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Modelling Cold Spray Splat Morphologies Using Smoothed Particle Hydrodynamics

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Modelling Cold Spray Splat Morphologies Using Smoothed Particle Hydrodynamics Modelling Cold Spray Splat Morphologies Using Smoothed Particle Hydrodynamics Luke Stephen Mason Submitted for the degree of Doctor of Philosophy Heriot-Watt University Institute of Mechanical, Process and Energy Engineering Engineering and Physical Sciences July 28, 2015 c British Crown Owned Copyright 2015/AWE. Any quotation from the thesis or use of any of the information contained in it must acknowledge this thesis as the source of the quotation or information. Abstract The small scale, short duration and hostile environment for instrumentation presented by cold spray coating makes experimental observations challenging, and therefore requires computational models capable of capturing the splat formation process. Current coating models are dominated by the Finite Element Method (FEM); whilst this has lead to significant improvements in understanding, the method is limited due to the reliance on a mesh coupled with the significant strains and strain rates involved. Eulerian methods have also been applied but retrieval of material histories and accurate interface track- ing remains challenging. The Smoothed Particle Hydrodynamics (SPH) method is a meshless method that combines the advantages of FEM and Eulerian approaches. The current work extends the work of applying SPH to solid mechanics with heat conduc- tion, improved tensile stability corrections and a novel zero impedance boundary. Solver performance is increased with the application of the multi-threading capabilities of the C++ 11 standard. The development of the SPH solver is described, validated and bench- marked against known analytical and experimental test cases. An in-depth investigation of parameters affecting splat morphologies is performed. Finally, a model of a coating formation process involoving multiple feedstock impact events is described and analysed in order to demonstrate the capabilities of the newly developed solver. i For Becky and Puff. ii Acknowledgements I would like to thank my supervisor Dr Yeaw Chu Lee for his help and support over the years as well as the members of our research group Alasdair Mckean, Odin Du’Plessis, Noel Ehigiamusoe and Samat Maxutov. I would also like to thank Dr Richard Fu for his supervision in the initial stages of the project and for his assistance with proof reading. This project was sponsored by AWE plc and I would like to thank Dr Robin Williams, David Youngs and Chris Batha who provided advice and invaluble feedback at our project review meetings. Finally I would like to thank my family and friends for their patience and support but most importantly my fianc´ee Rebecca without whom I would not have made it this far. iii iv Contents 1 Literature Review 1 1.1 ColdSpray .................................. 1 1.2 SmoothedParticleHydrodynamics . 15 1.2.1 KernelFunctions ........................... 16 1.2.2 Domain Boundaries and Boundary Conditions . 18 1.2.3 Elastic-Plastic Materials . 22 1.2.4 Stability . 24 1.2.5 SPH Application to Cold Spray . 27 1.3 Summary ................................... 28 1.4 AimsandObjectives ............................. 29 2 Methodology 31 2.1 GeneralFormulation ............................. 31 2.1.1 Errors in the SPH interpolation . 36 2.1.2 Kernels ................................ 37 2.2 GoverningEquations ............................. 39 2.2.1 EquationsofMotion ......................... 39 2.2.2 ArtificialCorrections ......................... 46 2.2.3 Constituitive Models . 47 2.3 NumericalMethodology ........................... 51 2.3.1 Variable smoothing . 51 2.3.2 NeighbourSearch........................... 52 2.3.3 InteractionPairs ........................... 54 2.3.4 TimeIntegration ........................... 54 v 2.3.5 BoundaryConditions......................... 56 2.4 SolverImplementation ............................ 59 2.4.1 ParameterFiles............................ 60 2.4.2 Particle Files . 60 2.4.3 Optimisations . 63 3 Verification Tests 68 3.1 VerificationTests ............................... 68 3.1.1 BoundaryRebound.......................... 68 3.1.2 Shock-Tube .............................. 74 3.1.3 Poiseuille Flow . 82 3.1.4 CouetteFlow ............................. 86 3.1.5 Heat Conduction in Slab . 89 3.1.6 Instability Correction . 92 3.1.7 TaylorBarImpact .......................... 96 3.2 Scalability and Performance Benchmarks . 103 4 Boundary Conditions for an Isolated Three-Dimensional Cold Spray Splat Morphology 108 4.1 ProblemDefinition .............................. 109 4.2 ParticleMassIndependence . 112 4.3 Boundary Condition Selection . 113 4.4 HighTemperatureZeroImpedanceDomain . 122 4.5 Summary ................................... 125 5 Three-Dimensional Single Splat Formation 132 5.0.1 ModelB................................ 134 5.1 Heat Conduction Effects on Splat Morphology and Properties . 144 5.1.1 Pair1 ................................. 144 5.1.2 Pair2 ................................. 147 5.1.3 Pair3 ................................. 152 5.1.4 Summary ............................... 155 5.2 Impact Velocity Effects on Splat Formation . 156 vi 5.3 FeedstockSizeEffectonSplatMorphology . 161 5.4 Feedstock Initial Temperature Effect on Splat Morphology . .... 165 5.5 Feedstock/SubstrateBonding . 171 6 Coating Formation 176 6.1 ProblemDefinition .............................. 176 6.2 CoatingAnalysis ............................... 177 6.3 Summary ................................... 190 7 Conclusions and Future Work 191 A Boundary Condition Selection Test Results 196 A.1 NoBoundaryCondition ........................... 196 A.2 PeriodicBoundaryCondition . 206 A.3 ZeroImpedanceBoundaryCondition . 213 B Single Splat Impact Test Results 220 B.1 AdiabaticModels ............................... 220 B.1.1 ModelA................................ 220 B.1.2 ModelC................................ 231 B.2 HeatConductionModels . .. .. 241 B.2.1 ModelD................................ 241 B.2.2 ModelE................................ 251 B.2.3 ModelF................................ 261 B.2.4 ModelG................................ 271 B.2.5 ModelH................................ 281 B.3 LargerConductiveFeedstocks . 291 B.3.1 ModelI ................................ 291 B.3.2 ModelJ ................................ 301 B.4 Smaller Conductive Feedstocks . 311 B.4.1 ModelK................................ 311 B.4.2 ModelL................................ 321 vii List of Tables 1.1 Typicalcoatingmethodparameters . 2 1.2 Typicalcoatingproperties........................... 9 2.1 Summary of continuous SPH approximations. 35 2.2 Summary of discrete SPH approximations. 36 2.3 Domain defintion section of parameters.txt. 61 2.4 Material defintion section of parameters.txt. ... 62 2.5 The particles.txt file associated with the example parameters.txt. .... 65 2.6 Solver functional description. 67 3.1 Boundary rebound initial conditions. 70 3.2 Shock-tube initial conditions. 75 3.3 Poiseuille flow initial conditions. 85 3.4 Couette flow initial conditions. 88 3.5 Heat conduction initial conditions. 91 3.6 Instability correction test initial conditions. .. 93 3.7 Taylor bar test initial conditions. 101 3.8 Taylorbarcomparisondata. 102 3.9 Material data for scalability tests. 104 4.1 Isolated splat formation boundary condition test material properties. 110 4.2 Comparison of levels of Von Mises stress in the feedstock. .... 120 5.1 Modelled feedstock initial conditions for three-dimensional single splat for- mation. .................................... 133 5.2 Model B flattening ratio, splat centre height and P ∗ dataat22ns.. 135 viii 5.3 Model pairing for comparison of adiabatic and conductive models. .... 144 5.4 Comparison of the effect of impact velocity on final flattening ratio and penetrationintothesubstrate. 157 5.5 Comparison of the effect of feedstock diameter on final flattening ratio and penetrationintothesubstrate. 162 5.6 Comparison of the effect of feedstock initial temperature on final flattening ratio and penetration into the substrate for a variety of feedstock diame- ters. ...................................... 167 B.1 Model A flattening ratio, P ∗ and splat centre height data at 24 ns. 221 B.2 Model C flattening ratio, P ∗ and splat centre height data at 24 ns. 231 B.3 Model D flattening ratio, P ∗ and splat centre height data at 24 ns. 241 B.4 Model E flattening ratio, P ∗ and splat centre height data at 22 ns. 251 B.5 Model F flattening ratio, P ∗ and splat centre height data at 23 ns. 261 B.6 Model G flattening ratio, P ∗ and splat centre height data at 26 ns. 273 B.7 Model H flattening ratio, P ∗ and splat centre height data at 24 ns. 283 B.8 Model I flattening ratio, P ∗ and splat centre height data at 38 ns. 292 B.9 Model J flattening ratio, P ∗ and splat centre height data at 40 ns. 302 B.10 Model K flattening ratio, P ∗ and splat centre height data at 3.8ns. 312 B.11 Model L flattening ratio, P ∗ and splat centre height data at 4.1ns. 322 ix List of Figures 1.1 Schematicofsplatregions. .......................... 3 1.2 Evolution of a typical splat impact. 3 1.3 Cross section of a copper particle bonded to a stainless steel substrate. Particlevelocitywas700m/s ........................ 4 1.4 Oxide cracking and interface cleaning . 6 1.5 Interface mixing between a copper feedstock and aluminium substrate.. 7 1.6 Etched copper coating showing splat interfaces . ... 8 1.7 Schematicofcoldspraynozzle. 10 1.8 Schematic of cold spray nozzle alginment . 12 1.9 Quintic kernel derivatives. 25 1.10 Tensile test initial condition. 26 1.11 Tensile test without correction at t = 0.001716 s. .. 27 2.1 Schematic of rigid
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