Mechanical Behaviour of Single-Crystal Nickel-Based · PDF file · 2008-02-04Nf Fatigue life n α Normal vector of ... 16 Flowchart of analysis done by ANSYS . . . . . . . . .

DEPARTMENT OF MANAGEMENT AND ENGINEERING

Mechanical Behaviour of Single-Crystal

Nickel-Based Superalloys

Master Thesis carried out at Division of Solid Mechanics Linköping University

January 2008

Daniel Leidermark

LIU-IEI-TEK-A--08/00283--SE

Institute of Technology, Dept. of Management and Engineering, SE-581 83 Linköping, Sweden

Framläggningsdatum

Presentation date2008-01-28Publiceringsdatum

Publication date2008-02-04

Avdelning, institutionDivision, department

Division of Solid MechanicsDept. of Management and EngineeringSE-581 83 LINKÖPING

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Language

Svenska/Swedish

Engelska/EnglishX

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Report category

Licentiatavhandling

Examensarbete

C-uppsats

D-uppsats

Övrig rapport

X

ISBN:

ISRN: LIU-IEI-TEK-A--08/00283--SE

Serietitel:

Title of series

Serienummer/ISSN:

Number of series

URL för elektronisk versionURL for electronic version

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-10722

Titel

TitleMechanical behaviour of single-crystal nickel-based superalloys

Författare

AuthorDaniel Leidermark

Sammanfattning

Abstract

In this paper the mechanical behaviour, both elastic and plastic, of single-crystalnickel-based superalloys has been investigated. A theoretic base has been establishedin crystal plasticity, with concern taken to the shearing rate on the slip systems.A model of the mechanical behaviour has been implemented, by using FORTRAN, as auser defined material model in three major FEM-programmes. To evaluate the modela simulated pole figure has been compared to an experimental one. These pole figuresmatch each other very well. Yielding a realistic behaviour of the model.

Nyckelord:Keyword

material model, single-crystal, superalloy, crystal plasticity, LS-DYNA, ABAQUS, ANSYS,FORTRAN, pole figure

V

Abstract

In this paper the mechanical behaviour, both elastic and plastic, of single-crystal nickel-based superalloys has been investigated. A theoretic base hasbeen established in crystal plasticity, with concern taken to the shearingrate on the slip systems. A model of the mechanical behaviour has beenimplemented, by using FORTRAN, as a user defined material model in threemajor FEM-programmes. To evaluate the model a simulated pole figure hasbeen compared to an experimental one. These pole figures match each othervery well. Yielding a realistic behaviour of the model.

— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

VII

Preface

This work was done during the autumn of 2007 as a master thesis at LinköpingUniversity. I would like to thank my two supervisors, Kjell Simonsson andSören Sjöström, for all their help and hints during the work of this masterthesis. A big appreciating for the support and interesting discussions with allthe PhD colleagues at the division. Also the financial support from the KMEprogramme is appreciated. A big thanks to Jonas Larsson at Medeso AB fortesting the material model in ANSYS. I would like to thank the people atSIEMENS in Finspång, for letting me be "one of the team" during the threeweeks I spent there. Big thanks to Johan Moverare and Ru Peng who solvedthe mystery of the pole figure, that had haunted me for weeks. A specialthanks to my family who have supported and pushed me all the way fromthe time that I was a child to now. And finally my girlfriend Maria, who Ilike to thank for always being there for me.

Daniel Leidermark

Linköping, January 2008


IX

Nomenclature

Variable DescriptionAk Associated thermodynamic forcesa Hardening parameterb Material parametera1, a2 Crystal orientationsCe Elastic tangent stiffness tensorcm Constants material arrayC1, C2 Material parameterc Material parameterD Rate of deformation tensorEe Elastic Green-Lagrange strain tensorE Modulus of elasticityF Total deformation gradientF e Elastic deformation gradientF p Plastic deformation gradientG Shear modulusGs Reference slip resistanceGα

r Slip resistance of each slip systemhαβ Strain hardening ratehβ Single slip hardening rateh0 Reference hardening rateI Unit tensorK Bulk modulusKI , KII , KIII Stress intensity factor in Mode I, II, IIIL Velocity gradientm Slip rate sensitivityM 1, M 2 Structural tensorsNf Fatigue lifenα Normal vector of each slip systemqαβ Latent-hardening matrixq Latent-hardening ratioq Heat fluxR Load ratios Specific entropysα Slip direction of each slip systemS 2:nd Piola-Kirchhoff stress tensorT TemperatureVk Internal state variablesW Spin tensor


X

Variable Descriptionα Slip systemγ0 Reference shearing rate∆γα Shearing rate of each slip system∆γαmax Maximum shearing rate∆ǫ Strain amplitude∆KI Range of the stress intensity factor in Mode I∆t Timestepεy Strain in y-directionη Elastic parameterJ Jacobian determinantλ Lamé constantµ Lamé constant

ω

Mandel stress tensor∇T Temperature gradientP int Internal powerρ Densityσ Cauchy stress tensorσu Ultimate stressσy Tensile load in y-directionτ Kirchhoff stress tensorτα Resolved shear stressΦ Thermodynamic dissipationφ Plastic lattice rotationΨ Helmholtz free energyΩ Current configurationΩ Intermediate configurationΩiso Isoclinic intermediate configurationΩ0 Reference configuration


XI

Contents

1 Introduction 1

1.1 Gas turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Superalloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Fatigue 5

2.1 Low-cycle fatigue . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Thermomechanical fatigue . . . . . . . . . . . . . . . . . . . . 5

3 Crack propagation 7

4 Crystal structure 9

5 Theory 11

5.1 Tangent stiffness tensor . . . . . . . . . . . . . . . . . . . . . . 115.2 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3 Crystal plasticity . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3.1 Virgin material . . . . . . . . . . . . . . . . . . . . . . 205.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 21

6 Implementation 23

6.1 Elastic material model . . . . . . . . . . . . . . . . . . . . . . 246.1.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Crystal plasticity material model . . . . . . . . . . . . . . . . 266.2.1 Validation . . . . . . . . . . . . . . . . . . . . . . . . . 27

6.3 Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

7 Discussion 33

7.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33


XII

List of Figures

1 The interior of a stationary power generating gas turbine . . . 22 The In-Phase thermomechanical fatigue cycle . . . . . . . . . 63 The Out-of Phase thermomechanical fatigue cycle . . . . . . . 64 Crack loaded in different Modes . . . . . . . . . . . . . . . . . 75 FCC crystal structure . . . . . . . . . . . . . . . . . . . . . . 96 The (111) plane in the unit cell . . . . . . . . . . . . . . . . . 107 Material description, with a plastic lattice rotation. . . . . . . 128 Material description, without a plastic lattice rotation. . . . . 139 Rotation of the crystal orientation . . . . . . . . . . . . . . . . 1710 The cube loaded uniaxially by a tensile load . . . . . . . . . . 2411 Stereographic projection and (001) standard pole figure . . . . 2812 Crystal orientations in correspondence to the global coordi-

nate system used by Kalidindi and (011) pole figure . . . . . . 2913 (001) pole figure of the deformed cube . . . . . . . . . . . . . 3014 Flowchart of analysis done by LS-DYNA . . . . . . . . . . . . 3115 Flowchart of analysis done by ABAQUS . . . . . . . . . . . . 3116 Flowchart of analysis done by ANSYS . . . . . . . . . . . . . 32

List of Tables

1 Slip systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Material parameters for pure nickel . . . . . . . . . . . . . . . 253 Material parameters for copper . . . . . . . . . . . . . . . . . 284 Initial slip hardening parameters for copper . . . . . . . . . . 29


1

1 Introduction

In gas turbines the operating temperature is very high. The temperature isso high that regular steels will melt. Therefore superalloys are often usedto manage the high temperature. The superalloys treated in this report aresingle-crystal superalloys, which have even better properties against temper-ature then their coarse-grained polycrystal cousins. The thermal efficiencyincreases with the operating temperature of a gas turbine and therefore thetemperature is increasing with every new turbine that is developed. Whenthe temperature is getting higher and higher the components of the turbinewill be more and more exposed to fatigue, which will limit the lifetime of theturbine components. At a certain point the turbine components will reachthe crack initiation point due to fatigue and the crack will then propagatethrough the single-crystal with little resistance. The designer wants to pro-duce better and more efficient gas turbines which can manage higher andhigher temperatures. This requires that under the development of new gasturbines there are tools and directives which take all of these aspects intoconsideration.

How do the components of the turbine handle certain temperatures and loadcycles? How long is the life of the components? When will a crack be ini-tialised and propagate? The first thing is to look into the material and seehow it behaves. This is done by developing a constitutive model of the su-peralloy that will handle all of these aspects.

SIEMENS Industrial Turbomachinery AB in Finspång, Sweden, developsand manufactures gas turbines for a wide range of applications. SIEMENS isparticipating in a research programme that aims at solving material relatedproblems associated with the production of electricity based on renewable fu-els and to contribute in the development of new materials for energy systemsof the future. This programme, called Konsortiet för Materialteknik för ter-miska energiprocesser (KME), was founded in 1997 and consists at presentof 8 industrial companies and 12 energy companies participating throughElforsk AB in the programme. Elforsk AB, owned jointly by Svensk Energi(Swedenergy) and Svenska Kraftnät (The Swedish National Grid), startedoperations in 1993 with the overall aim to coordinate the industry’s jointresearch and development.

The here presented master thesis has been carried out as a first step in


2 1 INTRODUCTION

one of the KME-projects, namely KME-410 Thermomechanical fatigue ofnotched components made of single-crystal nickel-base superalloys. The basicgoal of this project is to develop constitutive and life prediction models forsuperalloys in gas turbines. At this early stage of the project the lifetimeestimation has not yet been addressed and the focus has instead been placedon the the basic constitutive model, which so far has been made in twoversions: one elastic model and one crystal plasticity model.

1.1 Gas turbines

The function of a gas turbine is to supply electric power, to propel heavymachinery or transport vessels such as ships and aircrafts. A gas turbinebasically consists of a compressor, a combustor and a turbine, see Figure 1.The incoming air is compressed in the compressor to increase the pressure ofthe air. The compressed air then enters the combustion chamber, where it ismixed with the fuel and ignited. These hot gases will then flow through theturbine and by doing so make the turbine rotate. The temperature of theturbine components can range from 50C to 1500C [1]. The turbine drivesthe compressor by the jointly connected shaft. In jet engines the hot gases arethen passed through a nozzle, giving an increase in thrust as it is returned tonormal atmospheric pressure. For stationary power generating gas turbinesthere is an extra power turbine which generates electricity, instead of thenozzle.

CompressorCombustor

Turbine

Power turbine

Figure 1: The interior of a stationary power generating gas turbine, withpermission from SIEMENS


1.2 Superalloys 3

1.2 Superalloys

A superalloy is an alloy that exhibits excellent mechanical strength and creepresistance at high temperatures. A superalloy also has very good corrosionand oxidation resistance. The word alloy means to combine or bind together.In this context it means to combine materials with beneficial properties andin doing so receive a material that has a combination of these properties.Nickel-based superalloys are alloys based on nickel. Nickel is used as the basematerial on account of its face-centered cubic (FCC) crystal lattice structure,which is both ductile and tough, its moderate cost and low rates of thermallyactivated processes. Nickel is also stable in the FCC form when heated fromroom temperature to its melting point, i.e., there are no phase transforma-tions. Compared with other typical aerospace materials, like titanium andaluminium, nickel is a rather dense material, which is due to its small inter-atomic distances.

There are often more than 10 different alloying materials in a superalloy,each with their specific enhancing property. The alloying materials reside indifferent phases, which for a typical nickel-based superalloy are [2]

1. The gamma phase, γ. This phase exhibits the FCC crystal latticestructure and it forms a matrix phase, in which the other phases reside.Common materials of this phase are nickel, iron, cobalt, chromium,molybdenum, ruthenium and rhenium.

2. The gamma prime phase, γ′. This ordered phase is promoted by addi-tions of aluminium, titanium, tantalum, niobium and presents a barrierto dislocations. The role of this phase is to confer strength to the su-peralloy.

3. Carbides and borides. These segregate to the grain boundaries of theγ phase, as a grain boundary strengthening element. Carbon, boronand zirconium often reside in this phase.

There are also other phases in certain superalloys. However these should beavoided, because they do not promote the properties of the superalloys.

The historical development of superalloys started prior to the 1940s, thesesuperalloys were iron-based and cold wrought. In the 1940s the investmentcasting was introduced of cobalt-based superalloys, by which the operatingtemperature was raised significantly. These were mainly used in aircraft jet


4 1 INTRODUCTION

engines and land turbines. During the 1950s the vacuum melting technicwas developed allowing a fine control of the chemical composition of the su-peralloys, which in turn led to a revolution in processing techniques suchas directional solidification of alloys and of single-crystal superalloys. Inthe 1970s powder metallurgy was introduced to develop certain superalloys,leading to improved property uniformity due to the elimination of microseg-regation and the development of fine grains. In the later part of the 20thcentury the superalloys have become commonly used for many applications.

Single-crystal superalloys are alloys that only consist of one grain. Theyhave no grain-boundaries, hence grain-boundary strengtheners like carbonand boron are unnecessary. Grain-boundaries are easy diffusion paths andtherefore reduce the creep resistance of the superalloys. Due to the nonex-istence of grain-boundaries single-crystal superalloys possess the best creepproperties of all superalloys.

Nickel-based superalloys are used in aircraft and industrial gas turbines asblades, disks, vanes and combustors. Superalloys are also used in rocketengines, space vehicles, submarines, nuclear reactors, military electric motors,chemical processing vessels, and heat exchanger tubing.


5

2 Fatigue

The gas turbine will be exposed to load- and temperature cycling. This leadsto fatigue of the components.

2.1 Low-cycle fatigue

Low-cycle fatigue (LCF) will take place when the temperature is below thecreep regime or when a component is affected by isothermal cycling. Wherethe stress is high enough for plastic deformation to occur in a component,then the stress is not as useful as the strain when it comes to describingthis. Hence, low-cycle fatigue is usually characterised by the Coffin-Mansonrelation, expressed here in both elasticity and plasticity

∆ǫ =C2

EN b

f + C1Ncf (1)

where ∆ǫ is the strain amplitude, E is the modulus of elasticity, Nf is thefatigue life and C1, C2, b, c are material parameters. Equation (1) is alsoknown as universal slope .

2.2 Thermomechanical fatigue

When it comes to fatigue of high temperatures and temperature cycling itis called thermomechanical fatigue (TMF). This is taken into concern whenthe material is in the creep regime. There are two essential types of TMFcycles: In-Phase cycle and Out-of Phase cycle.

• In-Phase cycle

This is when the strain and the temperature are cycled in phase, seeFigure 2. A typical example is a cold spot in a hot environment, whichat high temperature will be loaded in tension and at low temperatureloaded in compression.


6 2 FATIGUE

σ

ε

Tmax

Tmin

Figure 2: The In-Phase thermomechanical fatigue cycle

• Out-of Phase cycle

This is when the strain and the temperature are cycled in counterphase,see Figure 3. A typical example is a hot spot in a cold environment,which at low temperature will be loaded in tension and at high tem-perature loaded in compression.

σ

ε

Tmin

Tmax

Figure 3: The Out-of Phase thermomechanical fatigue cycle


7

3 Crack propagation

After a number of load cycles, a fatigue crack may initiate and then propa-gate through the superalloy. In a coarse-grain polycrystal material the crackwill be slowed down due to the grain boundaries. In single-crystal superal-loys there are no grain boundaries, implying that the crack propagation willencounter very little resistance. There are three specific crack modes, whichare seen in Figure 4.

a) b) c)

Figure 4: Crack loaded in a) Mode I, b) Mode II and c) Mode III. From [3],with permission from Dahlberg T.

For each of the three crack modes there is one corresponding stress intensityfactor

KI = σyy

√πaf (2)

KII = τxy

√πag (3)

KIII = τyz

√πah (4)

where a is the crack length and f , g, h are functions of geometry and typeof loading. In cyclic loading the crack growth can be described by Paris’ law

da

dN= C (∆KI)

n (5)

where C and n are material properties and ∆KI is the range of the stress in-tensity factor due to crack growth in Mode I. For a more detailed descriptionof the crack growth, the effect of the load ratio has to be included. Paris’law is then modified by the R-value, which is the load ratio between theminimum stress σmin and the maximum stress σmax, expressed as

R =σmin

σmax

(6)

For more information the reader is referred to e.g. Suresh [4].


9

4 Crystal structure

The material used in gas turbines is nickel-based superalloys. The materialproperties of nickel in the high temperature domain are very good. Nickeldoes not change its properties with the temperature as much as other ma-terials, which makes it a good base material in superalloys. There is a widerange of other alloying materials present in superalloys as well. With nickelas the base material the superalloys have the same crystal lattice structureas nickel, namely face-centered cubic (FCC), see Figure 5.

Figure 5: FCC crystal structure

The FCC structure is a very close-packed structure with a coordination num-ber of 12 [5], which is the maximum. The coordination number is the numberof atoms surrounding each particular atom in the structure. Inelastically, thematerial deforms primarily along the planes which are most tightly packed,these are called close-packed planes. The FCC structure has 4 close-packedplanes which, in Miller indices, are of the family 111.

111

(111)(111)(111)(111)

The unit cell of the crystal structure, with plane (111), is seen in Figure 6.The axes of the unit cell, labelled a1, a2 and a3, define the crystal orientationwith respect to the global coordinate system. It is most likely that the crystalorientation do not coincide with the global coordinate system.

In each of these planes there are three slip directions, disregarding the nega-tive directions. These directions are the most close-packed directions in each


10 4 CRYSTAL STRUCTURE

a1

a2

a3

Figure 6: The (111) plane in the unit cell

planes. The slip directions are of the family < 110 >.

< 110 >

[110] [110][101] [101][011] [011]

These planes and their respective slip directions constitute 12 slip systems,which are shown in Table 1. Note that the respective slip directions andnormal directions are orthogonal.

Table 1: Slip systems

α sα nα α sα nα

11√2

[110]1√3

[111] 71√2

[101]1√3

[111]

21√2

[101]1√3

[111] 81√2

[011]1√3

[111]

31√2

[011]1√3

[111] 91√2

[110]1√3

[111]

41√2

[101]1√3

[111] 101√2

[110]1√3

[111]

51√2

[110]1√3

[111] 111√2

[101]1√3

[111]

61√2

[011]1√3

[111] 121√2

[011]1√3

[111]


11

5 Theory

Basic knowledge in continuum mechanics [6] and material mechanics [7] isessential for the understanding of the following section. All the theory is madein a large deformation context with general tensor notation and Cartesiancoordinates. Note: x = dx

dt

5.1 Tangent stiffness tensor

The tangent stiffness tensor has been adopted from Schröder et al. [8] andreworked to suit the behaviour of a superalloy. Nickel-based superalloys areelastically anisotropic when in single-crystal form. Hence the stiffness isdependent on the crystal orientation relative to the loading direction. Thetangent stiffness tensor yields

Ce =λI ⊗ I + µ(I⊗I + I⊗I) + 2η(M 1 ⊗ M 1 + M 2 ⊗ M 2

+ M 1 ⊗ M 2 − I ⊗ M 1 − I ⊗ M 2)(7)

where λ, µ are the Lamé constants, η is an additional third elastic parameterand M 1, M 2 are structural tensors that depend on the crystal orientationsa1, a2 accordingly to

M 1 = a1 ⊗ a1 (8)

M 2 = a2 ⊗ a2 (9)

The operator ⊗, called dyadic product, assembles two vectors to a 2:nd ordertensor, two 2:nd order tensors to a 4:th order tensor, etc. In Cartesiancoordinates the dyadic product takes the form

(a ⊗ a)ij = aiaj (10)

(M ⊗ M)ijkl = MijMkl (11)

(M⊗M )ijkl = MikMjl (12)

(M⊗M )ijkl

= MilMjk (13)


12 5 THEORY

5.2 Kinematics

When the body is deformed the material description is changed. As seen inFigure 7, the body undergoes a deformation from the reference configuration(Ω0) to the current configuration (Ω). Instead of taking the direct way,with the use of the total deformation gradient F , the other way through theisoclinic intermediate configuration (Ωiso) and the intermediate configuration(Ω) can be taken [9]. The first step is performed by shearing of the lattice,due to the plastic deformation gradient F p. The lattice then undergoes aplastic lattice rotation φ. Finally, the lattice is both elastically stretchedand rotated by the elastic deformation gradient F e.

nα

sαΩ0

nα

sαΩ

nα0

sα0

Ωiso

nα

sα

Ω

F

F p

φ

F e

Figure 7: Material description, with a plastic lattice rotation.

The total deformation gradient is thus divided into an elastic part and aplastic part, through the following multiplicative decomposition.

F = F eφF p (14)

In the subsequent discussion the plastic lattice rotation φ is set equal tothe unit tensor, so the material exhibits no plastic lattice rotation. There-


5.2 Kinematics 13

fore the material description in Figure 8 is used instead, hence the isoclinicintermediate configuration becomes the intermediate configuration.

nα

sαΩ0

nα

sαΩ

nα

sα

Ω

F

F p F e

Figure 8: Material description, without a plastic lattice rotation.

The multiplicative decomposition is then expressed as [10]

F = F eF p (15)

The velocity gradient can then also be expressed in an elastic part and aplastic part.

L = F F−1 = F eF e−1

+ F eF pF p−1

F e−1

(16)

From Equation (16) the following can be defined

Le = F eF e−1

(17)

Lp = F eF pF p−1

F e−1

(18)

Lp = F pF p−1

(19)


14 5 THEORY

where Le, Lp are the elastic and plastic velocity gradient, respectively, de-fined in the current configuration (Ω) while Lp is the plastic velocity gradientdefined in the intermediate configuration (Ω).

The velocity gradient can be divided into one symmetric part and one skew-symmetric part.

L =1

2

(

L + LT)

+1

2

(

L − LT)

= D + W (20)

where D is the rate of deformation tensor (symmetric) and W is the spintensor (skew-symmetric). These two can each be divided into an elastic partand a plastic part, accordingly to

D = De + Dp (21)

W = W e + W p (22)

where

De =1

2

(

Le + LeT

)

, Dp =1

2

(

Lp + LpT

)

(23)

W e =1

2

(

Le − LeT

)

, W p =1

2

(

Lp − LpT

)

(24)

The elastic Green-Lagrange strain tensor Ee measured relative to the inter-mediate configuration is defined as

Ee =1

2

(

F eT

F e − I)

(25)

The relationship between the elastic rate of deformation tensor De defined inthe current configuration and the elastic Green-Lagrange strain rate tensor˙Ee defined in the intermediate configuration is given by a push-forward or a

pull-back operation [11]

De = F e−T ˙EeF e−1

, ˙Ee = F eT

DeF e (26)

The 2:nd Piola-Kirchhoff stress tensor S defined in the intermediate configu-ration can be expressed by a pull-back of the Kirchhoff stress tensor τ fromthe current configuration by the following

S = Fe−1

τFe−T ⇒ τ = F

e

SFe

T

= Jσ (27)


5.2 Kinematics 15

where J = det F e. As can be seen, in order to receive the Cauchy stresstensor the Kirchhoff stress tensor is scaled by the Jacobian determinant [11].

The internal power P int, when a body is deformed, is defined as

P int =

∫

Ω

σ :DdV (28)

The internal power can be divided into an elastic part and a plastic part

P int =

∫

Ω

σ :DedV +

∫

Ω

σ:DpdV (29)

The elastic part is transformed by a regular pull-back to the intermediateconfiguration, accordingly to

∫

Ω

σ:DedV =

∫

Ω

S : ˙EedV (30)

and the plastic part is transformed, due to symmetry of σ and D, as

∫

Ω

σ :DpdV =

∫

Ω

σ:LpdV =

∫

Ω

S :(

F eT

LpF e)

dV =

∫

Ω

ω:LpdV (31)

where

ω

is the Mandel stress tensor. The Mandel stress tensor is a non-symmetric tensor and it is defined in the intermediate configuration. FromEquation (31) it can be seen that the Mandel stress tensor in relation to the2:nd Piola-Kirchhoff stress tensor is given by

ω

= F eT

F eS (32)

This can be further developed with the insertion of Equation (27), so that itrelates to the Kirchhoff stress tensor by

ω

= F eT

τF e−T

(33)


16 5 THEORY

5.3 Crystal plasticity

In a tension test of a single-crystal the axial load that initiates plastic flowdepends on the crystal orientation. A shear stress acting in the slip direction,on the slip plane, must be produced by the axial load. It is this shear stress,called the resolved shear stress, that initiates the plastic deformation. It isexpressed by Schmid’s law, with the Kirchhoff stress tensor, as

τα = nατsα

(34)

The slip occurs on the slip systems that exhibit the greatest resolved shearstress. If only one slip system is active, the other slip systems have a smallerresolved shear stress than the initial critical stress and due to this slip doesnot occur on these systems. This is called single slip. Secondary slip systemscan also be activated, this is then called multi slip, but these are not consid-ered in this report.

During deformation of a sample, either in tension or compression, the crystalorientation will rotate. As seen in Figure 9 the normal direction nα willrotate away from the axial axis in tension and toward it in compression.

The slip systems defined in Chapter 4 are defined in the intermediate configu-ration. The slip directions may be transformed into the current configurationby

sα = F esα (35)

Since sα and nα are unit vectors and orthogonal to each other it follows that

nα · sα = nα · sα = 0 (36)

Hence, the transformation for the normal vector can be defined as

nα = nαF e−1

(37)

where sα and nα are orthogonal to each other but no longer unit vectors.

In this work it has as, a first step, been decided to use a crystal plasticitymodel that has already been developed and which is used in the scientificsociety. This is done to receive results at an early stage in the project. The


5.3 Crystal plasticity 17

nα

sα

nα

sα

nα

sα

a) b)

Figure 9: Rotation of the crystal orientation in a) tension b) compression

model will be developed further in a more thoroughly study in the future.Hence, the following model is adopted from the work of Kalidindi [12] andHopperstad [13].

As mentioned above, plastic deformation occurs due to slip on the active slipsystems [14], in the current configuration this is expressed as

Lp =∑

α

γαsα ⊗ nα (38)

where γα is the shearing rate on the slip system α. With the use of Equation(35) and (37) the plastic deformation can be expressed in the intermediateconfiguration as

Lp =∑

α

γαsα ⊗ nα (39)

The plastic part of the internal power in the current configuration can be


18 5 THEORY

expressed with Equation (38), taking into account that dV = J dV , as

∫

Ω

σ:LpdV =

∫

Ω

∑

α

γατ :(sα ⊗ nα) dV =

∫

Ω

∑

α

γαταdV (40)

or in the intermediate configuration, with Equation (39), as

∫

Ω

ω

:LpdV =

∫

Ω

∑

α

γα ω

:(sα ⊗ nα) dV =

∫

Ω

∑

α

γαταdV (41)

where τα is the resolved shear stress. As Equation (40) and (41) yields thesame result, the resolved shear stress can be expressed from both of them, as

τα = τ :(sα ⊗ nα) = ω:(sα ⊗ nα) (42)

It can be shown from Equation (42) that the following is true

τα = sατnα = sα

ωnα (43)

which represents Schmid’s law in both the current- and the intermediate con-figuration.

The shearing rate on the slip system α is in this work assumed to obey thefollowing viscoplastic relation [12]

γα = γ0

( |τα|Gα

r

)1

m

sgn (τα) (44)

where γ0 is the reference shearing rate, m is the slip rate sensitivity and Gαr

is the slip resistance on the slip system α.

The hardening rate on slip system α is calculated by

Gαr =

∑

β

hαβ∣

∣

∣γβ

∣

∣

∣(45)

where hαβ is the strain hardening rate on slip system α due to shearing onslip system β, if α = β then hαα is the self-hardening rate of slip system αand if α 6= β then hαβ is the latent-hardening rate of slip system α caused


5.3 Crystal plasticity 19

by slip on system β.

The strain hardening rate is defined as

hαβ = qαβhβ (46)

where qαβ is the matrix that describes the latent-hardening of the single-crystal and hβ is the single slip hardening rate. In the latent-hardeningmatrix qαβ the systems 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, 11, 12 are copla-nar. The ratio between the latent-hardening rate and the self-hardening rateare unity, for coplanar slip systems. The non-coplanar systems depend onthe latent-hardening ratio parameter q (typically 1 ≤ q ≤ 1.4), which repre-sents a stronger latent-hardening effect in the intersecting slip systems [12].If q = 1, then only self-hardening is obtained. The latent-hardening matrixconsists of

qαβ =

A qA qA qAqA A qA qAqA qA A qAqA qA qA A

(47)

where

A =

1 1 11 1 11 1 1

(48)

The single slip hardening rate hβ is composed of

hβ = h0

(

1 − Gβr

Gs

)a

(49)

where h0 is the reference hardening rate, Gs is the reference slip resistanceand a is a slip system hardening parameter.


20 5 THEORY

5.3.1 Virgin material

For an initially virgin material with only one slip system activated, e.g. α =1, there is only one resolved shear stress, hence

τα = τ 1 (50)

There exists an initial slip resistance on each slip plane, prior to deformation.The slip resistance of the active slip system is

Gαr = G1

r (51)

The other slip systems are not updated due to slip on these systems, butthey grow because of the latent-hardening from the active slip system α.

The shearing rate on the active slip system can then be expressed as

γ1 = γ0

( |τ 1|G1

r

)1

m

sgn(

τ 1)

(52)

Because it is only one activated slip system the strain hardening rate, withβ = α, is expressed as

hαα = qααhα = qααh0

(

1 − Gαr

Gs

)a

(53)

The latent-hardening matrix qαα then yields unity in all positions, repre-senting that on the active slip system only self-hardening is obtained. Thehardening rate G1

r for this slip system then yields

G1r = h0

(

1 − G1r

Gs

)a∣

∣

∣

∣

∣

γ0

( |τ 1|G1

r

)1

m

sgn(

τ 1)

∣

∣

∣

∣

∣

(54)


5.4 Thermodynamics 21

5.4 Thermodynamics

The Helmholtz free energy is assumed given by

Ψ = Ψ(Ee, Vk) (55)

where Ee is the elastic Green-Lagrange strain tensor in the intermediateconfiguration and Vk are the internal state variables, which account for theloading history of the material. Differentiating this yields

dΨ

dt=

∂Ψ

∂Ee˙Ee +

∂Ψ

∂Vk

Vk (56)

The 2:nd principle of thermodynamics can be expressed with the use ofHelmholtz free energy, leading to the Clausius-Duhem inequality [7]

σ:D − ρ(

Ψ + sT)

− q∇T

T≥ 0 (57)

where ρ is the density of the current configuration, s is the specific entropy,T is the temperature, q is the heat flux and ∇T is the temperature gradient.For isothermal conditions motivating the adopted form of the Helmholtzfree energy and with a decoupled thermal and mechanical dissipation, withρ0 = J ρ, the following mechanical dissipation inequality is received

τ :D − ρ0dΨ

dt≥ 0 (58)

Equation (58) can be further developed with the insertion of Equation (56)and by separating D into an elastic part and a plastic part which can betaken from Equations (30) and (31), respectively. This gives

S : ˙Ee +

ω

:Lp − ρ0∂Ψ

∂Ee˙Ee − ρ0

∂Ψ

∂Vk

Vk ≥ 0 (59)

Considering only elastic deformations and since Clausius-Duhem inequality

holds for any particular ˙Ee it follows that

S = ρ0∂Ψ

∂Ee(60)

and, as a result the following relation is received

ω

:Lp − AkVk ≥ 0 (61)


22 5 THEORY

where the thermodynamic forces associated with the internal variables are

Ak = ρ0∂Ψ

∂Vk

(62)

From this the dissipation is received

Φ =

ω

:Lp − AkVk ≥ 0 (63)

By, as a specific case letting, the Helmholtz free energy take the form (nointernal variables)

Ψ =1

2ρ0Ce :Ee :Ee (64)

Equation (60) then yields

S = ρ0∂Ψ

∂Ee= Ce :Ee (65)

The adopted constitutive formulation must be associated with a positivedissipation. Under the given conditions requirement is

Φ =

ω

:Lp ≥ 0 (66)

With the insertion of Equation (41) and (44) this gives

Φ =

ω

:Lp =∑

α

γατα = γ0

∑

α

( |τα|Gα

r

)1

m

|τα| ≥ 0 (67)

which implies that the 2:nd principle of thermodynamics is fulfilled.

However, since the hardening of the material is surely associated with amicrostructural storage of energy, it follows that the use of the above modelin a thermomechanical analysis will overestimate the heat production, dueto the inelastic flow. Thus, in the future, a more thermomechanical realisticmodel will be developed.


23

6 Implementation

The material model has been implemented in the three FEM-programmesLS-DYNA [15], ABAQUS [16] and ANSYS [17], which are world wide spreadand extensively used in industry.

At first only an elastic material model was developed. This was later devel-oped further into a crystal plasticity material model. The elastic materialmodel has been implemented in all three FEM-programmes, while the crystalplasticity material model so far only has been implemented in LS-DYNA. Forthe two models, the following 11 parameters must be given as input data

cm(1) = K Bulk moduluscm(2) = G Shear moduluscm(3) = λ Lamé constantcm(4) = µ Lamé constantcm(5) = η Elastic parametercm(6) = a1(1) Crystal orientationcm(7) = a1(2) Crystal orientationcm(8) = a1(3) Crystal orientationcm(9) = a2(1) Crystal orientationcm(10)= a2(2) Crystal orientationcm(11)= a2(3) Crystal orientation

The bulk modulus and the shear modulus are calculated from λ and µ accord-ingly to an isotropic elastic behaviour (Hooke’s law). They are only neededin LS-DYNA for calculating an estimate of the critical time step (neededfor explicit analysis), and they are thus not used in the material models.When performing a simulation, the only thing to do is to give these 11 in-put parameters. Based on the given input the material models will calculatethe internal forces and for an implicit analysis the tangent stiffness tensortoo. The material parameters concerning the slip hardening are incorporatedin the implemented code. These will be input parameters in the input filein the near future, hence all parameters will be specified in the input data file.

The density of the material is also set in the input data file but is not usedin the material models. The total deformation gradient is calculated by thespecific FEM-programme in use. The material models were implemented inFORTRAN 77.


24 6 IMPLEMENTATION

6.1 Elastic material model

The elastic material model calculates the Cauchy stress tensor for the nexttime step (n+1). The deformation gradient consists only of the elastic part,thus the total deformation gradient equals the elastic deformation gradient.The analysis is done implicitly.

Pidgin code

En+1 =1

2

(

F Tn+1F n+1 − I

)

.

Sn+1 = CeEn+1.

σn+1 =1

det F n+1

F n+1Sn+1FTn+1.

6.1.1 Validation

To evaluate that the elastic material model calculates the right result a val-idation of the obtained modulus of elasticity was made. Different moduli ofelasticity will be received for different crystal orientations. A cube with sidesof length 1 m was uniaxially loaded in the y-direction by a tensile load ofσy = 1 · 106 Pa, see Figure 10.

σy

x

y

z

Figure 10: The cube loaded uniaxially by a tensile load


6.1 Elastic material model 25

Material properties of pure nickel were used, since its moduli of elasticity inthe crystal orientations [100], [110] and [111] are known, see e.g. [2]. Theseare

E[100] = 125 GPa

E[110] = 220 GPa

E[111] = 294 GPa

Other material parameters that were used in the analysis are specified inTable 2.

Table 2: Material parameters for pure nickel

η λ µ ρ-147 GPa 13.5 GPa 118.5 GPa 8.902 · 103 kg/m3

From the FEM-programme the following uniaxial strains were obtained forthe three crystal orientations

εy [100] = 7.98702 · 10−6

εy [110] = 4.52995 · 10−6

εy [111] = 3.33786 · 10−6

The moduli of elasticity were then calculated by

σy = Eεy (68)

yielding the following results

E[100] = 125.203 GPa

E[110] = 220.753 GPa

E[111] = 299.593 GPa

which match those from above very well.


26 6 IMPLEMENTATION

6.2 Crystal plasticity material model

The crystal plasticity material model calculates the Cauchy stress tensor forthe next time step, but also the slip resistance and the plastic deformationgradient are calculated. The crystal plasticity material model has only beenimplemented for explicit analysis in LS-DYNA.

Pidgin code

loop, k : 1 → 12

F en+1,k = F n+1,kF

p−1

n+1,k−1

Ee

n+1,k =1

2(F eT

n+1,kFen+1,k − I)

Sn+1,k = CeEe

n+1,k

ω

n+1,k = F eT

n+1,kFen+1,kSn+1,k

loop, α : 1 → 12

ταn+1,k = sα ω

n+1,knα

∆γαmax

n+1 = γαmax

n+1 ∆tn+1 = ∆tn+1γ0

(∣

∣ταmax

n+1,k

∣

∣

Gαmax

r,n

)1

m

sgn(

ταmax

n+1,k

)

end loop

Lpαmax

n+1 =

(

Fpn+1,k − F

pn+1,k−1

∆tn+1

)

Fp−1

n+1,k−1 = γαmax

n+1 sαmax ⊗ nαmax

⇒ Fpn+1,k =

(

I + ∆γαmax

n+1 sαmax ⊗ nαmax

)

Fpn+1,k−1

end loop

F en+1 = F n+1F

p−1

n+1

ω

n+1 = F eT

n+1Fen+1Sn+1

σn+1 =1

det F en+1

F e−T

n+1

ω

n+1FeT

n+1

sα = F esα

nα = nαF e−1

Gαr,n+1 = Gα

r,n +∑

β

qαβh0

(

1 −Gβ

r,n

Gs

)a∣

∣

∣γβ

n+1

∣

∣

∣∆tn+1

where n is the current time step. The first loop is over all the slip planes,where k symbolises which plane that is observed. In this loop the plastic


6.2 Crystal plasticity material model 27

deformation gradient is updated and the slip system that experience thegreatest resolved shear stress is determined. When the all planes have beendealt with the plastic deformation gradient reaches its final value. This isused to update the elastic deformation gradient and the Cauchy stress is cal-culated. The slip directions and the normal directions are also transformedinto the current configuration by the elastic deformation gradient and finallythe slip resistance is updated.

Initially the plastic deformation gradient Fpn+1,k−1 equals F

pn,k−1 and the

initial value for Fpn,k−1, with n = 0 and k = 1, is the unit tensor I. This

corresponds to that the intermediate configuration coincide with the referenceconfiguration initially.

6.2.1 Validation

The validation of the crystal plasticity material model is of a more compli-cated matter compared with the elastic material model. The slip systemsare transformed to the current configuration, which reflects that the bodyis deformed. The normal directions of the deformed slip systems can thenbe evaluated by a stereographic projection [18], and in doing so a so calledpole figure is received. The (001) pole figure means that the normal direc-tion (001) of the crystal is orientated in the centre of the pole figure. Thestereographic projection is created by letting all the normal directions of theplanes be extended to an imaginary reference sphere. A projection plane isplaced tangent to this sphere. At the other side of the sphere and orthogonalto the projection plane a point is marked. This point is called point of pro-jection, from this point lines are drawn to intersect the normal directions atthe sphere radius. These lines then cut the projection plane at a number ofplaces, that mark the relative position of the planes of the crystal structure,see Figure 11.

The simulated pole figure is then compared with an experimental pole figuredone by X-ray diffraction, of a sample of the same material that is deformedin the same way as in the simulation. These two should coincide to reflectthat the simulation gives the same result as the experiment.

A single-crystal copper cylinder was investigated by Kalidindi [12], whichwas compressed in room temperature by an axial strain rate of −0.001 s−1.The material properties and slip hardening parameters of copper, that were


28 6 IMPLEMENTATION

Projection plane

Reference sphere

Point ofprojection

PP ′

a) b)

Figure 11: a) Stereographic projection and b) (001) standard pole figure.From [18]

used in the experiment, can be seen in Table 3 and 4, respectively. The slipresistance Gα

r is initially set equal to Gr0, which is the initial value of the slip

resistance on each slip plane.

Table 3: Material parameters for copper

η λ µ ρ-104 GPa 20 GPa 75 GPa 8.93 · 103 kg/m3

The deformation of the copper cylinder was studied both computationallyand experimentally. Pole figures were drawn for both applications and com-pared in Kalidindi’s work, the received normal directions of the planes are(101), (101), (110), (110) and in the centre (011), see Figure 12.


6.2 Crystal plasticity material model 29

Table 4: Initial slip hardening parameters for copper

a Gr0Gs h0 m q γ0

2.5 16 MPa 190 MPa 250 MPa 0.012 1.4 0.001 s−1

a1,xy

z

a2a3

45o

(110) (110)

(101) (101)

(011)x

y

a) b)

Figure 12: a) Crystal orientations in correspondence to the global coordinatesystem used by Kalidindi b) (011) pole figure. From [12]

With the pole figures from Figure 12 as reference, the simulated pole figurewere to be duplicated. The same material properties for the copper wereused, but the geometry was a bit different. It was a cube with sides oflength 1 m, instead of a cylinder, and, further, it consisted of only oneelement. This would not change the slip plane deformation, due to the single-crystal structure and a state of homogeneous deformation. A stereographicprojection was made in MATLAB using the normal directions of the slipplanes from the deformed cube, with the global z-axis in the centre. Thecrystal orientations, in the simulation, are expressed in the global coordinatesystem, hence the use of the (001) pole figure to evaluate the directions. Thissimulated pole figure was then compared to the (001) standard pole figure inFigure 11b) to get the corresponding crystal orientations of the slip planes.These are (101), (101), (110) and (110), see Figure 13. To compare thispole figure with the one in Figure 12 one has to rotate it 45o around thea1-axis, due to that Kalidindi expresses the global coordinates in the crystalorientations, and use the opposite normal directions of (110) and (110). Asone can see they match each other very well, since (110) and (110) are thesame planes as (110) and (110) but with opposite normal directions. Theexception is the plane in the centre (011), this plane does not experience any


30 6 IMPLEMENTATION

slip because the load is parallel to the normal direction. Hence, by use ofSchmid’s law, the resolved shear stress is equal to zero on this plane and dueto this the shearing rate is also zero. So in the search for the slip system thatexhibit the greatest resolved shear stress, then this one is not included in thematerial model, hence no point corresponding to (011) is present in Figure13.

(101) (101)

(110)(110)

a1

a2

Figure 13: (001) pole figure of the deformed cube

6.3 Flowchart

The specific FEM-programmes have different interfaces with the two im-plemented material models. There are some "boxes" that are used for allFEM-programmes, and these are

• Neutral Material Model

Contains either the elastic material model or the crystal plasticity ma-terial model. Calculates the corresponding stress state and for theplasticity model also the plastic deformation gradient and the slip re-sistance.

• Const

This calculates the tangent stiffness tensor (not yet for the crystal plas-ticity material model).

• Subroutines

Contains all the subroutines used in the main programmes, e.g. calcu-lations of the inverse and dyadic product, Voight-transformations etc.


6.3 Flowchart 31

Furthermore, for the specific implementations it is to be noted that theflowcharts are structured as

• LS-DYNA

If the calculations are to be done implicitly then the material model iscalled by two routines. These are umat50 and utan50. If the analysisis to be done explicitly then only one command is used, umat50. Theutan50 is used to calculate the tangent stiffness tensor. When thecalculations are done implicitly then the timestepping algorithm is doneby Newton’s method, hence the need of the tangent stiffness tensor.In explicit analysis the timestepping is done by the central differencemethod. The flowchart for LS-DYNA can be found in Figure 14, wherethe dashed rectangle is only used for implicit analysis.

LS-DYNA

utan50

umat50 Neutral Material Model

Const Subroutines

Figure 14: Flowchart of analysis done by LS-DYNA

• ABAQUS

Only the elastic material model has been implemented for ABAQUS,thus the analysis is done implicitly. ABAQUS call the material modelby use of umat. The corresponding flowchart of the algorithms can befound in Figure 15.

ABAQUS umat

Neutral Material Model

Const Subroutines

Figure 15: Flowchart of analysis done by ABAQUS


32 6 IMPLEMENTATION

• ANSYS

Only the elastic material model has been implemented for ANSYS,thus the analysis is done implicitly. ANSYS call the material model byuse of usermat. The corresponding flowchart of the algorithms can befound in Figure 16.

ANSYS usermat

Neutral Material Model

Const Subroutines

Figure 16: Flowchart of analysis done by ANSYS


33

7 Discussion

The work presented in this report has been to model and implement a crystalplasticity material model for single-crystal superalloys. In the literature thereare many kinds of crystal plasticity models, some handle many aspects andothers less. The model used in this work focuses mainly on the hardening ofthe single-crystal due to deformation on the slip systems.

The first step in the modeling was to develop an elastic material model,which was evaluated against three given moduli of elasticity in certain crys-tal directions. The second step was to develop a crystal plasticity materialmodel, which depended on the crystal structure and how it deformed. Thecrystal plasticity material model was evaluated against an experimental polefigure. As can be seen from the validations, both material models (elasticand plastic) exhibits the correct behaviour of single-crystal superalloys.

These models can also be used for coarse-grain polycrystal materials. Themodels are then applied to each grain, each with their own crystal orientation.To get satisfying orientations these need to be randomised, so in the pre-processor there is to be some kind of randomiser for each grain.

7.1 Future work

This work is a project in the programme KME, as been said above, hencethis is the first step in studying the fatigue behaviour of single-crystal super-alloys. There are many more aspects, than those given in this report, to beconsidered. In the future the following steps will be carried out:

• The crystal plasticity material model will be implemented for ABAQUSand ANSYS. It will also be implemented as an implicit model, implyingthat also a consistent tangent stiffness tensor needs to be set up.

• The high temperature in gas turbines is an essential source to fatigueand thus the temperature dependency need to be implemented as anext step.

• Handle that secondary slip systems might become activated, overshoot-ing effect.


34 7 DISCUSSION

• Handle the back-stress on the slip planes, due to the Bauschinger effect,and incorporate this in the model.

• Handle LCF in the model and later TMF.

• Handle fatigue crack propagation.


35

References

[1] Stekovic S., 2007, Low Cylce Fatigue and Thermo-Mechanical Fatigueof Uncoated and Coated Nickel-base Superalloys, Linköping University,Linköping.

[2] Reed R.C., 2006, The Superalloys - Fundamentals and Applications,Cambridge University Press, Cambridge.

[3] Dahlberg T., Ekberg A., 2002, Failure Fracture Fatigue, Studentlitter-atur, Malmö.

[4] Suresh S., 1991,Fatigue of Materials, Cambridge University Press, Cam-bridge.

[5] Askeland D.R., 2001, The Science and Engineering of Material, NelsonThornes Ltd, Cheltenham.

[6] Mase G. T., Mase G. E., 1999, Continuum Mechanics for Engineers,CRC Press LLC, Boca Raton.

[7] Lemaitre J., Chaboche J.-L., 1990, Mechanics of Solid Materials, Cam-bridge University Press, Avon.

[8] Schröder J., Gruttmann F., Löblein J., 2002, A Simple Orthotropic Fi-nite Elasto-Plasticity Model Based on Generalized Stress-Strain Mea-sures, Computational Mechanics 30. p48-64.

[9] Haupt P., 2002, Continuum Mechanics and Theory of Materials,Springer-Verlag, Berlin Heidelberg.

[10] Khan A.S., Huang S., 1995, Continuum Theory of Plasticity, John Wiley& Sons Inc, New York.

[11] Belytschko T., Liu W.K., Moran B., 2000, Nonlinear Finite Elementsfor Continua and Structures, John Wiley & Sons Ltd, Chichester.

[12] Kalidindi S.R., 1992, Polycrystal Plasticity: Constitutive Modeling andDeformation Processing, Massachusetts Institute of Technology, Cam-bridge.

[13] Hopperstad O.S., Private communication.

[14] Peirce D., Asaro R.J., Needleman A., 1982, An Analysis of Nonuniformand Localized Deformation in Ductile Single Crystals, Acta Metall vol30. p1087-1119.


36

[15] LS-DYNA, 2008-01-25, http://www.ls-dyna.com/

[16] ABAQUS, 2008-01-25, http://www.simulia.com/products/abaqus_fea.html

[17] ANSYS, 2008-01-25, http://www.ansys.com/

[18] Hertzberg R.W., 1996, Deformation and Fracture Mechanics of Engi-neering Materials, John Wiley & Sons Inc.


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Mechanical Behaviour of Single-Crystal Nickel-Based · PDF file · 2008-02-04Nf Fatigue life n α Normal vector of ... 16 Flowchart of analysis done by ANSYS . . . . . . . . .