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 date

2008-01-28 Publiceringsdatum Publication date

2008-02-04 Språk Language

Avdelning, institution

Division, department

Division of Solid Mechanics Dept. of Management and Engineering SE-581 83 LINKÖPING

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Svenska/Swedish X Engelska/English

Licentiatavhandling

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

X Examensarbete C-uppsats D-uppsats

Serietitel: Title of series

Övrig rapport

Serienummer/ISSN: Number of series

URL för elektronisk version URL for electronic version

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

Mechanical behaviour of single-crystal nickel-based superalloys

Författare Author

Daniel Leidermark

Sammanfattning Abstract

In this paper the mechanical behaviour, both elastic and plastic, of single-crystal nickel-based superalloys has been investigated. A theoretic base has been established in 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 a user defined material model in three major FEM-programmes. To evaluate the model a simulated pole figure has been compared to an experimental one. These pole figures match 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 singlecrystal nickel-based superalloys has been investigated. A theoretic base has been established in 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 a user defined material model in three major FEM-programmes. To evaluate the model a simulated pole figure has been compared to an experimental one. These pole figures match each other very 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 Linköping University. I would like to thank my two supervisors, Kjell Simonsson and Sören Sjöström, for all their help and hints during the work of this master thesis. A big appreciating for the support and interesting discussions with all the PhD colleagues at the division. Also the financial support from the KME programme is appreciated. A big thanks to Jonas Larsson at Medeso AB for testing the material model in ANSYS. I would like to thank the people at SIEMENS in Finspång, for letting me be "one of the team" during the three weeks I spent there. Big thanks to Johan Moverare and Ru Peng who solved the mystery of the pole figure, that had haunted me for weeks. A special thanks to my family who have supported and pushed me all the way from the time that I was a child to now. And finally my girlfriend Maria, who I like to thank for always being there for me.

Daniel Leidermark Linköping, January 2008

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

IX

Nomenclature Variable Ak a b a1 , a2 Ce cm C1 , C2 c D Ee E F Fe Fp G Gs Gαr hαβ hβ h0 I K KI , KII , KIII L m M 1, M 2 Nf nα q αβ q q R s sα S T Vk W

Description Associated thermodynamic forces Hardening parameter Material parameter Crystal orientations Elastic tangent stiffness tensor Constants material array Material parameter Material parameter Rate of deformation tensor Elastic Green-Lagrange strain tensor Modulus of elasticity Total deformation gradient Elastic deformation gradient Plastic deformation gradient Shear modulus Reference slip resistance Slip resistance of each slip system Strain hardening rate Single slip hardening rate Reference hardening rate Unit tensor Bulk modulus Stress intensity factor in Mode I, II, III Velocity gradient Slip rate sensitivity Structural tensors Fatigue life Normal vector of each slip system Latent-hardening matrix Latent-hardening ratio Heat flux Load ratio Specific entropy Slip direction of each slip system 2:nd Piola-Kirchhoff stress tensor Temperature Internal state variables Spin tensor

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

X Variable α γ˙ 0 ∆γ α ∆γ αmax ∆ǫ ∆KI ∆t εy η J λ µ

ω ∇T P int ρ

σ σu σy

τ τα Φ φ Ψ Ω ¯ Ω ¯ iso Ω Ω0

Description Slip system Reference shearing rate Shearing rate of each slip system Maximum shearing rate Strain amplitude Range of the stress intensity factor in Mode I Timestep Strain in y-direction Elastic parameter Jacobian determinant Lamé constant Lamé constant Mandel stress tensor Temperature gradient Internal power Density Cauchy stress tensor Ultimate stress Tensile load in y-direction Kirchhoff stress tensor Resolved shear stress Thermodynamic dissipation Plastic lattice rotation Helmholtz free energy Current configuration Intermediate configuration Isoclinic intermediate configuration Reference configuration

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

XI

Contents 1 Introduction 1.1 Gas turbines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Superalloys . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3

2 Fatigue 2.1 Low-cycle fatigue . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Thermomechanical fatigue . . . . . . . . . . . . . . . . . . . .

5 5 5

3 Crack propagation

7

4 Crystal structure

9

5 Theory 5.1 Tangent stiffness tensor 5.2 Kinematics . . . . . . 5.3 Crystal plasticity . . . 5.3.1 Virgin material 5.4 Thermodynamics . . .

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11 11 12 16 20 21

6 Implementation 6.1 Elastic material model . . . . . . 6.1.1 Validation . . . . . . . . . 6.2 Crystal plasticity material model 6.2.1 Validation . . . . . . . . . 6.3 Flowchart . . . . . . . . . . . . .

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7 Discussion 33 7.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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

XII

List of Figures 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

The interior of a stationary power generating gas turbine . . The In-Phase thermomechanical fatigue cycle . . . . . . . . The Out-of Phase thermomechanical fatigue cycle . . . . . . Crack loaded in different Modes . . . . . . . . . . . . . . . . FCC crystal structure . . . . . . . . . . . . . . . . . . . . . The (111) plane in the unit cell . . . . . . . . . . . . . . . . Material description, with a plastic lattice rotation. . . . . . Material description, without a plastic lattice rotation. . . . Rotation of the crystal orientation . . . . . . . . . . . . . . . The cube loaded uniaxially by a tensile load . . . . . . . . . Stereographic projection and (001) standard pole figure . . . Crystal orientations in correspondence to the global coordinate system used by Kalidindi and (011) pole figure . . . . . (001) pole figure of the deformed cube . . . . . . . . . . . . Flowchart of analysis done by LS-DYNA . . . . . . . . . . . Flowchart of analysis done by ABAQUS . . . . . . . . . . . Flowchart of analysis done by ANSYS . . . . . . . . . . . .

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2 6 6 7 9 10 12 13 17 24 28

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29 30 31 31 32

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10 25 28 29

List of Tables 1 2 3 4

Slip systems . . . . . . . . . . . . . Material parameters for pure nickel Material parameters for copper . . Initial slip hardening parameters for

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

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1

1

Introduction

In gas turbines the operating temperature is very high. The temperature is so high that regular steels will melt. Therefore superalloys are often used to manage the high temperature. The superalloys treated in this report are single-crystal superalloys, which have even better properties against temperature then their coarse-grained polycrystal cousins. The thermal efficiency increases with the operating temperature of a gas turbine and therefore the temperature is increasing with every new turbine that is developed. When the temperature is getting higher and higher the components of the turbine will be more and more exposed to fatigue, which will limit the lifetime of the turbine components. At a certain point the turbine components will reach the crack initiation point due to fatigue and the crack will then propagate through the single-crystal with little resistance. The designer wants to produce better and more efficient gas turbines which can manage higher and higher temperatures. This requires that under the development of new gas turbines there are tools and directives which take all of these aspects into consideration. How do the components of the turbine handle certain temperatures and load cycles? How long is the life of the components? When will a crack be initialised and propagate? The first thing is to look into the material and see how it behaves. This is done by developing a constitutive model of the superalloy that will handle all of these aspects. SIEMENS Industrial Turbomachinery AB in Finspång, Sweden, develops and manufactures gas turbines for a wide range of applications. SIEMENS is participating in a research programme that aims at solving material related problems associated with the production of electricity based on renewable fuels and to contribute in the development of new materials for energy systems of the future. This programme, called Konsortiet för Materialteknik för termiska energiprocesser (KME), was founded in 1997 and consists at present of 8 industrial companies and 12 energy companies participating through Elforsk AB in the programme. Elforsk AB, owned jointly by Svensk Energi (Swedenergy) and Svenska Kraftnät (The Swedish National Grid), started operations in 1993 with the overall aim to coordinate the industry’s joint research and development. The here presented master thesis has been carried out as a first step in — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

2

1 INTRODUCTION one of the KME-projects, namely KME-410 Thermomechanical fatigue of notched components made of single-crystal nickel-base superalloys. The basic goal of this project is to develop constitutive and life prediction models for superalloys in gas turbines. At this early stage of the project the lifetime estimation has not yet been addressed and the focus has instead been placed on the the basic constitutive model, which so far has been made in two versions: 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 heavy machinery or transport vessels such as ships and aircrafts. A gas turbine basically consists of a compressor, a combustor and a turbine, see Figure 1. The incoming air is compressed in the compressor to increase the pressure of the air. The compressed air then enters the combustion chamber, where it is mixed with the fuel and ignited. These hot gases will then flow through the turbine and by doing so make the turbine rotate. The temperature of the turbine components can range from 50◦ C to 1500◦C [1]. The turbine drives the compressor by the jointly connected shaft. In jet engines the hot gases are then passed through a nozzle, giving an increase in thrust as it is returned to normal atmospheric pressure. For stationary power generating gas turbines there is an extra power turbine which generates electricity, instead of the nozzle. Compressor

Power turbine Combustor

Turbine Figure 1: The interior of a stationary power generating gas turbine, with permission from SIEMENS

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

1.2 Superalloys

1.2

Superalloys

A superalloy is an alloy that exhibits excellent mechanical strength and creep resistance at high temperatures. A superalloy also has very good corrosion and oxidation resistance. The word alloy means to combine or bind together. In this context it means to combine materials with beneficial properties and in 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 base material on account of its face-centered cubic (FCC) crystal lattice structure, which is both ductile and tough, its moderate cost and low rates of thermally activated processes. Nickel is also stable in the FCC form when heated from room temperature to its melting point, i.e., there are no phase transformations. Compared with other typical aerospace materials, like titanium and aluminium, nickel is a rather dense material, which is due to its small interatomic distances. There are often more than 10 different alloying materials in a superalloy, each with their specific enhancing property. The alloying materials reside in different phases, which for a typical nickel-based superalloy are [2] 1. The gamma phase, γ. This phase exhibits the FCC crystal lattice structure 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 additions of aluminium, titanium, tantalum, niobium and presents a barrier to dislocations. The role of this phase is to confer strength to the superalloy. 3. Carbides and borides. These segregate to the grain boundaries of the γ phase, as a grain boundary strengthening element. Carbon, boron and zirconium often reside in this phase. There are also other phases in certain superalloys. However these should be avoided, because they do not promote the properties of the superalloys. The historical development of superalloys started prior to the 1940s, these superalloys were iron-based and cold wrought. In the 1940s the investment casting was introduced of cobalt-based superalloys, by which the operating temperature was raised significantly. These were mainly used in aircraft jet — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

3

4

1 INTRODUCTION engines and land turbines. During the 1950s the vacuum melting technic was developed allowing a fine control of the chemical composition of the superalloys, which in turn led to a revolution in processing techniques such as directional solidification of alloys and of single-crystal superalloys. In the 1970s powder metallurgy was introduced to develop certain superalloys, leading to improved property uniformity due to the elimination of microsegregation and the development of fine grains. In the later part of the 20th century the superalloys have become commonly used for many applications. Single-crystal superalloys are alloys that only consist of one grain. They have no grain-boundaries, hence grain-boundary strengtheners like carbon and boron are unnecessary. Grain-boundaries are easy diffusion paths and therefore reduce the creep resistance of the superalloys. Due to the nonexistence of grain-boundaries single-crystal superalloys possess the best creep properties of all superalloys. Nickel-based superalloys are used in aircraft and industrial gas turbines as blades, disks, vanes and combustors. Superalloys are also used in rocket engines, space vehicles, submarines, nuclear reactors, military electric motors, chemical processing vessels, and heat exchanger tubing.

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

5

2

Fatigue

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

2.1

Low-cycle fatigue

Low-cycle fatigue (LCF) will take place when the temperature is below the creep regime or when a component is affected by isothermal cycling. Where the 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 describing this. Hence, low-cycle fatigue is usually characterised by the Coffin-Manson relation, expressed here in both elasticity and plasticity ∆ǫ =

C2 b Nf + C1 Nfc E

(1)

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

2.2

Thermomechanical fatigue

When it comes to fatigue of high temperatures and temperature cycling it is called thermomechanical fatigue (TMF). This is taken into concern when the material is in the creep regime. There are two essential types of TMF cycles: In-Phase cycle and Out-of Phase cycle. • In-Phase cycle This is when the strain and the temperature are cycled in phase, see Figure 2. A typical example is a cold spot in a hot environment, which at high temperature will be loaded in tension and at low temperature loaded in compression.

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

6

2 FATIGUE

σ Tmax

ε

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 temperature loaded in compression. σ

Tmin

ε

Tmax

Figure 3: The Out-of Phase thermomechanical fatigue cycle

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

7

3

Crack propagation

After a number of load cycles, a fatigue crack may initiate and then propagate through the superalloy. In a coarse-grain polycrystal material the crack will be slowed down due to the grain boundaries. In single-crystal superalloys there are no grain boundaries, implying that the crack propagation will encounter very little resistance. There are three specific crack modes, which are 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 intensity factor √ KI = σyy πaf (2) √ KII = τxy πag (3) √ (4) KIII = τyz πah where a is the crack length and f , g, h are functions of geometry and type of loading. In cyclic loading the crack growth can be described by Paris’ law da = C (∆KI )n dN

(5)

where C and n are material properties and ∆KI is the range of the stress intensity factor due to crack growth in Mode I. For a more detailed description of 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 the minimum 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]. — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

9

4

Crystal structure

The material used in gas turbines is nickel-based superalloys. The material properties of nickel in the high temperature domain are very good. Nickel does not change its properties with the temperature as much as other materials, which makes it a good base material in superalloys. There is a wide range of other alloying materials present in superalloys as well. With nickel as the base material the superalloys have the same crystal lattice structure as 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 number of 12 [5], which is the maximum. The coordination number is the number of atoms surrounding each particular atom in the structure. Inelastically, the material deforms primarily along the planes which are most tightly packed, these are called close-packed planes. The FCC structure has 4 close-packed planes which, in Miller indices, are of the family {111}.  (111)    ¯ (111) {111} (1¯11)    ¯¯ (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 orientation with respect to the global coordinate system. It is most likely that the crystal orientation do not coincide with the global coordinate system. In each of these planes there are three slip directions, disregarding the negative directions. These directions are the most close-packed directions in each — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

10

4 CRYSTAL STRUCTURE a3

a2 a1

Figure 6: The (111) plane in the unit cell planes. The slip directions are of the family < 110 >.   [110] [¯110] [101] [¯101] < 110 >  [011] [0¯11]

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

α 1 2 3 4 5 6

s¯α 1 √ [1¯10] 2 1 ¯ √ [101] 2 1 √ [01¯1] 2 1 √ [101] 2 1 ¯¯ √ [110] 2 1 √ [01¯1] 2

Table 1: Slip systems n¯α α s¯α 1 1 √ [111] 7 √ [¯101] 3 2 1 1 ¯¯ √ [111] 8 √ [011] 3 2 1 1 √ [111] 9 √ [110] 3 2 1 ¯ 1 ¯ √ [111] 10 √ [110] 3 2 1 ¯ 1 √ [111] 11 √ [101] 3 2 1 ¯ 1 ¯¯ √ [111] 12 √ [011] 3 2

n¯α 1 √ [1¯11] 3 1 ¯ √ [111] 3 1 ¯ √ [111] 3 1 ¯¯ √ [111] 3 1 ¯¯ √ [111] 3 1 ¯¯ √ [111] 3

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

11

5

Theory

Basic knowledge in continuum mechanics [6] and material mechanics [7] is essential for the understanding of the following section. All the theory is made in a large deformation context with general tensor notation and Cartesian coordinates. Note: x˙ = dx dt

5.1

Tangent stiffness tensor

The tangent stiffness tensor has been adopted from Schröder et al. [8] and reworked to suit the behaviour of a superalloy. Nickel-based superalloys are elastically anisotropic when in single-crystal form. Hence the stiffness is dependent on the crystal orientation relative to the loading direction. The tangent stiffness tensor yields C e =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 parameter and M 1 , M 2 are structural tensors that depend on the crystal orientations a1 , 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 order tensor, two 2:nd order tensors to a 4:th order tensor, etc. In Cartesian coordinates the dyadic product takes the form (a ⊗ a)ij = ai aj

(10)

(M ⊗ M )ijkl = Mij Mkl

(11)

(M ⊗M )ijkl = Mik Mjl

(12)

(M ⊗M )ijkl = Mil Mjk

(13)

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

12

5 THEORY

5.2

Kinematics

When the body is deformed the material description is changed. As seen in Figure 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 the ¯ iso ) and the intermediate configuration isoclinic intermediate configuration (Ω ¯ (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 a plastic lattice rotation φ. Finally, the lattice is both elastically stretched and rotated by the elastic deformation gradient F e .

n¯α nα

F

sα s¯α

Ω0

Ω Fe

Fp n¯α

n¯α0

φ s¯α0 ¯ Ω

s¯α

¯ iso Ω

Figure 7: Material description, with a plastic lattice rotation. The total deformation gradient is thus divided into an elastic part and a plastic part, through the following multiplicative decomposition. F = F e φF p

(14)

In the subsequent discussion the plastic lattice rotation φ is set equal to the unit tensor, so the material exhibits no plastic lattice rotation. There— Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

13

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

n¯α F

nα sα

s¯α

Ω0

Ω

Fp

Fe n¯α

¯ Ω

s¯α

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 a plastic part. −1 −1 −1 L = F˙ F −1 = F˙ e F e + F e F˙ p F p F e

(16)

From Equation (16) the following can be defined −1 Le = F˙ e F e

(17)

−1 −1 Lp = F e F˙ p F p F e

(18)

L¯p = F˙ p F

(19)

p−1

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

14

5 THEORY where Le , Lp are the elastic and plastic velocity gradient, respectively, defined in the current configuration (Ω) while L¯p is the plastic velocity gradient ¯ defined in the intermediate configuration (Ω). The velocity gradient can be divided into one symmetric part and one skewsymmetric part. L=


1
1 L + LT + L − LT = D + W 2 2

(20)

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

(21)

W = We + Wp

(22)

where   1 p 1 e T T L + Le , D p = L + Lp 2 2   1 e 1 p T T We = L − Le , W p = L − Lp 2 2 De =

(23) (24)

The elastic Green-Lagrange strain tensor E¯e measured relative to the intermediate configuration is defined as  1  eT e E¯e = F F −I (25) 2 The relationship between the elastic rate of deformation tensor D e defined in the current configuration and the elastic Green-Lagrange strain rate tensor E¯˙ e defined in the intermediate configuration is given by a push-forward or a pull-back operation [11] De = F e

−T

T −1 E¯˙ e F e , E¯˙ e = F e D e F e

(26)

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

¯ = Fe τFe S −1

−T

T

¯F e = J σ ⇒ τ = F eS

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

(27)

15

5.2 Kinematics where J = det F e . As can be seen, in order to receive the Cauchy stress tensor the Kirchhoff stress tensor is scaled by the Jacobian determinant [11]. The internal power P int , when a body is deformed, is defined as Z int P = σ :DdV

(28)

Ω

The internal power can be divided into an elastic part and a plastic part Z Z e int P = σ :D dV + σ :Dp dV (29) Ω

Ω

The elastic part is transformed by a regular pull-back to the intermediate configuration, accordingly to Z Z e ¯ E¯˙ e dV¯ σ :D dV = S: (30) ¯ Ω

Ω

and the plastic part is transformed, due to symmetry of σ and D, as Z Z Z Z  T  p p e p e ¯ ¯ σ :D dV = σ :L dV = S: F L F dV = : L¯p dV¯ (31) Ω

¯ Ω

¯ Ω

ω

Ω

ω

where is the Mandel stress tensor. The Mandel stress tensor is a nonsymmetric tensor and it is defined in the intermediate configuration. From Equation (31) it can be seen that the Mandel stress tensor in relation to the 2:nd Piola-Kirchhoff stress tensor is given by ¯ = F e F eS T

(32)

ω

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

= Fe τFe

−T

(33)

ω

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

16

5 THEORY

5.3

Crystal plasticity

In a tension test of a single-crystal the axial load that initiates plastic flow depends 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 is expressed by Schmid’s law, with the Kirchhoff stress tensor, as

τ α = nα τ sα

(34)

The slip occurs on the slip systems that exhibit the greatest resolved shear stress. If only one slip system is active, the other slip systems have a smaller resolved shear stress than the initial critical stress and due to this slip does not occur on these systems. This is called single slip. Secondary slip systems can also be activated, this is then called multi slip, but these are not considered in this report. During deformation of a sample, either in tension or compression, the crystal orientation will rotate. As seen in Figure 9 the normal direction n¯α will rotate away from the axial axis in tension and toward it in compression. The slip systems defined in Chapter 4 are defined in the intermediate configuration. The slip directions may be transformed into the current configuration by sα = F e s¯α

(35)

Since s¯α and n¯α are unit vectors and orthogonal to each other it follows that n¯α · s¯α = nα · sα = 0

(36)

Hence, the transformation for the normal vector can be defined as nα = n¯α F 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 plasticity model that has already been developed and which is used in the scientific society. This is done to receive results at an early stage in the project. The — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

17

5.3 Crystal plasticity

n¯α

nα

nα sα

s¯α

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] and Hopperstad [13]. As mentioned above, plastic deformation occurs due to slip on the active slip systems [14], in the current configuration this is expressed as X Lp = γ˙α sα ⊗ nα (38) α

where γ˙α 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 intermediate configuration as X L¯p = γ˙α s¯α ⊗ n¯α (39) α

The plastic part of the internal power in the current configuration can be — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

18

5 THEORY expressed with Equation (38), taking into account that dV = J dV¯ , as Z Z X Z X p α α α ¯ ˙ σ :L dV = γ τ :(s ⊗ n ) dV = γ˙α τ α dV¯ (40) Ω

Ω

Ω

α

α

or in the intermediate configuration, with Equation (39), as Z Z X Z X p : L¯ dV¯ = γ˙α :(s¯α ⊗ n¯α ) dV¯ = γ˙α τ α dV¯

ω

¯ Ω

ω

¯ Ω

¯ Ω

α

(41)

α

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

ω

τ α = τ :(sα ⊗ nα ) =

:(s¯α ⊗ n¯α )

(42)

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

τ α = sα τ nα = s¯α n¯α

(43)

ω

which represents Schmid’s law in both the current- and the intermediate configuration. The shearing rate on the slip system α is in this work assumed to obey the following viscoplastic relation [12] γ˙α = γ˙0



|τ α | Gαr

 m1

sgn (τ α )

(44)

where γ˙0 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 X G˙αr = hαβ γ˙β

(45)

β

where hαβ is the strain hardening rate on slip system α due to shearing on slip system β, if α = β then hαα is the self-hardening rate of slip system α and if α 6= β then hαβ is the latent-hardening rate of slip system α caused — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

19

5.3 Crystal plasticity by slip on system β. The strain hardening rate is defined as hαβ = q αβ hβ

(46)

where q αβ is the matrix that describes the latent-hardening of the singlecrystal and hβ is the single slip hardening rate. In the latent-hardening matrix q αβ the systems {1, 2, 3}, {4, 5, 6}, {7, 8, 9} and {10, 11, 12} are coplanar. The ratio between the latent-hardening rate and the self-hardening rate are unity, for coplanar slip systems. The non-coplanar systems depend on the latent-hardening ratio parameter q (typically 1 ≤ q ≤ 1.4), which represents a stronger latent-hardening effect in the intersecting slip systems [12]. If q = 1, then only self-hardening is obtained. The latent-hardening matrix consists of   A qA qA qA  qA A qA qA   (47) q αβ =   qA qA A qA  qA qA qA A where 

 1 1 1 A= 1 1 1  1 1 1 The single slip hardening rate hβ is composed of a  Gβr β h = h0 1 − Gs

(48)

(49)

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

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

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 = τ1

(50)

There exists an initial slip resistance on each slip plane, prior to deformation. The slip resistance of the active slip system is Gαr = G1r

(51)

The other slip systems are not updated due to slip on these systems, but they 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 γ˙1 = γ˙0



|τ 1 | G1r

 m1

sgn τ 1




(52)

Because it is only one activated slip system the strain hardening rate, with β = α, is expressed as  a Gαr αα αα α αα h = q h = q h0 1 − (53) Gs The latent-hardening matrix q αα then yields unity in all positions, representing that on the active slip system only self-hardening is obtained. The hardening rate G˙ 1r for this slip system then yields    1  m1 1 a
G |τ | r 1 ˙ 1 sgn τ (54) Gr = h0 1 − γ˙0 Gs G1r

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21

5.4 Thermodynamics

5.4

Thermodynamics

The Helmholtz free energy is assumed given by Ψ = Ψ(E¯e , Vk )

(55)

where E¯e is the elastic Green-Lagrange strain tensor in the intermediate configuration and Vk are the internal state variables, which account for the loading history of the material. Differentiating this yields dΨ ∂Ψ ∂Ψ ˙ = ¯e E¯˙ e + Vk dt ∂Vk ∂E

(56)

The 2:nd principle of thermodynamics can be expressed with the use of Helmholtz free energy, leading to the Clausius-Duhem inequality [7]   ∇T ˙ ˙ ≥0 (57) σ :D − ρ Ψ + sT − q T 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 Helmholtz free energy and with a decoupled thermal and mechanical dissipation, with ρ0 = J ρ, the following mechanical dissipation inequality is received

τ :D − ρ0

dΨ ≥0 dt

(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 be taken from Equations (30) and (31), respectively. This gives

ω

¯ E¯˙ e + S:

∂Ψ ∂Ψ ˙ : L¯p − ρ0 ¯e E¯˙ e − ρ0 Vk ≥ 0 ∂Vk ∂E

(59)

Considering only elastic deformations and since Clausius-Duhem inequality holds for any particular E¯˙ e it follows that ¯ = ρ0 ∂Ψ S ∂ E¯e

(60)

and, as a result the following relation is received : L¯p − Ak V˙k ≥ 0

(61)

ω

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

22

5 THEORY where the thermodynamic forces associated with the internal variables are Ak = ρ0

∂Ψ ∂Vk

(62)

From this the dissipation is received

ω

Φ=

: L¯p − Ak V˙k ≥ 0

(63)

By, as a specific case letting, the Helmholtz free energy take the form (no internal variables) Ψ=

1 e ¯e ¯e C :E :E 2ρ0

(64)

Equation (60) then yields ¯ = ρ0 ∂Ψ = C e : E¯e S ∂ E¯e

(65)

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

ω

Φ=

: L¯p ≥ 0

(66)

With the insertion of Equation (41) and (44) this gives 1

ω

Φ=

: L¯p =

X α

γ˙α τ α

= γ˙0

X  |τ α |  m α

Gαr

|τ α | ≥ 0

(67)

which implies that the 2:nd principle of thermodynamics is fulfilled. However, since the hardening of the material is surely associated with a microstructural storage of energy, it follows that the use of the above model in a thermomechanical analysis will overestimate the heat production, due to the inelastic flow. Thus, in the future, a more thermomechanical realistic model will be developed.

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23

6

Implementation

The material model has been implemented in the three FEM-programmes LS-DYNA [15], ABAQUS [16] and ANSYS [17], which are world wide spread and extensively used in industry. At first only an elastic material model was developed. This was later developed further into a crystal plasticity material model. The elastic material model has been implemented in all three FEM-programmes, while the crystal plasticity material model so far only has been implemented in LS-DYNA. For the two models, the following 11 parameters must be given as input data cm(1) = K cm(2) = G cm(3) = λ cm(4) = µ cm(5) = η cm(6) = a1 (1) cm(7) = a1 (2) cm(8) = a1 (3) cm(9) = a2 (1) cm(10)= a2 (2) cm(11)= a2 (3)

Bulk modulus Shear modulus Lamé constant Lamé constant Elastic parameter Crystal orientation Crystal orientation Crystal orientation Crystal orientation Crystal orientation Crystal orientation

The bulk modulus and the shear modulus are calculated from λ and µ accordingly to an isotropic elastic behaviour (Hooke’s law). They are only needed in LS-DYNA for calculating an estimate of the critical time step (needed for 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 input parameters. Based on the given input the material models will calculate the internal forces and for an implicit analysis the tangent stiffness tensor too. The material parameters concerning the slip hardening are incorporated in the implemented code. These will be input parameters in the input file in 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 used in the material models. The total deformation gradient is calculated by the specific FEM-programme in use. The material models were implemented in FORTRAN 77. — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

24

6 IMPLEMENTATION

6.1

Elastic material model

The elastic material model calculates the Cauchy stress tensor for the next time 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
1 E n+1 = F Tn+1 F n+1 − I . 2 S n+1 = C e E n+1 . 1 σ n+1 = F n+1 S n+1 F Tn+1 . det F n+1 6.1.1

Validation

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

σy

y x

Figure 10: The cube loaded uniaxially by a tensile load

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

25

6.1 Elastic material model Material properties of pure nickel were used, since its moduli of elasticity in the crystal orientations [100], [110] and [111] are known, see e.g. [2]. These are E[100] = 125 GP a E[110] = 220 GP a E[111] = 294 GP a Other material parameters that were used in the analysis are specified in Table 2. Table 2: Material parameters for pure nickel η λ -147 GP a 13.5 GP a

µ ρ 118.5 GP a 8.902 · 103 kg/m3

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

[100] [110] [111]

= 7.98702 · 10−6 = 4.52995 · 10−6 = 3.33786 · 10−6

The moduli of elasticity were then calculated by σy = Eεy

(68)

yielding the following results E[100] = 125.203 GP a E[110] = 220.753 GP a E[111] = 299.593 GP a which match those from above very well.

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

26

6 IMPLEMENTATION

6.2

Crystal plasticity material model

The crystal plasticity material model calculates the Cauchy stress tensor for the next time step, but also the slip resistance and the plastic deformation gradient are calculated. The crystal plasticity material model has only been implemented for explicit analysis in LS-DYNA. Pidgin code loop, k : 1 → 12

F en+1,k = F n+1,k F pn+1,k−1 1 eT e ¯e E n+1,k = (F n+1,k F n+1,k − I) 2 e ¯ n+1,k = C e E ¯ S n+1,k −1

¯ n+1,k = F en+1,k F en+1,k S T

ω

n+1,k

loop, α : 1 → 12

¯α n+1,k n

ω

α τn+1,k = s¯α

αmax αmax ∆γn+1 = γ˙ n+1 ∆tn+1 = ∆tn+1 γ˙ 0

end loop

α ! m1 τ max
n+1,k αmax sgn τ n+1,k α Gr,nmax

 F pn+1,k − F pn+1,k−1 −1 αmax αmax ¯ ¯ αmax = F pn+1,k−1 = γ˙ n+1 s ⊗n ∆tn+1
αmax αmax ¯ ¯ αmax F pn+1,k−1 ⇒ F pn+1,k = I + ∆γn+1 s ⊗n end loop 

α ¯ p max L n+1

F en+1 = F n+1 F pn+1 −1

¯ n+1 = F en+1 F en+1 S 1 −T σ n+1 = F en+1 e det F n+1 e ¯α α s =F s T

ω

n+1

eT n+1 F n+1

ω

nα = n¯α F e

−1

Gαr,n+1 = Gαr,n +

X β

q αβ h0

Gβr,n 1− Gs

!a

β γ˙ 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 — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

6.2 Crystal plasticity material model deformation gradient is updated and the slip system that experience the greatest resolved shear stress is determined. When the all planes have been dealt with the plastic deformation gradient reaches its final value. This is used to update the elastic deformation gradient and the Cauchy stress is calculated. The slip directions and the normal directions are also transformed into the current configuration by the elastic deformation gradient and finally the slip resistance is updated. Initially the plastic deformation gradient F pn+1,k−1 equals F pn,k−1 and the initial value for F pn,k−1, with n = 0 and k = 1, is the unit tensor I. This corresponds to that the intermediate configuration coincide with the reference configuration initially.

6.2.1

Validation

The validation of the crystal plasticity material model is of a more complicated matter compared with the elastic material model. The slip systems are transformed to the current configuration, which reflects that the body is deformed. The normal directions of the deformed slip systems can then be evaluated by a stereographic projection [18], and in doing so a so called pole figure is received. The (001) pole figure means that the normal direction (001) of the crystal is orientated in the centre of the pole figure. The stereographic projection is created by letting all the normal directions of the planes be extended to an imaginary reference sphere. A projection plane is placed tangent to this sphere. At the other side of the sphere and orthogonal to the projection plane a point is marked. This point is called point of projection, from this point lines are drawn to intersect the normal directions at the sphere radius. These lines then cut the projection plane at a number of places, 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 figure done by X-ray diffraction, of a sample of the same material that is deformed in the same way as in the simulation. These two should coincide to reflect that the simulation gives the same result as the experiment. A single-crystal copper cylinder was investigated by Kalidindi [12], which was 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 — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

27

28

6 IMPLEMENTATION

Projection plane

Reference sphere

P′ P

Point of projection

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 slip resistance Gαr 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 GP a 20 GP a 75 GP a 8.93 · 103 kg/m3 The deformation of the copper cylinder was studied both computationally and experimentally. Pole figures were drawn for both applications and compared in Kalidindi’s work, the received normal directions of the planes are (101), (¯101), (¯110), (110) and in the centre (011), see Figure 12.

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

29

6.2 Crystal plasticity material model

Table 4: Initial slip hardening parameters for copper a 2.5

Gr 0 16 MP a

Gs 190 MP a

h0 250 MP a

m q γ˙0 0.012 1.4 0.001 s−1 y

(¯110)

z a3

a2

(110) (011) x

45o a1 ,x a)

y b)

(¯101)

(101)

Figure 12: a) Crystal orientations in correspondence to the global coordinate system used by Kalidindi b) (011) pole figure. From [12] With the pole figures from Figure 12 as reference, the simulated pole figure were to be duplicated. The same material properties for the copper were used, but the geometry was a bit different. It was a cube with sides of length 1 m, instead of a cylinder, and, further, it consisted of only one element. This would not change the slip plane deformation, due to the singlecrystal structure and a state of homogeneous deformation. A stereographic projection was made in MATLAB using the normal directions of the slip planes from the deformed cube, with the global z-axis in the centre. The crystal orientations, in the simulation, are expressed in the global coordinate system, hence the use of the (001) pole figure to evaluate the directions. This simulated pole figure was then compared to the (001) standard pole figure in Figure 11b) to get the corresponding crystal orientations of the slip planes. These are (101), (¯101), (¯1¯10) and (1¯10), see Figure 13. To compare this pole figure with the one in Figure 12 one has to rotate it 45o around the a1 -axis, due to that Kalidindi expresses the global coordinates in the crystal orientations, and use the opposite normal directions of (¯1¯10) and (1¯10). As one can see they match each other very well, since (¯1¯10) and (1¯10) are the same planes as (110) and (¯110) but with opposite normal directions. The exception is the plane in the centre (011), this plane does not experience any — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

30

6 IMPLEMENTATION slip because the load is parallel to the normal direction. Hence, by use of Schmid’s law, the resolved shear stress is equal to zero on this plane and due to this the shearing rate is also zero. So in the search for the slip system that exhibit the greatest resolved shear stress, then this one is not included in the material model, hence no point corresponding to (011) is present in Figure 13. a2

(¯101)

(101) a1

(¯1¯10)

(1¯10)

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

6.3

Flowchart

The specific FEM-programmes have different interfaces with the two implemented material models. There are some "boxes" that are used for all FEM-programmes, and these are • Neutral Material Model Contains either the elastic material model or the crystal plasticity material model. Calculates the corresponding stress state and for the plasticity model also the plastic deformation gradient and the slip resistance. • Const This calculates the tangent stiffness tensor (not yet for the crystal plasticity material model). • Subroutines Contains all the subroutines used in the main programmes, e.g. calculations of the inverse and dyadic product, Voight-transformations etc. — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

31

6.3 Flowchart Furthermore, for the specific implementations it is to be noted that the flowcharts are structured as • LS-DYNA If the calculations are to be done implicitly then the material model is called by two routines. These are umat50 and utan50. If the analysis is to be done explicitly then only one command is used, umat50. The utan50 is used to calculate the tangent stiffness tensor. When the calculations are done implicitly then the timestepping algorithm is done by Newton’s method, hence the need of the tangent stiffness tensor. In explicit analysis the timestepping is done by the central difference method. The flowchart for LS-DYNA can be found in Figure 14, where the dashed rectangle is only used for implicit analysis. umat50

Neutral Material Model

LS-DYNA utan50

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 model by use of umat. The corresponding flowchart of the algorithms can be found in Figure 15. Neutral Material Model ABAQUS

umat Const

Subroutines

Figure 15: Flowchart of analysis done by ABAQUS

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

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 by use of usermat. The corresponding flowchart of the algorithms can be found in Figure 16. Neutral Material Model ANSYS

usermat Const

Subroutines

Figure 16: Flowchart of analysis done by ANSYS

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

33

7

Discussion

The work presented in this report has been to model and implement a crystal plasticity material model for single-crystal superalloys. In the literature there are many kinds of crystal plasticity models, some handle many aspects and others less. The model used in this work focuses mainly on the hardening of the 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 crystal directions. The second step was to develop a crystal plasticity material model, which depended on the crystal structure and how it deformed. The crystal plasticity material model was evaluated against an experimental pole figure. As can be seen from the validations, both material models (elastic and plastic) exhibits the correct behaviour of single-crystal superalloys. These models can also be used for coarse-grain polycrystal materials. The models are then applied to each grain, each with their own crystal orientation. To get satisfying orientations these need to be randomised, so in the preprocessor 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, hence this is the first step in studying the fatigue behaviour of single-crystal superalloys. There are many more aspects, than those given in this report, to be considered. In the future the following steps will be carried out: • The crystal plasticity material model will be implemented for ABAQUS and ANSYS. It will also be implemented as an implicit model, implying that also a consistent tangent stiffness tensor needs to be set up. • The high temperature in gas turbines is an essential source to fatigue and thus the temperature dependency need to be implemented as a next step. • Handle that secondary slip systems might become activated, overshooting effect. — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

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.

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References [1] Stekovic S., 2007, Low Cylce Fatigue and Thermo-Mechanical Fatigue of 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, Studentlitteratur, Malmö. [4] Suresh S., 1991,Fatigue of Materials, Cambridge University Press, Cambridge. [5] Askeland D.R., 2001, The Science and Engineering of Material, Nelson Thornes 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, Cambridge University Press, Avon. [8] Schröder J., Gruttmann F., Löblein J., 2002, A Simple Orthotropic Finite Elasto-Plasticity Model Based on Generalized Stress-Strain Measures, 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 Elements for Continua and Structures, John Wiley & Sons Ltd, Chichester. [12] Kalidindi S.R., 1992, Polycrystal Plasticity: Constitutive Modeling and Deformation Processing, Massachusetts Institute of Technology, Cambridge. [13] Hopperstad O.S., Private communication. [14] Peirce D., Asaro R.J., Needleman A., 1982, An Analysis of Nonuniform and Localized Deformation in Ductile Single Crystals, Acta Metall vol 30. p1087-1119. — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys —

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 Engineering Materials, John Wiley & Sons Inc.

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