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Finite Element Analysis Of 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 Avdelning, institution Presentation date Division, department 2008-01-28 Publiceringsdatum Division of Solid Mechanics Publication date Dept. of Management and Engineering 2008-02-04 SE-581 83 LINKÖPING Språk Rapporttyp ISBN: Language Report category Svenska/Swedish Licentiatavhandling ISRN: LIU-IEI-TEK-A--08/00283--SE X Engelska/English X Examensarbete C-uppsats Serietitel: D-uppsats 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 Mechanical behaviour of single-crystal nickel-based superalloys Title Författare Daniel Leidermark Author 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: material model, single-crystal, superalloy, crystal plasticity, LS-DYNA, ABAQUS, ANSYS, Keyword 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 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 Description Ak Associated thermodynamic forces a Hardening parameter b Material parameter a1, a2 Crystal orientations Ce Elastic tangent stiffness tensor cm Constants material array C1, C2 Material parameter c Material parameter D Rate of deformation tensor Ee Elastic Green-Lagrange strain tensor E Modulus of elasticity F Total deformation gradient F e Elastic deformation gradient F p Plastic deformation gradient G Shear modulus Gs Reference slip resistance α Gr Slip resistance of each slip system hαβ Strain hardening rate hβ Single slip hardening rate h0 Reference hardening rate I Unit tensor K Bulk modulus KI , KII , KIII Stress intensity factor in Mode I, II, III L Velocity gradient m Slip rate sensitivity M 1, M 2 Structural tensors Nf Fatigue life nα Normal vector of each slip system qαβ Latent-hardening matrix q Latent-hardening ratio q Heat flux R Load ratio s Specific entropy sα Slip direction of each slip system S 2:nd Piola-Kirchhoff stress tensor T Temperature Vk Internal state variables W Spin tensor — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys — X Variable Description α Slip system γ˙ 0 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 Timestep εy Strain in y-direction η Elastic parameter Jacobian determinant λJ Lamé constant µ Lamé constant ω Mandel stress tensor ∇T Temperature gradient int Internal power ρP Density σ Cauchy stress tensor σu Ultimate stress σy Tensile load in y-direction τ Kirchhoff stress tensor τ α Resolved shear stress Φ Thermodynamic dissipation φ Plastic lattice rotation Ψ Helmholtz free energy Ω Current configuration Ω¯ Intermediate configuration Ω¯ iso Isoclinic intermediate configuration Ω0 Reference configuration — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys — XI Contents 1 Introduction 1 1.1 Gasturbines............................ 2 1.2 Superalloys ............................ 3 2 Fatigue 5 2.1 Low-cyclefatigue ......................... 5 2.2 Thermomechanicalfatigue . 5 3 Crack propagation 7 4 Crystal structure 9 5 Theory 11 5.1 Tangentstiffnesstensor. 11 5.2 Kinematics ............................ 12 5.3 Crystalplasticity . .. .. 16 5.3.1 Virginmaterial . .. .. 20 5.4 Thermodynamics ......................... 21 6 Implementation 23 6.1 Elasticmaterialmodel . 24 6.1.1 Validation ......................... 24 6.2 Crystal plasticity material model . 26 6.2.1 Validation ......................... 27 6.3 Flowchart ............................. 30 7 Discussion 33 7.1 Futurework ............................ 33 — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys — XII List of Figures 1 The interior of a stationary power generating gas turbine . .. 2 2 The In-Phase thermomechanical fatigue cycle . 6 3 The Out-of Phase thermomechanical fatigue cycle . 6 4 CrackloadedindifferentModes . 7 5 FCCcrystalstructure ...................... 9 6 The (111) planeintheunitcell . 10 7 Material description, with a plastic lattice rotation. .... 12 8 Material description, without a plastic lattice rotation. .... 13 9 Rotationofthecrystalorientation. 17 10 The cube loaded uniaxially by a tensile load . 24 11 Stereographic projection and (001) standard pole figure . ... 28 12 Crystal orientations in correspondence to the global coordi- nate system used by Kalidindi and (011) pole figure . 29 13 (001)polefigureofthedeformedcube . 30 14 Flowchart of analysis done by LS-DYNA . 31 15 Flowchart of analysis done by ABAQUS . 31 16 Flowchart of analysis done by ANSYS . 32 List of Tables 1 Slipsystems............................ 10 2 Materialparametersforpurenickel . 25 3 Materialparametersforcopper . 28 4 Initial slip hardening parameters for copper . 29 — Mechanical Behaviour of Single-Crystal Nickel-Based Superalloys — 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 temper- ature 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 pro- duce 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 ini- tialised 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 su- peralloy 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 fu- els and to contribute in the development of new materials for energy systems of the future. This programme, called Konsortiet för Materialteknik för ter- miska 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
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