elastic fracture mechanics. The loads are based on residual stress state from production, thermo-mechanical stress from milling and mechanical cutting loads.

2 Aim

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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 common methods is milling. In milling a rotating multi-tooth cutter as seen in Fig. 1 is used to remove material from the workpiece. The cutter is rotated at high velocity and is equipped with three up to a hundred cutting inserts [2]. As the cutting inserts enters the workpiece segments of material are sheared off into chips.

Figure 1: Example of milling application: Milling of an engine block. The cutter is equipped with multiple cutting inserts. Picture taken from [5]. Image courtesy of AB Sandvik Coromant

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Flank face Rak eface 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, feed

depth fz and maximum chip thickness hex [2]. Other parameters that may also

effect the milling process are the rotational speed, material properties of the tool, workpiece material and the choice of cooling media. Furthermore, a typical cutting insert 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 Sandvik Coromant.

3.2 Materials design

Cemented carbides, such as the WC – Co system, are powder metallurgical materials commonly 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 materials combined offers properties which are desired for cutting tools [1]. Such properties include

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• 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 to cemented carbides, film-like thin layers of other materials can be added. Such coating layers were commercially introduced in the 1970’s. Coatings are usually added due to their property improving ability, leading to better wear and abrasive resistance of the cutting tool [1]. The outermost layer can also function as a identification layer to indicate the inserts type and model. Some of the materials that can be used as coatings are titanium nitride (TiN), titanium carbide (TiC), titanium carbonitride (Ti(C, N)), aluminum oxide (Al2O3), hafnium-nitride (HfN), titanium aluminum

nitride ((Ti, Al)N) and aluminum chromium nitride ((Al, Cr)N) [1].

Two methods used today to apply coatings are CVD (abbr. for Chemical Vapor Deposition) and PVD (abbr. for Physical Vapor Deposition) [1].

CVD is the most commonly used method [1]. In the CVD method coating layers are added by having the coating material in a gaseous form react with the solid substrate. This is often done at high temperatures between 700 to 1050 ◦C and yields excellent adhesion between the layers [5]. An example of a CVD coated milling 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 AB Sandvik Coromant.

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found to grow perpendicular to the rake face of the milling insert and are generally found completely fracturing the coating layers but not propagated into the substrate [2, 4].

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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 crack formation. Image courtesy of AB Sandvik Coromant

3.3 Wear mechanisms

3.3.1 Comb cracks

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(a) Illustration of a comb crack (b) Photograph of comb crack

Figure 6: Macroscopic detail of comb crack wear on the cutting edge. Image courtesy of 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 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

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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 and violet the coating. The substrate is not visible. Image courtesy of AB Sandvik Coromant.

3.4 Possible influence of chemical attack

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chemical attack can be seen in Fig. 11a, in contrast to a chemically unattacked microstructure as seen in Fig. 11b.

(b)

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Figure 10: Fractography of a comb crack. Region (b) indicates the comb crack surface, region (a) indicates a normal cemented carbide fracture surface. Image courtesy 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. The carbides are roughly in the size of 1 [µm]. Image courtesy of AB Sandvik Coromant.

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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 to the chemical attack in the principle crack [3]. It is possible that the dissolution of binder phase causes the substrates mechanical properties to be different than in its initial 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-approach to describe the materials mechanical behavior include [6, 9, 13, 14]. Examples of studies that have discretized the microstructural features include [15]. In a contin-uum approach the material is approximated to be homogeneous even though the microstructure have distinct constituents (i.e. WC-grains in a Co-matrix). Thus, individual properties of the grains and the matrix are not considered but instead their 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 according to [6] be estimated as

σtotal= ∆σmachining+ σres (1)

where ∆σmachining is the sum of stresses which arise during the cutting operation

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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 sizes and 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 in compression [16].

The cutting related stresses in ∆σmachining consists of both thermal and mechanical

related 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 cutting

insert. The cutting operations force components are illustrated in Fig. 13 according to [17].

Figure 13: Force components acting on the cutting edge according to [17]. Image courtesy of AB Sandvik Coromant.

In Fig. 13 Fc is the cutting direction force component and is calculated according

Kienzles equation [17] as:

Fc= fz· ap· kc (3)

where kc is the specific cutting force. The magnitude of kc is compensated for the

chip angle γ0 as

kc = kc1· a−mp c · (1 −

γ0

100) (4)

where kc1 and mc are constants. For e.g. cast iron workpiece material the specific

cutting force constant kc1 is listed in the magnitude of 750 − 1650 [M P a] according

to [5].

The radial component Fr and the axial component Fa are approximated according

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4 Theory

4.1 Stress and strain relation

Stresses (σ) are internal forces in a body which are defined as force per unit area [N

m2 = P a]. 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 in deformed state when the stresses are positive or negative. Negative stress (σ < 0) is commonly 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 undeformed length of the body. Strain theory is divided into small and large strain theory. Small strain 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 undeformed length L0 are significantly different. Large strain theory is according to [19] defined

as

ε = lnL L0

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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 the materials stiffness in terms of pressure [P a] [19].

In the FEM software ANSYS Mechanical APDL [20] stresses and strains are related as

{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

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where v is the Poisson’s ratio which relates the transverse strain to axial strain and G is the materials shear stiffness coefficient [20]. Furthermore, the stress vector component 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_{} = [ε}

x εy εz εxy εyz εxz]T (12)

The elastic strain vector also relates the total strain vector and the thermal strain vector as

{εel_{} = {ε} − {ε}th_} ₍₁₃₎

where the total strain vector is defined as

{ε} = {εth_{} + [D]}−1_{σ} ₍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 parameter thermal expansion coefficient α which has the dimension [_K1] and can be defined as

αSE = L − L0 L0(T − Tref)

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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 change according to [19] as

εth = (T − Tref)α = ∆T α (16)

Thermal strain in ANSYS Mechanical APDL [20] is expressed equivalent to Eq. 16 in 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 as}

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T ε

αSE

αIN

Figure 14: Illustration of instantaneous and secant definition of thermal expansion coefficient.

The secant coefficient is the average thermal strain resulting from a change in tem-perature from a strain-free temtem-perature. Thermal strain εth as a function of αSE is calculated as

εth = αSE(T ) · (T − Tref) (18)

The instantaneous coefficient is based on the thermal strain caused by an infinite small change in temperature. Thermal strain εth _{as a function of α}IN _{is calculated}

as εth = Z 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) ∈ R}2_{, namely}

• plane stress assumption • plane strain assumption

The two assumptions differ in the way of treating the third dimension (z). Plane stress is commonly applied for geometries where one dimension is much smaller than the others. Plane stress defines one of the principle stresses as zero [19]. This results in 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 considered much larger than the others [23]. Plane strain defines one of the principle strains as zero [19]. This results in that the out-of-plane strains are stated as

εz = γxz = γzy= 0 (21)

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Furthermore, a third variant called generalized plane strain exists in ANSYS Me-chanical APDL. Generalized plane strain is similar to plane strain but instead of setting 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 is done in the plane strain assumption [20].

4.1.3 Plasticity

Plastic deformation is defined as an irreversible deformation of a body. The opposite to plastic deformation is elastic deformation where a body recovers to its initial shape when unloading. Plasticity occurs when a body is subjected to stresses that exceeds a local yield criteria [24]. The criterion can be defined in different ways, e.g. as a function of state variables such as stress tensor or strain components. A common stress criteria is the material parameter yield strength σy. The yield strength of a

material is commonly an uniaxially measured property. Real case loading involves multiaxial loading. To relate uniaxial and multiaxial stress state a yield criterion is introduced. Plastic deformation is generally assumed to occur when the yield strength limit σy is reached [25], i.e. when σ ≥ σy. Crack initiation generally occur

in 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 solid crystalline materials involves slip planes and dislocation rearrangement in the crystal lattice 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 be rate-independent plasticity. Most metals experience rate-independent plasticity [26]. Creep plasticity is an example of time-dependent deformation as the material do not directly 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 effects occurs after loading beyond the yield point [19]. In ANSYS Mechanical APDL this can 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 by several ET-parameters (i.e. ET 1, ET 2, ...). This is due to the fact that the measured

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ε σ σy E ε σ σy σmax E ET ε σ σy σmax E ET 1ET 2 ET 3

Figure 15: Material models for stress-strain relation: Ideal plasticity (left ), bi-linear hardening (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]. Heat flow 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. The mechanisms 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 heat flow 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 transfer within a solid body. Regions with larger kinetic energy, i.e. motion energy, will exchange energy to regions with lower kinetic energy [25]. Heat flux by conduction is defined as

qconduction = −k

dT

dx (23)

where k is the thermal conductivity in [_m·KW ] and dT_dx is the temperature gradient [25].

4.2.2 Convection

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according to

qconvection = h∆T (24)

where h is the film coefficient which represent the heat transferability in [_mW2_·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 a typical forced convection and its heat transfer coefficient in the magnitude of 10000 [ W

m2_K] [27].

4.2.3 Thermal radiation

Thermal radiation is generated by charged particles that cause collisions between the atoms. This causes the atoms to exchange kinetic energy. Heat transfer by thermal 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 solve differential equations approximatively. Such equations are commonly found when mathematically stating the physical reality through boundary value problems (also known as field problems) and are commonly found in engineering problems [28]. These problems can be too complex to be solved using common analytical tools which is why FEM is used.

4.3.1 Discretization

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i j k l 1 2 3 4

Figure 16: Illustration of a two-dimensional element with four nodes (left ) and a mesh 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 allowed movements of each node are known as degree of freedom (DOF). The DOFs allow for reactions 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. which field variable the differential equations depends on. In structural analysis the DOF are seen as displacements and the forces are mechanical forces. In thermal analysis the 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 known as shape functions) [28]. These functions are typically chosen as polynomials and allow 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].

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4.3.2 Solving procedure

The principle of virtual work is a fundamental basis in structural mechanics and thus in FEM. The principle states that in order for a body as seen in Fig. 18, to remain in equilibrium the external virtual work must equal the internal virtual work due to applied load [20].

Γ Γt Γu Γc fb n ΩA ΩB ft ¯ 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 = ft_{, on external traction boundary Γ}

t

• 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 equations according to

Kuh = f (25)

where K is the coefficient matrix (also known as the stiffness matrix), uh is the total degree of freedom vector (containing the unknowns) and f is the applied force vector [20, 24, 29].

If the coefficient matrix consists of unknowns then Eq. (25) is non-linear. ANSYS Mechanical APDL utilize the Newton-Raphson method to solve such non-linear equations [20]. Newton-Raphson is a iterative solving process where the load is divided into increments in order to obtain a solution. Each increment yields a solution for which it is evaluated if equilibrium is achieved. If the incremental solution do not yield equilibrium the procedure returns to the last increment which did yield equilibrium and initiates a smaller increment. This procedure is repeated until all the increments result in convergence of the full solution.

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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 between or through elements. This is a violation of the previously stated requirement where no gaps or jumps between or in an elements field function is allowed. In order to solve a crack growth simulation using FEM, each crack growth increment must be followed by an updated discretization so that the mesh conforms to the propagating crack 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 ANSYS Mechanical APDL [32]. Cracks, voids, inclusions and bi-materials are examples of discontinuities [31]. XFEM offers a convenient method to solve discontinuities that convenient 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 defined as m set of functions fk(x) within a domain ΩP U [24] such that

m

X

k=1

fk(x) = 1 ∀x ∈ ΩP U (26)

PU is used to enrich solutions in XFEM by applying the PU condition to element shape functions Nj [24, 34]

n

X

j=1

Nj(x) = 1 (27)

Enrichments of solution is done to increase the accuracy of the solution. This is done by including information obtained from the analytical solution. In XFEM enrichments are done locally and extrinsic. Extrinsic enrichment means that extra enrichment functions are added to the standard approximation. A local enrichment means that only nodal points in the proximity of the discontinuity. This results in keeping the additional unknown constants at a minimum [24, 31]. In the case of enriched displacement field u(x) extrinsic enrichment is done as [31]

uh(x) = uF EM + uEnriched = n X j=1 Nj(x)uj+ m X k=1 Nk(x)ψ(x)ak (28)

where ψ are discontinuous enrichment function and ak unknown parameters that

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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, u}tip _{is the crack-tip displacement and}

umat _{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 displacement jumps across crack surfaces as

uh(x) = Nj(x)uj

| {z }

Regular

+ H(x)Nj(x)aj

| {z }

Displ. jump enr.

+ Nj(x)

X

F (x)bj

| {z }

Crack-tip sing. enr.

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where H(x) is a Heaviside function and takes the values ±1 depending on which side of the crack the current analyzing point is and aj is an enriched nodal degree of

freedom that accounts for the displacement jumps. F (x) is a crack-tip function and bj 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 | {z } Regular + H(x)Nj(x)aj | {z }

Displ. jump enr.

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= +

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 a function 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 X and time [35]. ui,1(x) and ui,2(x) are displacement vectors in sub-elements 1 and 2

in 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 f the Level-Set method (abbr. LSM) is a signed distance function. LSM complement XFEM with information of where and how to enrich the elements [31, 36, 37]. In order to reduce computational costs only regions close to the discontinuity are eval-uated [31, 36]. LSM captures information of the signed distance to the discontinuity boundary 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

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4.5 Fracture mechanics

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4.5.1 Stress-intensity factors

Stress-intensity factor (abbr. SIF) is a common method used to relate applied load on a body to the stress state in the crack tip. Stress-intensity factors K are linear relations 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 a

geometry-related function. Three types of SIFs (KI, KII and KIII) are defined in

order to characterize the stress state in a three dimensional body [38]. Each SIF corresponds 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 to KI, 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 yield catastrophic fracture. Tensile stresses act in a crack opening manner. If the stresses are compressive then the crack propagation halts. Load cycles in milling machining can be pure tensile or compressive, or vary between compressive and tensile state within 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 ratio R according to [38] as

R = KI,max KI,min

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log(∆KI)

log(_dNda)

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 threshold fatigue 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 be observed in the log-log diagram in stage two. This linear relation is described by Paris 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 P a}m/cycle√

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 by very rapid growth with little plastic deformation [25].

Lifetime-wise the three stages corresponds can be summarized to infinite lifetime in stage 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. An example 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 greater than 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 experiments where certain critical limits in terms of material parameters are measured and re-lated to mathematical functions that relates loading and fracture as

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where g is a function of the stress tensor σij in a certain point x [38]. This implies

that fracture occurs when the function is greater than the critical value σcr, which

is a material related parameter.

The fracture criteria is based on the circumferential stress state in XFEM-based crack growth modeling in ANSYS Mechanical APDL [20]. The maximum circum-ferential stress criterion (abbr. MCSC) defines an arc region in front of the crack tip. The arc boundary is used as stress evaluation points, illustrated in Fig. 23 where 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 σθθ

is 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. The MCSC criterion can also be based on shear stress τrθ [20]. The fracture direction is

then determined from

τrθ = 0 (43)

Paris law is used as FCG criteria in ANSYS Mechanical APDL [20]. If the FCG involves mixed load modes then the SIF range ∆KI in Eq. (39) is calculated as the

equivalent 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) a FCG criterion is expressed in terms of ∆Keqv so that crack growth initiates and

propagates when

∆Keqv > ∆Kth (45)

and catastrophic fracture occurs when

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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 defined as 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 the cycle increment dN is defined. Furthermore is fatigue crack growth in ANSYS Mechanical APDL based on singularity-based XFEM [20]. This results in that the crack propagates one element at a time. Crack propagation is only modified after an element is fully fractured[20].

4.5.4 Linear elastic fracture mechanics

Linear elastic fracture mechanics (abbr. LEFM) is a concept where the plastic deformation region at the crack tip is assumed to be very small compared to the rest of the body. Plastic deformation occurs in the crack tip regions as stresses which are generally many magnitudes larger than in the surroundings. Paris law in ANSYS Mechanical APDL is based on LEFM [20].

LEFM assumes that a semi-infinite crack within an infinite body is considered self similar to that of a crack tip in a finite body [38]. A criteria for when LEFM is applicable 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 cyclic

loading [18]. Furthermore, LEFM is generally not applied to short cracks. Short cracks are in the same lengths as the microstructural features. Therefor they are prone to interact with the microstructure. The transition of short crack to long crack 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 is

the cyclic yield strength [6].

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5 Theoretical reference frame

Crack analysis of WC – Co in milling inserts have previously reported to satisfy the LEFM conditions according to [6]. In this work a two-dimensional FEM model is used to evaluate thermal-stress and temperature distribution during cutting oper-ation. Furthermore, LEFM has been applied for crack growth in similar cemented carbide systems in [41].

FEM based residual and thermal-stress related to cutting as well as mechanical stresses have been previously been reported by [14] and [9]. The calculated residual stress state will be evaluated against values obtained from experimentally measured stresses and from analytically calculated stresses.

The analytical stress levels are based on one dimensional linear-elastic assumption based on [2, 6, 9] and stated as

σan_res,coating = ∆T Elayer 1 − vlayer (αsubst.− αlayer) (51) and σan_res,subst. = ∆T Elayer hsubst. Xh_layerE_layer 1 − vlayer (αsubst.− αlayer) (52)

where h is the layers thickness. The results in Tab. 1 are based on ∆T = 1030 and measurement at room-temperature.

The experimentally measured stress levels are synchrotron measured at BESSY Berlin 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] σexp

res [M P a] σanres [M P a]

Al2O3 5.5 613 801.3

TiCN 4 615 1003.6

WC – 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 Al2O3are limited as previously reported

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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 settings and 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-consistent unit system according to [50] where pressure is stated in [GP a], force in [N ], mass in [kg], length in [mm], energy in [J ], temperature in [K] and time in [ms].

6.2 FEM model

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6.2.1 Material data

The material data are taken from various literature sources [6, 9, 46, 47] and from internal databases at AB Sandvik Coromant. The used material properties are generally temperature dependent. All materials have non-temperature dependent plasticity data. The plasticity is modelled according to bi-linear isotropic material model. Extrapolation of the temperature dependent data have been done when sufficient 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 idealization of the material layers in Fig. 24. In Fig. 24b the geometry is idealized such that each rectangle represent a material layer corresponding to the layers in Fig. 4. All areas are glued together through a Boolean operation. This is equivalent to assume perfectly 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 actual microstructure and (b) an idealization.

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(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 layer according 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) 4

2. Coating layer (TiCN) 5.5 1. 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 been made to keep the aspect ratios close to 1:1. As the CAZ is modelled as a circle and quadrilateral elements are used a special meshing techniques is used to keep the elements quadrilateral shapes from being violated. The element size of the CAZ region is set to be 1

6 of the general element size which is set to 0.31 [µm]. The total

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Thermal solid PLANE55

Option Setting

Element behaviour Plane strain

Film coefficient evaluation Differential temperature between bulk and surface

Structural solid PLANE182

Option Setting

Element behaviour Plane strain Element 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

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(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 ∆σthermcalculation. All

sides except for the top are set to roller condition. This means that the bottom of the model is restrained from movement in the vertical direction. The sides of the model are set to be free to move in vertical direction but constrained from movement in the horizontal direction.

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6.2.5 Thermal-stress calculation from CVD deposition procedure

The residual stress state σreswhich is related to the CVD deposition procedure were

calculated using coupled thermal-structural analysis.

Coating layer creation is done using element kill and birth function of PLANE55 and PLANE182. All elements were initially assigned an uniform temperature as well as initial condition corresponding to room-temperature. All layers are de-activated before simulation initiation. Each coating layer is re-activated according to a vector which 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 holding rate 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 is used here. Each load step is divided into 50 sub-load steps. At the first load step of each 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 – Co 10165000 2.82 1143 TiCN 29965000 8.32 1143 32125000 8.92 1323 Al2O3 46525000 12.9 1323 108145000 30.04 293 108205000 30.05 293

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TiCN Al2O3 0 5 10 15 20 25 30 200 400 600 800 1000 1200 1400 Time [h] T emp erature [K]

Figure 29: Coating deposition time-line used in CVD deposition simulation. Each dot 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 DOF constraint. A new DOF constraint is applied at the activation of each new coating layer. The previous DOF constraint is simultaneously removed. Convection or radiation load is not included in this stage.

Temperature distribution is calculated in a thermal transient analysis with PLANE55 elements. The results are saved for each load step. A corresponding structural tran-sient analysis is performed with PLANE182 elements. In the structural analysis temperature results from the thermal analysis are applied as body forces. Large deformations and time effects are included in both analysis steps.

The strain-free temperature are set to deposition temperature for the coating layers and 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 a

coupled thermal-structural analysis.

Temperature loads are defined at the top surface. The temperature loads consist of convection cooling caused by use of cooling media as well as a thermal DOF constraint. 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 the}

cutting zone. This temperature is set to 1373 [K] [6].

Tool engagement and free-running time are set to tcontact= 13 [ms] and tf ree = 200

[ms] according to [6].

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6.3 XFEM model

Fatigue crack growth is performed using XFEM technique. The XFEM-dependent simulations are incorporated in the initial FEM model.

The fracture related material parameters used are listed in Tab. 5. The material parameters are based on fine-grained W C − 10.2 vol%Co composition Grade 10F according to [40]. The corresponding hardness is estimated from [1].

Parameter Description Value Source C Paris law constant 2.106 · 1010 [_{(GP a}mm/cycle√

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 [GP a √ mm] [40] KIC Fracture toughness 0.23717 [GP a √ 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]. All parameters are converted from their original state to match Eq. 39 and the [GP a]-[mm] consistent system.

Definition of initial crack and enrichment zone as well as choice of governing crack growth model are made in this stage. The full model domain is set as an enrichment applicable 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 the interface between substrate and the first coating layer, as seen in Fig. 30. This is a representation of the cooling crack found after CVD deposition. The initial cracks LSM enrichment is manually performed through a LSM calculating algorithm. LSM values of the propagating crack are calculated automatically [20].

σres is applied as initial state condition using results from the last load-step of the

σres 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 full

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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 according to [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

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6.3.1 Milling related mechanical forces

The mechanical loads presented in Eq. 2 are approximated to only consist of ∆σmech

and ∆σtherm. ∆σentry/exit is for simplicity not strictly modelled but its contribution

is assumed to be included in ∆σmech.

Fc and Fr are applied as element pressures. The following methods are evaluated

for 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 of the model is so small that the physical shear load can be translated into a normal pressure along the side in the depth direction - even though the actual load is on the surface.

Applying the load as a slope-decreasing pressure is based on the same assumption but where the stress decrease further into the model, i.e. close to the surface the pressure 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

[GP a] based on the magnitude of kc1 for a generic cast iron workpiece. The radial

component Fr is defined as a tensile pressure of 0.5 [GP a] 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 side

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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 assumption that the substrate is the most influential of all layers. Therefore only substrate material 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 · σB

at 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 tensile stress range. Tensile stresses in a milling tool are estimated to be in the magnitude from 700 to 1300 [M P a] [6]. A tensile stress value of 1200 [M P a] is assumed. These assumptions leads to that:

• Eq. 47 is satisfied as the initial crack length the width and height of the model all satisfies l > 7.461 [µm]. • Eq. 48 is satisfied as ∆KI = 6.737 [M P a √ m] and 1.1 · Kth = 6.49 [M P a √ m]. • Eq. 49 is satisfied if KI,max = ∆KI, i.e. KI,max < 0.9 · KIC = 6.75 [M P a

(51)

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-dimensional analysis is a limitation as the actual crack growth and especially comb crack growth in coated milling inserts occur in three dimensions [4]. Concerning plane approxima-tions only plane stress and plane strain condiapproxima-tions 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 could not be included as (coating) layer creation were done using element kill/birth func-tion in ANSYS Mechanical APDL. This funcfunc-tion de-actiavtes and re-activates ele-ments [20]. Convection condition must be applied to free edges [20]. De-activated coating layers are still defined in the model - why the free edges of the model are always the outermost exterior even when all coating layers are de-activated using element kill command.

A mesh sensitivity test was performed, where it was found that the stress levels in the crack tip varied with coarser mesh sizes (> 0.51 [µm]). However, the stresses in the 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 be evaluated 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 angle evaluation 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 physical measured crack increment length [20]. Comb cracks generally grow intergranularly [6] , why it is reasonable that the crack angle evaluation should be roughly every length equivalent to the mean grain diameter.

Simulation of fatigue crack growth in coated cemented carbide milling inserts

Simulation of fatigue crack growth

in coated cemented carbide milling

inserts

ANDRÉ TENGSTRAND

Acknowledgement

Abstract

Sammanfattning

List of Figures

List of Tables

Contents

Nomenclature

1

Background

2

Aim

3

Introduction

3.1

Milling

3.2

Materials design

3.3

Wear mechanisms

3.4

Possible influence of chemical attack

3.5

Modeling

4

Theory

4.1

Stress and strain relation

4.2

Heat transfer

4.3

Finite Element Method

4.4

Extended Finite Element Method

4.5

Fracture mechanics

5

Theoretical reference frame

6

Method

6.1

General procedure

6.2

FEM model

6.3

XFEM model

6.4

Limitations